Pattern-Mixture Models for MNAR Outcomes

Why Organize a Missing-Outcome Model by Pattern?

When outcomes are missing, the observed data include both the outcomes that were recorded and the patterns that indicate which outcomes were not recorded. A pattern-mixture model organizes the joint distribution around those patterns: first describe how common each pattern is, then describe the outcome distribution within each pattern. This makes a crucial question explicit: what must be assumed about outcomes that are missing within a pattern?

That question matters for causal effects. Randomization or adjustment for confounders can support a causal comparison of treatment groups, but neither reveals outcomes that were not measured. Under missing not at random (MNAR), the probability of observing an outcome may depend on its unobserved value even after conditioning on measured variables. A pattern-mixture model displays the extrapolation needed to complete the outcome distribution, rather than making that extrapolation disappear.

Definition: Let $A$ denote treatment, $X$ measured baseline covariates, $Y$ an outcome, and $R$ a missingness indicator, with $R=1$ when $Y$ is observed and $R=0$ when it is missing. A pattern-mixture factorization writes the joint distribution, conditional on treatment and covariates, as $$ p(Y,R\mid A,X)=p(R\mid A,X)\,p(Y\mid R,A,X). $$ For several outcomes, $R$ is a pattern label describing which components are observed. The factorization separates the frequency of each pattern from the outcome distribution within that pattern.

The factorization is a probability identity; it is not, by itself, an assumption that makes missing outcomes identifiable. For $R=1$, the outcome distribution $p(Y\mid R=1,A,X)$ is observed. For $R=0$, $Y$ is absent, so $p(Y\mid R=0,A,X)$ is not directly learned from the observed data. Under MNAR, the observed outcomes need not have the same distribution as the missing outcomes. A pattern-mixture analysis must therefore specify or otherwise justify the missing pattern's outcome distribution.

From Pattern-Specific Distributions to a Causal Mean

Suppose the target is the average treatment effect (ATE), $E[Y(1)-Y(0)]$, where $Y(a)$ is the potential outcome under treatment level $a$. The treatment comparison requires the usual causal conditions: consistency, conditional exchangeability given $X$, and treatment positivity. In particular, consistency connects the observed outcome to the potential outcome under the treatment received; exchangeability requires no unmeasured confounding after conditioning on $X$; and positivity requires each treatment level to be possible at covariate values in the target population.

Those conditions address treatment assignment, not missing outcomes. For the missingness problem, let $R$ indicate whether the outcome under the received treatment is observed. If the outcome is observed for everyone, the covariate-standardized arm mean is identified by the causal conditions. With missing outcomes, the pattern-mixture form of that mean is

E[Y(a)]=\int \left\{\sum_r P(R=r\mid A=a,X=x)\, E[Y\mid R=r,A=a,X=x]\right\}dF_X(x).

Here $F_X$ is the covariate distribution in the target population. For binary outcome response, the sum has the two terms $r=0$ and $r=1$. The response-pattern probabilities can be estimated from the observed data, as can the outcome distribution among those with observed outcomes. The conditional outcome distribution for a pattern in which $Y$ is missing must be supplied by an additional assumption or model. The ATE is the difference between the resulting arm means.

A short derivation shows where the extrapolation enters. Within a treatment and covariate stratum, the law of total expectation gives

E[Y\mid A=a,X=x] =\sum_r E[Y\mid R=r,A=a,X=x]P(R=r\mid A=a,X=x).

This equation holds whether or not outcomes are missing at random. But when $Y$ is missing for a pattern, its conditional expectation in that pattern cannot be computed from the observed outcomes there. Thus the identity alone does not identify the full mean. For example, changing the assumed mean $E[Y\mid R=0,A=a,X=x]$ while leaving all observed records unchanged changes the completed-data mean by

P(R=0\mid A=a,X=x)\times \text{the change in }E[Y\mid R=0,A=a,X=x].

This is the central feature of pattern-mixture modeling: conclusions depend on the specified outcome distribution in patterns with missing values, and the dependence is visible in the formula.

Worked Example: A Binary Outcome in a Randomized Trial

Hypothetical data. A randomized trial assigns 100 participants to treatment and 100 to control. In the treatment arm, 80 outcomes are observed and 48 are successes. In the control arm, 90 outcomes are observed and 45 are successes. The target is the risk difference, treatment risk minus control risk. Randomization supports exchangeability, and both treatments occur by design; assume consistency and that the trial population is the target population. These conditions do not determine the missing outcomes.

