Why the Conditioning Set Matters
In a pattern-mixture model, the outcome distribution is described separately for each observed missingness pattern. When an outcome is unobserved in one pattern, its pattern-specific distribution is not fully determined by the observed data. A restriction can link that distribution to distributions in other patterns, but the restriction must say which individuals or histories are being compared. That choice is the conditioning set.
For example, comparing patterns only among people with the same baseline covariates asserts one kind of similarity. Comparing them among people with the same baseline covariates and the same observed outcome history asserts a more detailed, generally different kind of similarity. Neither conditioning set is automatically correct. It encodes a substantive assumption about how the unobserved outcome distributions relate across patterns.
Here, \(C\) might consist of baseline covariates \(X\), observed outcome history \(H\), or both. The variables included in \(C\) must be defined and available in both patterns being compared. If a component of history is not observed in one of the patterns, a comparison conditional on that component requires additional modeling rather than a direct within-stratum comparison.
What a Restriction Says—and Does Not Say
A restriction conditional on baseline covariates says that the specified outcome feature is shared across patterns among individuals with the same baseline profile. It does not say that the overall pattern-specific means are equal: the patterns may contain different proportions of those baseline profiles. Adding observed history changes the claim. A restriction conditional on \(X\) and \(H\) says that the outcome feature is shared among individuals matching on both baseline information and the recorded history.
A richer conditioning set can express a more detailed assumption about the outcome within strata. But it also divides the data into smaller groups, which can make estimates unstable or leave some comparisons unsupported. It is therefore not enough to say that one restriction is “more adjusted.” The analyst must state the variables conditioned on, why they define relevant comparability, and where the comparison is supported.
Worked Example: Baseline Covariates Do Not Guarantee Equal Marginal Means
Suppose a hypothetical study has two patterns, \(G=A\) and \(G=B\), and a binary baseline covariate \(X\), coded low or high. The outcome \(Z\) is measured in both patterns for the purpose of this illustration. Assume the pattern-specific conditional means are 10 units for low \(X\) and 20 units for high \(X\) in both patterns. Thus the conditional mean restriction given \(X\) holds in each stratum.
Suppose 80% of pattern A has low \(X\), while 20% has high \(X\). Its marginal mean is $$ E(Z\mid G=A)=0.80(10)+0.20(20)=8+4=12\text{ units}. $$ Suppose pattern B has the reverse composition: 20% low and 80% high. Then $$ E(Z\mid G=B)=0.20(10)+0.80(20)=2+16=18\text{ units}. $$
The conditional means agree at each value of \(X\), but the marginal means differ by \(18-12=6\) units because the covariate compositions differ. This example shows why a conditional restriction must not be paraphrased as marginal equality. It also shows why the target distribution used to summarize conditional means matters.
Conditioning on Baseline Covariates Versus Outcome History
Let \(X\) denote baseline covariates and \(H\) a recorded outcome history preceding \(Z\). A restriction given \(X\) permits the relationship between \(H\) and \(Z\) to differ by pattern, as long as the means averaged over \(H\) within each \(X\)-stratum satisfy the restriction. A restriction given \((X,H)\) instead constrains the pattern-specific outcome means separately at each history value. The latter is not merely a different way of writing the former.
Worked Example: Adding History Changes the Assumption
Consider a hypothetical fixed baseline stratum \(X=x\), with a binary observed history \(H\). Each pattern has half its members at \(H=0\) and half at \(H=1\). The conditional means, in outcome units, are:
| Pattern | \(E(Z\mid G,H=0,X=x)\) | \(E(Z\mid G,H=1,X=x)\) |
|---|---|---|
| A | 8 | 12 |
| B | 10 | 10 |
Within pattern A, averaging over its history distribution gives $$ E(Z\mid G=A,X=x)=0.5(8)+0.5(12)=10\text{ units}. $$ For pattern B, the corresponding calculation is $$ E(Z\mid G=B,X=x)=0.5(10)+0.5(10)=10\text{ units}. $$ Therefore the restriction equating means conditional on \(X=x\) holds. However, the history-specific means are not equal: at \(H=0\), the means are 8 and 10 units; at \(H=1\), they are 12 and 10 units. The restriction conditional on \((X,H)\) fails.
