diff --git a/GraphicalRepresentation.png b/GraphicalRepresentation.png new file mode 100644 index 000000000..e3ea27433 Binary files /dev/null and b/GraphicalRepresentation.png differ diff --git a/Group 1 Members.md b/Group 1 Members.md new file mode 100644 index 000000000..e817538d1 --- /dev/null +++ b/Group 1 Members.md @@ -0,0 +1,7 @@ +GROUP 1 - Missing Data & Imputation: + +Elizabeth Subject-Scott + +Michael Underwood + +Javier Estrada diff --git a/Slides.html b/Slides.html new file mode 100644 index 000000000..8729daab7 --- /dev/null +++ b/Slides.html @@ -0,0 +1,4075 @@ + + + + + + + + + + + + + + + Missing Data and Imputation + + + + + + + + + + + + + + + + + + + +
+
+ +
+

Missing Data and Imputation

+

Michael Underwood

+

Elizabeth Subject-Scott

+

Javier Estrada

+
+ +
+

What is Missing Data?

+
    +
  • Missing data occurs when there are missing values in a dataset +
      +
    • Can be intentional or unintentional
      • +
    • +
  • Missing data is classified into 3 different categories: +
      +
    • Missing Completely At Random (MCAR)
      • +
    • Missing At Random (MAR)
      • +
    • Missing Not At Random (MNAR)
      • +
    • +
+
+
+

+ +

X are the completely observed variables.

+

Y are the partly missing variables.

+

Z is the component of the cause of missingness unrelated to X and Y.

+

R is the missingness.

+
+
+

Methods to Handle Missing Data

+
    +
  • Likelihood Bayesian Method +
      +
    • Predicts missing values based on a previous predictive distribution.
      • +
    • +
  • Weighting Method +
      +
    • Uses weights from available data to adjust for missing values.
      • +
    • +
  • Imputation Method +
      +
    • Uses estimates from original data to determine missing values
      • +
    • +
+
+
+

Deleting Missing Data

+
    +

  • When type is MCAR and the amount of missing data is small, deletion can be used.

    • +

  • 2 Types

    +
      +

    • Listwise deletion occurs when the entire observation is removed.

      • +

    • Pairwise deletion occurs when the variable of an observation is removed.

      • +
    • +

  • Deleting missing data can lead to the loss of important information regarding your dataset and is not recommended.

    • +
+
+
+

Preferred Method:

+
    +

  • Imputation

    +
      +

    • 2 Types

      +
        +

      • Single Imputation

        +
          +
        • Only one estimate is used to replace the missing data.
          • +
        • +

      • Multiple Imputation

        +
          +
        • Various estimates are used to replace the missing data by creating mulitple versions of the original dataset.
          • +
        • +
      • +
    • +
+
+
+

Single or Univariate Imputation

+
    +

  • Methods include:

    +
      +
    • Using the mean to replace a missing value. +
        +
      • The problem with this method is that it reduces the variance which leads to a smaller confidence interval.
        • +
      • +
    • Last Observation Carried Forward (LOCF) replaces a missing value with a previously observed value (the most recent value is carried forward). +
        +
      • The problem with this method is that it assumes that the previous observed value is perpetual, when in reality that may not be the case.
        • +
      • +
    • +
+
+
+

Multiple Imputation

+
    +
  • A set of m plausible values are generated for each unobserved data point, resulting in M complete data sets.
    • +
  • The new values are randomly drawn from predictive distributions either through joint modeling (JM) or fully conditional specification (FCS).
    • +
  • It is then analyzed and the results are combined, or pooled together, to obtain a single value for the missing data.
    • +
  • Multiple imputation by chained methods (MICE) is the most common and preferred method of multiple imputation.
    • +
+
+
+

Other Methods of Imputation

+
    +

  • Regression Imputation is based on a linear regression model. Missing data is randomly drawn from a conditional distribution when variables are continuous and from a logistic regression model when they are categorical.

    • +

  • Predictive Mean Matching is also based on a linear regression model. The approach is the same as regression imputation except instead of random draws from a conditional distribution, missing values are based on predicted values of the outcome variable.

    • +
+
+
+

+
    +

  • Hot Deck (HD) imputation is when a missing value is replaced by an observed response of a similar unit, also known as the donor. It can be either random or deterministic, which is based on a metric or value. It does not rely on model fitting.

    • +

  • Stochastic Regression (SR) Imputation is an extension of regression imputation. The process is the same but a residual term from the normal distribution of the regression of the predictor outcome is added to the imputed value. This maintains the variability of the data.

    • +
+
+
+

+
    +
  • Random Forest (RF) Imputation is based on machine learning algorithms. Missing values are first replaced with the mean or mode of that particular variable and then the dataset is split into a training set and a prediction set. The missing values are then replaced with predictions from these sets. This type of imputation can be used on continuous or categorical variables with complex interactions.
    • +
+
+
+

Methodology of Multiple Imputation

+
    +
  • Point of multiple imputation +
      +
    • Not to create artificial values for the missing data
      • +
    • Intended to be a tool in analyzing data with missing values
      • +
    • +
  • Developed by Donald Rubin, Harvard Professor in the 70s +
      +
    • Previous method was not correct +
        +
      • It is impossible to calculate missing data with certainty.
        • +
      • It did not take into account the estimates in error in the predictions.
        • +
      • He recommended creating multiple datasets, based on the Bayesian method.
        • +
      • It gives more precise estimates than using just one imputed dataset.
        • +
      • +
    • According to Rubin’s Rule, in multiple imputation missing values are replaced by m imputed values which result in M complete datasets. +
        +
      • For each of the M datasets, an estimate of \(\theta\) is acquired.
        • +
      • +
    • +
+
+
+

Pooling Estimates

+
    +
  • To calculate the pooled mean differences of \(\theta\):
    • +
+

\[{\overline{\theta}}_{M} = \displaystyle \frac{1}{M}\sum_{m = 1}^{M} {\hat{\theta}}_{m}\]

+
    +
  • This is the average of m estimates of M imputed datasets.
    • +
+
+
+

Pooling Standard Errors

+
    +
  • To calculate the proposed variance estimator of \({\hat{\theta}}_{m}\):
    • +
+

\[{\hat{\Phi}}_{M} = {\overline{\phi}}_{M} + (1+\displaystyle \frac{1}{M})B_{M}\]

+
    +

  • This is the total variance and is made up of two parts: the average within imputation variance and the between imputation variance (with a correction factor).

    • +

  • The average within imputation variance (normal standard error) is:

    • +
+

\[{\overline{\phi}}_{M} = \displaystyle \frac{1}{M}\sum_{m=1}^{M}{SE}^2\]

+

(It will be large in small samples and small in large samples.)

+
    +
  • The between imputation variance (how much extra variation there is due to the imputation process) is:
    • +
+

\[B_{M} = \displaystyle \frac{1}{M-1}\sum_{m = 1}^{M}({\hat{\theta}}_{m}-{\overline{\theta}}_{m})^{2}\]

+
    +

  • When the between variance is greater than the within variance, then greater efficiency occurs and more accurate estimates can be achieved by increasing M.

    • +

  • When the within variance is greater than the between variance, then little is gained by increasing M.

    • +

  • To calculate the total standard error:

    • +
+

\[SE_{pooled} = {\sqrt{\hat{\Phi}}_{M}}\]

+
+
+

Significance Testing

+

\[Wald_{pooled} = \frac{(\overline{\theta} - {\theta_{0}})^2}{{\hat{\Phi}}_{M}}\]

+
    +
  • The Wald test follows a t-distribution with the degrees of freedom as follows:
    • +
+

\[df = (M-1)(1+\frac{1}{M+1} * \frac{\overline{\theta}_{M}}{B_{M}})^2\]

+
    +
  • The 2 sided t-value is as follows:
    • +
+

\[t_{df, 1-\alpha/2}\]

+
    +
  • To test for significance using F-distribution:
    • +
+

\[ F_{1,df} = t^{2}_{df, 1-\alpha/2}\]

+
+
+

Assumptions for Multiple Imputation

+
    +

  • Observed data follow a normal distribution. If not normal, a transformation is first needed.

    • +

  • Missing data are classified as MAR, which is the probability that a missing value depends only on observed values and not unobserved values.

    • +

  • The parameters \({\theta}\) of the data model and the parameters \({\phi}\) of the model for the missing values are distinct. That is, knowing the values of \({\theta}\) does not provide any information about \({\phi}\).

    • +
+
+
+

Steps in Multiple Imputation:

+
+
+
+ +
+
+
    +

  • Step 1: Impute missing data

    • +

  • Step 2: Run regression models on all imputation sets

    • +

  • Step 3: Pool regression results into one regression result

    • +
+
+
+

MICE in R

+
    +

  • MICE = Multivariate Imputation by Chained Equations.

    • +

  • mice() method in R.

    • +

  • The method creates multiple imputations (replacement values) for multivariate missing data.

    • +

  • The MICE algorithm can impute mixes of continuous, binary, unordered categorical and ordered categorical data.

    • +

  • This can be done using the imputation regressions of the mice() method such as:

    • +

  • predictive mean matching (numeric data)

    • +

  • logistic regression imputation (binary data with two factorial levels)

    • +

  • polytomous regression imputation for unordered categorical data (factor > 2 levels)

    • +

  • proportional odds model for (ordered, > 2 levels).

    • +
+
+
+

Dataset

+
    +

  • King County, Seattle Home Sale Prices between 2014 and 2015

    +
      +

    • Contains the sale prices of 21,613 houses

      • +

    • The original dataset contained 21 columns with various selling attributes.

      +
        +

      • For the purpose of this project, we have condensed the variables to the following 4:

        +
          +
        • sales price of home
          • +
        • number of bedrooms
          • +
        • number of bathrooms
          • +
        • square footage of living space
          • +
        • +
      • +
    • +
+
+
# load library
+library(mice, warn.conflicts=FALSE)
+# load data
+original = read.csv("kc_house_data.csv")
+# first row
+head(original, 1)
+
+
   price bedrooms bathrooms sqft_living
+1 221900        3         1        1180
+
+
+
+
+

+
    +
  • First, we evaluate the dataset for missing values by using the is.na() function:
    • +
+
+
# check for missing data
+sapply(original, function(x) sum(is.na(x)))
+
+
      price    bedrooms   bathrooms sqft_living 
+          0           0           0           0
+
+
+
    +
  • There is no missing data; this is the original, complete dataset.
    • +
  • We will artificially create some missing values.
    • +
  • We do this by first duplicating the dataset. We want to preserve the original so that we can later check the accuracy of the imputation process.
    • +
+
+
# duplicate dataset
+house <- original
+
+
+
+

+
    +

  • Next, we randomly replace the desired amount of values with NA to mimic missing data.

    • +

  • We will assign 200 NA values to the following variables: - bedrooms, - bathrooms, - sqft_living

    • +

  • And 100 NA values to the price variable.

    • +
+
+
# replace some values with NA to create missing data
+set.seed(10)
+house[sample(1:nrow(house), 200), "sqft_living"] <- NA
+house[sample(1:nrow(house), 200), "bedrooms"] <- NA
+house[sample(1:nrow(house), 200), "bathrooms"] <- NA
+house[sample(1:nrow(house), 100), "price"] <- NA
+
+
    +
  • Using the is.na() function, we can check to confirm we have created missing data:
    • +
+
+
# check for missing data
+sapply(house, function(x) sum(is.na(x)))
+
+
      price    bedrooms   bathrooms sqft_living 
+        100         200         200         200
+
+
+
    +
  • Our dataset now has 700 NA/missing values. This equates to about 3% of the data (700/21,613).
    • +
+
+
+

Visualizations

+
+ +
+
+
+
# check the missingness patterns
+md.pattern(house, rotate.names = TRUE)
+
+

+
+
+
      price bedrooms bathrooms sqft_living    
+20920     1        1         1           1   0
+195       1        1         1           0   1
+196       1        1         0           1   1
+2         1        1         0           0   2
+197       1        0         1           1   1
+2         1        0         1           0   2
+1         1        0         0           1   2
+98        0        1         1           1   1
+1         0        1         1           0   2
+1         0        1         0           1   2
+        100      200       200         200 700
+
+
+
+
+
+
library(VIM)
+aggr_plot <- aggr(house, col=c('navyblue','red'), numbers=TRUE, sortVars=TRUE, labels=names(data), cex.axis=.7, gap=3, ylab=c("Histogram of missing data","Pattern"))
+
+

+
+
+

+ Variables sorted by number of missings: 
+    Variable       Count
+    bedrooms 0.009253690
+   bathrooms 0.009253690
+ sqft_living 0.009253690
+       price 0.004626845
+
+
+
+
+
    +
  • The red box plot on the left shows the distribution of sqft_living with price missing. The blue box plot shows the distribution of the remaining data points. The same is true for the price box plots on the bottom. We expect the red and blue boxplots to be very similar if our assumptions of MCAR data are true. In this case, they are.
    • +
+
+
marginplot(house[c(1,4)])
+
+

+
+
+
+
+
+
+
+

Imputation Process

+
    +

  • Since we are missing about 3% of our data, we need to perform at least 3 imputations. This will be done using the mice() function:

    • +

  • Since 5 is the default, we will use that (the m parameter can be used to adjust the the # of imputations)

    • +

  • The set.seed will be given the value 1337 (any number can be used here) to retrieve the same results each time the multiple imputation is performed.

    • +
+
+
# impute the data 5 times (default)
+imp = mice(data = house, set.seed = 1337, defaultMethod = c("pmm", "logreg", "polyreg",  "norm", "polr"))
+
+

+ iter imp variable
+  1   1  price  bedrooms  bathrooms  sqft_living
+  1   2  price  bedrooms  bathrooms  sqft_living
+  1   3  price  bedrooms  bathrooms  sqft_living
+  1   4  price  bedrooms  bathrooms  sqft_living
+  1   5  price  bedrooms  bathrooms  sqft_living
+  2   1  price  bedrooms  bathrooms  sqft_living
+  2   2  price  bedrooms  bathrooms  sqft_living
+  2   3  price  bedrooms  bathrooms  sqft_living
+  2   4  price  bedrooms  bathrooms  sqft_living
+  2   5  price  bedrooms  bathrooms  sqft_living
+  3   1  price  bedrooms  bathrooms  sqft_living
+  3   2  price  bedrooms  bathrooms  sqft_living
+  3   3  price  bedrooms  bathrooms  sqft_living
+  3   4  price  bedrooms  bathrooms  sqft_living
+  3   5  price  bedrooms  bathrooms  sqft_living
+  4   1  price  bedrooms  bathrooms  sqft_living
+  4   2  price  bedrooms  bathrooms  sqft_living
+  4   3  price  bedrooms  bathrooms  sqft_living
+  4   4  price  bedrooms  bathrooms  sqft_living
+  4   5  price  bedrooms  bathrooms  sqft_living
+  5   1  price  bedrooms  bathrooms  sqft_living
+  5   2  price  bedrooms  bathrooms  sqft_living
+  5   3  price  bedrooms  bathrooms  sqft_living
+  5   4  price  bedrooms  bathrooms  sqft_living
+  5   5  price  bedrooms  bathrooms  sqft_living
+
+
imputed <- complete(imp) # We need to create the imputed dataset using the complete() function
+
+
+
+

Imputation Process - Verifying Missing Data

+
    +
  • We now need to check the imputed dataset to make sure there is no missing data. Again, we use the is.na() function:
    • +
+
+
sapply(imputed, function(x) sum(is.na(x)))
+
+
      price    bedrooms   bathrooms sqft_living 
+          0           0           0           0
+
+
+
+
+

Post-Imputation Visualizations:

+
    +

  • We can check the quality of the imputations by running a strip plot, which is a single axis scatter plot.

    • +

  • It will show the distribution of the imputed values over the observed values.

    • +

  • Blue values are observed value and red values are imputed values. Ideally, we want the imputations to be values that could have been observed had the data not been missing.

    • +
+
+
# strip plots to check the quality of the imputations
+par(mfrow=c(2,2))
+stripplot(imp, price, pch = 19, cex=1, xlab = "Imputation number")
+
+

+
+
stripplot(imp, bedrooms, pch = 19, cex=1, xlab = "Imputation number")
+
+

+
+
stripplot(imp, bathrooms, pch = 19, cex=1, xlab = "Imputation number")
+
+

+
+
stripplot(imp, sqft_living, pch = 19, cex=1, xlab = "Imputation number")
+
+

+
+
+
+
+

Fitting and Pooling Imputed Datasets

+
    +
  • Next, we will run regression on our imputed datasets, so we will have 5 regression results. We do this with the with() function. For each regression, we get different results.
    • +
+
+
# fit complete-data model
+fit <- with(imp, lm(price~bedrooms+bathrooms+sqft_living))
+
+
    +

  • Next, we will pool the results from our regressions to arrive at estimates that will properly account for the missing data.

    • +

  • It will give us the estimate, standard error, test statistic, degrees of freedom, and the p-value for each variable.

    • +
+
+
# pool and summarize the results
+summary(pool(fit))
+
+
         term    estimate   std.error  statistic        df       p.value
+1 (Intercept)  75043.7775 6952.563287  10.793685 14431.278  4.677205e-27
+2    bedrooms -57930.4966 2357.322483 -24.574702  7683.745 1.936143e-128
+3   bathrooms   7624.2182 3531.505441   2.158914 11516.540  3.087739e-02
+4 sqft_living    309.7992    3.114475  99.470779  8431.335  0.000000e+00
+
+
+
+
+

Checking for Accuracy

+
    +
  • We will now compare the results from our pooled, imputed dataset to our original, complete dataset (before we removed values to create missing data) to check for accuracy of the imputation method by chained equations:
    • +
+
+
+
+
# fit original, complete dataset
+og_lm = lm(price~bedrooms+bathrooms+sqft_living, data = original)
+# compare imputed dataset to the original dataset
+summary(list("Imputed" = pool(fit), "Original" = og_lm), title = "Comparison", statistic = "p.value", estimate = "estimate", gof_omit = 'IC|Log|Adj|F|RMSE')
+
+
         Length Class Mode
+Imputed   4     mipo  list
+Original 12     lm    list
+
+
+
+
+

 

+
+
+
+
+

+
+
+
+
+
+
+

Results

+
    +

  • The estimates from the multiple imputation are very close to the estimates for the original dataset. The p-values are similar and the differences are minimal with exception of bathrooms which is off just slightly.

    • +

  • These results indicate a high accuracy of the imputation process. We can conclude that multiple imputation by chained equation is a reliable source to impute missing data.

