# Chapter 9 Statistical power in R

In this chapter we focus on effect size and statistical power.

## 9.1 Computing confidence intervals

### 9.1.1 Theoretical

### 9.1.2 Bootstrap

## 9.2 Effect Size

### 9.2.1 Cohen’s d

### 9.2.2 Pearson’s r

### 9.2.3 Odds ratio

## 9.3 Power analysis

We can compute a power analysis using functions from the `pwr`

package. Let’s focus on the power for a t-test in order to determine a difference in the mean between two groups. Let’s say that we think than an effect size of Cohen’s d=0.5 is realistic for the study in question (based on previous research) and would be of scientific interest. We wish to have 80% power to find the effect if it exists. We can compute the sample size needed for adequate power using the `pwr.t.test()`

function:

```
##
## Two-sample t test power calculation
##
## n = 63.76561
## d = 0.5
## sig.level = 0.05
## power = 0.8
## alternative = two.sided
##
## NOTE: n is number in *each* group
```

Thus, about 64 participants would be needed in each group in order to test the hypothesis with adequate power.

## 9.4 Power curves

We can also create plots that can show us how the power to find an effect varies as a function of effect size and sample size. We willl use the `crossing()`

function from the `tidyr`

package to help with this. This function takes in two vectors, and returns a tibble that contains all possible combinations of those values.

```
effect_sizes <- c(0.2, 0.5, 0.8)
sample_sizes = seq(10, 500, 10)
#
input_df <- crossing(effect_sizes,sample_sizes)
glimpse(input_df)
```

```
## Observations: 150
## Variables: 2
## $ effect_sizes <dbl> 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0…
## $ sample_sizes <dbl> 10, 20, 30, 40, 50, 60, 70, 80, 90, …
```

Using this, we can then perform a power analysis for each combination of effect size and sample size to create our power curves. In this case, let’s say that we wish to perform a two-sample t-test.

```
# create a function get the power value and
# return as a tibble
get_power <- function(df){
power_result <- pwr.t.test(n=df$sample_sizes,
d=df$effect_sizes,
type='two.sample')
df$power=power_result$power
return(df)
}
# run get_power for each combination of effect size
# and sample size
power_curves <- input_df %>%
do(get_power(.)) %>%
mutate(effect_sizes = as.factor(effect_sizes))
```

Now we can plot the power curves, using a separate line for each effect size.

## 9.5 Simulating statistical power

Let’s simulate this to see whether the power analysis actually gives the right answer. We will sample data for two groups, with a difference of 0.5 standard deviations between their underlying distributions, and we will look at how often we reject the null hypothesis.

```
nRuns <- 5000
effectSize <- 0.5
# perform power analysis to get sample size
pwr.result <- pwr.t.test(d=effectSize, power=.8)
# round up from estimated sample size
sampleSize <- ceiling(pwr.result$n)
# create a function that will generate samples and test for
# a difference between groups using a two-sample t-test
get_t_result <- function(sampleSize, effectSize){
# take sample for the first group from N(0, 1)
group1 <- rnorm(sampleSize)
group2 <- rnorm(sampleSize, mean=effectSize)
ttest.result <- t.test(group1, group2)
return(tibble(pvalue=ttest.result$p.value))
}
index_df <- tibble(id=seq(nRuns)) %>%
group_by(id)
power_sim_results <- index_df %>%
do(get_t_result(sampleSize, effectSize))
p_reject <-
power_sim_results %>%
ungroup() %>%
summarize(pvalue = mean(pvalue<.05)) %>%
pull()
p_reject
```

`## [1] 0.7998`

This should return a number very close to 0.8.