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This method was proposed by White et al. (2004) to address partial noncompliance by viewing survival outcome as a sequence of binary outcomes to provide an ‘approximate’ overall hazard ratio estimate which is adjusted for compliance. We note that this approach initially ignores survival time by considering the outcome as binary.

We define the following random variables for survival data to evaluate compliance with active treatment (i.e. compliance with active if offered) while ignoring compliance with placebo. Let Y represent survival status, for example, death (principal outcome of interest) so that Y = 1(0) if participant is dead (alive) by the end of the interval or study as the case may be. Let Z = 1(0) represent treatment or placebo arm while A = 1(0) represent observed treatment (placebo) receipt. Let S = 1(0) represent a participant’s latent (potential) compli- ance behaviour. Further let α represent the probability of noncompliance, i.e. α = Pr(S = 0). Given this setup, we can observe S as equal to A in the treatment arm but we cannot observe the latent compliance S in the placebo arm.

Assuming some form of compliance so that α > 0, let π0

Z and πZ1 respectively represent

the probabilities of death among noncompliers and compliers in the Z arm, i.e.

π0_Z = Pr(Y = 1|Z = z, S = 0) and π_Z1 = Pr(Y = 1|Z = z, S = 1),

where π0_Z and π1_Z may be assumed equal by randomization principle. Then the overall probability of death in arm Z may be given by

πZ = (1− α)π1z+ απ

0

z. (2.8)

We can use the above probabilities to define RRITT and RRCACE as quantities based on the

risk ratio scale where RRITT is simply the ratio of π1 and π0 while CACE may be considered

as the ratio of π1

1 and π01.

However, we note that while we can directly estimate RRITT, α, π10, π11 and π1, we may

not be able to directly estimate RRCACE, π00 and π01 because S is an unobservable (latent)

variable. For survival data, we can resolve this difficulty by using the exclusion restriction (ER) which assumes that the risk of death does not depend on the arm of randomization given the treatment received, i.e. Y ⊥ Z|A. The ER assumption then implies that

π0₀ = π₁0. (2.9)

We may use the ER assumption to find CACE estimates. Following White et al. (2004), we can subdivide survival data into small intervals and assume subjects on active treatment may only stop treatment at the start of that interval. A sufficiently small interval may allow us to assume constant hazard rates. Then by considering death as a sequence of binary outcomes and using T to denote the interval in which death occurred, White et al. obtained an ITT risk ratio in interval i as

ITTi =

Pr(T = i|T ≥ i, Z = 1)

Pr(T = i|T ≥ i, Z = 0), i = 1, . . . , I. (2.10) Using the counterfactuals framework, we can evaluate treatment effects by considering

participants in the placebo arm. Now let the random variable S (the compliance-type) define the interval in which stopping (death/censoring) occurs, or I + 1 if no stopping occurs. For participants on the control arm, S represents the counterfactual stopping interval, i.e. the interval in which stopping would have occurred had they been allocated to the treatment arm. White et al. (2004) then use an extended exclusion restriction assumption to obtain CACE estimate in each interval i. With the extended exclusion restriction, we assume that randomized allocation has no effect on any survivors to the start of interval i who would stop treatment by the start of interval i, i.e.

Pr(T = i|T ≥ i, S = s, Z = 1)

= Pr(T = i|T ≥ i, S = s, Z = 0) ∀ i = 1, . . . , I, s ≤ i. (2.11) We observe that the extended exclusion restriction assumption (2.11) mimics a version of the Markov property that the present is conditionally independent of the past. They then considered the CACE in interval i as the risk ratio among those who survive to interval i and who would not stop treatment before the end of interval i:

CACEi =

Pr(T = i|T ≥ i, S > i, Z = 1)

Pr(T = i|T ≥ i, S > i, Z = 0). (2.12) Both exclusion restriction (2.9) and extended exclusion restriction (2.11) assumptions may enable us use the proportion of non-compliers to estimate the proportion of compliers used in obtaining CACE estimate. White et al. then showed that

CACEi ≈

ITTi(1− θi)

1− θi ITTi

, (2.13)

where θi = Pr(Ai = 0|T = frm[o]−−i, Z = 1) is the probability that a participant in the

active treatment arm who experienced event (e.g. died) in interval i had previously stopped treatment, i.e. was non-complier by the time of her death.

Assuming short intervals in addition to EER assumption, White et al. (2004) showed that we can extend the CACE estimate given by equation (2.12) above for discrete time case to continuous (pooled) hazard rates at time t. By weighting the ITT(t) estimates from (2.10)

we then CACE estimate for proportional hazards as

CACEPH(t) =

ITT(t)(1− θ(t))

1− θ(t) ITT(t) , (2.14)

where θ = Pr (A = 0|Y =1, Z =1) , is the noncompliance probability among participants in the treatment arm who died. The CACEPH method is essentially a time-adjusted ITT estimate

that fits a Cox’s PH model (HRITT) to provide a Causal Hazard ratio Adjustment Regression

model (CHARM) where the linear predictor in the control arm is zero and in the active arm is a function of time, for example,

CHARM≡ h(t)=h0(t) exp[β0Z + β1f (t)Z], (2.15)

where f(t) can be specified as a combination of linear/quadratic function of time to event. White et al. (2004) proposed two ways to obtain a constant CHARM estimate (2.15) above:

(i) either allowing both HRITT and odds of noncompliance θ to vary over time or

(ii) assuming time invariance for both HRITT and θ so as to allow a simple CHARM ap-

proximation of (2.14) by CHARM≡ \CACEPH = d HRITT(1− ˆθ) 1− ˆθ dHRITT , (2.16)

where θ = Pr (A = 0|Z =1) represent the proportion of noncompliers randomized to the treatment arm who experienced event of interest (e.g death).

Assuming a single death at a time is experienced, estimation procedure (ii) may be imple- mented by estimating individual θis for the dead non-compliers and using a logistic regression

model with these values to examine evidence of change or trend in the θis (White et al., 2004).

For example,

would represent a linear trend of noncompliance. Here the outcome would be 1 if a participant is a noncomplier by the time of death and 0 if a complier by the time of her death, i.e.

θ =    1; A = 0|Y = 1 0; A = 1|Y = 1.

In our analysis we will implement the complex procedure (i) above using the Stata com- mand adjhr (White, 2002) which provides an ‘approximate’ constant (overall) CHARM estimate as a hazard ratio adjusted for compliance. We will also compare this with a sim- ple CHARM estimate obtained using a two-stage regression strategy: logistic regression to estimate θ which is then substituted in Equation (2.16) above.

2.7

Modelling noncompliance in two treatment arms