Thus, the baseline-only restriction allows pattern differences across history values that cancel when averaged within \(X\). The richer restriction rules out those differences at each history value. If \(H\) is scientifically important for predicting \(Z\), the richer comparison may be substantively appealing, but its validity is an additional assumption—not a mechanical improvement.
Standardizing Requires a Defined Target and Support
Conditional restrictions often motivate a standardized summary: average the conditional means over a chosen distribution of \(C\). This defines a target population or covariate mix. It does not generally recover either pattern's observed marginal mean, because the chosen weights need not match that pattern's covariate distribution.
There is an important support condition. Define the support for pattern \(g\) as the set of \(c\) values with positive probability under that pattern. A conditional mean in pattern \(g\) is only defined from its distribution at values in that support. A comparison between \(g\) and \(h\) therefore requires values in their common support, unless a model explicitly extrapolates beyond it. A standardization distribution \(q(c)\) used to compare the patterns must assign weight only to values where both conditional means are defined.
Worked Example: Restricting a Standardized Comparison to Common Support
In a hypothetical comparison, pattern A has support \(C\in\{0,1,2\}\), and pattern B has support \(C\in\{1,2,3\}\). Thus the common support is \(\{1,2\}\). Suppose the conditional means in outcome units are:
| \(C\) | Pattern A conditional mean | Pattern B conditional mean |
|---|---|---|
| 0 | 8 | Not defined from pattern B |
| 1 | 12 | 12 |
| 2 | 20 | 20 |
| 3 | Not defined from pattern A | 25 |
Choose a target distribution that assigns weight 0.25 to \(C=1\) and 0.75 to \(C=2\). This target lies entirely within common support. For pattern A, the standardized mean is $$ 0.25(12)+0.75(20)=3+15=18\text{ units}. $$ For pattern B, it is $$ 0.25(12)+0.75(20)=3+15=18\text{ units}. $$ The standardized means agree under these weights. It would be invalid to include \(C=0\) or \(C=3\) in a direct two-pattern standardization: the missing conditional mean is not supplied by the observed pattern distribution. Assigning it a value would require an explicit extrapolation model or a different target, not ordinary averaging.
A Short Justification: Fine- and Coarse-Conditioning Restrictions Differ
The difference between conditioning sets can be seen by averaging. Suppose \(C=(S,T)\), so that \(S\) is a coarser set and \(T\) is additional information. For pattern \(g\), the conditional mean given only \(S=s\) is $$ E(Z\mid G=g,S=s)=\int E(Z\mid G=g,S=s,T=t)\,dF_g(t\mid S=s). $$ This identity follows by conditioning first on \(T\) and then averaging over the distribution of \(T\) within pattern \(g\) and stratum \(S=s\).
Even if the fine conditional means agree across patterns at every shared \((s,t)\), the integral can differ between patterns because \(F_g(t\mid S=s)\) and \(F_h(t\mid S=s)\) may differ. Conversely, equality after averaging over \(T\) does not require equality at each \(t\): different history-specific means can balance, as in the preceding example. Thus neither conditional restriction generally implies the other without additional conditions, such as equal distributions of \(T\) across patterns within \(S=s\) for the first implication.
This reasoning also explains the role of standardization. If the same target distribution \(q(t\mid s)\) is used for both patterns and supported in both, then equal fine conditional means yield equal standardized means under that target. The result is about the chosen target distribution; it does not assert equality under each pattern's own distribution of \(T\).
Choosing and Checking a Conditioning Set
A defensible restriction begins with the scientific meaning of “comparable.” Baseline variables may capture measured differences established before follow-up. Observed outcome history may further distinguish individuals with different trajectories before the outcome of interest. Including history can make the assumption more specific, but it also changes its substantive content and can sharply reduce the number of comparable observations.
Specify which outcome distribution is being linked and which pattern labels are compared. State whether the restriction concerns a mean, another feature, or the full conditional distribution.
