80 CHAPTER 6 Hypothesis Testing1 − α. Confi dence levels are often expressed as a percentage, so the
confi dence level for the interval is 100(1 − α )%.
It should be noted that hypothesis tests are often constructed in a
way that the test statistic assumes the value of nuisance parameters
(e.g., the standard deviation of a normal distribution when testing that
the mean is different from 0) under the null hypothesis. This is done
because the test is designed to reject the null hypothesis, and such a
formulation generally leads to a more powerful test than the one you
would get by simply inverting the null hypothesis. Remember that
confi dence intervals have a different goal, namely to identify the most
plausible values for parameter based on the data, and the null hypoth-
esis has no relevance. For example, in hypothesis testing for a propor-
tion, when using a normal approximation, the unknown standard
deviation (which statisticians call a nuisance parameter) is replaced by
the value under the null hypothesis. Under the null hypothesis, let us
assume p = 1/2. Then if n is the sample size, the standard deviation for
the sample proportion is ppn() 1 − / , or, substituting p = 1/2, it is
12 /( n). But for the confi dence interval we would use p in place of
the unknown p making it ppnˆˆ()1/− , which will be different in
general.6.8 SAMPLE SIZE DETERMINATION
We will again look at the pacing leads example to demonstrate sample
size determination. Here we are only considering fi xed sample sizes.
Group sequential and adaptive designs allow the fi nal sample size to
depend on the data, and hence the sample size is unconditionally a
random integer N.
What is the required sample size for a test? It depends on how big
the treatment effect has to be. It also depends on the standard deviation
of the test statistic. Averaging sample values reduces the standard
deviation. If a random variable X has a standard deviation σ , then if
you average n , such variables that have the same mean and standard
deviation and are independent of each other the sample mean has stan-
dard deviation σ/ n. This explains why we get increasing power as
we increase n.
The sample standard deviation gets smaller and approaches 0 as
n → ∞. So the idea is to specify a power that you want to achieve, say