The Identification Gap in a Pattern-Mixture Model

What the Observed Data Leave Open

A pattern-mixture model describes outcomes separately within each missingness pattern, then combines those pattern-specific distributions using the pattern probabilities. Its attraction is that it makes the role of each pattern explicit. Its central identification question is equally direct: which parts of each pattern-specific outcome distribution are actually revealed by the observed outcomes?

Consider two outcome occasions, \(Y_1\) and \(Y_2\), with monotone missingness: a person observed at occasion 2 is also observed at occasion 1. Write the observation pattern as \(R=(R_1,R_2)\), where \(R_j=1\) means \(Y_j\) is observed and \(R_j=0\) means it is missing. The possible patterns are \(11\), \(10\), and \(00\). This setup allows dropout before the first occasion as well as between occasions. If the study guarantees that everyone has \(Y_1\), the \(00\) pattern is absent, and the corresponding gap discussed below does not arise in the population mixture.

Definition: A pattern-specific distribution is the conditional distribution of the full outcome vector given a pattern, \(f_r(y_1,y_2)=f(y_1,y_2\mid R=r)\). A distribution is identified from the observed data if no two full-data probability laws that agree on the observed-data law give different values for that distribution. Identification concerns what the data determine in principle, not how precisely a finite sample estimates it.

Let \(\pi_r=P(R=r)\) denote the probability of pattern \(r\). The observed data reveal pattern membership and the outcomes observed in that pattern. In particular, they reveal \(\pi_r\) and the conditional distribution of each observed outcome component given \(R=r\), subject to the usual population-level interpretation that the relevant pattern has positive probability. Without an additional restriction linking patterns or specifying missing outcomes, the observed data do not reveal the distributions of outcomes that are missing within a pattern.

Identify the Distribution, Not Just the Label

For pattern \(11\), both outcomes are recorded. The data therefore identify the full conditional joint distribution \(f_{11}(y_1,y_2)\). This includes its two margins and its within-pattern association.

For pattern \(10\), only \(Y_1\) is recorded. The data identify the margin \(f(Y_1\mid R=10)\), but not the full joint distribution \(f_{10}(y_1,y_2)\). A useful factorization makes the missing piece visible:

f_{10}(y_1,y_2) = f(y_1\mid R=10)\,f(y_2\mid y_1,R=10).

The first factor is identified from observed \(Y_1\) values in this pattern. The second factor describes the unobserved \(Y_2\) distribution among pattern-10 individuals, conditional on \(Y_1\). The observed data provide no direct observations from that factor. In particular, observing \(Y_1\) does not identify the association between \(Y_1\) and the missing \(Y_2\) within pattern 10.

For pattern \(00\), neither outcome is observed. Pattern membership reveals \(\pi_{00}\), but does not reveal the outcome distribution \(f_{00}(y_1,y_2)\). Both its marginal distributions and their joint association are unspecified by the observed outcomes.

Pattern	Outcomes observed	Identified from that pattern's observed outcomes	Not identified without further restrictions
11	\(Y_1,Y_2\)	Full joint distribution \(f_{11}(y_1,y_2)\)	None of the full outcome distribution
10	\(Y_1\)	Margin \(f(Y_1\mid R=10)\)	\(f(Y_2\mid Y_1,R=10)\)
00	Neither	No outcome distribution	Full joint distribution \(f_{00}(y_1,y_2)\)

Worked Example: Inventorying What Is Observed

Suppose a hypothetical two-visit study has 200 participants: 120 with pattern 11, 50 with pattern 10, and 30 with pattern 00. The pattern proportions are \(120/200=0.60\), \(50/200=0.25\), and \(30/200=0.15\). What does the observed data identify about the pattern-specific outcome distributions?

In the 120 pattern-11 participants, both outcomes are recorded. In a sufficiently large sample, the observed pairs provide information about the full distribution \(f_{11}(y_1,y_2)\). In the 50 pattern-10 participants, the 50 observed \(Y_1\) values provide information about \(f(Y_1\mid R=10)\), but none of these participants contributes an observed \(Y_2\) value. The 30 pattern-00 participants reveal that they belong to that pattern, but contribute neither outcome to an estimate of \(f_{00}(y_1,y_2)\).

Thus the three pattern probabilities are observed-data quantities, as is the \(Y_1\) margin for pattern 10 and the full joint distribution for pattern 11. The pattern-10 conditional distribution of \(Y_2\) given \(Y_1\), and the entire pattern-00 joint distribution, remain unidentified. The sample sizes affect estimation precision for observed quantities; increasing them does not by itself reveal outcomes that are never observed in those patterns.

