Selection and Pattern-Mixture Factorizations

Two Ways to Describe Outcomes and Missingness

When some components of a multivariate outcome are unobserved, a joint model must describe both the outcomes and which components are missing. There are two natural ways to factor this joint distribution. A selection model starts with the outcome distribution and models missingness conditional on outcomes. A pattern-mixture model starts with the probabilities of the missingness patterns and models outcomes conditional on pattern.

These are factorizations of the same joint law, not two different kinds of data. With unrestricted distributions, either can represent the same joint law as the other. In practice, however, analysts use restricted models or impose assumptions on quantities the observed data cannot determine. The factorization then makes those assumptions explicit in different ways.

Definition: Let \(Y=(Y_1,\ldots,Y_p)\) be the complete outcome vector and let \(R\) denote its missingness pattern. Write \(r\) for a possible pattern, \(Y_{\mathrm{obs}}(r)\) for the components observed under \(r\), and \(Y_{\mathrm{mis}}(r)\) for the remaining components. The selection factorization is \[ f(y,r)=f(y)\,P(R=r\mid Y=y), \] whereas the pattern-mixture factorization is \[ f(y,r)=P(R=r)\,f(y\mid R=r). \] These equations apply to densities or probability mass functions, with integrals replaced by sums for discrete outcomes.

The selection factorization separates the marginal distribution of complete outcomes from the rule determining the pattern. Its missingness model can depend on components of \(Y\) that are themselves unobserved. Such dependence is one way to express nonignorable missingness. The pattern-mixture factorization instead describes the outcome distribution separately within each pattern. It makes the distribution of outcomes among cases with missing components a distinct modeling target.

Equivalence of the Factorizations

The two representations are related by Bayes' rule. The conversion is exact when the relevant pattern has positive probability and the conditional distributions are defined on compatible supports.

Theorem: Suppose a joint law of \((Y,R)\) is given and \(P(R=r)>0\). Then \[ f(y\mid R=r)=\frac{f(y)P(R=r\mid Y=y)}{P(R=r)}. \] Conversely, given pattern probabilities \(p_r=P(R=r)\) and conditional densities or mass functions \(f_r(y)=f(y\mid R=r)\), define \[ f(y)=\sum_r p_r f_r(y),\qquad P(R=r\mid Y=y)=\frac{p_r f_r(y)}{\sum_s p_s f_s(y)} \] where the denominator is positive. These definitions give the same joint law \(f(y,r)=p_r f_r(y)\).

Proof. By the definition of conditional density or mass function, \(f(y\mid R=r)=f(y,r)/P(R=r)\). The selection factorization gives \(f(y,r)=f(y)P(R=r\mid Y=y)\), proving the first equation. For the converse, summing the proposed joint law over patterns gives the marginal \(f(y)=\sum_r p_r f_r(y)\). Dividing the joint mass or density \(p_r f_r(y)\) by this marginal yields the stated conditional probability of pattern \(r\) given \(y\), wherever the marginal is positive. Multiplication recovers \(p_r f_r(y)\), as required. \(\square\)

The theorem explains why choosing one factorization does not, by itself, settle whether missingness is ignorable. The substantive assumptions enter through restrictions on the factors. A selection model may specify a regression for \(P(R=r\mid Y=y)\), while a pattern-mixture model may specify distributions \(f_r\) across patterns. Restricted versions of these models need not correspond to one another: a convenient parametric family in one factorization may translate into a complicated or incompatible family in the other.

Worked Example: Converting a Binary Selection Model

Let \(Y\) be binary, with \(P(Y=1)=0.4\), and let \(R=1\) indicate that \(Y\) is observed. Suppose \(P(R=1\mid Y=1)=0.8\) and \(P(R=1\mid Y=0)=0.5\). Convert this selection model to pattern-mixture probabilities and distributions.

Joint probabilities. Multiplying each outcome probability by its response probability gives \(P(Y=1,R=1)=0.4(0.8)=0.32\) and \(P(Y=0,R=1)=0.6(0.5)=0.30\). For nonresponse, the corresponding probabilities are \(P(Y=1,R=0)=0.4(0.2)=0.08\) and \(P(Y=0,R=0)=0.6(0.5)=0.30\). They sum to \(1\).

