The Pattern-Mixture Factorization

Two Parts of One Joint Law

When follow-up outcomes are incomplete, the joint distribution describes both what outcomes occur and which outcomes are observed. The pattern-mixture factorization separates that joint law into two pieces: how likely each response pattern is, and the distribution of the outcome vector among units with that pattern. This separation is useful for describing how outcome distributions differ across patterns, but it does not by itself reveal the unobserved outcomes.

Let \(Y=(Y_1,\ldots,Y_T)\) denote the full outcome vector for one unit, \(R=(R_1,\ldots,R_T)\) its response-pattern vector, and \(X\) its baseline covariates. Use \(R_j=1\) when \(Y_j\) is observed and \(R_j=0\) when it is missing. A possible pattern is a particular vector \(r\) of zeros and ones. Write \(\pi_r(x)=P(R=r\mid X=x)\) for the probability of that pattern at covariate value \(x\), and write \(f_r(y\mid x)\) for the conditional distribution of the full outcome vector \(Y\) given \(R=r\) and \(X=x\). Here \(f\) denotes a probability mass function for discrete outcomes or a density for continuous outcomes.

Definition: The pattern-mixture factorization expresses the conditional joint law of the full outcome vector and response pattern as

f_{Y,R\mid X}(y,r\mid x) = \pi_r(x) f_r(y\mid x) = P(R=r\mid X=x) f_{Y\mid R,X}(y\mid r,x).

The first factor is the response-pattern distribution conditional on baseline covariates. The second is the full outcome distribution conditional on both the response pattern and baseline covariates.

The factors answer different questions. The first asks, “Among units with baseline covariates \(x\), how often does pattern \(r\) occur?” The second asks, “Among units with covariates \(x\) and pattern \(r\), what is the distribution of the full outcome vector?” The second question includes outcomes that are missing for that pattern: \(f_r\) is a distribution for \(Y\), not merely for the observed components.

The factorization is a probability identity, not a missingness assumption. It does not assert that patterns are independent of outcomes, nor does it assert that pattern membership causes outcomes. It is defined for patterns with \(\pi_r(x)>0\); if a pattern has zero probability at \(x\), its conditional outcome distribution at that covariate value is not defined by the conditional law.

Why the Factorization Holds

For any pattern \(r\) with positive probability given \(X=x\), the conditional probability rule gives the identity directly. For discrete outcomes, apply the rule to the event \(Y=y\); for continuous outcomes, the corresponding identity applies to densities.

Separate the joint event.
Condition on the baseline covariates and consider the joint event that \(Y=y\) and \(R=r\).

Apply the conditional probability rule.
The joint conditional probability equals the probability of the pattern times the probability of the outcome given that pattern.

Write the result as a distribution.
Replace the conditional probabilities by the corresponding mass functions or densities.

\begin{aligned} P(Y=y,R=r\mid X=x) &=P(R=r\mid X=x)P(Y=y\mid R=r,X=x),\\ f_{Y,R\mid X}(y,r\mid x) &=\pi_r(x) f_r(y\mid x). \end{aligned}

No independence condition is used in this derivation. In particular, the identity remains true when response patterns are strongly related to outcomes. To obtain an unconditional joint law, the conditional law can also be combined with the distribution of \(X\); the factorization here focuses on the law conditional on \(X\).

Worked Example: A Joint Probability From Two Factors

Suppose a hypothetical two-visit study has binary outcomes \(Y_1,Y_2\), and consider units with fixed baseline covariates \(X=x\). The possible patterns in this simplified example are \(r=(1,1)\), \(r=(1,0)\), and \(r=(0,0)\). Suppose their conditional probabilities are \(0.50\), \(0.30\), and \(0.20\), respectively. In pattern \((1,0)\), suppose the model for the full outcome vector assigns probability \(0.20\) to \(Y=(1,1)\).

The factorization gives the joint probability of pattern \((1,0)\) and full outcome vector \((1,1)\):

P(Y=(1,1),R=(1,0)\mid X=x) = P(R=(1,0)\mid X=x) P(Y=(1,1)\mid R=(1,0),X=x) = 0.30(0.20)=0.060.

Thus the joint probability is \(0.060\), or 6.0%, conditional on \(X=x\). This is a probability under the specified full-data law. Because \(Y_2\) is missing in pattern \((1,0)\), the conditional probability \(0.20\) is not directly observed for units in that pattern; it must come from a model or additional assumptions. The multiplication itself does not make that factor empirically identifiable.

Reading Each Factor in Context

The pattern probabilities \(\pi_r(x)\) describe the distribution of response patterns at covariate value \(x\). They are nonnegative and sum to one over patterns with positive support. The outcome distributions \(f_r(y\mid x)\) describe how the full outcome vector is distributed within each pattern. Pattern-specific means, variances, or other summaries can differ across \(r\), but those differences are descriptive features of the joint law; by themselves they do not establish a causal effect of response or dropout.

Worked Example: Including a Baseline Covariate

Consider a hypothetical study with a binary baseline covariate \(X\), where \(X=1\) denotes a specified baseline group. Suppose \(P(X=1)=0.40\) and, in that group, the probability of pattern \(R=(1,0)\) is \(0.50\). Suppose also that the pattern-specific model gives \(E(Y_2\mid R=(1,0),X=1)=7.0\) symptom-score points.

The probability that a randomly selected unit is in this baseline group and has pattern \((1,0)\) is

P(X=1,R=(1,0)) = P(X=1)P(R=(1,0)\mid X=1) = 0.40(0.50)=0.20.

