4.2 Longitudinal setting with last measured variable subject to non-response
4.2.3 Proof of Proposition 2
Let z = a, r index the randomised active arm or reference arm allocation for each patient i with follow-up outcome at time J denoted by YziJ. The outcome data at the final time point for the
reference patients are contained in the vector YrJ = (Yr1J, ...YrnJ) T
. The final visit outcome data for the non-deviating active patients are contained in the vector YaJ,o= {YaiJ; i ∈O}
T
.
We suppose that each deviating patient has two potential outcomes at time J : the one that would occur if they remain on active treatment post-deviation (de-jure) and the other that would occur off-treatment post-deviation (de-facto).
The potentially observable de-jure data for the nd deviating patients at time J are contained in
the vector YaJ,DJ,d and the alternative de-facto outcome data in the vector YaJ,DF,d. Define
Y = (YrJ, YaJ,o, YaJ,DJ,d, YaJ,DF,d)T as the collection of observed and potentially observable
outcome data, which has dimensions [(n + no+ 2nd) × 1].
For each deviating patient we can only observe one of the potential outcomes, either de-jure or de- facto. Consider two [(n + no+ 2nd) × (n + no+ 2nd)] matrices, DDJ and DDFof 0’s and 1’s such
that DDJY gives the [(n + no+ 2nd) × 1] de-jure data and DDFY gives the [(n + no+ 2nd) × 1]
de-facto data at time J .
Let a be a [(n + no+ 2nd) × 1] vector such that aTDDJY returns the de-jure treatment estimate
and aTDDFY returns the de-facto treatment estimate. When the deviating patients experience
de-jure behaviour post-deviation and are observed the expectation of the variance of the de-jure on-treatment estimand can be expressed as in (4.1). We consider settings where the expectation of the variance of our de-facto estimand can then be expressed as in (4.2).
We now suppose that post-deviation data are unobserved, i.e. the potentially observable de-facto and de-jure entries in Y are missing for the nd active patients. We alternatively impute these
outcomes, using de-jure imputation and de-facto imputation. This gives K ‘complete’ data samples Yk, of size [(n + no+ 2nd) × 1].
For this we need appropriate imputation distributions for the missing data under each scenario, with suitable posteriors for the included parameters. Under the de-jure assumption our imputation model is formed from the regression of YaJ,o on Pa,o where Pa,o is the design matrix for the
imputation model, which contains the values of the 1, ..., J − 1 outcomes and covariates included in the imputation model (excluding treatment), along with a vector of 1’s to include an intercept in the model for the no observed active patients.
The parameter estimates for the de-jure imputation model for the nd patients missing outcome
J are found using ˆβa,o = (PTa,oPa,o)−1PTa,oYaJ,o with assumed known covariance matrix Va,o =
(PTa,oPa,o)−1σ2.
We assume the large sample posterior for the parameters of the de-jure imputation model, denoted as ˆβDJ, is normal and centered on the ML estimator ˆβa,o with covariance matrix Va,o. That is,
ˆ
βDJ|YaJ,o∼ N ( ˆβa,o; Va,o).
The de-jure imputation model for active patient i deviating following time J − 1 and imputation k can therefore be expressed as,
˜
YaiJ,k|YaJ,o= Pa,d,ih ˆβa,o+ ba,o,k
i
+ ei,k for i ∈D,
where, ba,o,k∼ N (0, Va,o), ei,k∼ N (0, σ2) and Pa,d,icontains the values of the 1, ..., J −1 outcomes
and covariates included in the imputation model (plus a 1 for the intercept) for each deviating patient i, who deviates following time J − 1.
For de-facto imputation we assume the large sample posterior for the imputation parameters for the nd patients missing outcome J , ˆβDF, is normal and centered on the ML estimator ˆβDF,o with
known covariance matrix VDF,o, that is,
ˆ
βDF|YDF,J,o∼ N ( ˆβDF,o; VDF,o),
where YDF,J,oconsists of the relevant observed outcome data under the particular de-facto setting
of interest. The de-facto imputation model for active patient i deviating following time J − 1 and imputation k can therefore be expressed as,
˘
YaiJ,k|YDF,J,o= Pa,d,ih ˆβDF,o+ bDF,o,k
i
+ ei,k for i ∈D,
where, bDF,o,k ∼ N (0, VDF,o), and ei,k ∼ N (0, σ2). Under the assumption of equal variance-
covariance matrix of baseline and follow-up by treatment arm we consequently assume the same variance for the residuals in the de-jure and de-facto imputation models (σ2). In Subsection 4.2.5
we explore the impact of relaxing this assumption. We are interested in imputation inference for,
1 K PK k=1a TD DJYk or K1 PK k=1a TD DFYk.
Letting the number of imputations, K → ∞, the variance of our MI treatment estimate as esti- mated by Rubin’s rules is, VDJ, MI= ˆWDJ+ ˆBDJor VDF, MI= ˆWDF+ ˆBDFwhere under the conditions
required in the proposition, Eh ˆWDJ
i = EhK1 PK k=1a TD DJΣˆkDTDJa i → aTD DJΣDTDJa and Eh ˆWDF i = EhK1 PK k=1a TD DFΣˆkDTDFa i → aTD DJΣDTDJa + O(n−2). Under de-jure,
ˆ BDJ= 1 K − 1 K X k=1
πd(¯ek− ¯e) + πd P¯a,dba,o,k− ¯Pa,db¯a,o 2 , where ¯ek = n1 d P i∈Dei,k, ¯e = K1 P K k=1¯ek, ¯Pa,d = n1 d P
i∈DPa,d,i and ¯ba,o = K1 P
K
k=1ba,o,k.
