09-AnalyticPowerFunctions.Rmd

# Analytic Power Functions

For some designs it is possible to calculate power analytically, using closed functions. Within the `Superpower` package we have included a number of these closed functions. As you will see below, each analytic function only serves a very narrow scenario while the simulations functions are much more flexible. In addition, we will compare these functions to other packages/software. Please note, that the analytic power functions are designed to reject designs that are not appropriate for the functions (i.e., a `3w` design will be rejected by the `power_oneway_between` function).

## One-Way Between Subjects ANOVA

First, we can setup a one-way design with four levels, and perform a exact simulation power analysis with the `ANOVA_exact` function.

```{r}
string <- "4b"
n <- 60
mu <- c(80, 82, 82, 86) 

# Enter means in the order that matches the labels below.
sd <- 10
design_result <- ANOVA_design(design = string,
                   n = n, 
                   mu = mu, 
                   sd = sd)


exact_result <- ANOVA_exact(design_result,
                            alpha_level = alpha_level,
                            verbose = FALSE)
```

```{r echo = FALSE}

knitr::kable(exact_result$main_results,
             caption = "Exact ANOVA Result")%>%
  kable_styling(latex_options = "hold_position")
```

We can also calculate power analytically with a `Superpower` function.

```{r}
#using default alpha level of .05
power_oneway_between(design_result)$power 

```

This is a generalized function for one-way ANOVA's for any number of groups. It is in part based on code from the `pwr2ppl` [@R-pwr2ppl] package (but Aberson's code allows for different n per condition, and different sd per condition).

```{r}
pwr2ppl::anova1f_4(m1 = 80, m2 = 82, m3 = 82, m4 = 86,
          s1 = 10, s2 = 10, s3 = 10, s4 = 10,
          n1 = 60, n2 = 60, n3 = 60, n4 = 60,
          alpha = .05)
```

We can also use the function in the `pwr` package [@R-pwr]. Note that we need to calculate *f* to use this function, which is based on the means and sd, as illustrated in the formulas above. 

```{r}
pwr::pwr.anova.test(n = 60,
               k = 4,
               f = 0.2179449,
               sig.level = 0.05)
```

Finally, g\*Power [@faul2007g] provides the option to calculate `f` from the means, sd and n for the cells. It can then be used to calculate power.

![](screenshots/gpower_13.png)

## Two-way Between Subject Interaction

Now, we will setup a 2x2 between-subject ANOVA.

```{r}
string <- "2b*2b"
n <- 20
mu <- c(20, 20, 20, 25) 
# Enter means in the order that matches the labels below.
sd <- 5

design_result <- ANOVA_design(design = string,
                   n = n, 
                   mu = mu, 
                   sd = sd)

exact_result <- ANOVA_exact(design_result,
                            alpha_level = alpha_level,
                            verbose = FALSE)
```

```{r echo=FALSE}
knitr::kable(exact_result$main_results,
             caption = "Exact ANOVA Result")%>%
  kable_styling(latex_options = "hold_position")
```

Now, let's use the analytic function `power_twoway_between`.

```{r}
#using default alpha level of .05
power_res <- power_twoway_between(design_result) 
power_res$power_A
power_res$power_B
power_res$power_AB
```

We can compare these results to [@R-pwr2ppl], as well. 

```{r}
pwr2ppl::anova2x2(m1.1 = 20,
         m1.2 = 20,
         m2.1 = 20,
         m2.2 = 25,
         s1.1 = 5,
         s1.2 = 5,
         s2.1 = 5,
         s2.2 = 5,
         n1.1 = 20,
         n1.2 = 20,
         n2.1 = 20,
         n2.2 = 20,
         alpha = .05,
         all = "OFF")
```

## 3x3 Between Subject ANOVA

We can extend this function to a two-way design with 3 levels.

```{r}
string <- "3b*3b"
n <- 20
mu <- c(20, 20, 20, 20, 20, 20, 20, 20, 25) 
# Enter means in the order that matches the labels below.
sd <- 5

design_result <- ANOVA_design(design = string,
                   n = n, 
                   mu = mu, 
                   sd = sd)

exact_result <- ANOVA_exact(design_result,
                            alpha_level = alpha_level,
                            verbose = FALSE)
```

```{r echo=FALSE}

knitr::kable(exact_result$main_results,
             caption = "Exact ANOVA Result")%>%
  kable_styling(latex_options = "hold_position")
```

```{r}
#using default alpha level of .05
power_res <- power_twoway_between(design_result) 
power_res$power_A
power_res$power_B
power_res$power_AB
```