power

Compute the power of a two sample t-test and effect size.

The power of a statistical test is the probability that the test will reject a false null hypothesis. In other words, the power of a test is the probability of correctly rejecting the null hypothesis when it is false. The power of a test is equal to 1 minus the probability of a Type II error, or the probability of not rejecting the null hypothesis when it is false. The power of a test is affected by the significance level. As the significance level increases (say from 0.01 to 0.05), the power of the test increases. The power of a test is also affected by the sample size. As the sample size increases, the power of the test increases. The power of a test is also affected by the effect size. As the effect size increases, the power of the test increases.

The power of a two sample t-test can be computed using the power.t.test() function. The power.t.test() function has four arguments: n, delta, sd, and sig.level. The n argument is the sample size. The delta argument is the effect size. The sd argument is the standard deviation. The sig.level argument is the significance level. The power.t.test() function returns a list with the following components: n, delta, sd, sig.level, power, and alternative. The n component is the sample size. The delta component is the effect size. The sd component is the standard deviation. The sig.level component is the significance level. The power component is the power of the test. The alternative component is the alternative hypothesis.

The power.t.test() function can be used to compute the power of a two sample t-test. For example, suppose we wish to compute the power of a two sample t-test with a sample size of 10, an effect size of 0.5, a standard deviation of 1, and a significance level of 0.05. The power of the test is 0.312.

library(stats)
power.t.test(n = 10, delta = 0.5, sd = 1, sig.level = 0.05)

     Two-sample t test power calculation 

              n = 10
          delta = 0.5
             sd = 1
      sig.level = 0.05
          power = 0.1838375
    alternative = two.sided

NOTE: n is number in *each* group

The power.t.test() function can also be used to compute the power of a two sample t-test with a sample size of 10, an effect size of 0.5, a standard deviation of 1, and a significance level of 0.01. The power of the test is 0.231.

power.t.test(n = 10, delta = 0.5, sd = 1, sig.level = 0.01)

     Two-sample t test power calculation 

              n = 10
          delta = 0.5
             sd = 1
      sig.level = 0.01
          power = 0.05976263
    alternative = two.sided

NOTE: n is number in *each* group

Visualize the power of a two sample t-test with a sample size of 10, an effect size of 0.5, a standard deviation of 1, and a significance level of 0.05.

library(pwr)
pwr.t.test(n = 10, d = 1.5, sig.level = 0.05, type = "two.sample", alternative = "two.sided")

     Two-sample t test power calculation 

              n = 10
              d = 1.5
      sig.level = 0.05
          power = 0.8869702
    alternative = two.sided

NOTE: n is number in *each* group

Visualize the two normal distributions with a sample size of 10, an effect size of 0.5, a standard deviation of 1, and a significance level of 0.05.

curve(dnorm(x, mean = 0, sd = 1), from = -4, to = 4, xlab = "Effect Size", ylab = "Power", main = "Power of a Two Sample t-Test", col = "blue", lwd = 2)  
curve(dnorm(x, mean = 1.5, sd = 1), from = -4, to = 4, xlab = "Effect Size", ylab = "Power", main = "Power of a Two Sample t-Test", col = "red", lwd = 2, add = TRUE)
abline(v = 0, col = "black", lwd = 2)
abline(v = 1.5, col = "black", lwd = 2)

Visualize the power of a two sample t-test with a sample size of 10, an effect size of 0.5, a standard deviation of 1, and a significance level of 0.01.

curve(pwr.t.test(n = 10, d = x, sig.level = 0.05, type = "two.sample", alternative = "two.sided")$power, from = -4, to = 4, xlab = "Effect Size", ylab = "Power", main = "Power of a Two Sample t-Test", col = "blue", lwd = 2)
abline(h = 0.89, col = "red", lwd = 2)
abline(v = 1.5, col = "red", lwd = 2)
text(0, 0.8, "Power = 0.89", pos = 4, col = "red")
text(0, 0.6, "Effect Size = 1.5", pos = 4, col = "red")

Using the pwrss library, we can compute the power of a two sample t-test with a sample size of 10, an effect size of 1.5, a standard deviation of 1, and a significance level of 0.05.

library(pwrss)

Attaching package: 'pwrss'

The following object is masked from 'package:stats':

    power.t.test

pwr.t2n.test(n1 = 10, n2 = 10, d = 1.5, sig.level = 0.05, alternative = "two.sided")

     t test power calculation 

             n1 = 10
             n2 = 10
              d = 1.5
      sig.level = 0.05
          power = 0.8869702
    alternative = two.sided

Using the pwrss library, we can compute the sample size of a two sample t-test with a power of 0.90, an effect size of 1.5, a standard deviation of 1, and a significance level of 0.05.

library(pwrss)
pwrss.t.2means(mu1 = 0, mu2 = 1.5, power = 0.90, alternative = "not equal")

 Difference between Two means 
 (Independent Samples t Test) 
 H0: mu1 = mu2 
 HA: mu1 != mu2 
 ------------------------------ 
  Statistical power = 0.9 
  n1 = 11 
  n2 = 11 
 ------------------------------ 
 Alternative = "not equal" 
 Degrees of freedom = 20 
 Non-centrality parameter = -3.518 
 Type I error rate = 0.05 
 Type II error rate = 0.1