Pattern probabilities. Thus \(P(R=1)=0.32+0.30=0.62\), and \(P(R=0)=0.08+0.30=0.38\).

Outcome distributions within patterns. Among respondents, \(P(Y=1\mid R=1)=0.32/0.62=16/31\). Among nonrespondents, \(P(Y=1\mid R=0)=0.08/0.38=4/19\). These pattern probabilities and conditional distributions reproduce all four joint probabilities. The selection probabilities differ by the value of \(Y\), including a value that would be missing when \(R=0\); the model is nonignorable.

What the Observed Data Determine

For a pattern \(r\), the observed outcome distribution is obtained by integrating the pattern-specific complete-data distribution over components missing under \(r\). In density notation, it is \[ g_r(y_{\mathrm{obs}})=\int f_r(y_{\mathrm{obs}},y_{\mathrm{mis}})\,dy_{\mathrm{mis}}. \] The observed-data contribution for an individual with that pattern is the pattern probability \(p_r\) multiplied by this marginal density. In the selection representation, the same contribution is obtained by integrating \(f(y)P(R=r\mid Y=y)\) over the missing components. The two expressions agree because they are marginals of the same joint distribution.

This yields a key limitation: the observed data reveal pattern probabilities and distributions of outcomes actually observed within each pattern, but they do not generally reveal the distribution of the missing outcomes within that pattern. A model or sensitivity assumption is needed to supply that information. This is the source of nonidentification, not a defect of one particular factorization.

Result: For a fixed pattern \(r\), observed data identify the marginal distribution \(g_r\) of \(Y_{\mathrm{obs}}(r)\), not in general the full conditional distribution \(f_r\) of \((Y_{\mathrm{obs}}(r),Y_{\mathrm{mis}}(r))\).

Justification. The observed outcomes in pattern \(r\) have density \(g_r\), obtained by integrating \(f_r\) over \(y_{\mathrm{mis}}\). Integrating removes the information about how probability is allocated across missing values. For example, for binary \(Y\) that is always missing in pattern \(r\), the observed data in that pattern contain no \(Y\) value at all, so any conditional probability \(P(Y=1\mid R=r)\) produces the same observations in that pattern. Thus the full \(f_r\) cannot generally be recovered from \(g_r\) alone. \(\square\)

Worked Example: A Pattern-Mixture Sensitivity Parameter

Suppose \(Y\) is binary, \(R=1\) indicates it is observed, \(P(R=1)=0.7\), and the observed respondents have \(P(Y=1\mid R=1)=0.6\). Let \(q=P(Y=1\mid R=0)\), which the observed data do not determine. Compare \(q=0.2,\ 0.5,\) and \(0.8\).

The pattern-mixture factorization gives \(P(Y=1,R=1)=0.7(0.6)=0.42\) and \(P(Y=0,R=1)=0.7(0.4)=0.28\). For nonrespondents the joint probabilities are \(0.3q\) and \(0.3(1-q)\). Therefore the population prevalence is \(P(Y=1)=0.42+0.3q\), which equals \(0.48,\ 0.57,\) and \(0.66\) for the three choices of \(q\), respectively.

The same choices can be expressed in selection form. For \(q=0.2\), \(P(R=1\mid Y=1)=0.42/0.48=0.875\), while \(P(R=1\mid Y=0)=0.28/0.52\approx0.538\). For \(q=0.5\), the two probabilities are \(0.42/0.57\approx0.737\) and \(0.28/0.43\approx0.651\). For \(q=0.8\), they are \(0.42/0.66\approx0.636\) and \(0.28/0.34\approx0.824\).

All three models have the same response rate and the same observed respondent distribution. They differ in the unobserved nonrespondent distribution, and consequently in the population prevalence and the selection probabilities. Choosing \(q\) is a sensitivity assumption, not a quantity learned from these observed data.

