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A pattern-mixture model handles missing-not-at-random (MNAR) data by first sorting subjects into groups defined by which of their variables are actually observed — the “pattern” — and then modeling the outcome’s distribution separately within each pattern, before combining the patterns back into a single estimate. That combination step is where the method gets its name: the overall, full-data distribution is treated as a mixture of the pattern-specific distributions, weighted by how common each pattern is. It is one of two standard families for modeling MNAR data explicitly (the other is selection models), and it is the framework behind delta-adjustment, the most widely used practical technique for asking “how much would our conclusion change if the missing values were systematically worse (or better) than a standard model would assume?”
What makes a “pattern”
A missingness pattern is defined by the specific combination of variables observed and missing for a subject — not simply a binary complete/incomplete split. In a three-visit longitudinal trial (baseline, an interim visit, and a final visit), there are, at minimum, three patterns worth distinguishing: subjects observed at all three visits (completers), subjects who dropped out after the interim visit (baseline and interim observed, final missing), and subjects who dropped out before the interim visit (only baseline observed). Each pattern can carry a different amount of information about what a subject’s missing values would have looked like — a subject who was seen at the interim visit gives you something concrete to extrapolate from; a subject who vanished after baseline does not. A pattern-mixture model can assign each pattern its own model and its own sensitivity assumption instead of forcing every missing value through one shared rule, which is the property that makes it useful beyond a simple missing/observed split.
The core idea: a mixture of observed and missing distributions
The standard statement of the model, from the U.S. National Research Council’s panel report on missing data in clinical trials, is that the full-data distribution is written as a mixture — a weighted average — of the observed and missing data distributions, one component per pattern. Practically, that means: estimate (or assume) a distribution for the outcome within each pattern, weight each pattern by its observed proportion of the sample, and sum. For the patterns where the outcome actually was observed, that distribution comes directly from the data. For patterns with missing outcomes, it has to come from somewhere else — either an assumption that missing values behave like observed ones after conditioning on covariates (which collapses back to a standard missing-at-random, MAR-consistent estimate), or an explicit, stated departure from that assumption. Delta-adjustment is how that departure gets made explicit and quantitative.
Delta-adjustment: making the MNAR assumption a number
The mechanics, again per the NRC panel report: define a sensitivity parameter Δ that shifts the assumed mean for the missing/nonrespondent group relative to what a MAR-consistent model would predict, expressed as μ₀ = μ₁ + Δ (a nonrespondent mean offset from the respondent-consistent prediction by Δ). Δ = 0 reproduces the MAR assumption exactly — it is the boundary case, not a separate method. Moving Δ away from zero is what actually models MNAR: it says explicitly “subjects who dropped out would have scored Δ points worse (or better) than a model built only on who stayed in the study would predict for them,” and recomputes the estimate under that stated assumption. The NRC report’s own framing of why this particular parameterization is preferred over a selection-model equivalent is worth quoting directly: pattern-mixture sensitivity parameters offer a “transparent interpretation” because they typically represent a “difference in mean between respondents and nonrespondents,” a quantity a clinician or trial statistician can reason about directly, rather than a coefficient buried inside a missingness-probability model.
Pattern-mixture models vs. selection models
Both are legitimate, established ways of modeling MNAR data explicitly, and both appear in the same NRC chapter on sensitivity-analysis methods, but they factor the problem differently. A selection model specifies the full-data outcome distribution directly, then separately models the probability of being missing as a function of that outcome (observed and unobserved) — it answers “given a subject’s true outcome, how likely were they to drop out?” A pattern-mixture model inverts the factoring: it conditions on the pattern first, then asks what the outcome distribution looks like inside it — effectively specifying how the missing values are being “filled in” under a given set of assumptions, pattern by pattern. Selection models tend to require assumptions about the missingness mechanism that are hard to state in plain language and impossible to check from the observed data alone; pattern-mixture models trade that for the more legible, directly interpretable delta parameters above. Neither model type is testable against the data on the point where they actually differ — that’s the nature of MNAR — so the practical case for pattern-mixture over selection models is usually communicability: a stated Δ is something a clinical reviewer or regulator can evaluate for plausibility on its face.