Counting the Gap for Discrete Outcomes

A finite-category example lets us count unidentified components exactly. Suppose \(Y_1\) has \(K_1\) categories and \(Y_2\) has \(K_2\) categories. A probability distribution on \(K\) categories has \(K-1\) free probabilities, because the probabilities sum to one.

In pattern 11, the full joint distribution has \(K_1K_2\) cells, so it has \(K_1K_2-1\) free probabilities, all informed by observed outcome pairs. In pattern 10, the observed \(Y_1\) margin has \(K_1-1\) free probabilities. To complete the joint distribution, one also needs a distribution of \(Y_2\) for each of the \(K_1\) possible \(Y_1\) categories. Each such conditional distribution has \(K_2-1\) free probabilities. The unidentified part therefore has \(K_1(K_2-1)\) free probabilities. In pattern 00, the entire joint distribution has \(K_1K_2-1\) free probabilities that are not informed by outcomes.

\begin{aligned} \text{unidentified components in pattern }10 &=K_1(K_2-1),\\ \text{unidentified components in pattern }00 &=K_1K_2-1,\\ \text{total unidentified components} &=K_1(K_2-1)+(K_1K_2-1). \end{aligned}

This is a count of free probabilities in an unrestricted discrete model, not a count of every possible way to describe a distribution. It assumes each pattern has positive probability and that all outcome categories are possible. If some categories are structurally impossible, the appropriate count must be revised. If outcomes are continuous, the same identification distinction applies, but there is generally no finite number of free probabilities to count.

Worked Example: Counting Unidentified Probabilities

Suppose both outcomes are binary, so \(K_1=K_2=2\), and all three patterns have positive probability. Count the free probabilities in the full pattern-specific distributions and identify which are not determined by the observed outcomes.

The pattern-11 joint distribution has \(2\times2=4\) cells and \(4-1=3\) free probabilities. All three are informed by observed pairs. The pattern-10 joint distribution also has three free probabilities, but only its \(Y_1\) margin is observed. That margin has \(2-1=1\) free probability, leaving \(2(2-1)=2\) unidentified probabilities for the two conditional probabilities \(P(Y_2=1\mid Y_1=0,R=10)\) and \(P(Y_2=1\mid Y_1=1,R=10)\).

The pattern-00 joint distribution has \(4-1=3\) free probabilities, none determined by observed outcomes. Hence the total number of unidentified free probabilities is \(2+3=5\). The observed pattern-10 margin and the pattern frequencies do not reduce that count further: the margin concerns \(Y_1\), while the two missing conditional probabilities concern \(Y_2\) given \(Y_1\).

Why the Gap Affects Population Summaries

The overall outcome distribution is a mixture of the pattern-specific distributions:

f(y_1,y_2)=\sum_{r\in\{11,10,00\}}\pi_r f_r(y_1,y_2).

The weights \(\pi_r\) are identified by pattern membership. But knowing the weights does not identify a mixture if some of its component distributions are unknown. For example, the overall mean of \(Y_2\), when \(Y_2\) is numerical and has a finite mean, is

E(Y_2)= \pi_{11}E(Y_2\mid R=11) +\pi_{10}E(Y_2\mid R=10) +\pi_{00}E(Y_2\mid R=00).

The first conditional mean is informed by observed \(Y_2\) values in pattern 11. The second and third are not identified from outcomes: \(Y_2\) is missing for patterns 10 and 00. Consequently, the overall mean is not identified by the observed data alone when those patterns have positive probability, absent additional assumptions or restrictions.

Worked Example: Same Observations, Different Overall Means

Suppose a hypothetical study has \(\pi_{11}=0.60\), \(\pi_{10}=0.25\), and \(\pi_{00}=0.15\). Let \(Y_2\) be binary. Among pattern-11 participants, the observed proportion with \(Y_2=1\) is 0.40. What can be concluded about the overall mean \(E(Y_2)\) without specifying the missing-outcome distributions?

For a binary outcome, the mean equals the probability of outcome 1. The observed contribution from pattern 11 is \(0.60(0.40)=0.24\). Let \(q_{10}=P(Y_2=1\mid R=10)\) and \(q_{00}=P(Y_2=1\mid R=00)\). The observed data do not determine either quantity, but each must lie between 0 and 1. Therefore

E(Y_2)=0.24+0.25q_{10}+0.15q_{00}.

If both missing-pattern probabilities are set to 0, the mean is \(0.24+0.25(0)+0.15(0)=0.24\). If both are set to 1, it is \(0.24+0.25(1)+0.15(1)=0.64\). Both choices agree with the same observed pattern frequencies and pattern-11 outcomes, yet yield different overall means. Under only the stated information, the mean can range from 0.24 to 0.64. This is an identification range, not a confidence interval: it describes compatible values under the unrestricted model, rather than sampling uncertainty around one identified value.

