Available-Case Restrictions With Covariate Stratification

Why Stratify an Available-Case Restriction?

An available-case restriction uses outcomes observed at a target visit to specify outcomes missing at that visit. When baseline covariates are relevant, applying the restriction within covariate strata can preserve differences in outcome distributions across those strata. Applying it only after pooling may instead transfer the available group’s covariate composition to the missing group, even when those compositions differ.

The distinction is about the distribution being reconstructed. The target is the outcome distribution among participants whose target-visit outcome is missing. A restriction within strata first specifies that distribution separately in each stratum, then combines the stratum-specific distributions using the stratum proportions of the missing group. A pooled restriction uses the overall available-case distribution directly. These need not agree.

Definition: Let \(Y\) be the outcome at a target visit, \(A=1\) indicate that \(Y\) is observed, and \(A=0\) indicate that it is missing. Let \(S\) be a discrete baseline covariate stratum. An available-case restriction within strata is the assumption that, for every stratum \(s\) with both available and missing participants, \(P(Y\leq y\mid A=0,S=s)=P(Y\leq y\mid A=1,S=s)\) for every \(y\). Equivalently, the conditional outcome distributions agree within each stratum.

The assumption concerns the distribution of \(Y\) given \(S\), not just its mean. It also does not assert that the available and missing groups have the same distribution of \(S\). Define the missing-group stratum weight as \(w_s^{(0)}=P(S=s\mid A=0)\), and let \(F_{1s}(y)=P(Y\leq y\mid A=1,S=s)\). Under the restriction, the missing-outcome distribution is

F_0(y)=P(Y\leq y\mid A=0) =\sum_s w_s^{(0)}F_{1s}(y).

The weights in this mixture are the covariate-stratum proportions among participants with missing outcomes. They are not the proportions among available cases, nor necessarily the proportions in the full study population.

Deriving the Stratified Reconstruction

The reconstruction follows directly from conditioning on the stratum. For any outcome threshold \(y\), the events \(S=s\) partition the missing group. The law of total probability therefore gives

P(Y\leq y\mid A=0) =\sum_s P(Y\leq y\mid A=0,S=s)P(S=s\mid A=0).

Applying the available-case restriction replaces each conditional distribution in that sum by its available-case counterpart. This produces the stated formula. The same reasoning applies to event probabilities, category probabilities, and expectations when the expectations exist. For an integrable outcome, for example,

E(Y\mid A=0)=\sum_s P(S=s\mid A=0)E(Y\mid A=1,S=s).

In a sample, the unknown probabilities and conditional distributions are replaced by their empirical estimates: the fraction of missing cases in each stratum and the outcome distribution among available cases in that stratum. This plug-in calculation is meaningful only when each stratum receiving positive missing-group weight has available cases from which its conditional outcome distribution can be estimated. That is a stratum-specific positivity requirement.

Worked Example: Reconstructing a Binary Outcome Within Two Strata

Consider hypothetical data on 200 participants, split evenly between low- and high-risk baseline strata. At the target visit, \(Y=1\) denotes an event and \(Y=0\) denotes no event. The low-risk stratum has 80 available cases and 20 missing outcomes; among the available cases, 16 had the event. The high-risk stratum has 20 available cases and 80 missing outcomes; among those available cases, 16 had the event. Assume the available-case restriction holds within each stratum.

The stratum-specific available-case event probabilities are \(16/80=0.20\) in the low-risk stratum and \(16/20=0.80\) in the high-risk stratum. Among the 100 participants with missing outcomes, 20 are low risk and 80 are high risk. Thus the missing-group weights are \(20/100=0.20\) and \(80/100=0.80\). The reconstructed event probability is

P(Y=1\mid A=0) =0.20(0.20)+0.80(0.80) =0.04+0.64 =0.68.

This corresponds to 68 expected events among the 100 participants with missing outcomes under the restriction. The calculation uses 20 missing low-risk outcomes and 80 missing high-risk outcomes; the high-risk stratum has 100 participants in total, of whom 20 have available outcomes and 80 have missing outcomes.

Why Pooling Can Give a Different Distribution

A pooled available-case restriction assigns the overall available-case outcome distribution to the missing group without conditioning on \(S\). In the binary example, the pooled available cases include 32 events among 100 participants, so their event proportion is \(32/100=0.32\). Assigning that proportion to the missing group gives a pooled reconstruction of \(0.32\), rather than the stratified reconstruction of \(0.68\).

The difference arises because the available cases are mostly low risk, whereas the missing cases are mostly high risk. The pooled available-case event proportion weights the strata by their proportions among available cases; the stratified reconstruction weights the conditional event probabilities by their proportions among missing cases. Neither the formula nor the data imply that these two sets of weights are equal.

The difference also changes the reconstructed outcome distribution for the full study population. There are 32 observed events among available cases. The stratified restriction imputes an expected 68 events among the missing cases, giving \( (32+68)/200=0.50 \) as the reconstructed overall event proportion. Pooling imputes an expected 32 events among the missing cases, giving \( (32+32)/200=0.32 \). These are consequences of different restrictions, not two calculations of the same assumption.

Worked Example: Comparing Entire Distributions, Not Just Means

In another hypothetical study, \(Y\) is an outcome taking values 0 or 10 units. In stratum L, there are 25 available cases: 20 have \(Y=0\), and 5 have \(Y=10\). There are 75 participants with missing outcomes in L. In stratum H, there are 75 available cases: 15 have \(Y=0\), and 60 have \(Y=10\). There are 25 participants with missing outcomes in H. Assume the within-stratum restriction holds.

