3.4 Baseline adjusted setting
4.1.6 Extension for deviation in both arms
Up to here, the proof of Proposition 1 has focused on the two arm trial setting with deviation in one arm only. This is not a requirement for the approximation to hold, rather just a simplification for clarity. Below we explore the impact of deviation in both arms. We remain focused on settings where VDF, full= VDJ, full+O(n−2). If VDF, full= VDJ, full+O(n−2) with deviation in one arm then with
deviation in both arms we will also obtain VDF, full= VDJ, full+O(n
−2), Appendix B.1 illustrates this
for the baseline and single follow-up setting where the treatment effect is the mean difference in the follow-up outcome. Since an unequal variance structure by arm has a negligible effect in practice, we maintain the assumption of equal variance-covariance structure for baseline and follow-up in what follows.
Suppose among the n reference patients, only nr,o are actually observed, while the remaining nr,d
deviate post-baseline. LetRD and RO define the sets of indices for the deviators and completers in the reference arm respectively. Let πr,d= nr,d/n. The outcome data for the observed reference
patients are contained in the vector Yr,o= {Yri; i ∈RO} T
. The potentially observable de-jure data for the nr,d deviating reference patients are contained in the vector Yr,DJ,d and the alternative
de-facto outcome data in the vector Yr,DF,d. The full collection of observed and potentially
observable outcome data is now defined as Y = (Yr,o, Yr,DJ,d, Yr,DF,d, Ya,o, Ya,DJ,d, Ya,DF,d)T
which has dimensions [(nr,o+ 2nr,d+ no+ 2nd)]. We assume Y is normally distributed and has
known variance Σ.
We redefine the two matrices DDJ and DDF so that DDJY and DDFY now each give the de-
jure data or de-facto data across both treatment arms. We focus on settings where E [VDF, full] =
aTDDJΣDTDJa + O(n−2).
We follow the steps outlined in Section 4.1 to establish the de-jure imputation model for the deviating active arm patients. Under de-jure imputation, the imputation model for patient i and imputation k in the active arm is expressed as,
˜
Yai,k|Ya,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,iis the covariate data for each deviating active
patient i (excluding treatment group but including a 1), of dimensions [1 × p].
For the reference arm, under our de-jure assumption (on-treatment MAR), our imputation model is formed from the regression of Yr,o on Pr,o where Pr,o is the [nr,o× p] design matrix for the
imputation model, which contains the values of the (p-1) covariates included in the imputation model (including the baseline outcome but excluding the treatment indicator since we perform imputation separately by arm) with a vector of 1’s to include an intercept term in the model, for observed reference patients.
denoted as ˆβDJ,r, is normal and centered on the ML estimator ˆβr,o with covariance matrix Vr,o,
that is,
ˆ
βDJ,r|Yr,o∼ N ( ˆβr,o; Vr,o).
The de-jure imputation model for patient i and imputation k in the reference arm can therefore be expressed as,
˜
Yri,k|Yr,o= Pr,d,ih ˆβr,o+ br,o,k
i
+ ei,k for i ∈RD,
where br,o,k ∼ N (0, Vr,o), ei,k ∼ N (0, σ2) and Pr,d,i is the covariate data for each deviating
reference patient i (excluding treatment group but including a 1), of dimensions [1 × p].
Under de-facto imputation for patients in the active arm we assume the large sample posterior for the parameters of the imputation model, which we denote by ˆβDF,a, is normal and centered on the ML estimator ˆβDF,a,o with known covariance matrix VDF,a,o that is,
ˆ
βDF,a|YDF,a,o∼ N ˆβDF,a,o, VDF,a,o
,
where YDF,a,oconsists of the relevant observed outcome data under the particular de-facto setting.
The de-facto imputation model for patient i and imputation k in the active arm can therefore be expressed as,
˜
Yai,k|YDF,a,o= Pa,d,ih ˆβDF,a,o+ bDF,a,o,k
i
+ ei,k for i ∈D,
where bDF,a,o,k ∼ N (0, VDF,a,o), and ei,k ∼ N (0, σ2). Under de-facto imputation for patients
in the reference arm we assume the large sample posterior for the parameters of the imputation model, which we denote by ˆβDF,r, is normal and centered on the ML estimator ˆβDF,r,owith known covariance matrix VDF,r,o that is,
ˆ
βDF,r|YDF,r,o∼ N ˆβDF,r,o, VDF,r,o
,
where YDF,r,oconsists of the relevant observed outcome data under the particular de-facto setting.
