ST 520 Statistical Principles of Clinical Trials - NCSU Statistics ...

CHAPTER 6 ST 520, A. TSIATIS and D. Zhang
(b) The probability distribution of Tn can be evaluated (or at least approximated) at the
border between the null and alternative hypotheses; i.e. at ∆ = 0.
After we conduct the clinical trial and obtain the data, i.e. the realization (z1, . . .,zn) of
(Z1, . . .,Zn), we can compute tn = Tn(z1, . . .,zn) and then gauge this observed value against
the distribution of possible values that Tn can take under ∆ = 0 to assess the strength of
evidence for or against the null hypothesis. This is done by computing the p-value
P∆=0(Tn ≥ tn).
If the p-value is small, say, less than .05 or .025, then we use this as evidence against the null
hypothesis.
Note:
1. Most test statistics used in practice have the property that P∆(Tn ≥ x) increases as ∆
increases, for all x. In particular, this would mean that if the p-value P∆=0(Tn ≥ tn) were
less than α at ∆ = 0, then the probability P∆(Tn ≥ tn) would also be less than α for all ∆
corresponding to the null hypothesis H0 : ∆ ≤ 0.
2. Also, most test statistics are computed in such a way that the distribution of the test
statistic, when ∆ = 0, is approximately a standard normal; i.e.
Tn
(∆=0)
∼ N(0, 1),
regardless of the values of the nuisance parameters θ.
For the problem where we were comparing the mean response between two treatments, where
response was assumed normally distributed with equal variance by treatment, but possibly dif-
ference means, we would use the two-sample t-test; namely,
Tn =
sY
¯Y1 − ¯ Y2
� , 1/2
1
+ n1 n2
� 1
where ¯ Y1 denotes the sample average response among the n1 individuals assigned to treatment
1, ¯ Y2 denotes the sample average response among the n2 individuals assigned to treatment 2,
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