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

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CHAPTER 8 ST 520, A. TSIATIS and D. Zhang Also P(Y = 1|A = 1) = P(Y = 1|A = 1, C = 1)P(C = 1|A = 1) = P(Y ∗ 1 = P(Y ∗ 1 + P(Y = 1|A = 1, C = 0)P(C = 0|A = 1) = 1|A = 1, C = 1)P(C = 1|A = 1) + P(Y ∗ 0 = 1|C = 1)P(C = 1) + P(Y ∗ 0 = 1|C = 0)P(C = 0) = 1|A = 1, C = 0)P(C = 0|A = 1) = π COM 1 θ + π NC 0 (1 − θ) (8.6) P(Y = 1|A = 0) = P(Y ∗ 0 Subtracting (8.7) from (8.6) we get that or Recall that = 1|A = 0) = P(Y ∗ ∆ITT = P(Y = 1|A = 1) − P(Y = 1|A = 0) = (π COM 1 ∆ COM = P(Y ∗ 1 = 1|C = 1) − P(Y ∗ 0 0 = 1) = π0 = π COM 0 θ + π NC 0 (1 − θ). (8.7) − π COM 0 )θ, ∆ITT = ∆ COM θ. (8.8) = 1|C = 1) = E(Y ∗ 1 ∗ − Y0 |C = 1) is the difference in the mean counterfactual responses between treatment and placebo among patients that would comply with treatment. As such, ∆ COM is a causal parameter and some argue that it is the causal parameter of most interest since the patients that will benefit from a new treatment are those who will comply with the new treatment. Equation (8.8) makes it clear that the intention-to-treat analysis will yield an estimator which diminishes a causal treatment effect. In fact, since we are able to estimate the parameter θ, the probability of complying if offered the new treatment, then the causal parameter ∆ COM can be identified using parameters from the observable random variables; namely ∆ COM = P(Y = 1|A = 1) − P(Y = 1|A = 0) . (8.9) P(C = 1|A = 1) Since all the quantities on the right hand side of (8.9) are easily estimated from the data of a clinical trial, this means we can estimate the causal parameter ∆ COM . PAGE 128
CHAPTER 8 ST 520, A. TSIATIS and D. Zhang Remarks • If the null hypothesis of no treatment effect is true; namely H0 : ∆ COM = ∆ NC = ∆ = 0 – The intent to treat analysis, which estimates (∆ COM θ), gives an unbiased estimator of treatment difference (under H0) and can be used to compute a valid test of the null hypothesis • If we were interested in estimating the causal parameter ∆ COM , the difference in response rate between treatment and placebo among compliers only, then – the intent to treat analysis gives an underestimate of this population causal effect • Since there are no data available to estimate π NC 1 , we are not able to estimate ∆ NC or ∆ As-treated analysis In one version of an as treated analysis we compare the response rate of patients randomized to receive active drug who comply to all patients randomized to receive placebo. That is, we compute ∆AT = π COM 1 − π0. Since π0 = πCOM 0 θ + πNC 0 (1 − θ), after some algebra we get that ∆AT = ∆ + (π COM 1 − π NC 1 )(1 − θ), where ∆ denotes the average causal treatment effect. This makes clear that when there is noncompliance, (θ < 1), the as-treated analysis will yield an unbiased estimate of the average causal treatment effect only if π COM 1 = π NC 1 . As we’ve argued previously, this assumption is not generally true and hence the as-treated analysis can result in biased estimation even under the null hypothesis. Some Additional Remarks about Intention-to-Treat (ITT) Analyses • By not allowing any exclusions, we are preserving the integrity of randomization PAGE 129
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