The de-facto imputation model for patient i and imputation k in the reference arm can therefore be expressed as,
˜
Yri,k|YDF,r,o= Pr,d,ih ˆβDF,r,o+ bDF,r,o,k
i
+ ei,k for i ∈RD,
where bDF,r,o,k ∼ N (0, VDF,r,o), and ei,k ∼ N (0, σ2). We are interested in imputation inference
for, K1 PK k=1a TD DJYk or K1 PK k=1a TD
DFYk. For Rubin’s variance estimator, under the de-
scribed conditions, Eh ˆWDJ i = EhK1 PK k=1a TD DJΣˆkDTDJa i → aTD DJΣDTDJa and Eh ˆWDF i = Eh1 K PK k=1a TD DFΣˆkDTDFa i → aTD
DJΣDTDJa + O(n−2). Under de-jure,
ˆ BDJ= 1 K − 1 K X k=1
[ πd(¯ea,k− ¯ea) + πd P¯a,dba,o,k− ¯Pa,d¯ba,o − πr,d(¯er,k− ¯er)
−πr,d P¯r,dbr,o,k− ¯Pr,d¯br,o
2
,
where ¯ea,k = n1dPi∈Dei,k, ¯ea = K1 P K
k=1e¯a,k, ¯Pa,d = n1dPi∈DPa,d,i, ¯er,k = nr,d1 Pi∈RDei,k,
¯ er=K1 P K k=1e¯r,k, ¯Pr,d= n1 r,d P
i∈RDPr,d,i and ¯br,o= K1 P
K
k=1br,o,k. This has expectation,
Eh ˆBDJ
i = πd2
" σ2+ n
dP¯a,dVa,oP¯Ta,d
nd # + πr,d2 " σ2+ n r,dP¯r,dVr,oP¯Tr,d nr,d # . Under de-facto, ˆ BDF= 1 K − 1 K X k=1
[ πd(¯ea,k− ¯ea) + πd P¯a,dbDF,a,o,k− ¯Pa,db¯DF,a,o
−πr,d(¯er,k− ¯er) − πr,d P¯r,dbDF,r,o,k− ¯Pr,d¯bDF,r,o
2 ,
where ¯bDF,a,o= K1 P K
k=1bDF,a,o,k and ¯bDF,r,o= K1 P
K
k=1bDF,r,o,k. This has expectation,
Eh ˆBDF i =π2dσ 2 nd + πr,d2 σ 2 nr,d
+ π2dP¯a,dVDF,a,oP¯Ta,d+ π
2
r,dP¯r,dVDF,r,oP¯Tr,d
− 2πdπr,dP¯a,dCov (bDF,a,o,k, bDF,r,o,k) ¯PTr,d.
Eh ˆBDF
i
=Eh ˆBDJ
i
+ π2dP¯a,d[VDF,a,o− Va,o] ¯PTa,d+ π
2
r,dP¯r,d[VDF,r,o− Vr,o] ¯PTr,d
− 2πdπr,dP¯a,dCov (bDF,a,o,k, bDF,r,o,k) ¯PTr,d.
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 seen in the primary analysis under MAR then since Eh ˆWDJ
i = E [VDJ, full], 0 ≈ Eh ˆBDJ i − Eh ˆBDF i + Eh ˆBDJ i Eh ˆWDJ iO(n −2) . (4.10)
After substituting in the current results and simplifying this becomes,
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) . (4.11)
With deviation in both arms the difference between Rubin’s variance estimator and the information anchored variance remains a small quantity in comparison to the information anchored variance.