    • +
+
+
+

Conclusion

+
    +
  • Missing data can occur in research for a variety of reasons.
    • +
  • It is never a good idea to ignore it. Doing this will lead to biased estimates of parameters, loss of information, decreased statistical power, and weak reliability of findings.
    • +
  • The best course of action is to impute the missing data by using multiple imputation.
    • +
  • Performing multiple imputation will minimize the adverse effects caused by missing data on the analysis.
    • +
+
+ +
+
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/Slides.qmd b/Slides.qmd new file mode 100644 index 000000000..57370f7a2 --- /dev/null +++ b/Slides.qmd @@ -0,0 +1,455 @@ +--- +title: "Missing Data and Imputation" +author: + - Michael Underwood + - Elizabeth Subject-Scott + - Javier Estrada +format: + revealjs: + theme: sky + transition: slide + background-transition: fade +editor: visual +self-contained: true +--- + +## What is Missing Data? {.scrollable} + +- Missing data occurs when there are missing values in a dataset + - Can be intentional or unintentional +- Missing data is classified into 3 different categories: + - Missing Completely At Random (MCAR) + - Missing At Random (MAR) + - Missing Not At Random (MNAR) + +## + +![](GraphicalRepresentation.png){fig-align="center"} + +~X are the completely observed variables.~ + +~Y are the partly missing variables.~ + +~Z is the component of the cause of missingness unrelated to X and Y.~ + +~R is the missingness.~ + +## Methods to Handle Missing Data {.scrollable} + +- Likelihood Bayesian Method + - Predicts missing values based on a previous predictive distribution. +- Weighting Method + - Uses weights from available data to adjust for missing values. +- Imputation Method + - Uses estimates from original data to determine missing values + +## Deleting Missing Data + +- When type is MCAR and the amount of missing data is small, deletion can be used. + +- 2 Types + + - Listwise deletion occurs when the entire observation is removed. + + - Pairwise deletion occurs when the variable of an observation is removed. + +- Deleting missing data can lead to the loss of important information regarding your dataset and is not recommended. + +## Preferred Method: {.scrollable} + +- Imputation + + - 2 Types + + - Single Imputation + + - Only one estimate is used to replace the missing data. + + - Multiple Imputation + + - Various estimates are used to replace the missing data by creating mulitple versions of the original dataset. + +## Single or Univariate Imputation {.scrollable} + +- Methods include: + + - Using the mean to replace a missing value. + - The problem with this method is that it reduces the variance which leads to a smaller confidence interval. + - Last Observation Carried Forward (LOCF) replaces a missing value with a previously observed value (the most recent value is carried forward). + - The problem with this method is that it assumes that the previous observed value is perpetual, when in reality that may not be the case. + +## Multiple Imputation {.scrollable} + +- A set of *m* plausible values are generated for each unobserved data point, resulting in *M* complete data sets. +- The new values are randomly drawn from predictive distributions either through joint modeling (JM) or fully conditional specification (FCS). +- It is then analyzed and the results are combined, or pooled together, to obtain a single value for the missing data. +- Multiple imputation by chained methods (MICE) is the most common and preferred method of multiple imputation. + +## Other Methods of Imputation {.scrollable} + +- Regression Imputation is based on a linear regression model. Missing data is randomly drawn from a conditional distribution when variables are continuous and from a logistic regression model when they are categorical. + +- Predictive Mean Matching is also based on a linear regression model. The approach is the same as regression imputation except instead of random draws from a conditional distribution, missing values are based on predicted values of the outcome variable. + +## {.scrollable} + +- Hot Deck (HD) imputation is when a missing value is replaced by an observed response of a similar unit, also known as the donor. It can be either random or deterministic, which is based on a metric or value. It does not rely on model fitting. + +- Stochastic Regression (SR) Imputation is an extension of regression imputation. The process is the same but a residual term from the normal distribution of the regression of the predictor outcome is added to the imputed value. This maintains the variability of the data. + +## {.scrollable} + +- Random Forest (RF) Imputation is based on machine learning algorithms. Missing values are first replaced with the mean or mode of that particular variable and then the dataset is split into a training set and a prediction set. The missing values are then replaced with predictions from these sets. This type of imputation can be used on continuous or categorical variables with complex interactions. + +## Methodology of Multiple Imputation {.scrollable} + +- Point of multiple imputation + - Not to create artificial values for the missing data + - Intended to be a tool in analyzing data with missing values +- Developed by Donald Rubin, Harvard Professor in the 70s + - Previous method was not correct + - It is impossible to calculate missing data with certainty. + - It did not take into account the estimates in error in the predictions. + - He recommended creating multiple datasets, based on the Bayesian method. + - It gives more precise estimates than using just one imputed dataset. + - According to Rubin's Rule, in multiple imputation missing values are replaced by *m* imputed values which result in *M* complete datasets. + - For each of the *M* datasets, an estimate of $\theta$ is acquired. + +## Pooling Estimates {.scrollable} + +- To calculate the pooled mean differences of $\theta$: + +$${\overline{\theta}}_{M} = \displaystyle \frac{1}{M}\sum_{m = 1}^{M} {\hat{\theta}}_{m}$$ + +- This is the average of *m* estimates of *M* imputed datasets. + +## Pooling Standard Errors {.scrollable} + +- To calculate **the proposed variance estimator** of ${\hat{\theta}}_{m}$: + +$${\hat{\Phi}}_{M} = {\overline{\phi}}_{M} + (1+\displaystyle \frac{1}{M})B_{M}$$ + +- This is the total variance and is made up of two parts: the average within imputation variance and the between imputation variance (with a correction factor). + +- **The average within imputation variance** (normal standard error) is: + +$${\overline{\phi}}_{M} = \displaystyle \frac{1}{M}\sum_{m=1}^{M}{SE}^2$$ + +(It will be large in small samples and small in large samples.) + +- **The between imputation variance** (how much extra variation there is due to the imputation process) is: + +$$B_{M} = \displaystyle \frac{1}{M-1}\sum_{m = 1}^{M}({\hat{\theta}}_{m}-{\overline{\theta}}_{m})^{2}$$ + +- When the between variance is greater than the within variance, then greater efficiency occurs and more accurate estimates can be achieved by increasing *M*. + +- When the within variance is greater than the between variance, then little is gained by increasing *M*. + +- To calculate the *total standard error*: + +$$SE_{pooled} = {\sqrt{\hat{\Phi}}_{M}}$$ + +## Significance Testing {.scrollable} + +$$Wald_{pooled} = \frac{(\overline{\theta} - {\theta_{0}})^2}{{\hat{\Phi}}_{M}}$$ + +- The Wald test follows a t-distribution with the degrees of freedom as follows: + +$$df = (M-1)(1+\frac{1}{M+1} * \frac{\overline{\theta}_{M}}{B_{M}})^2$$ + +- The 2 sided t-value is as follows: + +$$t_{df, 1-\alpha/2}$$ + +- To test for significance using F-distribution: + +$$ F_{1,df} = t^{2}_{df, 1-\alpha/2}$$ + +## Assumptions for Multiple Imputation {.scrollable} + +- Observed data follow a normal distribution. If not normal, a transformation is first needed. + +- Missing data are classified as MAR, which is the probability that a missing value depends only on observed values and not unobserved values. + +- The parameters ${\theta}$ of the data model and the parameters ${\phi}$ of the model for the missing values are distinct. That is, knowing the values of ${\theta}$ does not provide any information about ${\phi}$. + +## Steps in Multiple Imputation: {.scrollable} + +```{r, warning=FALSE, echo=F, message=FALSE} +DiagrammeR::grViz("digraph { + +# initiate graph +graph [layout = dot, rankdir = LR, label = 'The MICE-Process\n\n',labelloc = t, fontcolor = DarkSlateBlue, fontsize = 45] + +# global node settings +node [shape = rectangle, style = filled, fillcolor = AliceBlue, fontcolor = DarkSlateBlue, fontsize = 35] +bgcolor = none + +# label nodes +incomplete [label = 'Incomplete data set'] +imputed1 [label = 'Imputed \n data set 1'] +estimates1 [label = 'Estimates from \n analysis 1'] +rubin [label = 'Rubin rules', shape = diamond] +combined [label = 'Combined results'] +imputed2 [label = 'Imputed \n data set 2'] +estimates2 [label = 'Estimates from \n analysis 2'] +imputedm [label = 'Imputed \n data set m'] +estimatesm [label = 'Estimates from \n anaalysis m'] + + +# edge definitions with the node IDs +incomplete -> imputed1 [arrowhead = vee, color = DarkSlateBlue] +imputed1 -> estimates1 [arrowhead = vee, color = DarkSlateBlue] +estimates1 -> rubin [arrowhead = vee, color = DarkSlateBlue] +incomplete -> imputed2 [arrowhead = vee, color = DarkSlateBlue] +imputed2 -> estimates2 [arrowhead = vee, color = DarkSlateBlue] +estimates2-> rubin [arrowhead = vee, color = DarkSlateBlue] +incomplete -> imputedm [arrowhead = vee, color = DarkSlateBlue] +imputedm -> estimatesm [arrowhead = vee, color = DarkSlateBlue] +estimatesm -> rubin [arrowhead = vee, color = DarkSlateBlue] +rubin -> combined [arrowhead = vee, color = DarkSlateBlue] +}") +``` + +- Step 1: Impute missing data + +- Step 2: Run regression models on all imputation sets + +- Step 3: Pool regression results into one regression result + +## MICE in R {.scrollable} + +- MICE = Multivariate Imputation by Chained Equations. + - mice() method in R. + +- The method creates multiple imputations (replacement values) for multivariate missing data. + +- The MICE algorithm can impute mixes of continuous, binary, unordered categorical and ordered categorical data. +- This can be done using the imputation regressions of the mice() method such as: + - predictive mean matching (numeric data) + - logistic regression imputation (binary data with two factorial levels) + - polytomous regression imputation for unordered categorical data (factor > 2 levels) + - proportional odds model for (ordered, > 2 levels). + +## Dataset {.scrollable} + +- King County, Seattle Home Sale Prices between 2014 and 2015 + + - Contains the sale prices of 21,613 houses + + - The original dataset contained 21 +columns with various selling attributes. + + - For the purpose of this project, we have condensed the variables to the following 4: + + - sales price of home + - number of bedrooms + - number of bathrooms + - square footage of living space + +```{r, warning=FALSE, echo=T, message=FALSE} +# load library +library(mice, warn.conflicts=FALSE) +# load data +original = read.csv("kc_house_data.csv") +# first row +head(original, 1) +``` + +## {.scrollable} + +- First, we evaluate the dataset for missing values by using the is.na() +function: + +```{r, warning=FALSE, echo=T, message=FALSE} +# check for missing data +sapply(original, function(x) sum(is.na(x))) +``` + +- There is no missing data; this is the original, complete dataset. +- We will artificially create some missing values. +- We do this by first duplicating the dataset. We want to preserve the original so that we can +later check the accuracy of the imputation process. + +```{r, warning=FALSE, echo=T, message=FALSE} +# duplicate dataset +house <- original +``` + +## {.scrollable} + +- Next, we randomly +replace the desired amount of values with NA to mimic missing data. + +- We will assign 200 NA values to the following variables: + - bedrooms, + - bathrooms, + - sqft_living + +- And 100 NA values to the price variable. + +```{r, warning=FALSE, echo=T, message=FALSE} +# replace some values with NA to create missing data +set.seed(10) +house[sample(1:nrow(house), 200), "sqft_living"] <- NA +house[sample(1:nrow(house), 200), "bedrooms"] <- NA +house[sample(1:nrow(house), 200), "bathrooms"] <- NA +house[sample(1:nrow(house), 100), "price"] <- NA +``` + +- Using the is.na() function, we can check to confirm we have created +missing data: + +```{r, warning=FALSE, echo=T, message=FALSE} +# check for missing data +sapply(house, function(x) sum(is.na(x))) +``` + +- Our dataset now has 700 NA/missing values. This equates to about 3% of the data (700/21,613). + +## Visualizations {.scrollable} + +::: {.panel-tabset} + +### Pattern +```{r, warning=FALSE, echo=T, message=FALSE} +#| echo: true +#| output-location: column +# check the missingness patterns +md.pattern(house, rotate.names = TRUE) +``` + +### Histogram of Missing Data + +```{r, warning=FALSE, echo=T, message=FALSE} +library(VIM) +aggr_plot <- aggr(house, col=c('navyblue','red'), numbers=TRUE, sortVars=TRUE, labels=names(data), cex.axis=.7, gap=3, ylab=c("Histogram of missing data","Pattern")) +``` + +### Margin Plot + +- The red box plot on the left shows the distribution of sqft_living with +price missing. The blue box plot shows the distribution of the remaining +data points. The same is true for the price box plots on the bottom. We +expect the red and blue boxplots to be very similar if our assumptions +of MCAR data are true. In this case, they are. + +```{r, warning=FALSE, echo=T, message=FALSE} +marginplot(house[c(1,4)]) +``` +::: + +## Imputation Process {.scrollable} + +- Since we are missing about 3% of our data, we need to perform at least 3 +imputations. This will be done using the mice() function: + +- Since 5 is the default, we will use that (the m parameter can be used to adjust the the # of imputations) + +- The set.seed will be given the value 1337 (any number can be used here) to +retrieve the same results each time the multiple imputation is +performed. + +```{r, warning=FALSE, echo=T, message=FALSE} +# impute the data 5 times (default) +imp = mice(data = house, set.seed = 1337, defaultMethod = c("pmm", "logreg", "polyreg", "norm", "polr")) +imputed <- complete(imp) # We need to create the imputed dataset using the complete() function +``` + +## Imputation Process - Verifying Missing Data {.scrollable} + +- We now need to check the imputed dataset to make sure there is no +missing data. Again, we use the is.na() function: + +```{r, warning=FALSE, echo=T, message=FALSE} +sapply(imputed, function(x) sum(is.na(x))) +``` + +## Post-Imputation Visualizations: {.scrollable} + +- We can check the quality of the imputations by running a strip plot, +which is a single axis scatter plot. + +- It will show the distribution of the imputed values over the observed values. + +- Blue values are observed value and red values are imputed values. Ideally, we want the +imputations to be values that could have been observed had the data not +been missing. + +```{r, warning=FALSE, echo=T, message=FALSE} +# strip plots to check the quality of the imputations +par(mfrow=c(2,2)) +stripplot(imp, price, pch = 19, cex=1, xlab = "Imputation number") +stripplot(imp, bedrooms, pch = 19, cex=1, xlab = "Imputation number") +stripplot(imp, bathrooms, pch = 19, cex=1, xlab = "Imputation number") +stripplot(imp, sqft_living, pch = 19, cex=1, xlab = "Imputation number") +``` + +## Fitting and Pooling Imputed Datasets {.scrollable} + +- Next, we will run regression on our imputed datasets, so we will have 5 +regression results. We do this with the with() function. For each +regression, we get different results. + +```{r, warning=FALSE, echo=T, message=FALSE} +# fit complete-data model +fit <- with(imp, lm(price~bedrooms+bathrooms+sqft_living)) +``` + +- Next, we will pool the results from our regressions to arrive at estimates that will properly account for the missing data. + +- It will give us the estimate, +standard error, test statistic, degrees of freedom, and the p-value for +each variable. + +```{r, warning=FALSE, echo=T, message=FALSE} +# pool and summarize the results +summary(pool(fit)) +``` + +## Checking for Accuracy {.scrollable} +- We will now compare the results from our pooled, imputed dataset to our original, complete dataset (before we removed values to create missing data) to check for accuracy of the imputation method by chained +equations: + +:::::: {.cols data-latex=""} + +::: {.col data-latex="{0.55\textwidth}"} + + +```{r, warning=FALSE, echo=T, message=FALSE} +# fit original, complete dataset +og_lm = lm(price~bedrooms+bathrooms+sqft_living, data = original) +# compare imputed dataset to the original dataset +summary(list("Imputed" = pool(fit), "Original" = og_lm), title = "Comparison", statistic = "p.value", estimate = "estimate", gof_omit = 'IC|Log|Adj|F|RMSE') +``` +::: + +::: {.col data-latex="{0.05\textwidth}"} +\ + +::: + +::: {.col data-latex="{0.4\textwidth}"} + +![](Summary Table.png){fig-align="center"} + +::: +:::::: + +## Results {.scrollable} + +- The estimates from the multiple imputation are very close to the estimates +for the original dataset. The p-values are similar and the differences are minimal with exception of bathrooms which is off just slightly. + +- These results indicate +a high accuracy of the imputation process. We can conclude that multiple +imputation by chained equation is a reliable source to impute missing +data. + +## Conclusion + +- Missing data can occur in research for a variety of reasons. +- It is never a good idea to ignore it. Doing this will lead to biased estimates of parameters, loss of information, decreased statistical power, and weak reliability of findings. +- The best course of action is to impute the missing data by using multiple imputation. +- Performing multiple imputation will minimize the adverse effects caused by missing data on the analysis. \ No newline at end of file diff --git a/Summary Table.png b/Summary Table.png new file mode 100644 index 000000000..79c8ae241 Binary files /dev/null and b/Summary Table.png differ diff --git a/index.html b/index.html index f8ab42114..4e4e010d7 100644 --- a/index.html +++ b/index.html @@ -2,21 +2,28 @@ - + - - + + + + -Report Sample +Missing Data and Imputation + + + + + + + + + + + + + + + + + + + + + - - - - - - - - - - - - + - - -
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Report Sample

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Missing Data and Imputation

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Report Sample

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Author
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Authors
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Student name

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Javier Estrada

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Michael Underwood

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Elizabeth Subject-Scott

Published
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October 25, 2022

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May 1, 2023

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Report Sample

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Website

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Slides

Introduction

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Kernel regression is a non-parametric estimator that estimates the conditional expectation of two variables which is random. The goal of a kernel regression is to discover the non-linear relationship between two random variables. To discover the non-linear relationship, kernel estimator or kernel smoothing is the main method to estimate the curve for non-parametric statistics. In kernel estimator, weight function is known as kernel function (Efromovich 2008). Cite this paper (Bro and Smilde 2014).

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Missing Data

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Missing data occurs when there are missing values in a dataset. There are many reasons why this happens. It can be intentional or unintentional, and oftentimes there is no way to know for sure what causes the missing data. There are three categories, otherwise known as missingness mechanisms, which measure the severity of the missingness (Mainzer et al. 2023):

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  • Missing completely at random (MCAR) is when the distribution of missing data is completely independent of any other variables. There is no relationship between the pattern of missingness and the observed data. This would be the best-case scenario.

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  • Missing at random (MAR) is when the distribution of missing data is dependent of the observed values. The pattern of missingness is related to the observed data and not the missing data.

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  • Missing not at random (MNAR) is when missing data is neither MCAR or MAR. The distribution of missing data is dependent on other variables. The pattern of missingness is related to the observed data, as well as the missing data. This would be the worst-case scenario and is nonignorable.

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Figure 1: Graphical Representation of Missingness Mechanisms (Schafer and Graham 2002)

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(X are the completely observed variables. Y are the partly missing variables. Z is the component of the cause of missingness unrelated to X and Y. R is the missingness.)

