Two Factorizations of One Joint Law
A selection model and a pattern-mixture model organize the same joint distribution in different ways. A selection model starts with the complete outcome distribution and describes how response depends on the outcome. A pattern-mixture model starts with the response pattern and describes the outcome distribution within each pattern. The distinction matters because outcomes are observed only for respondents: different assumptions about the missing outcomes can agree perfectly on everything observed.
Let \(Y\) denote an outcome defined for every unit, including those whose outcome is unobserved, and let \(R=1\) indicate that \(Y\) is observed and \(R=0\) that it is missing. Write \(f(y)\) for the marginal distribution of \(Y\), \(\pi_r(y)=P(R=r\mid Y=y)\) for the response-pattern probabilities conditional on the outcome, and \(p_r=P(R=r)\) for the pattern probabilities. When \(p_r>0\), write \(f_r(y)\) for the distribution of \(Y\) conditional on \(R=r\).
The two factorizations are related by Bayes' rule. For two response patterns, the conversions are
In the reverse direction, a selection representation gives
For discrete outcomes, the integrals are replaced by sums. If covariates \(X\) are included, the same identities apply within each covariate value \(x\): replace \(f(y)\), \(p_r\), and \(f_r(y)\) by their conditional versions given \(X=x\). These are algebraic identities, not additional assumptions that make the missing outcomes observable.
What the Observed Data Fix
With \(Y\) observed only when \(R=1\), the observed-data law identifies the response proportion \(p_1\) and the respondent outcome distribution \(f_1(y)\), subject to the usual sampling and support conditions. It does not reveal \(f_0(y)\), the distribution of outcomes among nonrespondents. Consequently, choosing different valid \(f_0\) distributions while holding \(p_1\) and \(f_1\) fixed changes the implied complete-data distribution and selection probabilities without changing the observed-data law.
This is a representation-level expression of nonidentification, not a contradiction between two accounts of the data. The selection model needs the complete-data distribution and the outcome-dependent response probabilities; the pattern-mixture model needs the response-pattern probabilities and the within-pattern outcome distributions. The unobserved part can be specified in either language. The earlier material on observational equivalence and nonidentification establishes why observed-data agreement alone cannot validate that specification.
Worked Example: Two Binary-Outcome Extrapolations
Suppose a hypothetical sample has response rate \(p_1=0.60\), and \(40\%\) of respondents have \(Y=1\). Thus \(P(Y=1\mid R=1)=0.40\), while \(P(Y=0\mid R=1)=0.60\). Compare two pattern-mixture assumptions: \(P(Y=1\mid R=0)=0.20\) and \(P(Y=1\mid R=0)=0.80\).
Under the first assumption, the implied population prevalence is
The selection probabilities follow from Bayes' rule. The joint probability of \(Y=1,R=1\) is \(0.60(0.40)=0.24\), so \(P(R=1\mid Y=1)=0.24/0.32=0.750\). For \(Y=0\), the corresponding joint probability is \(0.60(0.60)=0.36\), and \(P(Y=0)=0.68\), so \(P(R=1\mid Y=0)=0.36/0.68=0.529\).
Under the second assumption, the prevalence is
Now \(P(R=1\mid Y=1)=0.24/0.56=0.429\), while \(P(R=1\mid Y=0)=0.36/0.44=0.818\). Both constructions retain the same response rate and the same \(40\%\) success rate among respondents. They therefore imply the same observed-data law, even though their implied population prevalences are \(0.32\) and \(0.56\), respectively. One implies a higher response probability for \(Y=1\); the other implies a lower one.
Why the Translation Works
The conversion is a direct consequence of the two ways to factor a joint probability. In the selection representation, the joint density or probability for outcome \(y\) and pattern \(r\) is \(f(y)\pi_r(y)\). In the pattern-mixture representation, it is \(p_r f_r(y)\). Equating these joint probabilities gives both the identity and its conditional-probability interpretation.
Choose pattern probabilities \(p_r\) summing to one and proper within-pattern distributions \(f_r\). The implied complete-data distribution is \(f(y)=\sum_r p_r f_r(y)\).
For any \(y\) with \(f(y)>0\), set \(\pi_r(y)=p_r f_r(y)/f(y)\). These probabilities are nonnegative and sum to one because \(\sum_r p_r f_r(y)=f(y)\).
Multiplying gives \(f(y)\pi_r(y)=p_r f_r(y)\), so the recovered selection representation reproduces every pattern-specific joint probability.
The reverse construction follows by marginalizing the selection joint law over \(y\): \(p_r=\int f(y)\pi_r(y)\,dy\). Dividing the joint law by \(p_r\) then gives \(f_r(y)=f(y)\pi_r(y)/p_r\). This division requires \(p_r>0\). If a pattern has probability zero, its conditional outcome distribution is not defined by the joint law, because there are no units in that pattern.
For continuous outcomes, the selection probability formula is defined wherever the marginal density \(f(y)\) is positive. Values assigned at points where \(f(y)=0\) do not affect the joint law. For a pattern-mixture construction to translate into a selection representation, its pattern-specific distributions must combine into a proper marginal distribution, and the resulting conditional probabilities must be well-defined on its support. These conditions ensure a valid probability model; they do not ensure that the chosen missing-outcome distribution is scientifically credible.
