Why Covariate Composition Matters
A borrowed distribution can be well defined and still be a poor match for the individuals whose outcomes are missing. This can happen when the pattern used as a reference has a different mix of prognostic covariate values from the pattern receiving the borrowed outcomes. The pooled reference distribution then reflects the reference pattern’s composition, not necessarily the target pattern’s composition.
This distinction matters when the aim is to preserve the relationship between a covariate and the outcome. A marginal restriction compares distributions after covariate values have been mixed together. A covariate-stratified restriction compares distributions within covariate strata. If the patterns have different stratum proportions, these restrictions generally imply different borrowed distributions for the target pattern.
The difference is not automatically evidence that marginal borrowing is incorrect. A marginal restriction may be chosen deliberately for a marginal estimand. The key is to recognise which distribution the restriction supplies, and whether its covariate composition suits the intended use.
Two Different Restrictions
Let \(G\) denote a participant’s dropout pattern, \(Z\) a covariate measured before dropout, and \(Y\) the outcome at a visit that is missing for the target pattern \(g\). Let \(h\) be a reference pattern in which \(Y\) is observed. Write \(F_{g,z}(y)\) for the distribution of \(Y\) in pattern \(g\) and covariate stratum \(Z=z\), and \(F_{h,z}(y)\) for the corresponding distribution in pattern \(h\). These distributions for pattern \(g\) involve missing outcomes and are not directly observed.
The pooled reference distribution is a mixture over the reference pattern’s covariate composition. If \(p_h(z)=P(Z=z\mid G=h)\), then
Under the stratified restriction, the implied marginal distribution for the missing outcome in pattern \(g\) is instead
Thus, if \(p_h(z)\ne p_g(z)\) and the outcome distributions vary across strata, the two mixtures need not agree. This follows directly from their different weights; no difference in the within-stratum distributions is needed.
Worked Examples
Worked Example: Different Pattern Mixes Produce Different Borrowed Means
Consider two covariate strata, \(Z=0\) and \(Z=1\), and two dropout patterns. The hypothetical reference pattern \(h\) has 80 participants with \(Z=0\) and 20 with \(Z=1\). Their observed mean outcomes at the target visit are 10 and 20 points, respectively. The target pattern \(g\) has 20 participants with \(Z=0\) and 80 with \(Z=1\). Assume the within-stratum means from pattern \(h\) are the means used by the restriction.
The pooled reference mean is weighted by the composition of \(h\):
An unstratified restriction therefore assigns a marginal mean of 12 points to the missing outcomes in pattern \(g\). Under the covariate-stratified restriction, the borrowed means are 10 points for \(Z=0\) and 20 points for \(Z=1\). Weighting these by the composition of \(g\) gives
The 6-point difference is entirely due to the different covariate weights: both calculations use the same two stratum means. If the intended restriction is to transfer the reference outcome distribution within covariate strata, using the pooled reference mean as though it were the target pattern’s mean obscures that intended transfer. These calculations describe the implications of the restrictions; they do not show which restriction is substantively appropriate.
Worked Example: The Full Borrowed Distribution Can Change
Use the same hypothetical covariate proportions as above. Suppose the reference outcome distributions within strata are \(Y\mid Z=0,G=h\sim N(10,4)\) and \(Y\mid Z=1,G=h\sim N(20,4)\), where the second parameter is the variance, so the standard deviation in each stratum is 2 points. Under the unstratified restriction, the borrowed distribution is the mixture with weights 0.80 and 0.20. Under the stratified restriction for pattern \(g\), it is the mixture with weights 0.20 and 0.80.
For a concrete comparison, calculate the probability of an outcome at or below 15 points. Standardising within each stratum gives \(z=(15-10)/2=2.5\) for \(Z=0\), and \(z=(15-20)/2=-2.5\) for \(Z=1\). Using \(\Phi(2.5)\approx0.9938\) and \(\Phi(-2.5)\approx0.0062\), the unstratified borrowed distribution gives
The stratified restriction, mixed using pattern \(g\)’s proportions, gives
So the two borrowed distributions assign about 79.63% and 20.37%, respectively, to outcomes at or below 15 points. Equality of a marginal distribution is not a guarantee that the distribution obtained by borrowing within strata and then mixing for the target pattern will be the same. Here, the stratum-specific distributions are unchanged; only their weights differ.
Worked Example: What If a Covariate Stratum Has No Reference Cases?
Suppose the target pattern \(g\) contains 25 participants with \(Z=0\) and 75 with \(Z=1\). The reference pattern \(h\) contains 40 participants with \(Z=0\) and none with \(Z=1\). The observed mean outcome in the \(Z=0\) reference stratum is 10 points. Assume there are no observed reference outcomes in the \(Z=1\) stratum.