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Looking for patterns in the missing data can help us to determine which category they belong. These mechanisms are important in determining how to handle the missing data. MCAR would be the best-case scenario but seldom occurs. MAR and MNAR are more common. Most methods of dealing with missing data assume a MAR missingness. If data are found to be MNAR, worst-case scenario, then specialized expertise and additional steps are needed to model the missingness before handling the missing data. The problem with ignoring any missing values is that it does not give a true representation of the dataset and can lead to bias when analyzing. This reduces the statistical power of the analysis (van_Ginkel et al. 2020).

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According to Dong and Peng (2013), to enhance the quality of research, the following should be followed with regards to missing data:

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  • Determine under what conditions the missing data occur.
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  • Use principled methods to handle the missing data.
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  • Include and report the treatment of the missing data.
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Methods

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The common non-parametric regression model is \(Y_i = m(X_i) + \varepsilon_i\), where \(Y_i\) can be defined as the sum of the regression function value \(m(x)\) for \(X_i\). Here \(m(x)\) is unknown and \(\varepsilon_i\) some errors. With the help of this definition, we can create the estimation for local averaging i.e. \(m(x)\) can be estimated with the product of \(Y_i\) average and \(X_i\) is near to \(x\). In other words, this means that we are discovering the line through the data points with the help of surrounding data points. The estimation formula is printed below (R Core Team 2019):

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\[ -M_n(x) = \sum_{i=1}^{n} W_n (X_i) Y_i \tag{1} -\]

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\(W_n(x)\) is the sum of weights that belongs to all real numbers. Weights are positive numbers and small if \(X_i\) is far from \(x\).

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Methods to Deal with Missing Data

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There are three types of methods to deal with missing data, the likelihood and Bayesian method, weighting methods, or imputation methods (Cao et al. 2021).

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  • Likelihood Bayesian method is when information from a previous predictive distribution is combined with evidence obtained in a sample to predict a value. It requires technical coding and advanced statistical knowledge.

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  • The weighting method is a traditional approach when weights from available data are used to adjust for non-response in a survey. Inefficiency occurs when there are extreme weights or a need for many weights.

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  • The imputation method is when an estimate from the original dataset is used to estimate the missing value. There are two types of imputation: single and multiple.

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Analysis and Results

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Data and Vizualisation

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A study was conducted to determine how…

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Deleting Missing Data

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Another way to handle missing data is to simply delete it. Deleting missing values can lead to the loss of important information regarding your dataset, and is therefore not recommended. The two types of deletion methods are listwise and pairwise.

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Listwise deletion is when the entire observation is removed from the dataset. In certain cases when the amount of missing data is small and the type is MCAR, listwise deletion can be used. There usually won’t be bias but potentially important information may be lost.

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T-tests and chi-square tests can be used to assess pairs of predictor variables to determine whether the groups’ means differ significantly. According to van_Ginkel et al. (2020), if significant, the null hypothesis is rejected, therefore, indicating that the missing values are not randomly distributed throughout the data. This implies that the missing data is either MAR or MNAR. Conversely, if non-significant, this implies that the data cannot be MAR. This does not eliminate the possibility that it is not MNAR–other information about the population is needed to determine this.

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Whenever missing data is categorized as MAR or MNAR, listwise deletion would be wasteful, and the analysis biased. Alternate methods of dealing with the missing data is recommended: either pairwise deletion or imputation.

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Pairwise deletion is when only the missing variable of an observation is removed. It allows more data to be analyzed than listwise deletion but limits the ability to make inferences of the total sample. For this reason, it is recommended to use imputation to properly deal with missing data.

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Preferred Method to Handle Missing Data

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Imputation is the preferred method to handle missing data. It consists of replacing missing data with an estimate obtained from the original, available data. After imputation, there will be a full dataset to analyze. According to Wulff and Jeppesen (2017), 3-5 imputations are sufficient, and 10 imputations are more than enough. More recent research has indicated that to improve statistical power, the number of imputations created should be at least equal to the percent of missing data (5% equals 5 imputations, 10% equals 10 imputations, 20% equals 20 imputations, etc.) (Pedersen et al. 2017).

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Single, or univariate, imputation is when only one estimate is used to replace the missing data. Methods of single imputation include using the mean, the last observation carried forward, and random imputation. The following is a brief explanation of each:

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  • Using the mean to replace a missing value is a straight-forward process. The mean of the dataset is calculated, including the missing value. The mean is then multiplied by the number of observations in the study. Next, the known values are subtracted from the product, and this gives an estimate that can be used for any missing values. The problem with this method is that it can produce misleading results such as showing the variance and the confidence interval smaller than they actually are.

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  • Last Observation Carried Forward (LOCF) is a technique of replacing a missing value in longitudinal studies with a previously observed value, the most recent value is carried forward (Streiner 2008). The problem with this method is that it assumes that the previous observed value is perpetual when in reality that most likely is not the case.

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  • Random imputation is a method of randomly drawing an observation and using that observation for any of the missing values. The problem with this method is that it introduces additional variability.

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These single imputation methods are inaccurate. They often result in underestimation of standard errors or too small p-values (Dong and Peng 2013), which can cause bias in the analysis. Therefore, multiple imputation is the superior method because it handles missing data better and provides less biased results.

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Multiple, or multivariate, imputation is when various estimates are used to replace the missing data by creating multiple datasets from versions of the original dataset. According to Dong and Peng (2013), it can be done by using a “regression model, or a sequence of regression models, such as linear, logistic and Poison,” where a set of m plausible values are generated for each unobserved data point, resulting in M complete data sets. The new values are randomly drawn from predictive distributions either through joint modeling (JM, which is not used much anymore) or fully conditional specification (FCS) (Wongkamthong and Akande 2023). It is then analyzed and the results are combined to obtain a single value for the missing data.

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The purpose of multiple imputation is to create a pool of imputed data for analysis, but if the pooled results are lacking, then multiple imputation should not be done (Mainzer et al. 2023). Another reason not to use multiple imputation is if there are very few missing values; there may be no benefit in using it. Also worth noting is some statistical analyses software already have built-in features to deal with missing data.

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Multiple imputation by chained methods, otherwise known as MICE, is the most common and preferred method of multiple imputation (Wulff and Jeppesen 2017). It provides a more reliable way to analyze data with missing values. For this reason, this paper will focus on the methodology and application of the MICE process.

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Figure 2: Flowchart of the MICE-process based on procedures proposed by Rubin (Wulff and Jeppesen 2017)

Code -
# loading packages 
-library(tidyverse)
-library(knitr)
-library(ggthemes)
-library(ggrepel)
-library(dslabs)
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# load package
+library(DiagrammeR)
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+# flowchart of mice process
+DiagrammeR::grViz("digraph {
+
+# initiate graph
+graph [layout = dot, rankdir = LR, label = 'The MICE-Process\n\n',labelloc = t, fontcolor = DarkSlateBlue, fontsize = 45]
+
+# global node settings
+node [shape = rectangle, style = filled, fillcolor = AliceBlue, fontcolor = DarkSlateBlue, fontsize = 35]
+bgcolor = none
+
+# label nodes
+incomplete [label =  'Incomplete data set']
+imputed1 [label = 'Imputed \n data set 1']
+estimates1 [label = 'Estimates from \n analysis 1']
+rubin [label = 'Rubins Rules', shape = diamond]
+combined [label = 'Combined results']
+imputed2 [label = 'Imputed \n data set 2']
+estimates2 [label = 'Estimates from \n analysis 2']
+imputedm [label = 'Imputed \n data set m']
+estimatesm [label = 'Estimates from \n analysis m']
+
+# edge definitions with the node IDs
+incomplete -> imputed1 [arrowhead = vee, color = DarkSlateBlue]
+imputed1 -> estimates1 [arrowhead = vee, color = DarkSlateBlue]
+estimates1 -> rubin [arrowhead = vee, color = DarkSlateBlue]
+incomplete -> imputed2 [arrowhead = vee, color = DarkSlateBlue]
+imputed2 -> estimates2 [arrowhead = vee, color = DarkSlateBlue]
+estimates2-> rubin [arrowhead = vee, color = DarkSlateBlue]
+incomplete -> imputedm [arrowhead = vee, color = DarkSlateBlue]
+imputedm -> estimatesm [arrowhead = vee, color = DarkSlateBlue]
+estimatesm -> rubin [arrowhead = vee, color = DarkSlateBlue]
+rubin -> combined [arrowhead = vee, color = DarkSlateBlue]
+}")
+
+
+
+
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Rubin’s Rules: Donald Rubin was a Harvard professor in the seventies. He felt that the then current method of dealing with missing data (single imputation) was flawed. It was impossible to calculate missing values with absolute certainty because the method did not consider the estimates in error in the predictions of the missing data. For this reason, he recommended creating multiple datasets based on the Bayesian method and doing the following steps:

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  • Impute missing data by averaging the estimates across imputed datasets.
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  • Run regression on each of the imputed datasets.
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  • Find the standard errors and variance of the imputed estimates. Combine them using an adjustment term.
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He felt like this would provide more accurate results than using just one imputed dataset. His theory is referred to as Rubin’s Rules.

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Other Methods of Imputation

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There are other methods of imputation worth noting and are briefly described below.

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Regression Imputation is based on a linear regression model. Missing data is randomly drawn from a conditional distribution when variables are continuous and from a logistic regression model when they are categorical (van_Ginkel et al. 2020).

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Predictive Mean Matching (PMM) is also based on a linear regression model. The approach is the same as regression imputation when it comes to categorical missing values but different for continuous variables. Instead of random draws from a conditional distribution, missing values are based on predicted values of the outcome variable (van_Ginkel et al. 2020).

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Hot Deck (HD) imputation is when a missing value is replaced by a similar observed value, known as “the donor.” It can be either random or deterministic, which is based on a metric or value (Thongsri and Samart 2022). It does not rely on model fitting.

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Stochastic Regression (SR) Imputation is an extension of regression imputation. The process is the same but a residual term from the normal distribution of the regression of the predictor outcome is added to the imputed value (Thongsri and Samart 2022). This maintains the variability of the data.

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Random Forest (RF) Imputation is based on machine learning algorithms. Missing values are first replaced with the mean or mode of that particular variable and then the dataset is split into a training set and a prediction set (Thongsri and Samart 2022). The missing values are replaced with estimates from these sets. This type of imputation can be used on continuous or categorical variables with complex interactions.

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Methodology

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Multiple Imputation by Chained Equations (MICE)

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According to Rubin’s Rules, in multiple imputation m imputed values are created for each of the missing data and result in M complete datasets. For each of the M datasets, an estimate of \(\theta\) is acquired. Let \({\hat{\theta}}_{m}\) and \({\hat{\phi}}_{m}\) be an estimator of the variance of \({\hat{\theta}}_{m}\) based on the Mth complete dataset. The following formulas are used in the multiple imputation process (Arnab 2017) and (Heymans and Eekhout 2019).

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The combined estimator of \(\theta\), the pooled mean difference, is given by the following equation:

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\[{\overline{\theta}} = \displaystyle \frac{1}{M}\sum_{m = 1}^{M} {\hat{\theta}}_{m}\] This is the sum of the m estimators from each of the M imputed datasets which are pooled together and averaged.

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The proposed variance estimator of \({\hat{\theta}}_{M}\), the total variance, is given by the following equation:

+

\[{\hat{\Phi}} = {\overline{\phi}}_{M} + (1+\displaystyle \frac{1}{M})B_{M}\]

+

This formula is made up of two parts: the average within imputation variance and the between imputation variance (with a correction factor of \((1+\displaystyle \frac{1}{M})\)).

+

The imputation variance within, which is the normal standard error, is calculated as follows:

+

\[{\overline{\phi}} = \displaystyle \frac{1}{M}\sum_{m = 1}^{M}{SE}^2_m\]

+

This is the average of the sum of the standard errors squared of the imputed datasets. It will be large in small datasets and small in large datasets.

+

The between imputation variance, which is how much extra variation there is due to the imputation process, is calculated as follows:

+

\[B_{M} = \displaystyle \frac{1}{M-1}\sum_{m = 1}^{M}({\hat{\theta}}_{m}-{\overline{\theta}})^{2}\]

+

This is the sum of the sample variance for each imputed data set multiplied by 1 over the imputed datasets. It will be large when there is a high number of missing data and small when there is a small number of missing data (Heymans and Eekhout 2019). When the between variance is greater than the within variance, then greater efficiency occurs, and more accurate estimates can be achieved by increasing M. Conversely, when the within variance is greater than the between variance, then little is gained by increasing M.

+

To find the total standard error, take the square root of the total variance:

+

\[SE_{pooled} = {\sqrt{\hat{\Phi}}}\]

+

For significance testing, we use the Wald Test:

+

\[Wald_{pooled} = \frac{(\overline{\theta} - {\theta_{0}})^2}{{\hat{\Phi}}}\]

+

The Wald test follows a t-distribution with the degrees of freedom as follows:

+

\[df = (M-1)(1+\frac{1}{M+1} * \frac{\overline{\theta}}{B_{M}})^2\]

+

It results in a 2-sided t-test with the t-value as follows:

+

\[t_{df, 1-\alpha/2}\]

+

To test for significance using F-distribution, we square the t-value:

+

\[ F_{1,df} = t^{2}_{df, 1-\alpha/2}\]

+
+
+

Assumptions

+

All multiple imputation methods have the following assumptions:

+
    +

  • Observed data follow a normal multivariate distribution. If not normal, a transformation is first needed before imputation can be performed.

    • +

  • Missing data are classified as MAR, which is when the distribution of missing data is dependent of the observed values and not the unobserved values.

    • +

  • “The parameters \({\theta}\) of the data model and the parameters \({\phi}\) of the model for the missing values are distinct. That is, knowing the values of \({\theta}\) does not provide any information about \({\phi}\)” (“MI Procedure” 2019).

    • +
+

These assumptions are essential, otherwise we wouldn’t be able to predict a plausible replacement value for the missing data.

+
+
+

Steps:

+

The chained equation process has the following steps (Azur et al. 2011):

+

Step 1: Using simple imputation, replace the missing data with this value, referred to as the “place holder.”

+

Step 2: The “place holder” values for one variable are set back to missing.

+

Step 3: The observed values from this variable (dependent variable) are regressed on the other variables (independent variables) in the model, using the same assumptions when performing linear, logistic, or Poison regression.

+

Step 4: The missing values are replaced with predictions “m” from this newly created model.

+

Step 5: Repeat Steps 2-4 for each variable that have missing values until all missing values have been replaced.

+

Step 6: Repeat Steps 2-4, updating imputations each cycle for as many “M” cycles/imputations that are required.

+
+
+
+

Analysis and Results

+
+
MICE in R
+

Using the MICE (Multivariate Imputation by Chained Equations) package in R, a statistical programming software, we will be demonstrating how to impute missing data. We will run through the MICE process and provide results of the imputations. We will then check the accuracy of the imputations by comparing these results to the results of the original, complete data set.

+
+
+

Data and Visualizations

+
+
Load Data and Packages
Code -
# Load Data
-kable(head(murders))
+
# load libraries
+library(vtable)
+library(gtsummary)
+library(gt)
+library(ggplot2)
+library(VIM, warn.conflicts=FALSE)
+library(dplyr, warn.conflicts=FALSE)
+library(mice, warn.conflicts=FALSE)
+library(tidyverse)
+library(modelsummary)
+
+# load data
+original = read.csv("kc_house_data.csv")
-
- + + +
+
Description of Dataset
+

kc_house_data.csv

+
+
+
Details of Dataset
+

This dataset contains the sale prices of 21,613 houses in King County, Seattle between May 2014 to May 2015. The original dataset contained 21 columns with various selling attributes. For the purpose of this project, we have condensed the variables to the following four: sales price, number of bedrooms, number of bathrooms, and square footage of living space.

+
+
+
Definition of Data in Dataset
+
+++++ - - - - - + + + - - - - - - - - - - - - - - - - - - - + + + - - - - - + + + - - - - - + + + - - - - - + + +
stateabbregionpopulationtotalVariableTypeDescription
AlabamaALSouth4779736135
AlaskaAKWest71023119
ArizonaAZWest6392017232pricenumericSale price of homes sold between May 2014 and May 2015 in King County, Seattle; range from $75,000 to $7,700,000.
ArkansasARSouth291591893bedroomsintegerNumber of bedrooms in homes sold between May 2014 and May 2015 in King County, Seattle; range from 0 to 33.
CaliforniaCAWest372539561257bathroomsnumericNumber of bathrooms in homes sold between May 2014 and May 2015 in King County, Seattle; range from 0 to 8.
ColoradoCOWest502919665sqft_livingintegerNumber of square feet of living space in homes sold between May 2014 and May 2015 in King County, Seattle; range from 290 sf to 13,540 sf.
+
+
+
Summary of Dataset:
+
+
+Code +
# summary of data
+sumtable(original, col.align = c('left', 'center'))
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Summary Statistics
Variable N Mean Std. Dev. Min Pctl. 25 Pctl. 75 Max
price 21613 540182 367362 75000 321950 645000 7700000
bedrooms 21613 3.4 0.93 0 3 4 33
bathrooms 21613 2.1 0.77 0 1.8 2.5 8
sqft_living 21613 2080 918 290 1427 2550 13540
+ +
+
+
+
+
Plotting of Data
+
+
+Code +
# plot the data
+ggplot(original, aes(x=sqft_living, y=price)) +
+  geom_point(alpha=0.5, color='darkblue') +
+  labs(x="sqft_living", y="price")
+
+
+

+
+
+
+
+
Evaluate Dataset
+

First, we evaluate the dataset for missing values by using the is.na() function:

+
+
+Code +
# check for missing data
+sapply(original, function(x) sum(is.na(x)))
+
+
+
      price    bedrooms   bathrooms sqft_living 
+          0           0           0           0
+
+
+

There are no missing values; this is the original, complete dataset. Normally, this is where you would find out how much missing data your dataset contains and you would move on to the next step in imputation. Since our dataset does not contain any missing values, we will need to add an additional step to artificially create some missing data. We do this by first duplicating the dataset. We want to preserve the original dataset so that we can later check the accuracy of the imputation process.

+
+
+Code +
# duplicate dataset
+house <- original
+
+
+

Next, we randomly replace the desired amount of values with NA to mimic missing data. We will assign 200 NA values to the following variables: bedrooms, bathrooms, and sqft_living and 100 NA values to the price variable.

+
+
+Code +
# replace some values with NA to create missing data
+set.seed(10)
+house[sample(1:nrow(house), 200), "sqft_living"] <- NA
+house[sample(1:nrow(house), 200), "bedrooms"] <- NA
+house[sample(1:nrow(house), 200), "bathrooms"] <- NA
+house[sample(1:nrow(house), 100), "price"] <- NA
+
+
+

Using the is.na() function, we can check to confirm we have created missing data:

+
+
+Code +
# check for missing data
+sapply(house, function(x) sum(is.na(x)))
+
+
+
      price    bedrooms   bathrooms sqft_living 
+        100         200         200         200
+
+
+

Our dataset now has 700 NA/missing values. This equates to about 3% of the data (700/21,613).

