The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

6.1 Type I and Type II Errors 73

hypothesis, we are accepting the alternative. Whether we reject or

do not reject the null hypothesis, we are making a decision, and associ-

ated with that decision is a probability that we made the wrong

decision. These ideas will be discussed more thoroughly in the next

section.

6.1 TYPE I AND TYPE II ERRORS

The type I error or signifi cance level (denoted as α ) for a test is the
probability that our test statistic is in the rejection region for the null
hypothesis, but in fact the null hypothesis is true. The choice of a cutoff
that defi nes the rejection region determines the type I error, and can be
chosen for any sample size n ≥ 1.
The type II error (denoted as β ) depends on the cutoff value and
the true difference δ ≠ 0, when the null hypothesis is false. It is the
probability of not rejecting the null hypothesis when the null hypothesis
is false, and the true difference is actually δ. The larger delta is, the
lower the type II error becomes. The probability of correctly rejecting
at a given δ is called the power of the test. The power of the test is
1 − β. We can defi ne a power function f ( δ ) = 1 − β ( δ ). We use the
notation β ( δ ) to indicate the dependency of β on δ. When δ = 0 ,
f ( δ ) = α.
We can relate to these two types of errors by considering a real
problem. Suppose we are trying to show the effectiveness of a drug by
showing that it works better than placebo. The type I and type II errors
correspond to false claims. The type I error is the claim that the drug
is effective when it is not (i.e., is not better than placebo by more than
δ ). The type II error is the claim that the drug is not effective when it
really is effective (i.e., better than placebo by at least δ ).
However, it increases as | δ | increases (often in a symmetric fashion
about 0, i.e., f ( δ ) = f [ − δ ]). Figure 6.1 shows the power function for a
test that a normal population has a mean zero versus the alternative that
the mean is not zero for sample sizes n = 25 and 100 and a signifi cance
level of 0.05. The solid curve is for n = 25, and the dashed for n = 100.
We see that these power functions are both symmetric about 0, and
meet with a value of 0.05 at δ = 0. Since 100 is four times larger than
25, the power function increases more steeply for n = 100 compared to
n = 25.