76 CHAPTER 6 Hypothesis Testing
hypothesis of zero difference is true, and n is the number of patients in
the treatment group and also the number in the control group.
So then based on the t - distribution, we fi nd a critical value, call
it − C α. If the test statistic T ≤ − C α , we reject the null hypothesis. If
T > − C α , we cannot reject the null hypothesis. As we know, the power
function depends on the distribution of the test statistic under the alter-
native hypothesis and the chosen critical value − C α. This distribution
is a noncentral t - distribution. Just trust that statisticians can use such
distributions to compute power and required sample size. It is not
something that you need to learn.
In this test, we assume both samples come from normal populations
with the same variance and hence the same standard deviation. This is
a more realistic assumption for the pacing leads trial. Also, because
steroid - eluting leads had already been approved by the FDA for a com-
petitor, it is accepted that the steroid lead is preferred. Consequently,
the patients and the sponsor would both like to see more steroid leads
implanted during the trial, but still enough control leads so that the test
for difference in means will have the required statistical power (gener-
ally taken to be 0.80).
The test statistic t = ( m 1 − m 2 )/ SD , where m 1 is the sample mean
for the fi rst population with sample size n 1 and m 2 is the sample mean
for the second population with sample size n 2 and the pooled standard
deviation given by the following equation:
(^) SD= {}() 11 //n121 22+()n []()n−1 1+−()n s^2 /n 12 +−n 2.
Under the null hypothesis and the above conditions, t has Student ’ s t -
distribution with n 1 + n 2 − 2 df. We have seen the power function for
this test in Figure 6.1.
6.5 PAIRED T - TEST
The tests we have studied so far that involve two populations consid-
ered independent samples. With the paired t - test, we are deliberately
making the samples dependent, since we have matched pairs. The
pairing is used to create positive correlation that will reduce the vari-
ability of the estimate (say the difference of two sample means). One
common way to do this is to have the patient as the pairing variable.