+
+
+
Checking for Patterns
+

Next, we need to check the missingness by looking for patterns in the dataset by using the md.pattern() function. Of course our missing data was randomly created, so we know we have MCAR.

+
+
+Code +
# check the missingness patterns
+md.pattern(house, rotate.names = TRUE)
+
+
+

+
+
+
      price bedrooms bathrooms sqft_living    
+20920     1        1         1           1   0
+195       1        1         1           0   1
+196       1        1         0           1   1
+2         1        1         0           0   2
+197       1        0         1           1   1
+2         1        0         1           0   2
+1         1        0         0           1   2
+98        0        1         1           1   1
+1         0        1         1           0   2
+1         0        1         0           1   2
+        100      200       200         200 700
+
+
+

Blue are the observed/complete values, and red are the missing values. The numbers on the left indicate how many complete values there are and the numbers on the right show how many variables contain missing data. So for example, the first line shows there are 20,920 complete observations across all four variables. The next line indicates there are 195 complete observations in three of the variables with data missing in the sqft_living variable, so forth and so on all the way down the list. The total number of missing data for each variable is listed across the bottom. This visual gives a breakdown of where the missing data occurs.

+

Another way to look for patterns in the missing data is to use the aggr() function:

+
+
+Code +
# histogram, patterns and count for missing data
+aggr_plot <- aggr(house, col=c('navyblue','red'), numbers=TRUE, sortVars=TRUE, labels=names(data), cex.axis=.7, gap=3, ylab=c("Histogram of missing data","Pattern"))
+
+
+

+
+
+

+ Variables sorted by number of missings: 
+    Variable       Count
+    bedrooms 0.009253690
+   bathrooms 0.009253690
+ sqft_living 0.009253690
+       price 0.004626845
+
+
+

It provides a histogram of the missing values and shows each variable and the percent of missing data for each (as a decimal):

+
    +

  • bedrooms: 0.9%

    • +

  • bathrooms: 0.9%

    • +

  • sqft_living: 0.9%

    • +

  • price: 0.5%

    • +
+

Another visualization to check for the missingness is a margin plot:

+
+
+Code +
# margin plot for missingness
+marginplot(house[c(1,4)])
+
+
+

+
+
+

The red box plot on the left shows the distribution of sqft_living with price missing. The blue box plot shows the distribution of the remaining data. The same is true for the price box plots on the bottom. We expect the red and blue boxplots to be very similar if our assumptions of MCAR data are true. In this case, they are.

+
+
+
Imputation Process
+

Since we are missing about 3% of our data, we need to perform at least 3 imputations. This can be done by using the mice() function. Since 5 is the default, we will use that; if you need to use something different, you must set m equal to the number of imputations that you desire. The set.seed will be given the value 1337 (any number can be used here) to ensure that the same random values are generated each time.

+
+
+Code +
# impute the data 5 times (default)
+imp = mice(data = house, set.seed = 1337)
+
+
+

+ iter imp variable
+  1   1  price  bedrooms  bathrooms  sqft_living
+  1   2  price  bedrooms  bathrooms  sqft_living
+  1   3  price  bedrooms  bathrooms  sqft_living
+  1   4  price  bedrooms  bathrooms  sqft_living
+  1   5  price  bedrooms  bathrooms  sqft_living
+  2   1  price  bedrooms  bathrooms  sqft_living
+  2   2  price  bedrooms  bathrooms  sqft_living
+  2   3  price  bedrooms  bathrooms  sqft_living
+  2   4  price  bedrooms  bathrooms  sqft_living
+  2   5  price  bedrooms  bathrooms  sqft_living
+  3   1  price  bedrooms  bathrooms  sqft_living
+  3   2  price  bedrooms  bathrooms  sqft_living
+  3   3  price  bedrooms  bathrooms  sqft_living
+  3   4  price  bedrooms  bathrooms  sqft_living
+  3   5  price  bedrooms  bathrooms  sqft_living
+  4   1  price  bedrooms  bathrooms  sqft_living
+  4   2  price  bedrooms  bathrooms  sqft_living
+  4   3  price  bedrooms  bathrooms  sqft_living
+  4   4  price  bedrooms  bathrooms  sqft_living
+  4   5  price  bedrooms  bathrooms  sqft_living
+  5   1  price  bedrooms  bathrooms  sqft_living
+  5   2  price  bedrooms  bathrooms  sqft_living
+  5   3  price  bedrooms  bathrooms  sqft_living
+  5   4  price  bedrooms  bathrooms  sqft_living
+  5   5  price  bedrooms  bathrooms  sqft_living
+
+
+

We need to create the imputed dataset using the complete() function. We will not display the results since there are 21,613 rows.

+
+
+Code +
# create imputed dataset
+imputed <- complete(imp)
+
+
+

We now need to check the imputed dataset to make sure there is no missing data. Again, we use the is.na() function:

+
+
+Code +
# check imputed dataset for missing values
+sapply(imputed, function(x) sum(is.na(x)))
+
+
+
      price    bedrooms   bathrooms sqft_living 
+          0           0           0           0
+
+
+

We can check the quality of the imputations by running a strip plot, which is a single axis scatter plot. It will show the distribution of the imputed values (red) over the observed values (blue). We want the imputations to be values that could have been observed had the data not been missing. These look to be acceptable.

+
+
+Code +
# strip plots to check the quality of the imputations
+par(mfrow=c(2,2))
+stripplot(imp, price, pch = 19, cex=1, xlab = "Imputation number")
+
+
+

+
+
+Code +
stripplot(imp, bedrooms, pch = 19, cex=1, xlab = "Imputation number")
+
+
+

+
+
+Code +
stripplot(imp, bathrooms, pch = 19, cex=1, xlab = "Imputation number")
+
+
+

Code -
ggplot1 = murders %>% ggplot(mapping = aes(x=population/10^6, y=total)) 
-
-  ggplot1 + geom_point(aes(col=region), size = 4) +
-  geom_text_repel(aes(label=abb)) +
-  scale_x_log10() +
-  scale_y_log10() +
-  geom_smooth(formula = "y~x", method=lm,se = F)+
-  xlab("Populations in millions (log10 scale)") + 
-  ylab("Total number of murders (log10 scale)") +
-  ggtitle("US Gun Murders in 2010") +
-  scale_color_discrete(name = "Region")+
-      theme_wsj()
+
stripplot(imp, sqft_living, pch = 19, cex=1, xlab = "Imputation number")
-

+

+
+
+
+
Fitting and Pooling Imputed Datasets
+

Next, we will run regression on our imputed datasets, so we will have 5 regression results. We do this with the with() function. For each regression, we get different results.

+
+
+Code +
# fit complete-data model
+fit <- with(imp, lm(price~bedrooms+bathrooms+sqft_living))
+
+
+

Next, we will pool the results to arrive at estimates that will properly account for the missing data. It will give us the estimate, standard error, test statistic, degrees of freedom, and the p-value for each variable.

+
+
+Code +
# pool and summarize the results
+summary(pool(fit))
+
+
+
         term    estimate   std.error  statistic        df       p.value
+1 (Intercept)  75043.7775 6952.563287  10.793685 14431.278  4.677205e-27
+2    bedrooms -57930.4966 2357.322483 -24.574702  7683.745 1.936143e-128
+3   bathrooms   7624.2182 3531.505441   2.158914 11516.540  3.087739e-02
+4 sqft_living    309.7992    3.114475  99.470779  8431.335  0.000000e+00
+
+

We can now analyze our dataset as if there were no missing data!

-
-

Statistical Modeling

+
+
Checking for Accuracy
+

We will now compare the results from our pooled, imputed dataset to our original, complete dataset (before we removed values to create missing data) to check for accuracy of the imputation method by chained equations:

+
+
+Code +
# fit original, complete dataset
+og_lm = lm(price~bedrooms+bathrooms+sqft_living, data = original)
+
+# compare imputed dataset to the original dataset
+msummary(list("Imputed" = pool(fit), "Original" = og_lm), title = "Comparison of Results", statistic = "p.value", estimate = "estimate", gof_omit = 'IC|Log|Adj|F|RMSE')
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Comparison of Results
 Imputed Original
(Intercept) 75043.777 74662.099
(<0.001) (<0.001)
bedrooms −57930.497 −57906.631
(<0.001) (<0.001)
bathrooms 7624.218 7928.708
(0.031) (0.024)
sqft_living 309.799 309.605
(<0.001) (<0.001)
Num.Obs. 21613 21613
Num.Imp. 5
R2 0.507 0.507
+ +
+
+

The estimates from the multiple imputation are very close to the estimates for the original dataset, with the exception of bathrooms which is off just slightly. The p-values are all statistically significant in the original dataset, as well as the imputed dataset, with very minimal differences in their values. These results indicate a high accuracy of the imputation process. We can conclude that multiple imputation by chained equation is a reliable source to impute missing data.

-
-

Conlusion

-
+
+

Conclusion

+

In conclusion, missing data can occur in research for a variety of reasons. It is never a good idea to ignore it. Doing this will lead “to biased estimates of parameters, loss of information, decreased statistical power, and weak reliability of findings” (Dong and Peng 2013). When missing data is discovered, it is important to first identify it and look for missing data patterns. The best course of action is to impute the missing data by using multiple imputation. Next, define the variables in the dataset that are related to the missing values that will be used for imputation. Create the necessary number of complete data sets with imputed values. Run regression on these models and combine them using Rubin’s Rules. Finally, analyze the dataset. Performing these steps will minimize the adverse effects on the analysis caused by missing data (Pampka, Hutcheson, and Williams 2016).