Among observed outcomes, the treatment success proportion is $48/80=0.600$, and the control success proportion is $45/90=0.500$. For this illustrative pattern-mixture calculation, suppose the missing-outcome success proportion is 0.400 in the treatment arm and 0.300 in the control arm. These are explicit extrapolation assumptions, not quantities estimated from the missing records.

In the treatment arm, $P(R=1\mid A=1)=80/100=0.800$ and $P(R=0\mid A=1)=0.200$. Therefore the completed arm risk is

$$ E[Y(1)]=0.800(0.600)+0.200(0.400) =0.480+0.080=0.560. $$

In the control arm, $P(R=1\mid A=0)=90/100=0.900$ and $P(R=0\mid A=0)=0.100$, so

$$ E[Y(0)]=0.900(0.500)+0.100(0.300) =0.450+0.030=0.480. $$

The resulting risk difference is $0.560-0.480=0.080$, or 8.0 percentage points. Under the stated extrapolations and causal conditions, the estimated treatment risk is 8.0 percentage points higher than the control risk. This is a model-based result: the observed data alone do not establish the assumed missing-outcome success proportions.

Covariates and Pattern-Specific Extrapolation

When covariates matter, the pattern-mixture calculation should be made within covariate strata or through models conditional on $X$, then averaged over the target covariate distribution. This keeps the treatment standardization distinct from the missing-outcome extrapolation. For each arm and covariate value, the analysis needs both the pattern probability and the outcome distribution for every pattern that contributes to the mean.

In multivariate or longitudinal data, a pattern can reveal some outcomes but not others. The observed components may help predict missing components, but a prediction is still an extrapolation when the target component is absent in that pattern. A model may, for example, relate a missing later outcome to an earlier observed outcome within a pattern. Its parameters are informed by patterns where the relevant outcomes are jointly observed, but applying that relationship to a pattern with the later outcome missing requires a transport assumption across patterns.

Worked Example: Standardizing Across Two Covariate Strata

Hypothetical setup. A randomized study has two baseline strata, low and high risk, each comprising half of the target population. The outcome is binary. The table gives, for each treatment arm and stratum, the response probability, the observed-outcome success proportion, and the assumed missing-outcome success proportion.

Stratum	Treatment arm	Response probability	Success among observed	Assumed success among missing
Low risk	Treatment	0.900	0.700	0.500
Low risk	Control	0.800	0.600	0.400
High risk	Treatment	0.600	0.800	0.600
High risk	Control	0.500	0.700	0.500

Assume consistency, exchangeability from randomization, and treatment positivity. The missing-outcome proportions in the table are additional assumptions. In the low-risk stratum, the treatment mean is $0.900(0.700)+0.100(0.500)=0.630+0.050=0.680$. The control mean is $0.800(0.600)+0.200(0.400)=0.480+0.080=0.560$.

In the high-risk stratum, the treatment mean is $0.600(0.800)+0.400(0.600)=0.480+0.240=0.720$. The control mean is $0.500(0.700)+0.500(0.500)=0.350+0.250=0.600$. Standardize each arm mean to the target population's 50:50 stratum distribution:

$$ E[Y(1)]=0.500(0.680)+0.500(0.720)=0.700, $$

$$ E[Y(0)]=0.500(0.560)+0.500(0.600)=0.580. $$

The standardized risk difference is $0.700-0.580=0.120$, or 12.0 percentage points. The calculation uses the same target covariate weights for both arms. It also depends on the four stated assumptions for missing outcomes; randomization does not validate them.

More Than One Outcome and More Than One Pattern

For a vector of outcomes $Y=(Y_1,\ldots,Y_K)$, let pattern $r$ specify the subset $Y_{\mathrm{obs}}^r$ that is observed and the subset $Y_{\mathrm{mis}}^r$ that is missing. The corresponding factorization is

p(Y,R=r\mid A,X) =p(R=r\mid A,X)\, p(Y_{\mathrm{obs}}^r,Y_{\mathrm{mis}}^r\mid R=r,A,X).

The pattern probability is estimable from the observed pattern labels. The distribution of the observed components within that pattern is also estimable. The distribution involving missing components is not fully observed; a model must extrapolate it. Some patterns can have no observed value for a particular outcome, making that outcome's distribution within the pattern entirely dependent on assumptions. In other patterns, observed earlier measurements may inform a conditional prediction, but the validity of carrying that relationship over remains an assumption.

