44 PART 1 Getting Started with Biostatistics

likelihood of committing Type I and Type II errors — between the α and β error

rates. If you make α too small, β will become too large, and vice versa.

At this point, you may be wondering, “Is there any way to keep both Type I and

Type II error small?” The answer is yes, and it involves power, which is described

in the next section.

Grasping the power of a test

The power of a statistical test is the chance that it will come out statistically sig-

nificant when it should — that is, when the alternative hypothesis is really true.

Power is a probability and is very often expressed as a percentage. Beta (β) is the

chance of getting a nonsignificant result when the alternative hypothesis is true,

so you see that power and β are related mathematically: Power = 1 – β.

The power of any statistical test depends on several factors:»

» ^{The α level you’ve established for the test — that is, the chance you’re willing}

to accept making a Type I error (usually 0.05)»

» ^{The actual magnitude of the effect in the population, relative to the amount of}

noise in the data»

» ^{The size of your sample}

Power, sample size, effect size relative to noise, and α level can’t all be varied

independently. They’re interrelated, because they’re connected and constrained

by a mathematical relationship involving all four quantities.

This relationship between power, sample size, effect size relative to noise, and α

level is often very complicated, and it can’t always be written down explicitly as a

formula. But the relationship does exist. As evidence of this, for any particular

type of test, theoretically, you can determine any one of the four quantities if you

know the other three. So for each statistical test, there are four different ways to

do power calculations, with each way calculating one of the four quantities from

arbitrarily specified values of the other three. (We have more to say about this in

Chapter  5, where we describe practical issues that arise during the design of

research studies.) In the following sections, we describe the relationships between

power, sample size, and effect size, and briefly review how you can perform power

calculations.