How this complements tipping-point analysis
CASRAI’s tipping-point analysis guide covers a specific, widely used application of exactly this delta-adjustment machinery: it sweeps a single Δ across a full range of plausible values and reports the exact boundary where the study’s conclusion flips, without committing to any one value as “the” sensitivity analysis. Pattern-mixture modeling, as covered on this page, is the more general framework that sweep sits inside — and it supports something a single global sweep does not: different Δ values for different missingness patterns, and different Δ values for different treatment arms, in the same analysis. That distinction matters in practice. A single global Δ applied uniformly assumes every dropout, in every arm, deviates from the MAR prediction by the same fixed amount — a strong and often implausible assumption in its own right. A pattern-mixture setup lets an analyst instead say, for example: “dropouts who were seen at the interim visit get a small penalty, because we have real information on them; dropouts seen only at baseline get a larger one, because we have almost nothing to extrapolate from,” or “we specifically suspect informative dropout concealed treatment failures in the active arm, so only that arm’s missing data gets adjusted.” The worked example below shows both patterns of use, including a case where they produce meaningfully different conclusions from the same data.
A reproducible worked example
Every number in this section comes from an actual seeded simulation (Mulberry32 PRNG, seed fixed in the script) run for this page — not invented or approximated. It is a labeled, illustrative simulation, not a real trial.
Setup: 600 simulated subjects, 300 per arm, with a continuous functioning-score outcome (higher = better) measured at baseline, an interim visit, and a final visit two intervals later. True subject trajectories are linear: the control arm declines 1.0 point per visit-interval, the treatment arm improves 2.5 points per visit-interval, on top of a subject-level random effect (SD 6) and visit-level noise (SD 4). That gives a true final-visit treatment effect of exactly 7.00 points by construction (7.13 points measured empirically on this specific simulated cohort, since a finite sample never matches its generating formula exactly).
Dropout is generated to be genuinely MNAR: the probability of leaving the study before the interim visit, and separately the probability of leaving after the interim visit but before the final one, both depend on each subject’s own unobserved random effect — subjects on a worse-than-average trajectory are disproportionately more likely to drop out, and that tendency is not fully recoverable from what’s actually observed. This produced three patterns in the simulated data: 474 completers, 93 subjects who dropped after the interim visit, and 33 who dropped before it (counts differ slightly by arm: 239/46/15 in control, 235/47/18 in treatment).
Step 1 — naive complete-case estimate. Using only the 474 completers: treatment effect = 7.34. In this particular simulated cohort, that happens to be close to the 7.13 truth despite the dropout being genuinely MNAR by construction — a useful, slightly uncomfortable reminder that complete-case bias is not guaranteed to be large just because the missingness mechanism is technically MNAR; it depends on how much the dropout patterns actually differ between arms, which here was fairly similar.
Step 2 — pattern-mixture with symmetric delta-adjustment. A MAR-consistent reference model was fit on completers only (predicting the final-visit outcome from baseline and arm, plus the interim-visit value for the pattern that has one), then used to predict what each dropout’s final-visit value would have been under MAR. Applying the same Δ to both arms’ dropouts at increasing severity:
- Δ = 0 (pure MAR): effect = 7.39
- Δ = 3 for both patterns: effect = 7.35
- Δ = 8 for both patterns: effect = 7.29
- Pattern-specific Δ (3 for interim-dropouts, 8 for baseline-only dropouts, reflecting how much less is actually known about the second group): effect = 7.30
- Pattern-specific, more severe (6 and 14): effect = 7.23
The honest finding here is that none of this moved the treatment-effect estimate very much — even an 8-to-14-point delta penalty barely dented a 7-point effect. That is not a universal property of delta-adjustment; it is a direct consequence of applying the delta symmetrically to both arms when the two arms’ dropout patterns are similar in size. A penalty subtracted from both arms’ missing values cancels substantially in the between-arm subtraction — it depresses each arm’s estimated mean, but not the difference between them, unless the arms differ in how much of each pattern they contain or in how large a delta is applied to each.
Step 3 — arm-differential delta (the scenario that actually mattered). A more targeted MNAR concern in practice is that dropout conceals worse outcomes specifically in the treatment arm — e.g., unresponsive subjects quietly discontinuing rather than being recorded as treatment failures. Applying Δ only to the treatment arm’s dropouts and leaving the control arm at its MAR prediction:
- Δtreatment-only = 0: effect = 7.39
- Δtreatment-only = 5: effect = 6.31
- Δtreatment-only = 10: effect = 5.23
- Δtreatment-only = 15: effect = 4.14
- Δtreatment-only = 20: effect = 3.06
Against an illustrative minimal-clinically-important-difference threshold of 5.0 points, this version of the analysis crosses that line at Δtreatment-only ≈ 11.05 (found by bisection on the same model). The mixed, genuinely non-tidy conclusion from running both versions on the identical dataset: a symmetric, whole-sample delta-adjustment made this result look robust, while an arm-differential delta — testing a specific, clinically motivated MNAR hypothesis rather than a generic uniform one — showed the same data crossing a meaningful threshold at a plausible-sounding delta of about 11 points. Which framing is the right one to report depends entirely on which MNAR mechanism is actually plausible for the specific trial, not on which produces a more reassuring number — and a pattern-mixture setup is what makes it possible to test the arm-differential version at all, since a single global Δ swept the same way in both arms cannot represent it.
