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

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CHAPTER 10 ST 520, A. TSIATIS and D. Zhang versus HA : ∆ > 0. Remark This is the same framework that has been used throughout the course. At any interim analysis time t, our decision making will be based on the test statistic T(t) = ˆ ∆(t) se{ ˆ ∆(t)} , where ˆ ∆(t) is an estimator for ∆ and se{ ˆ ∆(t)} is the estimated standard error of ˆ ∆(t) using all the data that have accumulated up to time t. For two-sided tests we would reject the null hypothesis if the absolute value of the test statistic |T(t)| were sufficiently large and for one-sided tests if T(t) were sufficiently large. Example 1. (Dichotomous response) Let π1, π0 denote the population response rates for treatments 1 and 0 (say new treatment and control) respectively. Let the treatment difference be given by The test of the null hypothesis will be based on T(t) = ∆ = π1 − π0 p1(t) − p0(t) � ¯p(t){1 − ¯p(t)} � 1 1 + n1(t) n2(t) where using all the data available through time t, pj(t) denotes the sample proportion responding among the nj(t) individuals on treatment j = 0, 1. Example 2. (Time to event) Suppose we assume a proportional hazards model. Letting A denote treatment indicator, we consider the model λ1(t) = exp(−∆), λ0(t) and we want to test the null hypothesis of no treatment difference versus the two-sided alternative H0 : ∆ = 0 HA : ∆ �= 0, PAGE 168 �,
CHAPTER 10 ST 520, A. TSIATIS and D. Zhang or the one-sided test that treatment 1 does not improve survival H0 : ∆ ≤ 0 versus the alternative that it does improve survival HA : ∆ > 0. Using all the survival data up to time t (some failures and some censored observations), we would compute the test statistic T(t) = ˆ ∆(t) se{ ˆ ∆(t)} , where ˆ ∆(t) is the maximum partial likelihood estimator of ∆ that was derived by D. R. Cox and se{ ˆ ∆(t)} is the corresponding standard error. For the two-sided test we would reject the null hypothesis if |T(t)| were sufficiently large and for the one-sided test if T(t) were sufficiently large. Remark: The material on the use and the properties of the maximum partial likelihood estimator are taught in the classes on Survival Analysis. We note, however, that the logrank test computed using all the data up to time t is equivalent asymptotically to the test based on T(t). Example 3. (Parametric models) Any parametric model where we assume the underlying density of the data is given by p(z; ∆, θ), and use for ˆ ∆(t) the maximum likelihood estimator for ∆ and for se{ ˆ ∆(t)} compute the estimated standard error using the square-root of the inverse of the observed information matrix, with the data up to time t. In most important applications the test statistic has the property that the distribution when ∆ = ∆ ∗ follows a normal distribution, namely, T(t) = ˆ ∆(t) se{ ˆ ∆=∆ ∆(t)} ∗ ∼ N(∆ ∗ I 1/2 (t, ∆ ∗ ), 1), where I(t, ∆ ∗ ) denotes the statistical information at time t. Statistical information refers to Fisher information, but for those not familiar with these ideas, for all practical purposes, we can equate (at least approximately) information with the standard error of the estimator; namely, I(t, ∆ ∗ ) ≈ {se( ˆ ∆(t)} −2 . PAGE 169
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