Worked Example: A Continuous Outcome and Implied Logistic Selection
Consider a hypothetical setting with response rate \(p_1=0.70\). Suppose \(Y\mid R=1\sim N(10,4)\), where the second normal parameter is the variance. First assume \(Y\mid R=0\sim N(8,4)\). The pattern-mixture mean for the full population is
To express this same law as a selection model, use
The first term is the log odds of response. Because both normal distributions have variance \(4\), their density ratio has log
Since \(\log(0.70/0.30)=0.8473\), the implied model is \(\operatorname{logit}P(R=1\mid Y=y)=0.5y-3.653\). It gives higher response probabilities at larger \(y\). If instead \(Y\mid R=0\sim N(12,4)\), the full-population mean becomes \(0.70(10)+0.30(12)=10.6\), and the density-ratio log is \(5.5-0.5y\). The implied logit is \(6.347-0.5y\), giving lower response probabilities at larger \(y\). In both cases the response rate and the outcome distribution among respondents are unchanged; only the assumed unobserved distribution, and therefore its selection representation, has changed.
Worked Example: Translating a Three-Category Outcome
Suppose a hypothetical dataset has \(p_1=p_0=0.50\), with respondent probabilities \(P(Y=0,1,2\mid R=1)=(0.20,0.50,0.30)\). Consider missing-outcome probabilities \((0.60,0.30,0.10)\). The implied marginal probabilities are
Thus the marginal mean is \(0(0.40)+1(0.40)+2(0.20)=0.80\). The selection probabilities for the three outcome categories are, respectively,
Now change only the missing-outcome probabilities to \((0.10,0.20,0.70)\). The marginal probabilities become \((0.15,0.35,0.50)\), giving mean \(0(0.15)+1(0.35)+2(0.50)=1.35\). The response probabilities become \((0.667,0.714,0.300)\), calculated as \(0.10/0.15\), \(0.25/0.35\), and \(0.15/0.50\). Each construction still has response rate \(0.50\) and the same respondent distribution. The category-specific selection probabilities change because the extrapolation for nonrespondents changed.
Interpretation, Diagnostics, and Limits
The useful diagnostic question is whether a proposed representation actually reproduces the observed quantities. For a pattern-mixture specification, check that the pattern probabilities sum to one, each within-pattern distribution is proper, and the observed-pattern probability and distribution match the data. For a selection specification, check that the response probabilities lie between zero and one and that integrating \(f(y)\pi_1(y)\) yields the observed response rate and respondent distribution. A failure of these checks is a mathematical or implementation problem, not evidence about the unobserved outcomes.
Passing the checks does not distinguish among compatible extrapolations. A selection model can fit the observed respondent data and response rate while its outcome-dependent response function remains unsupported by the observed outcomes among nonrespondents. In pattern-mixture form, the same issue appears as an unobserved \(f_0\). Good fit to the observed-data law is therefore not a test of which missing-outcome distribution is correct.
| Representation | Specifies directly | Unobserved part that needs assumptions |
|---|---|---|
| Selection | Complete-outcome distribution \(f(y)\) and \(P(R=1\mid Y=y)\) | Outcome-dependent response behavior where outcomes are missing |
| Pattern-mixture | Response-pattern probabilities and \(f_r(y)\) within each pattern | The outcome distribution \(f_0(y)\) among nonrespondents |
The representations are interchangeable for a specified joint law, but not necessarily for a particular parametric model. For example, a simple linear-logit restriction on selection probabilities may translate into a complicated pattern-mixture distribution; a simple pattern-mixture restriction may translate into a nonlinear selection function. Thus two model classes that use different functional-form restrictions need not contain the same joint laws. Equivalence of factorizations should not be mistaken for equivalence of every restricted model, estimation method, or prior specification.
A common mistake is to say, “The selection model fits the observed response rate and respondent outcomes, so it verifies its predicted outcomes among nonrespondents.” That reasoning is wrong: those observed quantities are shared by many complete-data laws. The correction is to separate an observed-data fit check from an assumption about the missing outcomes, state the latter explicitly, and examine how conclusions change under defensible alternatives. Neither factorization alone supplies information that was not observed.
Check Your Understanding
Answer each question, then compare with the solutions below.
- If \(p_1=0.80\), \(P(Y=1\mid R=1)=0.25\), and \(P(Y=1\mid R=0)=0.50\), what is the implied marginal prevalence?
- In a continuous-outcome selection representation, what does \(\pi_1(y)=p_1f_1(y)/f(y)\) mean?
- Why can two models with different response probabilities conditional on \(Y\) have identical observed-data laws?
- What condition is needed to define \(f_0(y)\) from a joint law by dividing by \(p_0\)?
Solutions. (1) \(P(Y=1)=0.80(0.25)+0.20(0.50)=0.20+0.10=0.30\). (2) It is the probability of response for outcome value \(y\), obtained by dividing the joint density of \(Y=y,R=1\) by the marginal density of \(Y=y\), wherever that marginal density is positive. (3) The observed-data law fixes the response rate and the respondent outcome distribution, but not the outcome distribution among nonrespondents; different choices for the latter induce different conditional response probabilities while preserving the observed quantities. (4) \(p_0>0\); otherwise there is no positive-probability missing pattern on which to define a conditional outcome distribution.