Power curve of the Student t-test

Background

Let us draw two samples of $r_1$ and $r_2$ elements from two normal distributions $N(\mu_1, \sigma)$ and $N(\mu_2,\sigma)$, i.e. the variance is assumed to be equal. The $t$-statistic is:

$$ t=\frac{\bar{x}_1-\bar{x}_2}{\sqrt{\frac{1}{r_1}+\frac{1}{r_2}} \cdot \sqrt{\frac{(r_1-1)s_1^2+(r_2-1)s_2^2}{r_1+r_2-2}}}$$

where $\bar{x}_1$ and $\bar{x}_2$ are sample means, and $s_1^2$, $s_2^2$ are the unbiased sample variances. The $t$-statistic has Student $t$-distribution with $\nu=r_1+r_2-2$ degrees of freedom.

The power

The significance level is given and equals $1-\alpha$, where $\alpha$ is the probability of type I error. The difference is regarded as significant when $|\mu_1-\mu_2|\geq \delta$, where $\delta$ is given. The parameter $\delta$ does not appear directly but only as the noncentrality parameter (ncp) given by the formula:

$$ ncp = \frac{\delta}{\sigma\sqrt{\frac{1}{r_1}+\frac{1}{r_2}}}. $$

The power of the test is one less the probability of the type II error. The power curve is the graph of the power vs. $\delta$.

Significance level (alpha):

Degrees of freedom (nu):

Maximum ncp (maxncp):

## The code assumes that the following values are set
##  alpha - 1-significance level
##  nu    - the number of degrees of freedom
##  maxncp - the maximum noncentrality parameter (plot range)

##
## Function yielding the power of the t-test for a given:
## ncp - non-centrality parameter
## df  - the number of degrees of freedom
## alpha - 1- significance level
##
power.of.t.Student <- function(ncp = 1, df = 20, alpha = 0.05)
  {
    ## The critical value of t
    t.alpha.half <- qt(alpha/2, df = df, ncp = 0, lower.tail = F);

    ## Probability of type II error
    beta <- pt(t.alpha.half, df = df, ncp = ncp) -
      pt(-t.alpha.half, df = df, ncp = ncp);

    ## Power
    return(1 - beta);
  }

## Plot the power curve
plot(function(ncp){power.of.t.Student(ncp = ncp, df = nu, alpha = alpha)},
     0,
     maxncp,
     type='l',
     xlab='noncentrality parameter',
     ylab='power')

Copyright(©) Marek Rychlik, 2009.