+

References

-
-Bro, Rasmus, and Age K Smilde. 2014. “Principal Component Analysis.” Analytical Methods 6 (9): 2812–31. +
+Arnab, R. 2017. Survey Sampling Theory and Applications. Academic Press. https://www.sciencedirect.com/topics/mathematics/imputation-method. +
+
+Azur, M. J., E. A. Stuart, C. Frangakis, and P. J. Leaf. 2011. “Multiple Imputation by Chained Equations: What Is It and How Does It Work?” Int J Methods Psychiatr Res. 20 (1): 40–49. https://onlinelibrary.wiley.com/doi/epdf/10.1002/mpr.329. +
+
+Cao, Y., H. Allore, B. V. Wyk, and Gutman R. 2021. “Review and Evaluation of Imputation Methods for Multivariate Longitudinal Data with Mixed-Type Incomplete Variables.” Statistics in Medicine 41 (30): 5844–76. https://doi-org.ezproxy.lib.uwf.edu/10.1002/sim.9592. +
+
+Dong, Y., and C. J. Peng. 2013. “Principled Missing Data Methods for Researchers.” SpringerPlus 2 (222). https://doi.org/10.1186/2193-1801-2-222. +
+
+Heymans, M. W., and I. Eekhout. 2019. Applied Missing Data Analysis with SPSS and (r) Studio. Heymans & Eekhout. https://bookdown.org/mwheymans/bookmi/. +
+
+Mainzer, R., M. Moreno-Betancur, C. Nguyen, J. Simpson, J. Carlin, and K. Lee. 2023. “Handling of Missing Data with Multiple Imputation in Observational Studies That Address Causal Questions: Protocol for a Scoping Review.” BMJ Open 13: 1–6. http://dx.doi.org/10.1136/bmjopen-2022-065576. +
+
+“MI Procedure.” 2019. https://documentation.sas.com/doc/en/pgmsascdc/9.4_3.4/statug/statug_mi_details03.htm. +
+
+Pampka, M., G. Hutcheson, and J. Williams. 2016. “Handling Missing Data: Analysis of a Challenging Data Set Using Multiple Imputation.” International Journal of Research & Method in Education 39 (1): 19–37. https://doi.org/10.1080/1743727X.2014.979146. +
+
+Pedersen, A. B., E. M. Mikkelsen, D. Cronin-Fenton, N. R. Kristensen, T. M. Pham, L. Pedersen, and I. Petersen. 2017. “Missing Data and Multiple Imputation in Clinical Epidemiological Research.” Clinical Epidemiology 9: 157–66. https://www.tandfonline.com/doi/full/10.2147/CLEP.S129785. +
+
+Schafer, J. L., and J. W. Graham. 2002. “Missing Data: Our View of the State of the Art.” Psychological Methods 7 (2): 147–77. https://psycnet.apa.org/doi/10.1037/1082-989X.7.2.147.
-
-Efromovich, S. 2008. Nonparametric Curve Estimation: Methods, Theory, and Applications. Springer Series in Statistics. Springer New York. https://books.google.com/books?id=mdoLBwAAQBAJ. +
+Streiner, D. L. 2008. “Missing Data and the Trouble with LOCF.” EBMH 11 (1): 1–5. http://dx.doi.org/10.1136/ebmh.11.1.3-a.
-
-R Core Team. 2019. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org. +
+Thongsri, T., and K. Samart. 2022. “Composite Imputation Method for the Multiple Linear Regression with Missing at Random Data.” International Journal of Mathematics and Mathematics and Computer Science 17 (1): 51–62. http://ijmcs.future-in-tech.net/17.1/R-Samart.pdf. +
+
+van_Ginkel, J. R., M. Linting, R. C. Rippe, and A. van der Voort. 2020. “Rebutting Existing Misconceptions about Multiple Imputation as a Method for Handling Missing Data.” Journal of Personality Assessment 102 (3): 2812–31. https://doi.org/10.1080/00223891.2018.1530680. +
+
+Wongkamthong, C., and O. Akande. 2023. “A Comparative Study of Imputation Methods for Multivariate Ordinal Data.” Journal of Survey Statistics and Methodology 11 (1): 189–212. https://doi.org/10.1093/jssam/smab028. +
+
+Wulff, J. N., and L. E. Jeppesen. 2017. “Multiple Imputation by Chained Equations in Praxis: Guidelines and Review.” Electronics Journal of Business Research Methods 15 (1): 41–56. https://vbn.aau.dk/ws/files/257318283/ejbrm_volume15_issue1_article450.pdf.
+
diff --git a/index.qmd b/index.qmd index 080275f87..d0206f146 100644 --- a/index.qmd +++ b/index.qmd @@ -1,73 +1,684 @@ --- -title: "Report Sample" -author: "Student name" +title: "Missing Data and Imputation" +author: + - Javier Estrada + - Michael Underwood + - Elizabeth Subject-Scott date: '`r Sys.Date()`' -format: +format: html: code-fold: true -course: STA 6257 - Advance Statistical Modeling -bibliography: references.bib # file contains bibtex for references -#always_allow_html: true # this allows to get PDF with HTML features + code-tools: true + css: styles.css + theme: morph + toc: true + toc-location: left +course: STA 6257 - Advanced Statistical Modeling +bibliography: references.bib +editor: + markdown: + wrap: 72 +self-contained: true --- +[Website](https://missingdata-imputation.github.io/STA6257_Missing_Data_and_Imputation/) + +[Slides](https://missingdata-imputation.github.io/STA6257_Missing_Data_and_Imputation/Slides.html) + ## Introduction -Kernel regression is a non-parametric estimator that estimates the conditional expectation of two variables which is random. The goal of a kernel regression is to discover the non-linear relationship between two random variables. To discover the non-linear relationship, kernel estimator or kernel smoothing is the main method to estimate the curve for non-parametric -statistics. In kernel estimator, weight function is known as kernel function [@efr2008]. Cite this paper [@bro2014principal]. +#### Missing Data + +Missing data occurs when there are missing values in a dataset. There +are many reasons why this happens. It can be intentional or +unintentional, and oftentimes there is no way to know for sure what +causes the missing data. There are three categories, otherwise known as +missingness mechanisms, which measure the severity of the missingness +[@mai2023]: + +- Missing completely at random (MCAR) is when the distribution of + missing data is completely independent of any other variables. There + is no relationship between the pattern of missingness and the + observed data. This would be the best-case scenario. + +- Missing at random (MAR) is when the distribution of missing data is + dependent of the observed values. The pattern of missingness is + related to the observed data and not the missing data. + +- Missing not at random (MNAR) is when missing data is neither MCAR or + MAR. The distribution of missing data is dependent on other + variables. The pattern of missingness is related to the observed + data, as well as the missing data. This would be the worst-case + scenario and is nonignorable. + +Figure 1: Graphical Representation of Missingness Mechanisms [@sch2002] + +![](GraphicalRepresentation.png) + +(X are the completely observed variables. Y are the partly missing +variables. Z is the component of the cause of missingness unrelated to X +and Y. R is the missingness.) + +Looking for patterns in the missing data can help us to determine which +category they belong. These mechanisms are important in determining how +to handle the missing data. MCAR would be the best-case scenario but +seldom occurs. MAR and MNAR are more common. Most methods of dealing +with missing data assume a MAR missingness. If data are found to be +MNAR, worst-case scenario, then specialized expertise and additional +steps are needed to model the missingness before handling the missing +data. The problem with ignoring any missing values is that it does not +give a true representation of the dataset and can lead to bias when +analyzing. This reduces the statistical power of the analysis +[@van2020]. + +According to @don2013, to enhance the quality of research, the following +should be followed with regards to missing data: + +- Determine under what conditions the missing data occur. +- Use principled methods to handle the missing data. +- Include and report the treatment of the missing data. + +#### Methods to Deal with Missing Data + +There are three types of methods to deal with missing data, the +likelihood and Bayesian method, weighting methods, or imputation methods +[@cao2021]. + +- Likelihood Bayesian method is when information from a previous + predictive distribution is combined with evidence obtained in a + sample to predict a value. It requires technical coding and advanced + statistical knowledge. + +- The weighting method is a traditional approach when weights from + available data are used to adjust for non-response in a survey. + Inefficiency occurs when there are extreme weights or a need for + many weights. + +- The imputation method is when an estimate from the original dataset + is used to estimate the missing value. There are two types of + imputation: single and multiple. + +#### Deleting Missing Data + +Another way to handle missing data is to simply delete it. Deleting +missing values can lead to the loss of important information regarding +your dataset, and is therefore not recommended. The two types of +deletion methods are listwise and pairwise. + +*Listwise deletion* is when the entire observation is removed from the +dataset. In certain cases when the amount of missing data is small and +the type is MCAR, listwise deletion can be used. There usually won't be +bias but potentially important information may be lost. + +T-tests and chi-square tests can be used to assess pairs of predictor +variables to determine whether the groups' means differ significantly. +According to @van2020, if significant, the null hypothesis is rejected, +therefore, indicating that the missing values are not randomly +distributed throughout the data. This implies that the missing data is +either MAR or MNAR. Conversely, if non-significant, this implies that +the data cannot be MAR. This does not eliminate the possibility that it +is not MNAR--other information about the population is needed to +determine this. + +Whenever missing data is categorized as MAR or MNAR, listwise deletion +would be wasteful, and the analysis biased. Alternate methods of dealing +with the missing data is recommended: either pairwise deletion or +imputation. + +*Pairwise deletion* is when only the missing variable of an observation +is removed. It allows more data to be analyzed than listwise deletion +but limits the ability to make inferences of the total sample. For this +reason, it is recommended to use imputation to properly deal with +missing data. + +#### Preferred Method to Handle Missing Data + +Imputation is the preferred method to handle missing data. It consists +of replacing missing data with an estimate obtained from the original, +available data. After imputation, there will be a full dataset to +analyze. According to @wul2017, 3-5 imputations are sufficient, and 10 +imputations are more than enough. More recent research has indicated +that to improve statistical power, the number of imputations created +should be at least equal to the percent of missing data (5% equals 5 +imputations, 10% equals 10 imputations, 20% equals 20 imputations, etc.) +[@ped2017]. + +*Single, or univariate, imputation* is when only one estimate is used to +replace the missing data. Methods of single imputation include using the +mean, the last observation carried forward, and random imputation. The +following is a brief explanation of each: + +- Using the mean to replace a missing value is a straight-forward + process. The mean of the dataset is calculated, including the + missing value. The mean is then multiplied by the number of + observations in the study. Next, the known values are subtracted + from the product, and this gives an estimate that can be used for + any missing values. The problem with this method is that it can + produce misleading results such as showing the variance and the + confidence interval smaller than they actually are. + +- Last Observation Carried Forward (LOCF) is a technique of replacing + a missing value in longitudinal studies with a previously observed + value, the most recent value is carried forward [@str2008]. The + problem with this method is that it assumes that the previous + observed value is perpetual when in reality that most likely is not + the case. + +- Random imputation is a method of randomly drawing an observation and + using that observation for any of the missing values. The problem + with this method is that it introduces additional variability. + +These single imputation methods are inaccurate. They often result in +underestimation of standard errors or too small p-values [@don2013], +which can cause bias in the analysis. Therefore, multiple imputation is +the superior method because it handles missing data better and provides +less biased results. + +*Multiple, or multivariate, imputation* is when various estimates are +used to replace the missing data by creating multiple datasets from +versions of the original dataset. According to @don2013, it can be done +by using a "regression model, or a sequence of regression models, such +as linear, logistic and Poison," where a set of *m* plausible values are +generated for each unobserved data point, resulting in *M* complete data +sets. The new values are randomly drawn from predictive distributions +either through joint modeling (JM, which is not used much anymore) or +fully conditional specification (FCS) [@won2023]. It is then analyzed +and the results are combined to obtain a single value for the missing +data. + +The purpose of multiple imputation is to create a pool of imputed data +for analysis, but if the pooled results are lacking, then multiple +imputation should not be done [@mai2023]. Another reason not to use +multiple imputation is if there are very few missing values; there may +be no benefit in using it. Also worth noting is some statistical +analyses software already have built-in features to deal with missing +data. + +Multiple imputation by chained methods, otherwise known as MICE, is the +most common and preferred method of multiple imputation [@wul2017]. It +provides a more reliable way to analyze data with missing values. For +this reason, this paper will focus on the methodology and application of +the MICE process. + +Figure 2: Flowchart of the MICE-process based on procedures proposed by +Rubin [@wul2017] + +```{r, warning=FALSE, echo=T, message=FALSE} +# load package +library(DiagrammeR) + +# flowchart of mice process +DiagrammeR::grViz("digraph { + +# initiate graph +graph [layout = dot, rankdir = LR, label = 'The MICE-Process\n\n',labelloc = t, fontcolor = DarkSlateBlue, fontsize = 45] + +# global node settings +node [shape = rectangle, style = filled, fillcolor = AliceBlue, fontcolor = DarkSlateBlue, fontsize = 35] +bgcolor = none + +# label nodes +incomplete [label = 'Incomplete data set'] +imputed1 [label = 'Imputed \n data set 1'] +estimates1 [label = 'Estimates from \n analysis 1'] +rubin [label = 'Rubins Rules', shape = diamond] +combined [label = 'Combined results'] +imputed2 [label = 'Imputed \n data set 2'] +estimates2 [label = 'Estimates from \n analysis 2'] +imputedm [label = 'Imputed \n data set m'] +estimatesm [label = 'Estimates from \n analysis m'] + +# edge definitions with the node IDs +incomplete -> imputed1 [arrowhead = vee, color = DarkSlateBlue] +imputed1 -> estimates1 [arrowhead = vee, color = DarkSlateBlue] +estimates1 -> rubin [arrowhead = vee, color = DarkSlateBlue] +incomplete -> imputed2 [arrowhead = vee, color = DarkSlateBlue] +imputed2 -> estimates2 [arrowhead = vee, color = DarkSlateBlue] +estimates2-> rubin [arrowhead = vee, color = DarkSlateBlue] +incomplete -> imputedm [arrowhead = vee, color = DarkSlateBlue] +imputedm -> estimatesm [arrowhead = vee, color = DarkSlateBlue] +estimatesm -> rubin [arrowhead = vee, color = DarkSlateBlue] +rubin -> combined [arrowhead = vee, color = DarkSlateBlue] +}") +``` + +*Rubin's Rules*: Donald Rubin was a Harvard professor in the seventies. +He felt that the then current method of dealing with missing data +(single imputation) was flawed. It was impossible to calculate missing +values with absolute certainty because the method did not consider the +estimates in error in the predictions of the missing data. For this +reason, he recommended creating multiple datasets based on the Bayesian +method and doing the following steps: + +- Impute missing data by averaging the estimates across imputed + datasets. +- Run regression on each of the imputed datasets. +- Find the standard errors and variance of the imputed estimates. + Combine them using an adjustment term. + +He felt like this would provide more accurate results than using just +one imputed dataset. His theory is referred to as Rubin's Rules. + +#### Other Methods of Imputation + +There are other methods of imputation worth noting and are briefly +described below. + +*Regression Imputation* is based on a linear regression model. Missing +data is randomly drawn from a conditional distribution when variables +are continuous and from a logistic regression model when they are +categorical [@van2020]. + +*Predictive Mean Matching (PMM)* is also based on a linear regression +model. The approach is the same as regression imputation when it comes +to categorical missing values but different for continuous variables. +Instead of random draws from a conditional distribution, missing values +are based on predicted values of the outcome variable [@van2020]. + +*Hot Deck (HD) imputation* is when a missing value is replaced by a +similar observed value, known as "the donor." It can be either random or +deterministic, which is based on a metric or value [@tho2022]. It does +not rely on model fitting. + +*Stochastic Regression (SR) Imputation* is an extension of regression +imputation. The process is the same but a residual term from the normal +distribution of the regression of the predictor outcome is added to the +imputed value [@tho2022]. This maintains the variability of the data. + +*Random Forest (RF) Imputation* is based on machine learning algorithms. +Missing values are first replaced with the mean or mode of that +particular variable and then the dataset is split into a training set +and a prediction set [@tho2022]. The missing values are replaced with +estimates from these sets. This type of imputation can be used on +continuous or categorical variables with complex interactions. + +## Methodology + +#### Multiple Imputation by Chained Equations (MICE) + +According to Rubin's Rules, in multiple imputation *m* imputed values +are created for each of the missing data and result in *M* complete +datasets. For each of the *M* datasets, an estimate of $\theta$ is +acquired. Let ${\hat{\theta}}_{m}$ and ${\hat{\phi}}_{m}$ be an +estimator of the variance of ${\hat{\theta}}_{m}$ based on the *M*th +complete dataset. The following formulas are used in the multiple +imputation process [@arn2017] and [@hey2019]. + +The combined estimator of $\theta$, the pooled mean difference, is given +by the following equation: + +$${\overline{\theta}} = \displaystyle \frac{1}{M}\sum_{m = 1}^{M} {\hat{\theta}}_{m}$$ +This is the sum of the *m* estimators from each of the *M* imputed +datasets which are pooled together and averaged. + +The proposed variance estimator of ${\hat{\theta}}_{M}$, the total +variance, is given by the following equation: + +$${\hat{\Phi}} = {\overline{\phi}}_{M} + (1+\displaystyle \frac{1}{M})B_{M}$$ + +This formula is made up of two parts: the average within imputation +variance and the between imputation variance (with a correction factor +of $(1+\displaystyle \frac{1}{M})$). + +The imputation variance within, which is the normal standard error, is +calculated as follows: + +$${\overline{\phi}} = \displaystyle \frac{1}{M}\sum_{m = 1}^{M}{SE}^2_m$$ + +This is the average of the sum of the standard errors squared of the +imputed datasets. It will be large in small datasets and small in large +datasets. + +The between imputation variance, which is how much extra variation there +is due to the imputation process, is calculated as follows: + +$$B_{M} = \displaystyle \frac{1}{M-1}\sum_{m = 1}^{M}({\hat{\theta}}_{m}-{\overline{\theta}})^{2}$$ + +This is the sum of the sample variance for each imputed data set +multiplied by 1 over the imputed datasets. It will be large when there +is a high number of missing data and small when there is a small number +of missing data [@hey2019]. When the between variance is greater than +the within variance, then greater efficiency occurs, and more accurate +estimates can be achieved by increasing *M*. Conversely, when the within +variance is greater than the between variance, then little is gained by +increasing *M*. + +To find the total standard error, take the square root of the total +variance: + +$$SE_{pooled} = {\sqrt{\hat{\Phi}}}$$ + +For significance testing, we use the Wald Test: + +$$Wald_{pooled} = \frac{(\overline{\theta} - {\theta_{0}})^2}{{\hat{\Phi}}}$$ + +The Wald test follows a t-distribution with the degrees of freedom as +follows: + +$$df = (M-1)(1+\frac{1}{M+1} * \frac{\overline{\theta}}{B_{M}})^2$$ + +It results in a 2-sided t-test with the t-value as follows: + +$$t_{df, 1-\alpha/2}$$ + +To test for significance using F-distribution, we square the t-value: + +$$ F_{1,df} = t^{2}_{df, 1-\alpha/2}$$ + +#### Assumptions + +All multiple imputation methods have the following assumptions: + +- Observed data follow a normal multivariate distribution. If not + normal, a transformation is first needed before imputation can be + performed. + +- Missing data are classified as MAR, which is when the distribution + of missing data is dependent of the observed values and not the + unobserved values. + +- "The parameters ${\theta}$ of the data model and the parameters ${\phi}$ of the model for the missing values are distinct. That is, knowing the values of ${\theta}$ does not provide any information about ${\phi}$" [@mip2019]. + +These assumptions are essential, otherwise we wouldn't be able to +predict a plausible replacement value for the missing data. + +#### Steps: + +The chained equation process has the following steps [@azu2011]: + +*Step 1:* Using simple imputation, replace the missing data with this +value, referred to as the "place holder." + +*Step 2:* The "place holder" values for one variable are set back to +missing. -## Methods +*Step 3:* The observed values from this variable (dependent variable) +are regressed on the other variables (independent variables) in the +model, using the same assumptions when performing linear, logistic, or +Poison regression. +*Step 4:* The missing values are replaced with predictions "m" from this +newly created model. -The common non-parametric regression model is $Y_i = m(X_i) + \varepsilon_i$, where -$Y_i$ can be defined as the sum of the regression function value $m(x)$ for $X_i$. -Here $m(x)$ is unknown and $\varepsilon_i$ some errors. With the help of this definition, we can create the estimation for -local averaging i.e. $m(x)$ can be estimated with the product of $Y_i$ average -and $X_i$ is near to $x$. In other words, this means that we are discovering -the line through the data points with the help of surrounding data points. -The estimation formula is printed below [@R-base]: +*Step 5:* Repeat Steps 2-4 for each variable that have missing values +until all missing values have been replaced. -$$ -M_n(x) = \sum_{i=1}^{n} W_n (X_i) Y_i \tag{1} -$$ -$W_n(x)$ is the sum of weights that belongs to all real numbers. Weights -are positive numbers and small if $X_i$ is far from $x$. +*Step 6:* Repeat Steps 2-4, updating imputations each cycle for as many +"M" cycles/imputations that are required. ## Analysis and Results -### Data and Vizualisation -A study was conducted to determine how... + +##### MICE in R + +Using the MICE (Multivariate Imputation by Chained Equations) package in +R, a statistical programming software, we will be demonstrating how to +impute missing data. We will run through the MICE process and provide +results of the imputations. We will then check the accuracy of the +imputations by comparing these results to the results of the original, +complete data set. + +### Data and Visualizations + +##### Load Data and Packages ```{r, warning=FALSE, echo=T, message=FALSE} -# loading packages +# load libraries +library(vtable) +library(gtsummary) +library(gt) +library(ggplot2) +library(VIM, warn.conflicts=FALSE) +library(dplyr, warn.conflicts=FALSE) +library(mice, warn.conflicts=FALSE) library(tidyverse) -library(knitr) -library(ggthemes) -library(ggrepel) -library(dslabs) +library(modelsummary) + +# load data +original = read.csv("kc_house_data.csv") +``` + +##### Description of Dataset + +kc_house_data.csv + +##### Details of Dataset + +This dataset contains the sale prices of 21,613 houses in King County, +Seattle between May 2014 to May 2015. The original dataset contained 21 +columns with various selling attributes. For the purpose of this +project, we have condensed the variables to the following four: sales +price, number of bedrooms, number of bathrooms, and square footage of +living space. + +##### Definition of Data in Dataset + +| Variable | Type | Description | +|------------------|------------------|------------------------------------| +| price | numeric | Sale price of homes sold between May 2014 and May 2015 in King County, Seattle; range from \$75,000 to \$7,700,000. | +| bedrooms | integer | Number of bedrooms in homes sold between May 2014 and May 2015 in King County, Seattle; range from 0 to 33. | +| bathrooms | numeric | Number of bathrooms in homes sold between May 2014 and May 2015 in King County, Seattle; range from 0 to 8. | +| sqft_living | integer | Number of square feet of living space in homes sold between May 2014 and May 2015 in King County, Seattle; range from 290 sf to 13,540 sf. | + +##### Summary of Dataset: + +```{r, warning=FALSE, echo=T, message=FALSE} +# summary of data +sumtable(original, col.align = c('left', 'center')) +``` + +##### Plotting of Data + +```{r, warning=FALSE, echo=T, message=FALSE} +# plot the data +ggplot(original, aes(x=sqft_living, y=price)) + + geom_point(alpha=0.5, color='darkblue') + + labs(x="sqft_living", y="price") +``` + +##### Evaluate Dataset + +First, we evaluate the dataset for missing values by using the is.na() +function: + +```{r, warning=FALSE, echo=T, message=FALSE} +# check for missing data +sapply(original, function(x) sum(is.na(x))) +``` + +There are no missing values; this is the original, complete dataset. Normally, this is where you would find out how much missing data your dataset contains and you would move on to the next step in imputation. Since our dataset does not contain any missing values, we +will need to add an additional step to artificially create some missing data. We do this by first +duplicating the dataset. We want to preserve the original dataset so +that we can later check the accuracy of the imputation process. + +```{r, warning=FALSE, echo=T, message=FALSE} +# duplicate dataset +house <- original +``` + +Next, we randomly replace the desired amount of values with NA to mimic +missing data. We will assign 200 NA values to the following variables: +bedrooms, bathrooms, and sqft_living and 100 NA values to the price +variable. + +```{r, warning=FALSE, echo=T, message=FALSE} +# replace some values with NA to create missing data +set.seed(10) +house[sample(1:nrow(house), 200), "sqft_living"] <- NA +house[sample(1:nrow(house), 200), "bedrooms"] <- NA +house[sample(1:nrow(house), 200), "bathrooms"] <- NA +house[sample(1:nrow(house), 100), "price"] <- NA +``` + +Using the is.na() function, we can check to confirm we have created +missing data: + +```{r, warning=FALSE, echo=T, message=FALSE} +# check for missing data +sapply(house, function(x) sum(is.na(x))) +``` + +Our dataset now has 700 NA/missing values. This equates to about 3% of +the data (700/21,613). + +##### Checking for Patterns + +Next, we need to check the missingness by looking for patterns in the +dataset by using the md.pattern() function. Of course our missing data +was randomly created, so we know we have MCAR. + +```{r, warning=FALSE, echo=T, message=FALSE} +# check the missingness patterns +md.pattern(house, rotate.names = TRUE) +``` + +Blue are the observed/complete values, and red are the missing values. +The numbers on the left indicate how many complete values there are and +the numbers on the right show how many variables contain missing data. +So for example, the first line shows there are 20,920 complete +observations across all four variables. The next line indicates there +are 195 complete observations in three of the variables with data +missing in the sqft_living variable, so forth and so on all the way down +the list. The total number of missing data for each variable is listed +across the bottom. This visual gives a breakdown of where the missing +data occurs. + +Another way to look for patterns in the missing data is to use the +aggr() function: + +```{r, warning=FALSE, echo=T, message=FALSE} +# histogram, patterns and count for missing data +aggr_plot <- aggr(house, col=c('navyblue','red'), numbers=TRUE, sortVars=TRUE, labels=names(data), cex.axis=.7, gap=3, ylab=c("Histogram of missing data","Pattern")) +``` + +It provides a histogram of the missing values and shows each variable +and the percent of missing data for each (as a decimal): + +- bedrooms: 0.9% + +- bathrooms: 0.9% + +- sqft_living: 0.9% + +- price: 0.5% + +Another visualization to check for the missingness is a margin plot: + +```{r, warning=FALSE, echo=T, message=FALSE} +# margin plot for missingness +marginplot(house[c(1,4)]) ``` +The red box plot on the left shows the distribution of sqft_living with +price missing. The blue box plot shows the distribution of the remaining +data. The same is true for the price box plots on the bottom. We expect +the red and blue boxplots to be very similar if our assumptions of MCAR +data are true. In this case, they are. + +##### Imputation Process -```{r, warning=FALSE, echo=TRUE} -# Load Data -kable(head(murders)) +Since we are missing about 3% of our data, we need to perform at least 3 +imputations. This can be done by using the mice() function. Since 5 is +the default, we will use that; if you need to use something different, +you must set m equal to the number of imputations that you desire. The +set.seed will be given the value 1337 (any number can be used here) to +ensure that the same random values are generated each time. + +```{r, warning=FALSE, echo=T, message=FALSE} +# impute the data 5 times (default) +imp = mice(data = house, set.seed = 1337) +``` -ggplot1 = murders %>% ggplot(mapping = aes(x=population/10^6, y=total)) +We need to create the imputed dataset using the complete() function. We will not display the results since there are 21,613 rows. - ggplot1 + geom_point(aes(col=region), size = 4) + - geom_text_repel(aes(label=abb)) + - scale_x_log10() + - scale_y_log10() + - geom_smooth(formula = "y~x", method=lm,se = F)+ - xlab("Populations in millions (log10 scale)") + - ylab("Total number of murders (log10 scale)") + - ggtitle("US Gun Murders in 2010") + - scale_color_discrete(name = "Region")+ - theme_wsj() - +```{r, warning=FALSE, echo=T, message=FALSE} +# create imputed dataset +imputed <- complete(imp) +``` + +We now need to check the imputed dataset to make sure there is no +missing data. Again, we use the is.na() function: + +```{r, warning=FALSE, echo=T, message=FALSE} +# check imputed dataset for missing values +sapply(imputed, function(x) sum(is.na(x))) +``` + +We can check the quality of the imputations by running a strip plot, +which is a single axis scatter plot. It will show the distribution of +the imputed values (red) over the observed values (blue). We want the +imputations to be values that could have been observed had the data not +been missing. These look to be acceptable. + +```{r, warning=FALSE, echo=T, message=FALSE} +# strip plots to check the quality of the imputations +par(mfrow=c(2,2)) +stripplot(imp, price, pch = 19, cex=1, xlab = "Imputation number") +stripplot(imp, bedrooms, pch = 19, cex=1, xlab = "Imputation number") +stripplot(imp, bathrooms, pch = 19, cex=1, xlab = "Imputation number") +stripplot(imp, sqft_living, pch = 19, cex=1, xlab = "Imputation number") +``` + +##### Fitting and Pooling Imputed Datasets + +Next, we will run regression on our imputed datasets, so we will have 5 +regression results. We do this with the with() function. For each +regression, we get different results. + +```{r, warning=FALSE, echo=T, message=FALSE} +# fit complete-data model +fit <- with(imp, lm(price~bedrooms+bathrooms+sqft_living)) +``` + +Next, we will pool the results to arrive at estimates that will properly +account for the missing data. It will give us the estimate, standard +error, test statistic, degrees of freedom, and the p-value for each +variable. + +```{r, warning=FALSE, echo=T, message=FALSE} +# pool and summarize the results +summary(pool(fit)) + + +``` + +We can now analyze our dataset as if there were no missing data! + +##### Checking for Accuracy + +We will now compare the results from our pooled, imputed dataset to our +original, complete dataset (before we removed values to create missing +data) to check for accuracy of the imputation method by chained +equations: + +```{r, warning=FALSE, echo=T, message=FALSE} +# fit original, complete dataset +og_lm = lm(price~bedrooms+bathrooms+sqft_living, data = original) +# compare imputed dataset to the original dataset +msummary(list("Imputed" = pool(fit), "Original" = og_lm), title = "Comparison of Results", statistic = "p.value", estimate = "estimate", gof_omit = 'IC|Log|Adj|F|RMSE') ``` -### Statistical Modeling +The estimates from the multiple imputation are very close to the +estimates for the original dataset, with the exception of bathrooms +which is off just slightly. The p-values are all statistically +significant in the original dataset, as well as the imputed dataset, +with very minimal differences in their values. These results indicate a +high accuracy of the imputation process. We can conclude that multiple +imputation by chained equation is a reliable source to impute missing +data. -### Conlusion +## Conclusion -## References +In conclusion, missing data can occur in research for a variety of +reasons. It is never a good idea to ignore it. Doing this will lead "to +biased estimates of parameters, loss of information, decreased +statistical power, and weak reliability of findings" [@don2013]. When +missing data is discovered, it is important to first identify it and +look for missing data patterns. The best course of action is to impute +the missing data by using multiple imputation. Next, define the +variables in the dataset that are related to the missing values that +will be used for imputation. Create the necessary number of complete +data sets with imputed values. Run regression on these models and +combine them using Rubin's Rules. Finally, analyze the dataset. +Performing these steps will minimize the adverse effects on the analysis +caused by missing data [@pam2016]. diff --git a/index_files/figure-html/unnamed-chunk-2-1.png b/index_files/figure-html/unnamed-chunk-2-1.png deleted file mode 100644 index 623942f8e..000000000 Binary files a/index_files/figure-html/unnamed-chunk-2-1.png and /dev/null differ diff --git a/index_files/libs/bootstrap/bootstrap-icons.css b/index_files/libs/bootstrap/bootstrap-icons.css deleted file mode 100644 index f51d04bc9..000000000 --- a/index_files/libs/bootstrap/bootstrap-icons.css +++ /dev/null @@ -1,1704 +0,0 @@ -@font-face { - font-family: "bootstrap-icons"; - src: -url("./bootstrap-icons.woff?524846017b983fc8ded9325d94ed40f3") format("woff"); -} - -.bi::before, -[class^="bi-"]::before, -[class*=" bi-"]::before { - display: inline-block; - font-family: bootstrap-icons !important; - font-style: normal; - font-weight: normal !important; - font-variant: normal; - text-transform: 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content: "\f580"; } -.bi-spellcheck::before { content: "\f581"; } -.bi-square-fill::before { content: "\f582"; } -.bi-square-half::before { content: "\f583"; } -.bi-square::before { content: "\f584"; } -.bi-stack::before { content: "\f585"; } -.bi-star-fill::before { content: "\f586"; } -.bi-star-half::before { content: "\f587"; } -.bi-star::before { content: "\f588"; } -.bi-stars::before { content: "\f589"; } -.bi-stickies-fill::before { content: "\f58a"; } -.bi-stickies::before { content: "\f58b"; } -.bi-sticky-fill::before { content: "\f58c"; } -.bi-sticky::before { content: "\f58d"; } -.bi-stop-btn-fill::before { content: "\f58e"; } -.bi-stop-btn::before { content: "\f58f"; } -.bi-stop-circle-fill::before { content: "\f590"; } -.bi-stop-circle::before { content: "\f591"; } -.bi-stop-fill::before { content: "\f592"; } -.bi-stop::before { content: "\f593"; } -.bi-stoplights-fill::before { content: "\f594"; } -.bi-stoplights::before { content: "\f595"; } -.bi-stopwatch-fill::before { 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document.documentElement&&[].concat(...document.body.children).forEach((t=>j.off(t,"mouseover",d))),this._popper&&this._popper.destroy(),this._menu.classList.remove(Je),this._element.classList.remove(Je),this._element.setAttribute("aria-expanded","false"),U.removeDataAttribute(this._menu,"popper"),j.trigger(this._element,"hidden.bs.dropdown",t))}_getConfig(t){if(t={...this.constructor.Default,...U.getDataAttributes(this._element),...t},a(Ue,t,this.constructor.DefaultType),"object"==typeof t.reference&&!o(t.reference)&&"function"!=typeof t.reference.getBoundingClientRect)throw new TypeError(`${Ue.toUpperCase()}: Option "reference" provided type "object" without a required "getBoundingClientRect" method.`);return t}_createPopper(t){if(void 0===Fe)throw new TypeError("Bootstrap's dropdowns require Popper (https://popper.js.org)");let e=this._element;"parent"===this._config.reference?e=t:o(this._config.reference)?e=r(this._config.reference):"object"==typeof this._config.reference&&(e=this._config.reference);const i=this._getPopperConfig(),n=i.modifiers.find((t=>"applyStyles"===t.name&&!1===t.enabled));this._popper=qe(e,this._menu,i),n&&U.setDataAttribute(this._menu,"popper","static")}_isShown(t=this._element){return t.classList.contains(Je)}_getMenuElement(){return V.next(this._element,ei)[0]}_getPlacement(){const t=this._element.parentNode;if(t.classList.contains("dropend"))return ri;if(t.classList.contains("dropstart"))return ai;const e="end"===getComputedStyle(this._menu).getPropertyValue("--bs-position").trim();return t.classList.contains("dropup")?e?ni:ii:e?oi:si}_detectNavbar(){return null!==this._element.closest(".navbar")}_getOffset(){const{offset:t}=this._config;return"string"==typeof t?t.split(",").map((t=>Number.parseInt(t,10))):"function"==typeof t?e=>t(e,this._element):t}_getPopperConfig(){const t={placement:this._getPlacement(),modifiers:[{name:"preventOverflow",options:{boundary:this._config.boundary}},{name:"offset",options:{offset:this._getOffset()}}]};return"static"===this._config.display&&(t.modifiers=[{name:"applyStyles",enabled:!1}]),{...t,..."function"==typeof this._config.popperConfig?this._config.popperConfig(t):this._config.popperConfig}}_selectMenuItem({key:t,target:e}){const i=V.find(".dropdown-menu .dropdown-item:not(.disabled):not(:disabled)",this._menu).filter(l);i.length&&v(i,e,t===Ye,!i.includes(e)).focus()}static jQueryInterface(t){return this.each((function(){const e=hi.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}static clearMenus(t){if(t&&(2===t.button||"keyup"===t.type&&"Tab"!==t.key))return;const e=V.find(ti);for(let i=0,n=e.length;ie+t)),this._setElementAttributes(di,"paddingRight",(e=>e+t)),this._setElementAttributes(ui,"marginRight",(e=>e-t))}_disableOverFlow(){this._saveInitialAttribute(this._element,"overflow"),this._element.style.overflow="hidden"}_setElementAttributes(t,e,i){const n=this.getWidth();this._applyManipulationCallback(t,(t=>{if(t!==this._element&&window.innerWidth>t.clientWidth+n)return;this._saveInitialAttribute(t,e);const s=window.getComputedStyle(t)[e];t.style[e]=`${i(Number.parseFloat(s))}px`}))}reset(){this._resetElementAttributes(this._element,"overflow"),this._resetElementAttributes(this._element,"paddingRight"),this._resetElementAttributes(di,"paddingRight"),this._resetElementAttributes(ui,"marginRight")}_saveInitialAttribute(t,e){const i=t.style[e];i&&U.setDataAttribute(t,e,i)}_resetElementAttributes(t,e){this._applyManipulationCallback(t,(t=>{const i=U.getDataAttribute(t,e);void 0===i?t.style.removeProperty(e):(U.removeDataAttribute(t,e),t.style[e]=i)}))}_applyManipulationCallback(t,e){o(t)?e(t):V.find(t,this._element).forEach(e)}isOverflowing(){return this.getWidth()>0}}const pi={className:"modal-backdrop",isVisible:!0,isAnimated:!1,rootElement:"body",clickCallback:null},mi={className:"string",isVisible:"boolean",isAnimated:"boolean",rootElement:"(element|string)",clickCallback:"(function|null)"},gi="show",_i="mousedown.bs.backdrop";class bi{constructor(t){this._config=this._getConfig(t),this._isAppended=!1,this._element=null}show(t){this._config.isVisible?(this._append(),this._config.isAnimated&&u(this._getElement()),this._getElement().classList.add(gi),this._emulateAnimation((()=>{_(t)}))):_(t)}hide(t){this._config.isVisible?