So 20% of units are in that group-pattern combination under the stated probabilities. The conditional mean of 7.0 points has a different interpretation: it is the mean of \(Y_2\) under the specified full-outcome distribution for units with \(X=1\) and \(R=(1,0)\). Since \(Y_2\) is missing in this pattern, that mean is not a directly observed sample mean. The pattern probability and the pattern-specific outcome mean are separate model components, even though both condition on \(X\).

Conditioning on baseline covariates permits the pattern probabilities and outcome distributions to vary with those covariates. Omitting \(X\) would instead describe distributions averaged over its values, which can differ from the conditional distributions. Whether to model these factors with covariates, and how to specify the models, are substantive and statistical choices rather than consequences of the factorization.

Pattern-Specific Outcome Distributions

A pattern-specific distribution can be summarized or modeled in many ways. For example, a model could assign a separate mean and variance to an outcome in each pattern. Any such parametric form is an additional modeling choice; the factorization itself does not require normality, equal variances, or a particular regression structure.

Worked Example: A Model-Based Pattern-Specific Probability

Suppose, hypothetically, that under a proposed model \(Y_2\mid R=(1,0),X=x\) follows a normal distribution with mean 7 symptom-score points and standard deviation 3 points. This is an explicit modeling assumption, not a consequence of the pattern-mixture factorization. Suppose also that \(P(R=(1,0)\mid X=x)=0.30\). Under this model, calculate the joint probability of pattern \((1,0)\) and \(Y_2<8\).

First standardize the outcome threshold using the assumed conditional distribution:

P(Y_2<8\mid R=(1,0),X=x) = \Phi\left(\frac{8-7}{3}\right) = \Phi(0.3333) \approx 0.6306.

Then multiply by the pattern probability:

P(R=(1,0),Y_2<8\mid X=x) = 0.30(0.6306) \approx 0.1892.

The modeled joint probability is about 0.189, or 18.9%, conditional on \(X=x\). This calculation is internally valid if the stated normal distribution and pattern probability are the model being used. But \(Y_2\) is missing whenever \(R=(1,0)\), so the conditional distribution of \(Y_2\) in that pattern cannot be checked directly from those units’ observed \(Y_2\) values. The numerical result should therefore be understood as model-based, not as a directly observed frequency.

What the Data Identify—and What They Do Not

For a given pattern, the observed outcomes are the components of \(Y\) for which \(R_j=1\); the remaining components are missing. Observed data can inform the pattern probabilities and the distributions of the observed components within each pattern. In general, they do not alone determine the distribution of the full \(Y\) within a pattern when that pattern leaves some components unobserved. This is why a pattern-mixture specification may need assumptions or sensitivity parameters for the unseen portions of \(f_r\).

The factorization is an alternative organization of the same joint law represented by the selection factorization, which writes an outcome distribution conditional on covariates together with a response-pattern distribution conditional on outcomes and covariates. Neither organization, merely by being written down, imposes missing at random or missing not at random. Claims about what is identified from observed data, or about what assumptions permit estimation, require more than choosing a factorization.

A useful diagnostic is to ask whether every factor being interpreted is supported by observed data in the relevant pattern. If a model reports precise estimates for a distribution of outcomes that are all missing in a pattern, check which restrictions or assumptions supply that information. Also check whether fitted pattern probabilities are valid probabilities—nonnegative and summing to one—and whether the pattern-specific outcome distributions are plausible over the outcome scale. Sparse or absent patterns can make estimates unstable or conditional factors undefined.

Common Mistake and Limitations

Confusing the two factors. The pattern probability is not an outcome mean, and the conditional outcome distribution is not the probability of dropout. State the conditioning variables and pattern explicitly.
Treating the identity as an identifying assumption. The factorization always follows from the conditional probability rule on its support. It does not, by itself, identify outcomes that are unobserved within a pattern.
Reading a pattern difference causally. Different outcome distributions across patterns do not show that response status caused the outcome difference. The groups may differ in other ways, and some pattern-specific outcomes may be modeled rather than observed.
Ignoring support. A conditional distribution given a pattern cannot be estimated as a within-pattern empirical distribution when that pattern is absent or contains no observations for the relevant covariate values.

The pattern-mixture representation is valuable because it makes comparisons among response patterns explicit. Its limitation is equally important: when outcomes are missing by pattern, part of the full outcome distribution can remain unobserved. The representation organizes the problem; it does not solve that information gap automatically.

Key takeaway: Conditional on baseline covariates, the pattern-mixture factorization is the response-pattern probability multiplied by the full outcome distribution within that pattern. Interpret the factors separately, and distinguish model-based claims about missing outcomes from quantities directly supported by observed data.

Check Your Understanding

Answer each question, then compare with the solutions below.

In \(f_{Y,R\mid X}(y,r\mid x)=\pi_r(x)f_r(y\mid x)\), what does each factor represent?
If \(\pi_r(x)=0.25\) and \(P(Y_1=2\mid R=r,X=x)=0.40\), what is \(P(Y_1=2,R=r\mid X=x)\)?
Why may the distribution of an outcome missing throughout pattern \(r\) require assumptions or a model?
Does writing the pattern-mixture factorization assert that missingness is independent of the outcomes? Explain.

Solutions. (1) \(\pi_r(x)\) is the probability of response pattern \(r\) conditional on baseline covariates \(x\); \(f_r(y\mid x)\) is the distribution of the full outcome vector conditional on that pattern and those covariates. (2) By the conditional probability rule, \(0.25(0.40)=0.10\), so the joint conditional probability is 0.10. (3) The outcome values are not observed in that pattern, so the observed data do not directly reveal their distribution there; a model or additional assumptions must provide the missing information. (4) No. The factorization is an identity and imposes no independence condition between outcomes and response patterns.