Question

Factors Affecting Power in the t Test

After completing this week's assigned readings, discuss one of the following questions:

1. What things can a researcher do to try to increase the magnitude of the d effect size? Suppose that you can increase the d effect size while holding group sizes n₁ and n₂ constant. How will an increase in d influence the magnitude of t ?
2. Several factors influence statistical power for a one-sample t test. How does statistical power change (increase or decrease) for each of the following changes?
   • When d (effect size) increases.
   • When N (sample size) increases.
   • When the alpha level is made smaller.
   Explain your answer. For example, if we know ahead of time that the effect size d is very small, what does this tell us about the N we will need in order to have adequate statistical power? (We assume that all other terms included in the r ratio remain the same.)

Answer 1

The effect size d is given as:

$d = \frac{M_{1}-M_{2}}{s}=\frac{t}{\sqrt{n}}$

Here, M₁ and M₂ are the means of the two groups respectively, and s is the pooled standard deviation of the two groups given by:

$s = \sqrt{\frac{(n_{1}-1){s_{1}}^2+(n_{2}-1){s_{2}}^2}{n_{1}+n_{2}-2}}$

where ${s_{1}}^2 \& {s_{2}}^2$ are the individual standard deviations of each group.

From the first equation, we can see that d increases as the difference between the two groups increases, but this is not in the hands of the researcher. Also, d increases as s decreases. As we know that s is inversely proportional to the precision of an experiment, so by increasing the precision, the effect size d can be increased.

The first equation also clearly shows that an increase in d will result in an increase in the magnitude of t, when the group sizes n₁ and n₂ are kept constant.