(this._getElement().classList.remove(gi),this._emulateAnimation((()=>{this.dispose(),_(t)}))):_(t)}_getElement(){if(!this._element){const t=document.createElement("div");t.className=this._config.className,this._config.isAnimated&&t.classList.add("fade"),this._element=t}return this._element}_getConfig(t){return(t={...pi,..."object"==typeof t?t:{}}).rootElement=r(t.rootElement),a("backdrop",t,mi),t}_append(){this._isAppended||(this._config.rootElement.append(this._getElement()),j.on(this._getElement(),_i,(()=>{_(this._config.clickCallback)})),this._isAppended=!0)}dispose(){this._isAppended&&(j.off(this._element,_i),this._element.remove(),this._isAppended=!1)}_emulateAnimation(t){b(t,this._getElement(),this._config.isAnimated)}}const vi={trapElement:null,autofocus:!0},yi={trapElement:"element",autofocus:"boolean"},wi=".bs.focustrap",Ei="backward";class Ai{constructor(t){this._config=this._getConfig(t),this._isActive=!1,this._lastTabNavDirection=null}activate(){const{trapElement:t,autofocus:e}=this._config;this._isActive||(e&&t.focus(),j.off(document,wi),j.on(document,"focusin.bs.focustrap",(t=>this._handleFocusin(t))),j.on(document,"keydown.tab.bs.focustrap",(t=>this._handleKeydown(t))),this._isActive=!0)}deactivate(){this._isActive&&(this._isActive=!1,j.off(document,wi))}_handleFocusin(t){const{target:e}=t,{trapElement:i}=this._config;if(e===document||e===i||i.contains(e))return;const n=V.focusableChildren(i);0===n.length?i.focus():this._lastTabNavDirection===Ei?n[n.length-1].focus():n[0].focus()}_handleKeydown(t){"Tab"===t.key&&(this._lastTabNavDirection=t.shiftKey?Ei:"forward")}_getConfig(t){return t={...vi,..."object"==typeof t?t:{}},a("focustrap",t,yi),t}}const Ti="modal",Oi="Escape",Ci={backdrop:!0,keyboard:!0,focus:!0},ki={backdrop:"(boolean|string)",keyboard:"boolean",focus:"boolean"},Li="hidden.bs.modal",xi="show.bs.modal",Di="resize.bs.modal",Si="click.dismiss.bs.modal",Ni="keydown.dismiss.bs.modal",Ii="mousedown.dismiss.bs.modal",Pi="modal-open",ji="show",Mi="modal-static";class Hi extends B{constructor(t,e){super(t),this._config=this._getConfig(e),this._dialog=V.findOne(".modal-dialog",this._element),this._backdrop=this._initializeBackDrop(),this._focustrap=this._initializeFocusTrap(),this._isShown=!1,this._ignoreBackdropClick=!1,this._isTransitioning=!1,this._scrollBar=new fi}static get Default(){return Ci}static get NAME(){return Ti}toggle(t){return this._isShown?this.hide():this.show(t)}show(t){this._isShown||this._isTransitioning||j.trigger(this._element,xi,{relatedTarget:t}).defaultPrevented||(this._isShown=!0,this._isAnimated()&&(this._isTransitioning=!0),this._scrollBar.hide(),document.body.classList.add(Pi),this._adjustDialog(),this._setEscapeEvent(),this._setResizeEvent(),j.on(this._dialog,Ii,(()=>{j.one(this._element,"mouseup.dismiss.bs.modal",(t=>{t.target===this._element&&(this._ignoreBackdropClick=!0)}))})),this._showBackdrop((()=>this._showElement(t))))}hide(){if(!this._isShown||this._isTransitioning)return;if(j.trigger(this._element,"hide.bs.modal").defaultPrevented)return;this._isShown=!1;const t=this._isAnimated();t&&(this._isTransitioning=!0),this._setEscapeEvent(),this._setResizeEvent(),this._focustrap.deactivate(),this._element.classList.remove(ji),j.off(this._element,Si),j.off(this._dialog,Ii),this._queueCallback((()=>this._hideModal()),this._element,t)}dispose(){[window,this._dialog].forEach((t=>j.off(t,".bs.modal"))),this._backdrop.dispose(),this._focustrap.deactivate(),super.dispose()}handleUpdate(){this._adjustDialog()}_initializeBackDrop(){return new bi({isVisible:Boolean(this._config.backdrop),isAnimated:this._isAnimated()})}_initializeFocusTrap(){return new Ai({trapElement:this._element})}_getConfig(t){return t={...Ci,...U.getDataAttributes(this._element),..."object"==typeof t?t:{}},a(Ti,t,ki),t}_showElement(t){const e=this._isAnimated(),i=V.findOne(".modal-body",this._dialog);this._element.parentNode&&this._element.parentNode.nodeType===Node.ELEMENT_NODE||document.body.append(this._element),this._element.style.display="block",this._element.removeAttribute("aria-hidden"),this._element.setAttribute("aria-modal",!0),this._element.setAttribute("role","dialog"),this._element.scrollTop=0,i&&(i.scrollTop=0),e&&u(this._element),this._element.classList.add(ji),this._queueCallback((()=>{this._config.focus&&this._focustrap.activate(),this._isTransitioning=!1,j.trigger(this._element,"shown.bs.modal",{relatedTarget:t})}),this._dialog,e)}_setEscapeEvent(){this._isShown?j.on(this._element,Ni,(t=>{this._config.keyboard&&t.key===Oi?(t.preventDefault(),this.hide()):this._config.keyboard||t.key!==Oi||this._triggerBackdropTransition()})):j.off(this._element,Ni)}_setResizeEvent(){this._isShown?j.on(window,Di,(()=>this._adjustDialog())):j.off(window,Di)}_hideModal(){this._element.style.display="none",this._element.setAttribute("aria-hidden",!0),this._element.removeAttribute("aria-modal"),this._element.removeAttribute("role"),this._isTransitioning=!1,this._backdrop.hide((()=>{document.body.classList.remove(Pi),this._resetAdjustments(),this._scrollBar.reset(),j.trigger(this._element,Li)}))}_showBackdrop(t){j.on(this._element,Si,(t=>{this._ignoreBackdropClick?this._ignoreBackdropClick=!1:t.target===t.currentTarget&&(!0===this._config.backdrop?this.hide():"static"===this._config.backdrop&&this._triggerBackdropTransition())})),this._backdrop.show(t)}_isAnimated(){return this._element.classList.contains("fade")}_triggerBackdropTransition(){if(j.trigger(this._element,"hidePrevented.bs.modal").defaultPrevented)return;const{classList:t,scrollHeight:e,style:i}=this._element,n=e>document.documentElement.clientHeight;!n&&"hidden"===i.overflowY||t.contains(Mi)||(n||(i.overflowY="hidden"),t.add(Mi),this._queueCallback((()=>{t.remove(Mi),n||this._queueCallback((()=>{i.overflowY=""}),this._dialog)}),this._dialog),this._element.focus())}_adjustDialog(){const t=this._element.scrollHeight>document.documentElement.clientHeight,e=this._scrollBar.getWidth(),i=e>0;(!i&&t&&!m()||i&&!t&&m())&&(this._element.style.paddingLeft=`${e}px`),(i&&!t&&!m()||!i&&t&&m())&&(this._element.style.paddingRight=`${e}px`)}_resetAdjustments(){this._element.style.paddingLeft="",this._element.style.paddingRight=""}static jQueryInterface(t,e){return this.each((function(){const i=Hi.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===i[t])throw new TypeError(`No method named "${t}"`);i[t](e)}}))}}j.on(document,"click.bs.modal.data-api",'[data-bs-toggle="modal"]',(function(t){const e=n(this);["A","AREA"].includes(this.tagName)&&t.preventDefault(),j.one(e,xi,(t=>{t.defaultPrevented||j.one(e,Li,(()=>{l(this)&&this.focus()}))}));const i=V.findOne(".modal.show");i&&Hi.getInstance(i).hide(),Hi.getOrCreateInstance(e).toggle(this)})),R(Hi),g(Hi);const Bi="offcanvas",Ri={backdrop:!0,keyboard:!0,scroll:!1},Wi={backdrop:"boolean",keyboard:"boolean",scroll:"boolean"},$i="show",zi=".offcanvas.show",qi="hidden.bs.offcanvas";class Fi extends B{constructor(t,e){super(t),this._config=this._getConfig(e),this._isShown=!1,this._backdrop=this._initializeBackDrop(),this._focustrap=this._initializeFocusTrap(),this._addEventListeners()}static get NAME(){return Bi}static get Default(){return Ri}toggle(t){return this._isShown?this.hide():this.show(t)}show(t){this._isShown||j.trigger(this._element,"show.bs.offcanvas",{relatedTarget:t}).defaultPrevented||(this._isShown=!0,this._element.style.visibility="visible",this._backdrop.show(),this._config.scroll||(new fi).hide(),this._element.removeAttribute("aria-hidden"),this._element.setAttribute("aria-modal",!0),this._element.setAttribute("role","dialog"),this._element.classList.add($i),this._queueCallback((()=>{this._config.scroll||this._focustrap.activate(),j.trigger(this._element,"shown.bs.offcanvas",{relatedTarget:t})}),this._element,!0))}hide(){this._isShown&&(j.trigger(this._element,"hide.bs.offcanvas").defaultPrevented||(this._focustrap.deactivate(),this._element.blur(),this._isShown=!1,this._element.classList.remove($i),this._backdrop.hide(),this._queueCallback((()=>{this._element.setAttribute("aria-hidden",!0),this._element.removeAttribute("aria-modal"),this._element.removeAttribute("role"),this._element.style.visibility="hidden",this._config.scroll||(new fi).reset(),j.trigger(this._element,qi)}),this._element,!0)))}dispose(){this._backdrop.dispose(),this._focustrap.deactivate(),super.dispose()}_getConfig(t){return t={...Ri,...U.getDataAttributes(this._element),..."object"==typeof t?t:{}},a(Bi,t,Wi),t}_initializeBackDrop(){return new bi({className:"offcanvas-backdrop",isVisible:this._config.backdrop,isAnimated:!0,rootElement:this._element.parentNode,clickCallback:()=>this.hide()})}_initializeFocusTrap(){return new Ai({trapElement:this._element})}_addEventListeners(){j.on(this._element,"keydown.dismiss.bs.offcanvas",(t=>{this._config.keyboard&&"Escape"===t.key&&this.hide()}))}static jQueryInterface(t){return this.each((function(){const e=Fi.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t](this)}}))}}j.on(document,"click.bs.offcanvas.data-api",'[data-bs-toggle="offcanvas"]',(function(t){const e=n(this);if(["A","AREA"].includes(this.tagName)&&t.preventDefault(),c(this))return;j.one(e,qi,(()=>{l(this)&&this.focus()}));const i=V.findOne(zi);i&&i!==e&&Fi.getInstance(i).hide(),Fi.getOrCreateInstance(e).toggle(this)})),j.on(window,"load.bs.offcanvas.data-api",(()=>V.find(zi).forEach((t=>Fi.getOrCreateInstance(t).show())))),R(Fi),g(Fi);const Ui=new Set(["background","cite","href","itemtype","longdesc","poster","src","xlink:href"]),Vi=/^(?:(?:https?|mailto|ftp|tel|file|sms):|[^#&/:?]*(?:[#/?]|$))/i,Ki=/^data:(?:image\/(?:bmp|gif|jpeg|jpg|png|tiff|webp)|video\/(?:mpeg|mp4|ogg|webm)|audio\/(?:mp3|oga|ogg|opus));base64,[\d+/a-z]+=*$/i,Xi=(t,e)=>{const i=t.nodeName.toLowerCase();if(e.includes(i))return!Ui.has(i)||Boolean(Vi.test(t.nodeValue)||Ki.test(t.nodeValue));const n=e.filter((t=>t instanceof RegExp));for(let t=0,e=n.length;t{Xi(t,r)||i.removeAttribute(t.nodeName)}))}return n.body.innerHTML}const Qi="tooltip",Gi=new Set(["sanitize","allowList","sanitizeFn"]),Zi={animation:"boolean",template:"string",title:"(string|element|function)",trigger:"string",delay:"(number|object)",html:"boolean",selector:"(string|boolean)",placement:"(string|function)",offset:"(array|string|function)",container:"(string|element|boolean)",fallbackPlacements:"array",boundary:"(string|element)",customClass:"(string|function)",sanitize:"boolean",sanitizeFn:"(null|function)",allowList:"object",popperConfig:"(null|object|function)"},Ji={AUTO:"auto",TOP:"top",RIGHT:m()?"left":"right",BOTTOM:"bottom",LEFT:m()?"right":"left"},tn={animation:!0,template:'',trigger:"hover focus",title:"",delay:0,html:!1,selector:!1,placement:"top",offset:[0,0],container:!1,fallbackPlacements:["top","right","bottom","left"],boundary:"clippingParents",customClass:"",sanitize:!0,sanitizeFn:null,allowList:{"*":["class","dir","id","lang","role",/^aria-[\w-]*$/i],a:["target","href","title","rel"],area:[],b:[],br:[],col:[],code:[],div:[],em:[],hr:[],h1:[],h2:[],h3:[],h4:[],h5:[],h6:[],i:[],img:["src","srcset","alt","title","width","height"],li:[],ol:[],p:[],pre:[],s:[],small:[],span:[],sub:[],sup:[],strong:[],u:[],ul:[]},popperConfig:null},en={HIDE:"hide.bs.tooltip",HIDDEN:"hidden.bs.tooltip",SHOW:"show.bs.tooltip",SHOWN:"shown.bs.tooltip",INSERTED:"inserted.bs.tooltip",CLICK:"click.bs.tooltip",FOCUSIN:"focusin.bs.tooltip",FOCUSOUT:"focusout.bs.tooltip",MOUSEENTER:"mouseenter.bs.tooltip",MOUSELEAVE:"mouseleave.bs.tooltip"},nn="fade",sn="show",on="show",rn="out",an=".tooltip-inner",ln=".modal",cn="hide.bs.modal",hn="hover",dn="focus";class un extends B{constructor(t,e){if(void 0===Fe)throw new TypeError("Bootstrap's tooltips require Popper (https://popper.js.org)");super(t),this._isEnabled=!0,this._timeout=0,this._hoverState="",this._activeTrigger={},this._popper=null,this._config=this._getConfig(e),this.tip=null,this._setListeners()}static get Default(){return tn}static get NAME(){return Qi}static get Event(){return en}static get DefaultType(){return Zi}enable(){this._isEnabled=!0}disable(){this._isEnabled=!1}toggleEnabled(){this._isEnabled=!this._isEnabled}toggle(t){if(this._isEnabled)if(t){const e=this._initializeOnDelegatedTarget(t);e._activeTrigger.click=!e._activeTrigger.click,e._isWithActiveTrigger()?e._enter(null,e):e._leave(null,e)}else{if(this.getTipElement().classList.contains(sn))return void this._leave(null,this);this._enter(null,this)}}dispose(){clearTimeout(this._timeout),j.off(this._element.closest(ln),cn,this._hideModalHandler),this.tip&&this.tip.remove(),this._disposePopper(),super.dispose()}show(){if("none"===this._element.style.display)throw new Error("Please use show on visible elements");if(!this.isWithContent()||!this._isEnabled)return;const t=j.trigger(this._element,this.constructor.Event.SHOW),e=h(this._element),i=null===e?this._element.ownerDocument.documentElement.contains(this._element):e.contains(this._element);if(t.defaultPrevented||!i)return;"tooltip"===this.constructor.NAME&&this.tip&&this.getTitle()!==this.tip.querySelector(an).innerHTML&&(this._disposePopper(),this.tip.remove(),this.tip=null);const n=this.getTipElement(),s=(t=>{do{t+=Math.floor(1e6*Math.random())}while(document.getElementById(t));return t})(this.constructor.NAME);n.setAttribute("id",s),this._element.setAttribute("aria-describedby",s),this._config.animation&&n.classList.add(nn);const o="function"==typeof this._config.placement?this._config.placement.call(this,n,this._element):this._config.placement,r=this._getAttachment(o);this._addAttachmentClass(r);const{container:a}=this._config;H.set(n,this.constructor.DATA_KEY,this),this._element.ownerDocument.documentElement.contains(this.tip)||(a.append(n),j.trigger(this._element,this.constructor.Event.INSERTED)),this._popper?this._popper.update():this._popper=qe(this._element,n,this._getPopperConfig(r)),n.classList.add(sn);const l=this._resolvePossibleFunction(this._config.customClass);l&&n.classList.add(...l.split(" ")),"ontouchstart"in document.documentElement&&[].concat(...document.body.children).forEach((t=>{j.on(t,"mouseover",d)}));const c=this.tip.classList.contains(nn);this._queueCallback((()=>{const t=this._hoverState;this._hoverState=null,j.trigger(this._element,this.constructor.Event.SHOWN),t===rn&&this._leave(null,this)}),this.tip,c)}hide(){if(!this._popper)return;const t=this.getTipElement();if(j.trigger(this._element,this.constructor.Event.HIDE).defaultPrevented)return;t.classList.remove(sn),"ontouchstart"in document.documentElement&&[].concat(...document.body.children).forEach((t=>j.off(t,"mouseover",d))),this._activeTrigger.click=!1,this._activeTrigger.focus=!1,this._activeTrigger.hover=!1;const e=this.tip.classList.contains(nn);this._queueCallback((()=>{this._isWithActiveTrigger()||(this._hoverState!==on&&t.remove(),this._cleanTipClass(),this._element.removeAttribute("aria-describedby"),j.trigger(this._element,this.constructor.Event.HIDDEN),this._disposePopper())}),this.tip,e),this._hoverState=""}update(){null!==this._popper&&this._popper.update()}isWithContent(){return Boolean(this.getTitle())}getTipElement(){if(this.tip)return this.tip;const t=document.createElement("div");t.innerHTML=this._config.template;const e=t.children[0];return this.setContent(e),e.classList.remove(nn,sn),this.tip=e,this.tip}setContent(t){this._sanitizeAndSetContent(t,this.getTitle(),an)}_sanitizeAndSetContent(t,e,i){const n=V.findOne(i,t);e||!n?this.setElementContent(n,e):n.remove()}setElementContent(t,e){if(null!==t)return o(e)?(e=r(e),void(this._config.html?e.parentNode!==t&&(t.innerHTML="",t.append(e)):t.textContent=e.textContent)):void(this._config.html?(this._config.sanitize&&(e=Yi(e,this._config.allowList,this._config.sanitizeFn)),t.innerHTML=e):t.textContent=e)}getTitle(){const t=this._element.getAttribute("data-bs-original-title")||this._config.title;return this._resolvePossibleFunction(t)}updateAttachment(t){return"right"===t?"end":"left"===t?"start":t}_initializeOnDelegatedTarget(t,e){return e||this.constructor.getOrCreateInstance(t.delegateTarget,this._getDelegateConfig())}_getOffset(){const{offset:t}=this._config;return"string"==typeof t?t.split(",").map((t=>Number.parseInt(t,10))):"function"==typeof t?e=>t(e,this._element):t}_resolvePossibleFunction(t){return"function"==typeof t?t.call(this._element):t}_getPopperConfig(t){const e={placement:t,modifiers:[{name:"flip",options:{fallbackPlacements:this._config.fallbackPlacements}},{name:"offset",options:{offset:this._getOffset()}},{name:"preventOverflow",options:{boundary:this._config.boundary}},{name:"arrow",options:{element:`.${this.constructor.NAME}-arrow`}},{name:"onChange",enabled:!0,phase:"afterWrite",fn:t=>this._handlePopperPlacementChange(t)}],onFirstUpdate:t=>{t.options.placement!==t.placement&&this._handlePopperPlacementChange(t)}};return{...e,..."function"==typeof this._config.popperConfig?this._config.popperConfig(e):this._config.popperConfig}}_addAttachmentClass(t){this.getTipElement().classList.add(`${this._getBasicClassPrefix()}-${this.updateAttachment(t)}`)}_getAttachment(t){return Ji[t.toUpperCase()]}_setListeners(){this._config.trigger.split(" ").forEach((t=>{if("click"===t)j.on(this._element,this.constructor.Event.CLICK,this._config.selector,(t=>this.toggle(t)));else if("manual"!==t){const e=t===hn?this.constructor.Event.MOUSEENTER:this.constructor.Event.FOCUSIN,i=t===hn?this.constructor.Event.MOUSELEAVE:this.constructor.Event.FOCUSOUT;j.on(this._element,e,this._config.selector,(t=>this._enter(t))),j.on(this._element,i,this._config.selector,(t=>this._leave(t)))}})),this._hideModalHandler=()=>{this._element&&this.hide()},j.on(this._element.closest(ln),cn,this._hideModalHandler),this._config.selector?this._config={...this._config,trigger:"manual",selector:""}:this._fixTitle()}_fixTitle(){const t=this._element.getAttribute("title"),e=typeof this._element.getAttribute("data-bs-original-title");(t||"string"!==e)&&(this._element.setAttribute("data-bs-original-title",t||""),!t||this._element.getAttribute("aria-label")||this._element.textContent||this._element.setAttribute("aria-label",t),this._element.setAttribute("title",""))}_enter(t,e){e=this._initializeOnDelegatedTarget(t,e),t&&(e._activeTrigger["focusin"===t.type?dn:hn]=!0),e.getTipElement().classList.contains(sn)||e._hoverState===on?e._hoverState=on:(clearTimeout(e._timeout),e._hoverState=on,e._config.delay&&e._config.delay.show?e._timeout=setTimeout((()=>{e._hoverState===on&&e.show()}),e._config.delay.show):e.show())}_leave(t,e){e=this._initializeOnDelegatedTarget(t,e),t&&(e._activeTrigger["focusout"===t.type?dn:hn]=e._element.contains(t.relatedTarget)),e._isWithActiveTrigger()||(clearTimeout(e._timeout),e._hoverState=rn,e._config.delay&&e._config.delay.hide?e._timeout=setTimeout((()=>{e._hoverState===rn&&e.hide()}),e._config.delay.hide):e.hide())}_isWithActiveTrigger(){for(const 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a/index_files/libs/quarto-html/quarto-syntax-highlighting.css +++ /dev/null @@ -1,167 +0,0 @@ -/* quarto syntax highlight colors */ -:root { - --quarto-hl-ot-color: #003B4F; - --quarto-hl-at-color: #657422; - --quarto-hl-ss-color: #20794D; - --quarto-hl-an-color: #5E5E5E; - --quarto-hl-fu-color: #4758AB; - --quarto-hl-st-color: #20794D; - --quarto-hl-cf-color: #003B4F; - --quarto-hl-op-color: #5E5E5E; - --quarto-hl-er-color: #AD0000; - --quarto-hl-bn-color: #AD0000; - --quarto-hl-al-color: #AD0000; - --quarto-hl-va-color: #111111; - --quarto-hl-bu-color: inherit; - --quarto-hl-ex-color: inherit; - --quarto-hl-pp-color: #AD0000; - --quarto-hl-in-color: #5E5E5E; - --quarto-hl-vs-color: #20794D; - --quarto-hl-wa-color: #5E5E5E; - --quarto-hl-do-color: #5E5E5E; - --quarto-hl-im-color: #00769E; - --quarto-hl-ch-color: #20794D; - --quarto-hl-dt-color: #AD0000; - --quarto-hl-fl-color: #AD0000; - --quarto-hl-co-color: #5E5E5E; - --quarto-hl-cv-color: #5E5E5E; - --quarto-hl-cn-color: #8f5902; - --quarto-hl-sc-color: #5E5E5E; - --quarto-hl-dv-color: #AD0000; - --quarto-hl-kw-color: #003B4F; -} - -/* other quarto variables */ -:root { - --quarto-font-monospace: SFMono-Regular, Menlo, Monaco, Consolas, "Liberation Mono", "Courier New", monospace; -} - -code span { - color: #003B4F; -} - -code.sourceCode > span { - color: #003B4F; -} - -div.sourceCode, -div.sourceCode pre.sourceCode { - color: #003B4F; -} - -code span.ot { - color: #003B4F; -} - -code span.at { - color: #657422; -} - -code span.ss { - color: #20794D; -} - -code span.an { - color: #5E5E5E; -} - -code span.fu { - color: #4758AB; -} - -code span.st { - color: #20794D; -} - -code span.cf { - color: #003B4F; -} - -code span.op { - color: #5E5E5E; -} - -code span.er { - color: #AD0000; -} - -code span.bn { - color: #AD0000; -} - -code span.al { - color: #AD0000; -} - -code span.va { - color: #111111; -} - -code span.pp { - color: #AD0000; -} - -code span.in { - color: #5E5E5E; -} - -code span.vs { - color: #20794D; -} 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sectionIndex - 1; - if (current !