# How to compute the power of a test comparing two population means

## Description

When creating a factorial design, it is important that it has adequate power to detect significant main effects and interaction effects of interest. How can we calculate the power of a two-sample $t$ test that we aim to perform in such a situation?

## Using statsmodels, in Python

View this solution alone.

From statsmodels, we use the solve_power function in the TTestIndPower class. That function embodies a relationship among five variables; you provide any four of them and it will compute the fifth to be consistent with the first four, regarding the two-sample $t$-test you plan to perform. Let’s get started by importing the package and create a TTestIndPower object.

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from statsmodels.stats.power import TTestIndPower
analysis = TTestIndPower()


For this example, let’s say that:

• You plan to create a balanced $4\times2$ factorial experiment with 32 subjects.
• You expect the effect size for the main effect of factor A to be medium (0.25 according to Cohen’s 1988 text).
• You want to know the expected power for the test of a main effect of factor A.
• Your significance level is $\alpha=0.05$.

We proceed as follows.

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obs = 32       # number of subjects (or observations)
effect = 0.25  # effect size
alpha = 0.05   # significance level
ratio = 1      # ratio of the number of observations in one sample to the other

# We leave power unspecified, so that solve_power will compute it for us:
analysis.solve_power( effect_size=effect, power=None, nobs1=obs, ratio=ratio, alpha=alpha )

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0.1662985260871502


The power is 0.1663, which means that the probability of rejecting the null hypothesis when in fact it is false OR the probability of avoiding a Type II error is 0.1663.

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## Solution, in R

View this solution alone.

We use the power.t.test function in R. It embodies a relationship among five variables; you provide any four of them and it will compute the fifth to be consistent with the first four, regarding the two-sample $t$-test you plan

For this example, let’s say that:

• You plan to create a balanced $4\times2$ factorial experiment with 32 subjects.
• You wish to be able to detect a difference
• You want to know the expected power for the test of a main effect of factor A.
• Your significance level is $\alpha=0.05$.

We proceed as follows.

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# install.packages('pwr') # if you have not already installed it
library(pwr)

obs <- 32       # number of subjects (or observations)
effect <- 0.25  # effect size
alpha <- 0.05   # significance level
ratio <- 1      # ratio of the number of observations in one sample to the other

# We leave power unspecified, so that power.t2n.test will compute it for us:
pwr.t2n.test(n1=obs, n2=obs, d=effect, sig.level=alpha, power=NULL)

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t test power calculation

n1 = 32
n2 = 32
d = 0.25
sig.level = 0.05
power = 0.1662985
alternative = two.sided


The power is 0.1663, which means that the probability of rejecting the null hypothesis when in fact it is false OR the probability of avoiding a Type II error is 0.1663.