Multivariate Patterns and Nonignorability

With a vector of outcomes, patterns specify which coordinates are observed. For example, with \(Y=(Y_1,Y_2)\), pattern \(r=(1,0)\) means that \(Y_1\) is observed and \(Y_2\) is missing. A pattern-mixture model must describe the joint distribution of both coordinates within that pattern, including the unobserved coordinate. A selection model instead describes the probability of each pattern as a function of the complete vector.

Worked Example: One Missing Coordinate in a Two-Outcome Study

Let \(Y_1,Y_2\) be binary. Suppose half the subjects have pattern \(R=(1,1)\), with \(Y_1\) and \(Y_2\) independent and probabilities \(P(Y_1=1\mid R=(1,1))=0.6\) and \(P(Y_2=1\mid R=(1,1))=0.7\). The other half have pattern \(R=(1,0)\), with conditionally independent outcomes, \(P(Y_1=1\mid R=(1,0))=0.6\), and \(P(Y_2=1\mid R=(1,0))=q=0.2\).

The observed distribution of \(Y_1\) is Bernoulli with probability \(0.6\) in both patterns, so its overall observed distribution is also Bernoulli with probability \(0.6\). In pattern \((1,1)\), the observed \(Y_2\) distribution has probability \(0.7\) of a one. There is no observed \(Y_2\) in pattern \((1,0)\), so the value \(q=0.2\) is an assumption about that pattern.

The marginal probability that \(Y_2=1\) is \(0.5(0.7)+0.5(0.2)=0.45\). The selection probability for pattern \((1,1)\), conditional on \(Y_2=1\), is \(P(R=(1,1)\mid Y_2=1)=0.5(0.7)/0.45=7/9\). Conditional on \(Y_2=0\), it is \(0.5(0.3)/0.55=3/11\). Thus the pattern probability depends on \(Y_2\), which is missing for pattern \((1,0)\). The pattern-mixture specification and the selection probabilities describe the same joint law, but foreground different parts of it.

Interpreting Assumptions and Avoiding a Common Mistake

In a selection model, missingness is ignorable in the familiar likelihood setting when its conditional mechanism depends only on observed components (together with the usual parameter-distinctness conditions). Dependence on unobserved components is nonignorable. In a pattern-mixture model, the corresponding concern appears as the need to specify or relate distributions for outcomes that are not observed in a given pattern. The two descriptions should be judged by their implied joint distributions and restrictions, not by their labels alone.

Selection perspective: specify how the probability of each pattern varies with complete outcomes. Nonignorability may be represented by dependence on missing outcomes.
Pattern-mixture perspective: specify pattern probabilities and outcome distributions within each pattern. Nonignorability is reflected in differences between pattern-specific distributions, including assumptions about unobserved components.
Observed-data information: use the observed pattern proportions and within-pattern observed outcome distributions; acknowledge that missing outcome distributions require additional assumptions.

Common mistake: “A pattern-mixture model is automatically nonignorable because it conditions outcomes on the missingness pattern.” Conditioning is simply part of the factorization and is valid for any joint law. A missingness mechanism can be ignorable even when the joint distribution is written in pattern-mixture form. The correction is to ask what restrictions are imposed: does the implied probability of a pattern depend on unobserved outcomes, or do the model assumptions require unsupported distinctions between missing and observed values?

A practical analysis should state which quantities are identified by observed data, which are modeled, and which are varied as sensitivity parameters. A selection parameter such as a slope for an unobserved outcome and a pattern-mixture parameter such as \(q\) can encode related departures from ignorability, but their numerical values are not interchangeable without specifying the full model and doing the conversion.

Check Your Understanding

Use the factorizations and examples above to check your understanding.

State the selection and pattern-mixture factorizations of the joint law of \(Y\) and \(R\).
Why does the pattern-mixture distribution \(f(y\mid R=r)\) generally contain quantities that cannot be estimated from the observed data in pattern \(r\)?
In the binary sensitivity example, which observed quantities remain fixed as \(q\) changes, and which population quantity changes?
For two outcomes, what does pattern \(R=(1,0)\) mean, and which outcome distribution within that pattern requires an assumption?
Why does writing a model in pattern-mixture form not, by itself, establish that missingness is nonignorable?