# Power in R

# RClub_Power_11.10.15

*Jessica Kosie*

*November 10, 2015*

Modified from: R CLUB 5/15/14: Power Analyses in R

Original creator: Brendan Ostlund

REFRESHER: POWER is the probability of correctly rejecting a null hypothesis that is actually false.

P(rejecting NULL hypothesis|NULL hypothesis is false)

In other words, its the prob. of finding an effect that is really present.

Power = 1 – P(Type II error)

Power, effect size, sample size, and the significance level are inter-related, and if you know 3 of these quantities you can calculate the fourth (exciting eh?). Let’s get to it.

```
#First, we'll need to install the "pwr" package.
# install.packages("pwr")
library(pwr)
```

`## Warning: package 'pwr' was built under R version 3.2.2`

```
#POWER ANALYSIS FUNCTIONS!
# T-TESTS!
```

Function:

pwr.t.test(n = , d = , sig.level = , power = , type = c(“two.sample”, “one.sample”, “paired”), alternative= c(“two.sided”, “less”, “greater”))

Remember, power, effect size, sample size, and the significance level are related so we can play around with the information we have to determine the missing quantity.

```
#For example, what is the power for a two-sided ind. samples t-test given n=25 and a medium effect size?
pwr.t.test(n=25, d=.5, sig.level= .05)
```

```
##
## Two-sample t test power calculation
##
## n = 25
## d = 0.5
## sig.level = 0.05
## power = 0.4101003
## alternative = two.sided
##
## NOTE: n is number in *each* group
```

```
#Another example: What is the sample size needed to achieve a pwr=.80 and a small effect size?
pwr.t.test(d=.2, sig.level= .05, power=.8)
```

```
##
## Two-sample t test power calculation
##
## n = 393.4057
## d = 0.2
## sig.level = 0.05
## power = 0.8
## alternative = two.sided
##
## NOTE: n is number in *each* group
```

```
# Note: if you add "$n" to the end of the command only the necessary sample size will be displayed.
pwr.t.test(d=.2, sig.level= .05, power=.8)$n
```

`## [1] 393.4057`

```
# R can also calculate pwr for two samples with unequal sample sizes:
pwr.t2n.test(n1= 35, n2= 17, d=.8, sig.level= .05, alternative = "greater")
```

```
##
## t test power calculation
##
## n1 = 35
## n2 = 17
## d = 0.8
## sig.level = 0.05
## power = 0.8471659
## alternative = greater
```

```
# ANOVA!
# what's Cohen's rules of thumb for anova effect size (i.e., "f")?
cohen.ES(test = c("anov"), size = c("small"))
```

```
##
## Conventional effect size from Cohen (1982)
##
## test = anov
## size = small
## effect.size = 0.1
```

`cohen.ES(test = c("anov"), size = c("medium"))`

```
##
## Conventional effect size from Cohen (1982)
##
## test = anov
## size = medium
## effect.size = 0.25
```

`cohen.ES(test = c("anov"), size = c("large"))`

```
##
## Conventional effect size from Cohen (1982)
##
## test = anov
## size = large
## effect.size = 0.4
```

One-way ANOVA pwr analysis function:

pwr.anova.test(k = , n = , f = , sig.level = , power = )

…where k is the number of groups and n is the number of obs. IN EACH GROUP.

`pwr.anova.test(k =4, n=32, f=.3, sig.level =.05)`

```
##
## Balanced one-way analysis of variance power calculation
##
## k = 4
## n = 32
## f = 0.3
## sig.level = 0.05
## power = 0.8101954
##
## NOTE: n is number in each group
```

CORRELATION!

Function:

pwr.r.test(n = , r = , sig.level = , power = )

CHI-SQUARE

Function:

pwr.chisq.test(w =, N = , df = , sig.level =, power = )

…where “w”= effect size and “n”=number of observations.

```
# Quick example:
pwr.chisq.test(w=.3, N=120, df=9, sig.level=.05)
```

```
##
## Chi squared power calculation
##
## w = 0.3
## N = 120
## df = 9
## sig.level = 0.05
## power = 0.6045631
##
## NOTE: N is the number of observations
```

REGRESSION!

Function:

pwr.f2.test(u =, v = , f2 = , sig.level = , power = )

where “u”= numerator degrees of freedom (number of continuous variables + number of dummy codes – 1)

…and “v”=denominator (error) degrees of freedom

For example, find the power for a multiple regression test with 2 continuous predictors and 1 categorical

```
# predictor (i.e. Marital status with k=3 so 3-1=2 dummy codes) that has a large effect size and a sample size of 30.
pwr.f2.test(u =3, v =30, f2 =.35, sig.level =.05)
```

```
##
## Multiple regression power calculation
##
## u = 3
## v = 30
## f2 = 0.35
## sig.level = 0.05
## power = 0.7807726
```

```
# Generating a table of sample sizes
# What is the required sample size for an independent samples t-test with pwr=.80?
seq <- seq(.1,.8, .1)
FindN <- rep(0,8)
for(i in 1:8) FindN[i]=pwr.t.test(d=seq[i],power=.8,sig.level=.05,type="two.sample")$n
data.frame(d=seq , N=ceiling(FindN))
```

```
## d N
## 1 0.1 1571
## 2 0.2 394
## 3 0.3 176
## 4 0.4 100
## 5 0.5 64
## 6 0.6 45
## 7 0.7 34
## 8 0.8 26
```

```
# can swap in other functions, just make sure arguments are relevant to test
seq <- seq(.1,.8, .1)
FindN <- rep(0,8)
for(i in 1:8) FindN[i]=pwr.anova.test(k=4, f=seq[i], power=.8,sig.level=.05)$n
data.frame(f=seq , N=ceiling(FindN))
```

```
## f N
## 1 0.1 274
## 2 0.2 70
## 3 0.3 32
## 4 0.4 19
## 5 0.5 12
## 6 0.6 9
## 7 0.7 7
## 8 0.8 6
```

```
# NOTE: "ceiling" is an argument for rounding numbers so the values are slightly different than if we
# computed the sample size based on specified values. For example...
pwr.anova.test(k =4,f=.3, sig.level =.05, power=.8)
```

```
##
## Balanced one-way analysis of variance power calculation
##
## k = 4
## n = 31.27917
## f = 0.3
## sig.level = 0.05
## power = 0.8
##
## NOTE: n is number in each group
```

```
#HELPFUL RESOURCE:
# Quick-R blog: http://www.statmethods.net/stats/power.html
```