Which has expectation,
Eh ˆBDJ
i = π2d
" σ2+ n
dP¯a,dVa,oP¯Ta,d
nd # . Under de-facto, ˆ BDF= 1 K − 1 K X k=1
πd(¯ek− ¯e) + πd P¯a,dbDF,o,k− ¯Pa,d¯bDF,o
2 ,
where ¯bDF,o= K1 PKk=1bDF,o,k. Which has expectation,
Eh ˆBDF
i = πd2
" σ2+ n
dP¯a,dVDF,oP¯Ta,d
nd
# .
The information anchored variance is,
E [Vanchored] = a TD DJΣDTDJa + O(n−2) + Eh ˆBDJ i + Eh ˆBDJ i Eh ˆWDJ i O(n −2).
If Rubin’s variance estimator is information anchoring and preserves the information loss in the primary analysis under MAR (4.4) holds, which in this setting is,
0 ≈ π2dP¯a,d(Va,o− VDF,o) ¯PTa,d+
Eh ˆBDJ
i Eh ˆWDJ
iO(n
−2) . (4.12)
This gives the required result.
The difference between Rubin’s variance estimator and the information anchored variance is a small quantity, under the described conditions. We see the result corresponds with that obtained in the baseline and single follow-up setting. ¯Pa,d, Va,o and VDF,o will be of greater dimensions
here, but the order of these terms will remain the same i.e. the overall magnitude of these terms will be very small. The additional follow-up data does not have a marked impact when only the last measured variable is subject to non-response and the last measured time point is the focus of analysis. Rubin’s variance estimator provides an excellent approximation of the information anchored variance.
4.2.4
Implementation for improved information anchoring
As in the baseline and single follow-up setting, if the variance of the parameters in the de-facto imputation model corresponds to the variance of the parameters in the de-jure imputation model (Va,o= VDF,o) then Rubin’s variance estimator will better preserve the loss of information in the
primary design based analysis in the current longitudinal trial setting with deviation in one arm. The first term on the RHS of (4.12) will disappear.
As described in Subsection 4.1.4 when this is not the case we can alter the reference based MI procedure by bootstrapping the observed reference sample, then drawing the required sample size from the estimated reference distribution to achieve Va,o= VDF,o. Alternatively we could re-scale
the variance of the parameters in the de-facto imputation distribution to achieve Va,o = VDF,o.
However as discussed in Subsection 4.1.3 the first term on the RHS of (4.12) is very small, relative to the information anchored variance we desire. Thus even without this additional step we will still see an excellent approximation between Rubin’s variance estimator and the desired information anchored variance.
4.2.5
Relaxing the equal variance assumption
As in Subsection 4.1.5 we can consider the impact of relaxing the assumption of equal variance- covariance matrix for baseline and follow-up across trial arms. Similar to the baseline and single follow-up setting when we relax the equal variance assumption we can no longer assume the variance of the residuals in the de-jure imputation model matches the variance of the residuals in the de-facto imputation model. In this case we denote the variance of the residuals in the de-jure imputation model as σ2
DJ and in the de-facto imputation model as σDF2 . Then under de-facto imputation, the
requirement for information anchoring becomes,
0 ≈ π2d σ
2
DJ− σDF2
nd
+ ¯Pa,d(Va,o− VDF,o) ¯PTa,d
+ Eh ˆBDJ i Eh ˆWDJ iO(n −2) . (4.13)
difference in the variance structure by trial arm. But this additional component is multiplied by πd/n i.e. is O(πd/n), thus will always be relatively small. The difference between Va,o− VDF,o will
also be slightly larger when the variance structures differ by trial arm. However as discussed in the single follow-up outcome setting in Subsection 4.1.5, the additional difference can be expressed as a component with the same order as VDF,oin the equal variance setting and Va,o− VDF,oremains
multiplied by πd2. Thus the order of this component will be the same as when variance-covariance structures were equal.
So we see that —similar to the baseline and single follow-up setting— relaxing the equal variance assumption does not greatly effect the approximation between Rubin’s variance estimator and the information anchored variance in the longitudinal setting, where the last measured variable is subject to non-response in one arm. Especially since we do not expect the variance structure to differ markedly by arm.
4.2.6
Extension for deviation in both arms
Following the approach outlined in Subsection 4.1.6 for the current longitudinal setting with devi- ation in both arms, we obtain a result which corresponds with that obtained in the baseline and single follow-up setting (4.11). The difference between Rubin’s variance estimator and information anchored variance is,
0 ≈π2dP¯a,d[Va,o− VDF,a,o] ¯PTa,d+ π
2
r,dP¯r,d[Vr,o− VDF,r,o] ¯PTr,d
+ 2πdπr,dP¯a,dCov (bDF,a,o,k, bDF,r,o,k) ¯PTr,d+
Eh ˆBDJ
i Eh ˆWDJ
iO(n
−2) .
Rubin’s variance estimator still provides an excellent approximation of the information anchored variance. This is unsurprising given that the additional follow-up data did not have a marked impact when only the last measured variable was subject to non-response in one treatment arm.