Why Compare These Representations?
In a longitudinal study, the follow-up outcomes that are missing cannot be checked against their predictions. That makes it important to distinguish two questions: does a model describe the observed data adequately, and what does it assume about outcomes after dropout? Selection and pattern-mixture models organize the joint distribution differently, so restrictions that seem natural in one representation can imply a particular extrapolation in the other.
The comparison here uses two outcome occasions. Let \(Y_1\) be the outcome at the first occasion, \(Y_2\) the follow-up outcome, and \(R=1\) indicate that \(Y_2\) is observed; \(R=0\) indicates dropout before that outcome is recorded. Assume \(Y_1\) is observed for everyone. This is a simple monotone-dropout setting. All numerical examples are hypothetical.
These are both factorizations of the same joint distribution when left unrestricted. In practice, models impose restrictions: for example, a particular form for the outcome-dependent response probability in a selection model, or a particular relationship between respondents' and dropouts' outcome distributions in a pattern-mixture model. The restrictions, not the names of the factorizations alone, determine the extrapolated predictions.
What the Observed Data Do—and Do Not—Determine
Because \(Y_1\) is observed for everyone, the observed data can describe its distribution and the probability of follow-up observation conditional on \(Y_1\). Among those with \(R=1\), they also reveal the distribution of \(Y_2\) conditional on \(Y_1\) and response. They do not reveal \(f(y_2\mid y_1,R=0)\), the follow-up outcome distribution among dropouts.
A selection model expresses the missing part through \(P(R\mid y_1,y_2)\). If this probability depends on the unobserved \(Y_2\), its dependence cannot generally be learned from observed follow-up outcomes alone. A pattern-mixture model instead expresses the missing part through \(f(y_2\mid y_1,R=0)\). That distribution also requires an assumption or sensitivity parameter. Thus, either representation needs an extrapolation restriction for an MNAR analysis; changing representation does not by itself solve the identification problem.
The connection follows from conditioning the selection representation on the observed history and response status. For values where the relevant probabilities and densities are positive,
This identity shows how outcome-dependent selection reweights the outcome distribution within each response pattern. Conversely, the pattern-specific distributions and the response-pattern probabilities determine the joint distribution. An unrestricted representation in either form can therefore be translated into the other. A restricted model in one form need not translate into the same restricted model in the other.
Worked Example: Translating Selection Probabilities Into a Dropout Prediction
Suppose a hypothetical study has one baseline stratum, so \(Y_1\) is held fixed. Follow-up outcome \(Y_2\), measured in points, takes values 40 or 60. Among 500 respondents, 200 have outcome 60 and 300 have outcome 40; 300 participants dropped out. Consider a selection model that assumes \(P(R=1\mid Y_2=60)=0.50\) and \(P(R=1\mid Y_2=40)=0.75\).
The model implies the numbers with each outcome before response selection by dividing the observed respondent counts by their assumed response probabilities:
The implied total is 800, comprising 500 respondents and 300 dropouts, consistent with the stated response count. The implied dropout counts are \(400-200=200\) with outcome 60 and \(400-300=100\) with outcome 40. Therefore, the selection model predicts a dropout mean of
The observed respondent mean is \(\{200(60)+300(40)\}/500=48.00\) points. Under this selection assumption, dropouts have a higher mean than respondents. The prediction follows from the assumed response probabilities: the model gives the high-outcome group a lower chance of being observed. It is not a conclusion verified by the observed outcomes of the dropouts.
Comparing Predictions on a Shared Observed-Data Basis
A useful comparison holds the observed-data components fixed: the same distribution of \(Y_1\), response rate, and observed follow-up outcomes among respondents. Then predictions for dropouts can differ only because the models extrapolate differently. This prevents a disagreement about missing outcomes from being confused with a difference in how the candidates fit information that is actually observed.
Worked Example: Same Observed Counts, Different Pattern-Mixture Prediction
Keep the data from the previous example: 200 respondents have \(Y_2=60\), 300 have \(Y_2=40\), and 300 participants dropped out. A pattern-mixture specification could predict 100 dropouts with outcome 60 and 200 with outcome 40. Its predicted dropout mean is
This is different from the selection model's 53.33-point prediction, although both preserve the same observed respondent counts and the same total number of dropouts. The pattern-mixture specification implies 300 people with outcome 60 and 500 with outcome 40 overall. Its response probabilities, as implied by these completed counts, are \(200/300\approx0.6667\) for outcome 60 and \(300/500=0.6000\) for outcome 40. These differ from the selection model's assumed probabilities of 0.50 and 0.75.
Thus the candidates cannot both describe the same full joint distribution, even though they agree on the observed summaries. The example does not establish which dropout prediction is correct. It demonstrates that the observed data alone do not choose between these extrapolations.
A continuous pattern-mixture sensitivity model often makes the extrapolation explicit by relating the missing-outcome mean to the observed-outcome mean. For example, within a baseline stratum, one might set the mean among dropouts to the respondent mean plus a specified shift \(\delta\), in outcome units. This is a modeling assumption, not a quantity estimated from the missing outcomes.
Worked Example: A Pattern-Mixture Mean Shift
Suppose, hypothetically, that 80 of 100 participants have observed follow-up, with respondent mean 52 points. The remaining 20 have missing \(Y_2\). Assume a pattern-mixture sensitivity specification in which the mean among dropouts is \(\delta=-6\) points relative to the respondent mean. The assumed dropout mean is \(52-6=46\) points. The corresponding mean for all 100 participants is
If instead \(\delta=0\), the dropout mean is 52 points and the overall mean is \(\{80(52)+20(52)\}/100=52.00\) points. The observed response rate and respondent outcomes are unchanged; the difference in overall means comes entirely from the alternative pattern-mixture assumptions. A shift is interpretable only relative to the specified conditioning variables and population. If the shift is meant to apply within \(Y_1\) strata, it should be applied and then averaged over the appropriate stratum distribution rather than treated as a universal marginal shift without justification.