An additional component which depends on the difference between the variance of the imputation parameters in the de-jure imputation model and de-facto imputation model for the reference arm, multiplied by π2r,d, is now included. The exact size of this additional piece depends on the specific de-facto scenario. But it will be similar in size to the first component on the RHS of (4.11), which we have already discussed is of a small order for realistic proportions of missing data. The covariance between the parameters of the active and reference arm de-facto imputation models also contributes to the sharpness of the approximation between Rubin’s variance estimator and the information anchored variance. This is intuitive because the imputations in the arms may now be correlated. The exact size of this additional piece again depends on the specific de-facto scenario. But it is always multiplied by πdπr,d thus will be of a relatively small order in practice.
We now explore the implications for each de-facto option.
behaviour. The imputation model for the deviating patients in the reference arm remains the same as under de-jure imputation, hence VDF,r,o = Vr,o. The de-jure imputation model for the
reference patients also becomes the imputation model for the deviating active patients, that is
VDF,a,o= Vr,o. Therefore in the CR setting result (4.11) is,
0 ≈π2dP¯a,d[Va,o− Vr,o] ¯PTa,d+ 2πdπr,dP¯a,dVr,oP¯Tr,d+
Eh ˆBDJ
i Eh ˆWDJ
iO(n
−2) .
With deviation in both arms under CR, the sharpness of the approximation depends on the differ- ence between Va,oand Vr,o, as well as the exact size of Vr,o. When the variance of the parameters
in the de-jure imputation model for the active patients matches the variance of the parameters in the de-jure imputation model for the reference patients, and both are small, improved information anchoring will be obtained.
Under LMCF imputation the covariance between the imputation parameters of the active and reference de-facto imputation models is zero. Hence in the LMCF setting result (4.11) is,
0 ≈π2dP¯a,d[Va,o− VDF,a,o] ¯PTa,d+ π2r,dP¯r,d[Vr,o− VDF,r,o] ¯PTr,d
+ Eh ˆBDJ i Eh ˆWDJ iO(n −2) .
With deviation in both arms under LMCF, the sharpness of the approximation depends on the difference between Va,o and VDF,a,o, as well as the difference between Vr,o and VDF,r,o. That
is when the variance of the parameters in the de-jure imputation model for the active patients matches the variance of the parameters in LMCF imputation model for the active patients and the variance of the parameters in the de-jure imputation model for the reference patients matches the variance of the parameters in LMCF imputation model for the reference patients, improved information anchoring will be obtained.
Under J2R or CIR imputation, the de-facto imputation model for patients in the refernce arm is the same as under de-jure imputation, i.e. VDF,r,o= Vr,o. Result (4.11) in the J2R/CIR settings
becomes,
0 ≈πd2P¯a,d[Va,o− VDF,a,o] ¯PTa,d+ 2πdπr,dP¯a,dCov (bDF,a,o,k, br,o,k) ¯PTr,d
+ Eh ˆBDJ i Eh ˆWDJ iO(n −2) .
jure imputation model for the active patients (Va,o) matches the variance of the parameters in
the de-facto imputation model (VDF,a,o), improved information anchoring will be obtained. The
approximation will also be improved when the covariance between the parameters of the de-facto imputation model for the active patients and the parameters of the de-jure imputation model for the reference patients is small.
Implementation for improved information anchoring
With deviation in both arms the covariance between the parameters of the active and reference imputation models will additionally affect the sharpness of the approximation of the information anchored variance by Rubin’s variance estimator. Thus we could not simply rescale the variance parameters in the imputation models to obtain perfect information anchoring. We would need to also adjust for the covariance. The bootstrap approach, as outlined in Subsection 4.1.4, would also not achieve this goal. To remove the covariance terms, after bootstrapping the data to estimate the observed data distributions, we would need to draw separate data for each treatment arm to construct the imputation models.
We do not develop this approach further here, but highlight that the procedure described in Sub- section 4.1.4 is not immediately applicable with deviation in both arms. Generally, in all de-facto settings (one of CR, J2R, CIR or LMCF) the approximation of the information anchored vari- ance by Rubin’s variance estimator will be excellent, since the order of the terms in the difference between these quantities are relatively small in comparison to the information anchored variance itself. In Section 4.5 we explore the impact of deviation in both arms via simulation to demonstrate this.