The paired t-test shares the same built-in function in R with the one-sample t-test. In this notebook we will illustrate how to perform the paired t-test using the built-in function, as well as performing the test by basic calculations.
In our notebook for the one-sample t-test, we described the usage of the t.test() function. The t.test() function can be used for both one-sample and two-sample tests. Here we will focus on the two-sample case, and in particular, paired data. To see the details of the function, we can type ?t.test in the console.
To use this function, we need to provide it some data. Let's start by creating the data:
before = c(164, 137, 133, 157, 173, 143, 144, 180, 178, 154) #Data collected before treatment
after = c(129, 127, 122, 120, 129, 122, 125, 139, 145, 139) #Data collected after treatment
D0 = 0 #hypothesized differencetest = t.test(x = before, y = after, alternative = 'greater', mu = D0, paired = TRUE)
print(test)##
## Paired t-test
##
## data: before and after
## t = 6.5764, df = 9, p-value = 5.1e-05
## alternative hypothesis: true difference in means is greater than 0
## 95 percent confidence interval:
## 19.18552 Inf
## sample estimates:
## mean of the differences
## 26.6
Since we are testing if the before treatment data is larger than the after treatment data, we set the alternative to 'greater'. Note that we now set the argument paired to TRUE. This is to tell R that the two data sets we inputted are paired data.
Suppose we want to extract the various numerical results from the test, such as the p-value or the test statistics. We can do so by extracting the corresponding values from the output:
test$statistic #Test statistics## t
## 6.576427
test$parameter #Degrees of freedom## df
## 9
test$p.value #p-value## [1] 5.10044e-05
Of course, we can also perform the hypothesis test via simple math. To calculate the test statistics, we can just directly use the test statistics formula:
di = before - after #Calculate the differences for each pair
d.bar = mean(di) #Mean of differences
s.d = sd(di) #Sample standard deviation of the differences
n = length(before) #Number of data pairs
alpha = 0.01 #Alpha value
test.stat = (d.bar - D0)/(s.d/sqrt(n))
print(test.stat)## [1] 6.576427
To get the critical value, we can use the qt() function (for the t-distribution):
crit.val = qt(p = alpha, df = n - 1, lower.tail = F)
print(crit.val)## [1] 2.821438
The argument df is for the degrees of freedom, where the argument lower.tail = F is to tell R that we want to find the critical value which gives a probability of 0.01 on the upper tail. If we were to do a lower-tailed test, the command will be qt(p = 0.01, df = n - 1, lower.tail = T) (or by default, lower.tail is TRUE if there is no input for this argument).
To obtain the p-value, we can use the pt() function:
p.val = pt(test.stat, df = n - 1, lower.tail = F)
print(p.val)## [1] 5.10044e-05
If this were a two-tailed test, remember we need to multiply the p-value by 2:
p.val2 = 2*pt(abs(test.stat), df = n - 1, lower.tail = F) #If we were to do a two-tailed test
print(p.val2)## [1] 0.0001020088