Consequences for a Treatment Contrast
Different missing-outcome predictions can affect an estimand that combines observed and missing outcomes, including a treatment-group mean contrast. To compare candidates meaningfully, define the target population and use the same observed-data components in each arm. The following calculation illustrates point estimates under stated shifts; it does not include sampling uncertainty.
Worked Example: Treatment Contrast Under Two Extrapolations
In a hypothetical two-arm study, 80 of 100 participants assigned to treatment have observed follow-up, with respondent mean 54 points. In control, 90 of 100 have observed follow-up, with respondent mean 50 points. Consider first a pattern-mixture assumption that dropout means are 5 points below the respondent means in treatment and 2 points below in control.
The assumed missing means are \(54-5=49\) points for treatment and \(50-2=48\) points for control. The treatment mean for all 100 assigned participants is
The control mean is
The resulting treatment-minus-control contrast is \(53.00-49.80=3.20\) points. Under an alternative assumption, suppose treatment dropouts average 10 points below treatment respondents, while control dropouts average the respondent mean. The resulting means are \(\{80(54)+20(44)\}/100=52.00\) points and \(\{90(50)+10(50)\}/100=50.00\) points, giving a contrast of 2.00 points. Neither calculation identifies the appropriate shifts from observed outcomes. They show how assumptions about each arm's missing outcomes can change the estimated contrast.
Assumptions, Diagnostics, and Failure Modes
A comparison is interpretable only when the candidates target the same outcome, time point, population, and observed-data components. The assumptions also need to be stated at the level at which they operate. A response probability conditional on both \(Y_1\) and \(Y_2\) is not interchangeable with one conditional only on \(Y_1\); a marginal shift is not automatically the same as a shift conditional on baseline outcome.
Several checks can identify problems in the observable part of a candidate analysis:
- Check that fitted response rates reproduce the observed response pattern, overall and across relevant \(Y_1\) values or other observed covariates.
- Check whether the model reproduces the observed \(Y_2\) distribution among respondents, including its relationship with \(Y_1\).
- Verify that implied pattern-specific probabilities are valid and that selection probabilities are between zero and one. Extreme or nearly zero response probabilities can make implied completed-data counts highly unstable.
- Compare dropout predictions across a prespecified range of plausible assumptions, and report how conclusions change. A single selected extrapolation can conceal important uncertainty.
These checks assess compatibility with observed information and mathematical coherence. They cannot confirm an MNAR assumption about \(Y_2\) for participants whose \(Y_2\) is missing. Good observed-data fit is not evidence that one dropout prediction is true. If a model fails to reproduce observed response rates or respondent outcomes, its extrapolation is difficult to defend; if it passes, the unobserved part still needs sensitivity analysis.
| Feature | Selection representation | Pattern-mixture representation |
|---|---|---|
| Primary factorization | Outcome distribution times response probability conditional on outcomes | Response-pattern probability times outcome distribution conditional on pattern |
| Where the extrapolation appears | In how response probability depends on unobserved outcomes | In the assumed outcome distribution among dropouts |
| Useful question to ask | How does the assumed selection process reweight outcomes? | How do dropout outcomes differ from respondent outcomes? |
| Key limitation | Outcome-dependent response parameters are not generally identified by observed outcomes alone | The missing-pattern outcome distribution is not observed and needs an assumption |
A common mistake is to say, “The selection and pattern-mixture models fit the observed data equally well, so their predictions for dropouts should agree.” That does not follow. The observed likelihood can leave the missing-outcome distribution unrestricted by the data, while candidate restrictions impose different extrapolations. The correction is to compare the predictions explicitly, holding the observed-data components fixed and stating the assumptions that generate each prediction.
A second pitfall is to read the pattern-mixture factorization as \(P(R\mid y_1)f(y_1,y_2\mid R)\). These factors use incompatible conditioning structures and do not generally form the joint distribution. The valid chain-rule forms are \(P(R)f(y_1,y_2\mid R)\) and \(f(y_1)P(R\mid y_1)f(y_2\mid y_1,R)\).
Check Your Understanding
Answer each question, then compare with the solutions below.
- In the two-occasion setup, which follow-up outcome distribution is not directly observed?
- Among 40 respondents, 10 have outcome 70 and 30 have outcome 50. A selection model assumes response probabilities 0.50 at outcome 70 and 0.75 at outcome 50. What dropout counts and dropout mean does it imply?
- With 75% response, respondent mean 20 units, and a pattern-mixture shift of \(-4\) units, what overall mean follows from the shift assumption?
- Why can two models have the same observed-data fit but different dropout predictions?
Solutions. (1) The unobserved distribution is \(f(y_2\mid y_1,R=0)\), the follow-up outcome distribution among dropouts conditional on the first outcome. (2) The implied total counts are \(10/0.50=20\) at 70 and \(30/0.75=40\) at 50. Thus there are 10 missing at 70 and 10 missing at 50; the implied dropout mean is \(\{10(70)+10(50)\}/20=60\) outcome units. (3) The assumed dropout mean is \(20-4=16\), so the overall mean is \(0.75(20)+0.25(16)=19\) units. (4) Observed-data fit constrains response information and outcomes among respondents, but does not determine the outcomes of dropouts. Different restrictions on selection or on the dropout distribution can preserve the observed components while implying different missing-outcome distributions.