List baseline covariates and observed history explicitly. Verify that each component is defined and observed for the patterns being compared.
Tabulate counts or inspect covariate distributions within pattern and conditioning strata. A broad overlap in \(X\) does not guarantee overlap in \((X,H)\).
If standardizing, state the target distribution and ensure it places weight only where the needed pattern-specific conditional quantities are defined.
Compare scientifically plausible conditioning sets or departures from the imposed restriction. Such comparisons assess sensitivity; observed data alone do not verify an assumption about an unobserved outcome distribution.
Warning signs include empty or very small pattern-by-history cells, extreme estimates driven by a few observations, and a target distribution that assigns substantial weight to values seen in only one pattern. In those cases, a conditional restriction may be formally written but weakly supported by data. Combining categories may improve stability but changes the conditioning set and therefore changes the assumption; it should not be treated as a purely computational fix.
How This Choice Relates to Other Approaches
| Approach | What is compared | Main implication |
|---|---|---|
| Unrestricted pattern-specific distributions | No equality restriction links the patterns for the unobserved outcome. | Leaves more pattern-specific features unconstrained, so the missing distribution is not identified without further assumptions. |
| Restriction conditional on baseline \(X\) | Patterns are linked within baseline-covariate strata. | Allows differences associated with observed history to remain within a baseline stratum. |
| Restriction conditional on \((X,H)\) | Patterns are linked within finer baseline-and-history strata. | Makes a more detailed conditional claim, but needs adequate overlap at that finer resolution. |
| Selection model | Models outcome and pattern membership through a different factorization. | Provides an alternative parameterization; it does not remove the need to justify assumptions about unobserved outcomes. |
These approaches are not interchangeable merely because they can produce similar numerical summaries in one data set. A conditional restriction is most transparent when its conditioning set and target population are reported alongside the result. The introductory material on pattern-mixture models motivates why restrictions are needed; the present focus is how the conditioning variables determine what those restrictions claim.
Common Pitfall and Limitations
- Wrong reasoning: “The conditional means agree after adjusting for \(X\), so the pattern-specific marginal means must agree.” Correction: Marginal means also depend on each pattern's distribution of \(X\). Use an explicitly chosen common target distribution if a standardized comparison is intended.
- Wrong reasoning: “Adding observed history only controls more variables, so it preserves the original assumption.” Correction: Conditioning on history changes the restriction from equality averaged over history to equality within each history stratum.
- Wrong reasoning: “A conditional mean for a pattern can be used at every covariate value.” Correction: It is defined from that pattern's distribution only on its support. Outside common support, a comparison needs a declared extrapolation model or a target that avoids those values.
No diagnostic can prove a restriction about an outcome that is unobserved in a pattern. Data can reveal sparse support, unstable estimates, or contradictions with observed portions of a model, but they cannot establish that the missing conditional distributions truly satisfy the assumed link. Conclusions should therefore be presented as conditional on the stated restriction, and sensitivity to plausible alternatives should be considered.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In one baseline stratum, the means agree across patterns after averaging over history. Does this establish equality at each history value?
- Pattern A and pattern B have common support only at \(C=1\). Can a direct standardized comparison assign positive weight to \(C=2\)? What would be required to do so?
- Why can equal conditional means given \((X,H)\) coexist with unequal marginal pattern-specific means?
- What substantive change occurs when a restriction is changed from conditioning on \(X\) to conditioning on \((X,H)\)?
Solutions. (1) No. History-specific differences can cancel when averaged; the worked example with means 8 and 12 in one pattern and 10 and 10 in the other demonstrates this. (2) Not as a direct comparison based on both pattern distributions, because the conditional quantity is unsupported in at least one pattern at \(C=2\). One must restrict the target to common support or specify and justify an extrapolation model. (3) The patterns can have different distributions of \((X,H)\). Averaging equal conditional means over different distributions can yield different marginal means. (4) The baseline-only restriction equates the specified outcome feature after averaging over history within baseline strata; the richer restriction equates it separately at each history value as well, which is a different and generally stronger conditional claim.