The pooled reference mean is 10 points, so a marginal restriction can specify a borrowed marginal mean of 10 points for pattern \(g\). But a stratified restriction requires the \(Z=1\) reference distribution \(F_{h,1}\), which is unavailable: there are no reference cases with \(Z=1\). The information given supports calculation of the pooled reference mean, but does not support a data-based estimate of that missing stratum-specific distribution.
It would be unjustified to report the stratified target mean as \(0.25(10)+0.75(10)=10\) points unless an additional assumption supplies the \(Z=1\) distribution or its mean. That calculation silently extends the \(Z=0\) distribution to \(Z=1\). The lack of overlap is a limitation of the available reference data, not a problem solved merely by choosing a stratified formula.
Why the Mixture Formula Gives This Result
The contrast can be shown at the level of expectations. Suppose each stratum-specific outcome has a finite mean \(\mu_{h,z}\). Under the stratified restriction, the mean for the missing outcomes in pattern \(g\) is the expectation of a mixture:
The first equality follows by conditioning on \(Z\) and summing the conditional expectations weighted by the target pattern’s stratum proportions. The second uses the stratified restriction to replace the unobserved conditional mean in pattern \(g\) with the observed conditional mean in pattern \(h\). By contrast, the pooled reference mean is \(\sum_z p_h(z)\mu_{h,z}\). Subtracting the two shows exactly when a composition difference matters:
Because the stratum proportions in each pattern sum to one, adding the same constant to every \(\mu_{h,z}\) does not change this difference. The difference is zero if the proportions match, or if the stratum means are all equal. Otherwise it can be nonzero. In the first worked example it is \(0.20(10-12)+0.80(20-12)=6\) points.
Practical Checks and Choices
Before applying a restriction, specify what is being borrowed and how it will be used. In particular, distinguish a marginal distributional restriction from a restriction intended to preserve the covariate-conditional outcome distributions. The latter requires a reference distribution within every relevant target stratum, along with a defensible reason for transferring it.
For each covariate stratum, compare its proportion in the target pattern with its proportion in the proposed reference pattern. Large differences signal that their pooled distributions use different mixtures.
If the observed reference outcome distributions vary little across strata, changing the mixture weights may have little impact. If they differ substantially, calculate or otherwise examine the implications of both sets of weights.
A conditional restriction cannot be informed by a reference stratum with no observations. Sparse strata may also make the estimated distributions unstable; a calculation can be defined mathematically while still relying on weak empirical information.
If the analysis needs outcome distributions conditional on \(Z\), a pooled distribution may discard the intended stratum-specific relationship. If the target is explicitly marginal and pooled borrowing is the chosen assumption, report that choice rather than presenting it as conditional borrowing.
The main alternatives are not interchangeable. Pooled borrowing uses the reference pattern’s natural covariate mix. Stratified borrowing transfers distributions within strata and then, for a target-pattern summary, weights them using the target pattern’s covariate mix. Standardization can also express the latter calculation by applying target-pattern weights to observed reference stratum summaries. Each approach depends on assumptions about which distributions can be transferred; stratification does not by itself establish that those assumptions are true.
A common mistake is to say, “The reference and target patterns have equal borrowed distributions, so their outcomes are equal within each covariate stratum.” That conclusion does not follow: a marginal equality concerns pooled distributions, which can use different covariate weights and can conceal different stratum-specific distributions. The correction is to state whether the restriction is marginal or conditional, check the pattern compositions, and calculate the implied mixture using the weights specified by that restriction.
These calculations concern the distributional implications of a chosen restriction. They do not identify the unobserved outcomes from observed data alone. When covariate overlap is poor or the strata are small, the implied stratified distributions may be uncertain or unavailable; when a marginal restriction is used, it may intentionally replace covariate-specific structure with a pooled distribution. Report the restriction and its covariate weighting clearly so that the resulting analysis can be interpreted in context.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A reference pattern has covariate proportions 0.60 and 0.40, with stratum means 5 and 15 points. What is its pooled mean?
- A target pattern has proportions 0.20 and 0.80 in those same strata. What mean follows from the stratified restriction, and how does it compare with the pooled reference mean?
- Name two conditions under which the stratified and pooled means in Question 2 would be equal.
- What problem arises if a target pattern contains a covariate stratum with no participants in the reference pattern?
Solutions. (1) The pooled reference mean is \(0.60(5)+0.40(15)=3+6=9\) points. (2) The stratified target mean is \(0.20(5)+0.80(15)=1+12=13\) points, which is 4 points above the pooled reference mean. (3) They are equal if the pattern proportions match, or if the stratum means are equal; more generally, the weighted difference can also cancel. (4) The reference distribution for that stratum is unavailable, so a stratified restriction cannot be filled in from the observed reference data alone. An additional assumption or source of information would be needed.