Choosing delta values without inventing them
Because Δ is an assumption, not something estimable from the observed data, the standard advice — consistent with the NRC panel’s framing of pattern-mixture parameters as clinically interpretable by design — is to ground candidate values in something outside the dataset itself: a natural-history or untreated-cohort estimate of how the outcome tends to move for subjects who stop participating, expert clinical elicitation of a plausible worst-case difference, or the boundary values a tipping-point sweep already identified as the point where conclusions change. Reporting a single delta value chosen to produce a particular result, without a stated rationale for why that value is clinically plausible, defeats the purpose of the method — the value of delta-adjustment is the transparency of the assumption, not the number it happens to produce.
Reporting a pattern-mixture sensitivity analysis
In a regulated clinical-trial context, missing-data sensitivity analysis is an explicit expectation of the ICH estimand framework (see CASRAI’s ICH E9 entry) rather than an optional add-on: the primary estimand is defined and estimated under a stated set of assumptions about intercurrent events including dropout, and supplementary sensitivity analyses — pattern-mixture and delta-adjustment among the standard tools — are expected to show whether the primary conclusion holds up under plausible departures from that assumption. A pattern-mixture analysis should be reported with: the patterns actually used and their sample sizes per arm, the reference model each pattern’s missing values were extrapolated from, the specific delta values applied and the stated rationale for each, and whether the analysis was symmetric across arms or arm-differential and why. It is a sensitivity analysis in support of a primary estimate — typically one produced by FIML or multiple imputation pooled with Rubin’s rules — not a replacement for one, and it should never be substituted for those primary methods or for single-imputation shortcuts like LOCF, which make an unstated, generally indefensible assumption about missing trajectories rather than a stated, examinable one.
Frequently Asked Questions
What is a pattern-mixture model, in plain terms?
A way of handling missing-not-at-random data that groups subjects by which of their variables are actually observed (their “pattern”), models the outcome separately within each group, and combines the group-specific estimates — weighted by how common each pattern is — into one overall estimate. Where a pattern’s outcome is missing, that group’s contribution comes from an explicit, stated assumption (delta-adjustment) rather than from the observed data alone.
How is a pattern-mixture model different from a selection model?
They are the two standard ways of modeling MNAR data explicitly, and they factor the same problem in opposite directions. A selection model specifies the outcome distribution first, then models the probability of being missing as a function of that outcome. A pattern-mixture model conditions on the missingness pattern first, then models the outcome distribution inside each pattern. Pattern-mixture parameters (typically a mean difference between respondents and nonrespondents) tend to be easier to state and evaluate in plain clinical language than a selection model’s missingness-probability coefficients.
Is pattern-mixture analysis the same thing as tipping-point analysis?
No — tipping-point analysis is one specific, widely used application built on the same delta-adjustment idea: it sweeps a single global delta across a range and reports the exact value where a study’s conclusion flips. Pattern-mixture modeling is the broader framework, and it additionally supports different delta values for different missingness patterns and different treatment arms in the same analysis, which a single global sweep cannot represent.
What does Δ = 0 mean in a delta-adjustment analysis?
It means no MNAR departure is being assumed — the missing values are assumed to follow the same distribution the MAR-consistent reference model predicts for them. Δ = 0 is the boundary case that reproduces a standard MAR-based estimate (the kind FIML or multiple imputation would already produce); moving away from zero is what actually introduces an MNAR assumption into the analysis.
How many missingness patterns should I define?
As many as are needed to distinguish groups that genuinely carry different amounts of information about their own missing values — in practice, this usually tracks a study’s monotone dropout structure (which visit a subject was last seen at) rather than an arbitrary number chosen in advance. Defining more patterns than the sample size can support separately estimated deltas for is a real tradeoff; collapsing very small dropout groups into a coarser pattern is a reasonable, commonly made compromise.
The worked example above is a labeled, seeded simulation built for this page, not a real trial or published dataset.
