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

CHAPTER 6 ST 520, A. TSIATIS and D. Zhang
we conduct a clinical trial and collect data from a sample of individuals. The data from n
individuals in our sample will be denoted generically as (z1, . . .,zn) and represent realizations of
random vectors (Z1, . . .,Zn). The Zi may represent a vector of random variables for individual
i; e.g. response, treatment assignment, other covariate information.
As statisticians, we posit a probability model that describes the distribution of (Z1, . . .,Zn) in
terms of population parameters which includes ∆ (treatment difference) as well as other param-
eters necessary to describe the probability distribution. These other parameters are referred to
as nuisance parameters. We will denote the nuisance parameters by the vector θ. As a simple
example, let the data for the i-th individual in our sample be denoted by Zi = (Yi, Ai), where
Yi denotes the response (taken to be some continuous measurement) and Ai denotes treatment
assignment, where Ai can take on the value of 1 or 2 depending on the treatment that patient i
was assigned. We assume the following statistical model: let
and
(Yi|Ai = 2) ∼ N(µ2, σ 2 )
(Yi|Ai = 1) ∼ N(µ2 + ∆, σ 2 ),
i.e. since ∆ = µ1 − µ2, then µ1 = µ2 + ∆. The parameter ∆ is the test parameter (treatment
difference of primary interest) and θ = (µ2, σ 2 ) are the nuisance parameters.
Suppose we are interested in testing H0 : ∆ ≤ 0 versus HA : ∆ > 0. The way we generally
proceed is to combine the data into a summary test statistic that is used to discriminate between
the null and alternative hypotheses based on the magnitude of its value. We refer to this test
statistic by
Tn(Z1, . . ., Zn).
Note: We write Tn(Z1, . . .,Zn) to emphasize the fact that this statistic is a function of all the
data Z1, . . .,Zn and hence is itself a random variable. However, for simplicity, we will most often
refer to this test statistic as Tn or possibly even T.
The statistic Tn should be constructed in such a way that
(a) Larger values of Tn are evidence against the null hypothesis in favor of the alternative
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