The available-case probabilities of \(Y=10\) are \(5/25=0.20\) in L and \(60/75=0.80\) in H. The missing-group stratum weights are \(75/100=0.75\) for L and \(25/100=0.25\) for H. Thus the stratified probability of \(Y=10\) among missing cases is

P_{\mathrm{strat}}(Y=10\mid A=0) =0.75(0.20)+0.25(0.80) =0.15+0.20 =0.35.

Because the only possible values are 0 and 10, the stratified reconstructed mean is \(0(0.65)+10(0.35)=3.5\) units. For comparison, the pooled available cases have \(5+60=65\) values equal to 10 among \(25+75=100\) cases. The pooled reconstruction assigns probability \(65/100=0.65\) to \(Y=10\), giving mean \(0(0.35)+10(0.65)=6.5\) units. The pooled and stratified distributions differ at the outcome value 10, so this is not merely a difference in how the same distribution’s mean is summarized.

Assumptions, Diagnostics, and Limits

The restriction is an assumption about unobserved outcomes. It is not verified simply because the available cases show different outcome distributions across strata, and agreement of observed distributions across strata would not establish the restriction either. Its use requires careful definition of the target outcome, the observation indicator, and the strata.

Consistent outcome definition. The available and missing outcomes must refer to the same target visit and measurement. A value measured at a different visit is not automatically a value of the target outcome.
Strata defined before observing the target outcome. Covariates should be available for both available and missing participants and should define the conditioning groups used in the restriction.
Stratum-specific support. Every stratum with missing participants needs available cases to estimate its conditional outcome distribution. If a stratum has no available cases, its distribution is not supplied by this restriction.
Credible conditional equality. The assumption says that, within each stratum, availability does not distinguish the outcome distribution. Important outcome-related differences among available and missing participants remaining within strata undermine its plausibility.

Useful diagnostics include tabulating available and missing counts by stratum, comparing the covariate composition of those groups, and inspecting available-case outcome distributions within each stratum. Sparse cells indicate unstable estimates; an empty available-case cell prevents direct estimation. These checks reveal data limitations and possible concerns, but they cannot test equality of the missing and available outcome distributions because the missing outcomes are unobserved.

Stratification is not guaranteed to make a restriction true. It conditions on the chosen strata, not on every possible predictor of the outcome and missingness. Conversely, dividing a modest sample into many strata can leave little information to estimate each available-case distribution. The result is also sensitive to the choice and granularity of strata. If a stratum has missing cases but no available cases, borrowing an outcome distribution from another stratum is an additional assumption, not a calculation justified by the within-stratum restriction.

Worked Example: An Empty Available-Case Stratum

Suppose a hypothetical study has 40 participants in stratum L, all with observed outcomes, and 60 participants in stratum H, all with missing outcomes. In L, 12 of the 40 observed outcomes are events, so the observed event proportion is \(12/40=0.30\). There are no available outcomes in H.

The missing group consists entirely of stratum H, so \(P(S=H\mid A=0)=1\). The stratified reconstruction would require \(P(Y=1\mid A=1,S=H)\), but its denominator is zero: there are no available H cases from which to estimate this probability. The within-stratum available-case restriction therefore does not yield a numerical reconstruction here. Applying L’s 0.30 event proportion to H would be a pooled or cross-stratum assumption, not the stated stratified restriction. The observed data alone do not determine the missing-group event proportion.

The main alternatives answer different questions or impose different restrictions. A pooled available-case restriction uses the marginal available-case outcome distribution, which can be inappropriate when stratum composition differs. A complete-case restriction uses only participants observed at all relevant visits; it may exclude available outcomes from participants who are incomplete elsewhere. A regression-based approach can model outcome variation across covariates, but its estimates then depend on the chosen model and its assumptions. Stratification makes the borrowed distribution explicit within defined groups, while requiring adequate available cases in every relevant group.

Common Mistakes

Using available-group weights for the missing-group mixture. The target is \(P(Y\leq y\mid A=0)\), so the weights are \(P(S=s\mid A=0)\).
Treating pooled and stratified restrictions as interchangeable. They agree only under additional conditions, such as matching stratum compositions or identical conditional outcome distributions across strata.
Calling the restriction empirically confirmed. The equality concerns outcomes that are missing and cannot be checked from the observed outcomes alone.
Reporting an event probability as a CDF value at the largest binary outcome. For binary \(Y\in\{0,1\}\), \(P(Y=1)\) is the event probability, whereas \(P(Y\leq 1)=1\). State the event probability directly.

Key takeaway: Reconstruct the missing-outcome distribution by combining stratum-specific available-case distributions with the missing group’s stratum weights. Pooling first changes the restriction unless the relevant stratum distributions or compositions make the two results coincide.

Check Your Understanding

Answer each question, then compare with the solutions below.

In a stratum, 30 of 50 available participants have an event. There are 40 missing participants in that stratum. Under the restriction, what is the estimated event probability among those 40?
There are two strata. Among missing cases, 25% are in L and 75% in H. Their available-case event proportions are 0.10 and 0.50. Calculate the stratified missing-group event probability.
Why can the overall available-case event proportion differ from the stratified reconstruction?
A stratum has missing cases but no available cases. Does the within-stratum restriction provide an estimated outcome distribution for those missing cases? Explain.

Solutions. (1) The available-case proportion is \(30/50=0.60\); under the restriction, the estimated event probability among the 40 missing participants is 0.60. (2) Weight by the missing-group proportions: \(0.25(0.10)+0.75(0.50)=0.025+0.375=0.40\). (3) The pooled available-case proportion weights strata by their shares among available cases, while the stratified reconstruction weights conditional event proportions by the missing group’s stratum shares; those weights or conditional proportions can differ. (4) No. The required conditional distribution among available cases in that stratum cannot be estimated from an empty available-case cell; borrowing from another stratum adds an assumption.