== currentActive) { - removeAllActive(); - currentActive = current; - makeActive(current); - if (init) { - window.dispatchEvent(sectionChanged); - } - init = true; - } - } - }; - - const inHiddenRegion = (top, bottom, hiddenRegions) => { - for (const region of hiddenRegions) { - if (top <= region.bottom && bottom >= region.top) { - return true; - } - } - return false; - }; - - const categorySelector = "header.quarto-title-block .quarto-category"; - const activateCategories = (href) => { - // Find any categories - // Surround them with a link pointing back to: - // #category=Authoring - try { - const categoryEls = window.document.querySelectorAll(categorySelector); - for (const categoryEl of categoryEls) { - const categoryText = categoryEl.textContent; - if (categoryText) { - const link = `${href}#category=${encodeURIComponent(categoryText)}`; - const linkEl = window.document.createElement("a"); - linkEl.setAttribute("href", link); - for (const child of 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one - const nearestListing = findNearestParentListing( - offsetAbsoluteUrl(window.location.pathname), - listingHrefs - ); - if (nearestListing) { - activateCategories(nearestListing); - } else { - // See if the referrer is a listing page for this item - const referredRelativePath = offsetAbsoluteUrl(document.referrer); - const referrerListing = listingHrefs.find((listingHref) => { - const isListingReferrer = - listingHref === referredRelativePath || - listingHref === referredRelativePath + "index.html"; - return isListingReferrer; - }); - - if (referrerListing) { - // Try to use the referrer if possible - activateCategories(referrerListing); - } else if (listingHrefs.length > 0) { - // Otherwise, just fall back to the first listing - activateCategories(listingHrefs[0]); - } - } - }); - } - } - if (hasTitleCategories()) { - findAndActivateCategories(); - } - - const findNearestParentListing = (href, listingHrefs) => { - if (!href || !listingHrefs) { - return undefined; - } - // Look up the tree for a nearby linting and use that if we find one - const relativeParts = href.substring(1).split("/"); - while (relativeParts.length > 0) { - const path = relativeParts.join("/"); - for (const listingHref of listingHrefs) { - if (listingHref.startsWith(path)) { - return listingHref; - } - } - relativeParts.pop(); - } - - return undefined; - }; - - const manageSidebarVisiblity = (el, placeholderDescriptor) => { - let isVisible = true; - - return (hiddenRegions) => { - if (el === null) { - return; - } - - // Find the last element of the TOC - const lastChildEl = el.lastElementChild; - - if (lastChildEl) { - // Find the top and bottom o the element that is being managed - const elTop = el.offsetTop; - const elBottom = - elTop + lastChildEl.offsetTop + lastChildEl.offsetHeight; - - // Converts the sidebar to a menu - const convertToMenu = () => { - for (const child of el.children) { - child.style.opacity = 0; - child.style.display = "none"; - } - - const toggleContainer = 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window.document.createElement("div"); - toggleContents.classList = el.classList; - toggleContents.classList.add("zindex-over-content"); - toggleContents.classList.add("quarto-sidebar-toggle-contents"); - for (const child of el.children) { - if (child.id === "toc-title") { - continue; - } - - const clone = child.cloneNode(true); - clone.style.opacity = 1; - clone.style.display = null; - toggleContents.append(clone); - } - toggleContents.style.height = "0px"; - toggleContainer.append(toggleContents); - el.parentElement.prepend(toggleContainer); - - // Process clicks - let tocShowing = false; - // Allow the caller to control whether this is dismissed - // when it is clicked (e.g. sidebar navigation supports - // opening and closing the nav tree, so don't dismiss on click) - const clickEl = placeholderDescriptor.dismissOnClick - ? toggleContainer - : toggleTitle; - - const closeToggle = () => { - if (tocShowing) { - toggleContainer.classList.remove("expanded"); - toggleContents.style.height = "0px"; - tocShowing = false; - } - }; - - const positionToggle = () => { - // position the element (top left of parent, same width as parent) - const elRect = el.getBoundingClientRect(); - toggleContainer.style.left = `${elRect.left}px`; - toggleContainer.style.top = `${elRect.top}px`; - toggleContainer.style.width = `${elRect.width}px`; - }; - - // Get rid of any expanded toggle if the user scrolls - window.document.addEventListener( - "scroll", - throttle(() => { - closeToggle(); - }, 50) - ); - - // Handle positioning of the toggle - window.addEventListener( - "resize", - throttle(() => { - positionToggle(); - }, 50) - ); - positionToggle(); - - // Process the click - clickEl.onclick = () => { - if (!tocShowing) { - toggleContainer.classList.add("expanded"); - toggleContents.style.height = null; - tocShowing = true; - } else { - closeToggle(); - } - }; - }; - - // Converts a sidebar from a menu back to a sidebar - const convertToSidebar = () => { - for (const child of el.children) { - child.style.opacity = 1; - clone.style.display = null; - } - - const placeholderEl = window.document.getElementById( - placeholderDescriptor.id - ); - if (placeholderEl) { - placeholderEl.remove(); - } - - el.classList.remove("rollup"); - }; - - if (isReaderMode()) { - convertToMenu(); - isVisible = false; - } else { - if (!isVisible) { - // If the element is current not visible reveal if there are - // no conflicts with overlay regions - if (!inHiddenRegion(elTop, elBottom, hiddenRegions)) { - convertToSidebar(); - isVisible = true; - } - } else { - // If the element is visible, hide it if it conflicts with overlay regions - // and insert a placeholder toggle (or if we're in reader mode) - if (inHiddenRegion(elTop, elBottom, hiddenRegions)) { - convertToMenu(); - isVisible = false; - } - } - } - } - }; - }; - - // Find any conflicting margin elements and add margins to the - // top to prevent overlap - const marginChildren = window.document.querySelectorAll( - ".column-margin.column-container > * " - ); - let lastBottom = 0; - for (const marginChild of marginChildren) { - const top = marginChild.getBoundingClientRect().top; - if (top < lastBottom) { - const margin = lastBottom - top; - marginChild.style.marginTop = `${margin}px`; - } - lastBottom = top + marginChild.getBoundingClientRect().height; - } - - // Manage the visibility of the toc and the sidebar - const marginScrollVisibility = manageSidebarVisiblity(marginSidebarEl, { - id: "quarto-toc-toggle", - titleSelector: "#toc-title", - dismissOnClick: true, - }); - const sidebarScrollVisiblity = manageSidebarVisiblity(sidebarEl, { - id: "quarto-sidebarnav-toggle", - titleSelector: ".title", - dismissOnClick: false, - }); - let tocLeftScrollVisibility; - if (leftTocEl) { - tocLeftScrollVisibility = manageSidebarVisiblity(leftTocEl, { - id: "quarto-lefttoc-toggle", - titleSelector: "#toc-title", - dismissOnClick: true, - }); - } - - // Find the first element that uses formatting in special columns - const conflictingEls = window.document.body.querySelectorAll( - '[class^="column-"], [class*=" column-"], aside, [class*="margin-caption"], [class*=" margin-caption"], [class*="margin-ref"], [class*=" margin-ref"]' - ); - - // Filter all the possibly conflicting elements into ones - // the do conflict on the left or ride side - const arrConflictingEls = Array.from(conflictingEls); - const leftSideConflictEls = arrConflictingEls.filter((el) => { - if (el.tagName === "ASIDE") { - return false; - } - return Array.from(el.classList).find((className) => { - return ( - className !== "column-body" && - className.startsWith("column-") && - !className.endsWith("right") && - !className.endsWith("container") && - className !== "column-margin" - ); - }); - }); - const rightSideConflictEls = arrConflictingEls.filter((el) => { - if (el.tagName === "ASIDE") { - return true; - } - - const hasMarginCaption = Array.from(el.classList).find((className) => { - return className == "margin-caption"; - }); - if (hasMarginCaption) { - return true; - } - - return Array.from(el.classList).find((className) => { - return ( - className !== "column-body" && - !className.endsWith("container") && - className.startsWith("column-") && - !className.endsWith("left") - ); - }); - }); - - const kOverlapPaddingSize = 10; - function toRegions(els) { - return els.map((el) => { - const top = - el.getBoundingClientRect().top + - document.documentElement.scrollTop - - kOverlapPaddingSize; - return { - top, - bottom: top + el.scrollHeight + 2 * kOverlapPaddingSize, - }; - }); - } - - const hideOverlappedSidebars = () => { - marginScrollVisibility(toRegions(rightSideConflictEls)); - sidebarScrollVisiblity(toRegions(leftSideConflictEls)); - if (tocLeftScrollVisibility) { - tocLeftScrollVisibility(toRegions(leftSideConflictEls)); - } - }; - - window.quartoToggleReader = () => { - // Applies a slow class (or removes it) - // to update the transition speed - const slowTransition = (slow) => { - const manageTransition = (id, slow) => { - const el = document.getElementById(id); - if (el) { - if (slow) { - el.classList.add("slow"); - } else { - el.classList.remove("slow"); - } - } - }; - - manageTransition("TOC", slow); - manageTransition("quarto-sidebar", slow); - }; - - const readerMode = !isReaderMode(); - setReaderModeValue(readerMode); - - // If we're entering reader mode, slow the transition - if (readerMode) { - slowTransition(readerMode); - } - highlightReaderToggle(readerMode); - hideOverlappedSidebars(); - - // If we're exiting reader mode, restore the non-slow transition - if (!readerMode) { - slowTransition(!readerMode); - } - }; - - const highlightReaderToggle = (readerMode) => { - const els = document.querySelectorAll(".quarto-reader-toggle"); - if (els) { - els.forEach((el) => { - if (readerMode) { - el.classList.add("reader"); - } else { - el.classList.remove("reader"); - } - }); - } - }; - - const setReaderModeValue = (val) => { - 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url={https://books.google.com/books?id=mdoLBwAAQBAJ}, +@article{alr2021, + title={Study on the Missing Data Mechanisms and Imputation Methods}, + author={Alruhaymi, A.Z. and Kim, C.J.}, + journal={Open Journal of Statistics}, + volume={11}, + number={4}, + pages={477-492}, + year={2021}, + publisher={Scientific Research Publishing}, + url={https://doi.org/10.4236/ojs.2021.114030} +} + +@book{arn2017, + title={Survey Sampling Theory and Applications}, + author={Arnab, R.}, + isbn={9780128118481}, + url={https://www.sciencedirect.com/topics/mathematics/imputation-method}, + year={2017}, + publisher={Academic Press} +} + +@article{azu2011, + title={Multiple imputation by chained equations: what is it and how does it work?}, + author={Azur, M. J. and Stuart, E. A. and Frangakis, C. and Leaf, P. J.}, + journal={Int J Methods Psychiatr Res.}, + volume={20}, + number={1}, + pages={40-49}, + year={2011}, + publisher={Wiley Online Library}, + url={https://onlinelibrary.wiley.com/doi/epdf/10.1002/mpr.329} +} + +@article{cao2021, + title={Review and evaluation of imputation methods for multivariate longitudinal data with mixed-type incomplete variables}, + author={Cao, Y. and Allore, H. and Wyk, B. V. and Gutman R.}, + journal={Statistics in Medicine}, + volume={41}, + number={30}, + pages={5844-5876}, + year={2021}, + publisher={John Wiley & Sons, Ltd}, + url={https://doi-org.ezproxy.lib.uwf.edu/10.1002/sim.9592} +} + +@article{car2021, + title={Missing data: A statistical framework for practice}, + author={Carpenter, J. R. and Smuk, M.}, + journal={Biometrical Journal}, + volume={63}, + number={5}, + pages={915-947}, + year={2021}, + publisher={John Wiley & Sons, Ltd}, + url={https://doi.org/10.1002/bimj.202000196} +} + +@article{cha2023, + title={Binned Data Provide Better Imputation of Missing Time Series Data from Wearables}, + author={Chakrabarti, S. and Biswas, N. and Karnani, K. and Padul, V. and Jones, L. D. and Kesari, S. and Ashili, S.}, + journal={Sensors}, + volume={23}, + number={3}, + pages={1454}, + year={2023}, + publisher={MDPI}, + url={https://doi.org/10.3390/s23031454} +} + +@article{don2006, + title={Review: A gentle introduction to imputation of missing values}, + author={Donders, A. R. and van der Heijden, G. J. and Stijnen, T. and Moons, K. G.}, + journal={Journal of Clinical Epidemiology}, + volume={59}, + number={10}, + pages={1087-1091}, + year={2006}, + url={https://doi.org/10.1016/j.jclinepi.2006.01.014} +} + +@article{don2013, + title={Principled missing data methods for researchers}, + author={Dong, Y. and Peng, C. J.}, + journal={SpringerPlus}, + volume={2}, + number={222}, + year={2013}, + url={https://doi.org/10.1186/2193-1801-2-222} +} + +@article{end2001, + title={The Performance of the Full Information Maximum Likelihood Estimator in Multiple Regression Models with Missing Data}, + author={Enders, C. K.}, + journal={Sage Journals}, + volume={61}, + number={5}, + year={2001}, + publisher={Educational and Psychological Measurement (EPM)}, + url={https://doi.org/10.1177/0013164401615001} +} + +@article{fan2021, + title={Imputation and likelihood methods for matrix-variate logistic regression with response misclassification}, + author={Fang, J. and Yi, G. Y.}, + journal={The Canadian Journal of Statistics}, + volume={49}, + number={4}, + pages={1298-1316}, + year={2021}, + publisher={John Wiley & Sons, Ltd}, + url={https://doi.org/10.1002/cjs.11620} +} + +@article{fie2008, + title={A review of RCTs in four medical journals to assess the use of imputation to overcome missing data in quality of life outcomes}, + author={Fielding, S. and Maclennan, G. and Cook, J. A. and Ramsay, C. R.}, + journal={Trials}, + volume={9}, + number={51}, + pages={1-6}, year={2008}, - publisher={Springer New York} + publisher={BioMed Central}, + url={https://doi.org/10.1186/1745-6215-9-51} } -@article{bro2014principal, - title={Principal component analysis}, - author={Bro, Rasmus and Smilde, Age K}, - journal={Analytical methods}, - volume={6}, + +@article{ham2021, + title={A Comparison of Multiple Imputation Methods for Recovering Missing Data in Hydrological Studies}, + author={Hamzah, F. B. and Hamzah, F. M. and Razali, S. R. and Samad, H.}, + journal={Civil Engineering Journal}, + volume={7}, number={9}, - pages={2812--2831}, - year={2014}, - publisher={Royal Society of Chemistry} + pages={1608-1619}, + year={2021}, + url={https://pdfs.semanticscholar.org/e875/5b85e63669c6b3fd9ba6f380400f087b39fc.pdf} +} + +@book{hey2019, + title={Applied Missing Data Analysis With SPSS and (R) Studio}, + author={Heymans, M. W. and Eekhout, I.}, + url={https://bookdown.org/mwheymans/bookmi/}, + year={2019}, + publisher={Heymans & Eekhout} +} + +@article{hug2019, + title={Accounting for missing data in statistical analyses: multiple imputation is not always the answer}, + author={Hughes, R. A. and Heron, J. and Sterne, J. A. and Tilling, K.}, + journal={International Journal of Epidemiology}, + volume={48}, + number={4}, + pages={1294-1304}, + year={2019}, + url={https://doi.org/10.1093/ije/dyz032} +} + +@article{jav2021, + title={Evaluation of Four Multiple Imputation Methods for Handling Missing Binary Outcome Data in the Presence of an Interaction between a Dummy and a Continuous Variable}, + author={Javadi, S. and Bahrampour, A. and Saber, M. M. and Garrusi, B. and Baneshi, M. R.}, + journal={Journal of Probability and Statistic}, + volume={2021}, + number={6668822}, + pages={1-14}, + year={2021}, + publisher={Hindawi}, + url={https://doi.org/10.1155/2021/6668822} +} + +@article{kan2013, + title={The prevention and handling of the missing data}, + author={Kang, H.}, + journal={Korean Journal of Anesthesiology}, + volume={64}, + number={5}, + pages={402-406}, + year={2013}, + url={https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3668100/} +} + +@article{mai2023, + title={Handling of missing data with multiple imputation in observational studies that address causal questions: protocol for a scoping review}, + author={Mainzer, R. and Moreno-Betancur, M. and Nguyen, C. and Simpson, J. and Carlin, J. and Lee, K.}, + journal={BMJ Open}, + volume={13}, + pages={1-6}, + year={2023}, + url={http://dx.doi.org/10.1136/bmjopen-2022-065576} } + +@article{mip2019, + title={MI Procedure}, + year={2019}, + publisher={SAS}, + url={https://documentation.sas.com/doc/en/pgmsascdc/9.4_3.4/statug/statug_mi_details03.htm} +} + +@article{oli2003, + title={The comparative efficacy of imputation methods for missing data in structural equation modeling}, + author={Olinsky, A. and Chen, S. and Harlow, L.}, + journal={European Journal of Operational Research}, + volume={151}, + number={1}, + pages={53-79}, + year={2003}, + url={https://doi.org/10.1016/S0377-2217(02)00578-7} +} + +@article{pam2016, + title={Handling missing data: analysis of a challenging data set using multiple imputation}, + author={Pampka, M. and Hutcheson, G. and Williams, J.}, + journal={International Journal of Research & Method in Education}, + volume={39}, + number={1}, + pages={19-37}, + year={2016}, + url={https://doi.org/10.1080/1743727X.2014.979146} +} + +@article{ped2017, + title={Missing data and multiple imputation in clinical epidemiological research}, + author={Pedersen, A. B. and Mikkelsen, E. M. and Cronin-Fenton, D. and Kristensen, N. R. and Pham, T. M. and Pedersen, L. and Petersen, I.}, + journal={Clinical Epidemiology}, + volume={9}, + pages={157-166}, + year={2017}, + url={https://www.tandfonline.com/doi/full/10.2147/CLEP.S129785} +} + +@article{sch2002, + title={Missing Data: Our View of the State of the Art}, + author={Schafer, J. L. and Graham, J. W.}, + journal={Psychological Methods}, + volume={7}, + number={2}, + pages={147-177}, + year={2002}, + publisher={American Psychological Association, Inc.}, + url={https://psycnet.apa.org/doi/10.1037/1082-989X.7.2.147} +} + +@article{str2008, + title={Missing data and the trouble with LOCF}, + author={Streiner, D. L.}, + journal={EBMH}, + volume={11}, + number={1}, + pages={1-5}, + year={2008}, + publisher={BMJ}, + url={http://dx.doi.org/10.1136/ebmh.11.1.3-a} +} + +@article{tho2022, + title={Composite Imputation Method for the Multiple Linear Regression with Missing at Random Data}, + author={Thongsri, T. and Samart, K.}, + journal={International Journal of Mathematics and Mathematics and Computer Science}, + volume={17}, + number={1}, + pages={51-62}, + year={2022}, + url={http://ijmcs.future-in-tech.net/17.1/R-Samart.pdf} +} + +@article{ura2020, + title={A multiple regression imputation method with application to sensitivity analysis under intermittent missingness}, + author={Uranga, R. and Molenberghs, G. and Allende, S.}, + journal={Communications in Statistics - Theory and Methods}, + volume={51}, + number={15}, + pages={5146-5161}, + year={2020}, + publisher={Taylor & Francis Online}, + url={https://doi.org/10.1080/03610926.2020.1834581} +} + +@article{van2011, + title={mice: Multivariate Imputation by Chained Equations in R}, + author={van_Buuren, Groothuis-Oudshoorn, K.}, + journal={Journal of Statistical Software}, + volume={45}, + pages={1–67}, + year={2011}, + url={https://doi.org/10.18637/jss.v045.i03} +} + +@article{van2020, + title={Rebutting Existing Misconceptions About Multiple Imputation as a Method for Handling Missing Data}, + author={van_Ginkel, J. R. and Linting, M. and Rippe, R. C. and van der Voort, A.}, + journal={Journal of Personality Assessment}, + volume={102}, + number={3}, + pages={2812-2831}, + year={2020}, + url={https://doi.org/10.1080/00223891.2018.1530680}, +} + +@article{won2023, + title={A Comparative Study of Imputation Methods for Multivariate Ordinal Data}, + author={Wongkamthong, C. and Akande, O.}, + journal={Journal of Survey Statistics and Methodology}, + volume={11}, + number={1}, + pages={189-212}, + year={2023}, + url={https://doi.org/10.1093/jssam/smab028} +} + +@article{wul2017, + title={Multiple imputation by chained equations in praxis: guidelines and review}, + author={Wulff, J. N. and Jeppesen, L. E.}, + journal={Electronics Journal of Business Research Methods}, + volume={15}, + number={1}, + pages={41-56}, + year={2017}, + url={https://vbn.aau.dk/ws/files/257318283/ejbrm_volume15_issue1_article450.pdf} +} + +@article{yan2005, + title={Imputation and Variable Selection in Linear Regression Models with Missing Covariates}, + author={Yang, W. and Belin, T. R. and Boscardin, W. J.}, + journal={Biometrics}, + volume={61}, + pages={498-506}, + year={2005}, + publisher={Blackwell Publishing Limited}, + url={https://web-s-ebscohost-com.ezproxy.lib.uwf.edu/ehost/pdfviewer/pdfviewer?vid=0&sid=a4db468b-e77f-45ae-958e-935ed046e8be%40redis} +} + +