Justification by Observational Equivalence

The identification claim can be justified by constructing two full-data laws with the same observed-data distribution but different values for a target. Keep the pattern probabilities and all observed outcome distributions fixed. In pattern 10, keep \(f(Y_1\mid R=10)\) fixed, but choose two different conditional distributions for \(Y_2\mid Y_1,R=10\). In pattern 00, choose two different joint distributions for \((Y_1,Y_2)\). The observed data in patterns 10 and 00 contain no values of \(Y_2\), and pattern 00 contains no outcome values at all. Thus these choices change no observable probabilities. If they change a target, that target cannot be uniquely recovered from the observed-data law.

Hold the observed law fixed.
Retain the same pattern probabilities, the same \(f_{11}\), and the same observed margin \(f(Y_1\mid R=10)\).

Vary only unobserved components.
Change \(f(Y_2\mid Y_1,R=10)\) or \(f_{00}(y_1,y_2)\), leaving all observed components unchanged.

Compare the target.
If the overall distribution or a summary such as \(E(Y_2)\) changes, multiple full-data laws fit the same observed law. The target is therefore not identified without a restriction that rules out some of those laws.

This argument establishes nonidentification; it does not say that every target is unidentified. Pattern probabilities and distributions of observed outcomes are identified. Nor does nonidentification mean that estimation software cannot return a number. A number can be produced by imposing modeling choices or restrictions, but it is then not determined by the observed data alone.

Diagnostics, Alternatives, and Limits

The most useful diagnostic is an observation-pattern inventory: for each pattern, list which outcome components are observed, then compare that list with every distribution required for the target analysis. A fitted model that reports a pattern-specific distribution for unobserved outcomes is using some extrapolation, constraint, or prior information, whether or not that choice is prominent in the output. Inspecting the model specification and data contribution by pattern helps reveal which quantities are supplied by observations and which depend on assumptions.

Keep structural identification separate from finite-sample instability. A pattern may have positive probability and its observed distribution may be identified, yet a small number of participants can make its estimate noisy or unstable. Conversely, a very large sample cannot identify a distributional component that contributes no observed outcomes. If a pattern has probability zero in the population, its conditional distribution is not defined as a population conditional distribution and its mixture weight is zero; it does not contribute to the population mixture.

A pattern-mixture model makes the unknown outcome distributions within patterns explicit. A selection model instead expresses missingness probabilities conditional on outcomes. These are alternative parameterizations of the relationship between outcomes and missingness; changing parameterization does not make unobserved outcomes automatically identified. The identifying information still comes from observed data plus restrictions or assumptions. The earlier material on missing-data concepts provides background terminology, but the accounting here concerns the specific components of a two-visit pattern-mixture model.

The component count also has limits. It assumes an unrestricted distribution over the stated categories; parametric families or shared-parameter structures may reduce the number of free parameters, but that reduction comes from model assumptions, not from additional observed outcomes. Likewise, a restriction may identify a target or selected components without identifying every feature of every pattern-specific distribution. The relevant question is always: which unknown components does this particular target depend on, and what information or restriction determines each one?

Common Mistake

Common mistake: “The pattern frequencies are known, and we observe \(Y_1\) for pattern 10, so weighting the pattern-specific results identifies the overall distribution.”

Correction: The weights identify how much each pattern contributes, not the outcomes missing within that pattern. Pattern 10 supplies only its \(Y_1\) margin, and pattern 00 supplies no outcome distribution. A weighted overall distribution still depends on those unknown components. State which quantities are observed and which are imposed or extrapolated before interpreting a weighted result.

Check Your Understanding

Answer each question by distinguishing observed quantities from missing pattern-specific distributions. Brief solutions follow.

With patterns 11, 10, and 00, which full pattern-specific distribution is identified directly from observed outcomes?
For binary \(Y_1\) and \(Y_2\), how many free probabilities in the pattern-10 joint distribution remain unidentified after observing its \(Y_1\) margin?
Why do the observed pattern proportions alone not identify the overall mean of a binary \(Y_2\)?
If pattern 00 has probability zero, does its conditional outcome distribution contribute to the population mixture?

Solutions. (1) \(f_{11}(y_1,y_2)\), because both outcomes are observed in that pattern. (2) Two: one conditional probability \(P(Y_2=1\mid Y_1=y_1,R=10)\) for each of the two \(Y_1\) categories. (3) The proportions supply mixture weights, but the mean also depends on \(P(Y_2=1\mid R=10)\) and \(P(Y_2=1\mid R=00)\), which the outcomes in those patterns do not reveal. (4) No. A zero-weight pattern contributes nothing to the mixture, and its population conditional distribution is not defined when the pattern has probability zero.