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

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 .
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