Worked Example: A Later Outcome Missing in Several Patterns

Hypothetical setup. In a two-visit study, let $R$ record whether the outcomes $Y_1$ and $Y_2$ are observed, in that order. Suppose the pattern probabilities are $P(R=11)=0.500$, $P(R=10)=0.200$, $P(R=01)=0.200$, and $P(R=00)=0.100$. The target is the mean of the binary later outcome $Y_2$. Suppose its observed mean is 0.600 in pattern 11 and 0.400 in pattern 01.

The data do not reveal $Y_2$ in patterns 10 and 00. For illustration, assume its mean in pattern 10 is 0.500, based on an extrapolation that uses the earlier outcome available in that pattern, and assume its mean in pattern 00 is 0.450. The latter pattern has neither outcome observed, so its $Y_2$ mean is wholly extrapolated. These assumptions, rather than the observed means alone, complete the pattern-specific distribution needed for this calculation.

Applying the law of total expectation across the four patterns gives

$$ E[Y_2] =0.500(0.600)+0.200(0.500)+0.200(0.400)+0.100(0.450). $$

The four contributions are $0.300$, $0.100$, $0.080$, and $0.045$. Thus $E[Y_2]=0.300+0.100+0.080+0.045=0.525$. The mean is 0.525 under the stated pattern-specific assumptions. The observed data identify the pattern weights and the means for patterns 11 and 01, but not the two means assigned to patterns 10 and 00.

What Is Identified, and What Must Be Extrapolated?

A practical pattern-mixture analysis separates three kinds of information:

Pattern frequencies: probabilities such as $P(R=r\mid A,X)$, estimated from whether outcomes were recorded.
Observed components within patterns: distributions of the outcomes that are actually recorded in each pattern.
Missing components within patterns: distributions that require an extrapolation whenever the relevant outcome is not observed there.

The causal assumptions and the missingness assumptions should be reported separately. Consistency, exchangeability, and treatment positivity support interpreting a treatment contrast causally. A model for the outcome distribution in missing patterns addresses a different problem. Even in a perfectly randomized trial, observed-data evidence cannot by itself determine an outcome distribution among participants whose outcomes were not recorded under MNAR.

A useful diagnostic question is: which exact conditional distribution in the estimand is not observed? Naming it makes the substantive assumption inspectable. For a scalar outcome, it may be the mean among nonresponders in one treatment arm. For longitudinal outcomes, it may be the distribution of a later measurement among people with an earlier measurement but no later one. If a covariate is included, the extrapolation may need to hold within covariate levels, not only on average.

Common Mistakes

Equating the factorization with identification. The factorization is true by the probability chain rule, but it does not reveal $p(Y\mid R=0,A,X)$. The correction is to state the additional model or assumption used for that distribution.
Assuming randomization solves missingness. Randomization addresses treatment assignment, not whether unobserved outcomes resemble observed ones. State the treatment assumptions and missing-outcome assumptions separately.
Calling an assumed missing-pattern mean an observed estimate. In an MNAR analysis, that mean is not learned directly from the missing records. Label it as an extrapolation and show how it enters the final estimate.
Reporting a treatment effect without its target population. With covariates, arm means must be averaged over a stated target covariate distribution. Otherwise the reported contrast may not correspond to the intended estimand.

Key takeaway: Pattern-mixture models factor the joint distribution into missingness-pattern probabilities and pattern-specific outcome distributions. Observed data reveal the patterns and their recorded outcomes, but causal treatment effects under MNAR require explicit extrapolations for outcomes missing within those patterns.

Check Your Understanding

Use the factorization and examples above to answer the following.

For a binary outcome with response indicator $R$, which pieces of $p(Y,R\mid A,X)$ are directly estimable when $R=0$ outcomes are never recorded?
Why do consistency, conditional exchangeability, and treatment positivity not, by themselves, identify a causal mean when outcomes are MNAR?
If the assumed mean among missing outcomes increases by 0.10 in a treatment arm with a 20% missing-outcome rate, by how much does that arm's completed mean change?
In a two-visit study, why does an observed first-visit outcome not automatically identify the distribution of the missing second-visit outcome?
When covariate strata have unequal proportions in the target population, what weights should be used to combine stratum-specific arm means?