Concept Architecture
Concept
Theoretically, Pattern Mixture Model is a statistical modelling approach for analysing incomplete data by explicitly modelling the distribution of outcomes conditional on observed missing data patterns. It forms part of the likelihood-based framework for handling missing data, particularly when data may be Missing Not at Random (MNAR). Rather than modelling the missingness mechanism directly, the method stratifies observations according to missing data patterns and models each pattern separately before combining results across patterns.
Mathematically, Pattern Mixture Models decompose the joint distribution of the data and missingness indicators into the conditional distribution of the outcome given the missingness pattern and the marginal distribution of the missingness pattern. Estimation is typically performed using maximum likelihood or Bayesian methods, with identifying restrictions or sensitivity parameters introduced because the observed data alone generally do not identify the full MNAR model.
In practice, Pattern Mixture Models are primarily used in clinical trials and longitudinal health economic evaluations to assess the robustness of treatment effect estimates to different assumptions about missing outcome or cost data. They are widely employed in sensitivity analyses where departures from the Missing At Random assumption are considered plausible.
Purpose
Used to model incomplete data by estimating outcome distributions within distinct missing data patterns, allowing sensitivity analyses under Missing Not at Random assumptions and improving the assessment of uncertainty arising from missing observations.
Mathematical Formulae
Primary Formula
f(Y, R) = f(Y | R) ? P(R)
where:
- Y = outcome data
- R = missingness pattern indicator
Supporting Formulae
For pattern r:
f(Y) = ? P(R = r) ? f(Y | R = r)
Likelihood:
L(?) = ? f(Yi | Ri; ?) ? P(Ri)
Related Mathematical Methods
- Maximum Likelihood Estimation
- Bayesian Estimation
- Sensitivity Analysis
- Multiple Imputation
- Selection Models
- Shared Parameter Models
- Longitudinal Mixed Models
Example
A health economic evaluation alongside a clinical trial measures quality-adjusted life years (QALYs) over 24 months. Participants are classified into three missing data patterns: complete follow-up, withdrawal after 12 months, and withdrawal after 6 months. Separate regression models estimate mean QALYs within each pattern. Overall estimates are obtained by weighting each pattern according to its observed frequency, with sensitivity analyses applying alternative assumptions to the unobserved outcomes of participants who withdrew early.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| IF | =IF(B2="",1,0) | Identify missing observations and assign missingness patterns. |
| COUNTIFS | =COUNTIFS(PatternRange,"Pattern 1") | Count observations within each missing data pattern. |
| AVERAGEIFS | =AVERAGEIFS(QALYRange,PatternRange,"Pattern 1") | Estimate mean outcomes within each pattern. |
| SUMPRODUCT | =SUMPRODUCT(PatternProbabilities,PatternMeans) | Calculate weighted overall outcome estimates across patterns. |
| LOG | =LOG(Likelihood) | Support likelihood-based estimation procedures where appropriate. |
VBA (Optional)
Automate classification of observations into missing data patterns and generate weighted summary estimates for predefined sensitivity scenarios.
Sources
- Little RJA. Pattern-Mixture Models for Multivariate Incomplete Data. Journal of the American Statistical Association. 1993.
- Little RJA, Rubin DB. Statistical Analysis with Missing Data. 3rd ed.
- Molenberghs G, Kenward MG. Missing Data in Clinical Studies.
- Carpenter JR, Kenward MG. Multiple Imputation and its Application.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- ISPOR Good Practice Reports on handling missing data in economic evaluation.
Related Concepts (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is a pattern-mixture model?
A missing data modelling approach stratifying analysis by the observed pattern of missingness, rather than assuming one model applies to everyone.
Source: Little 1993
How does a pattern-mixture model handle different groups of dropouts?
A pattern-mixture model handles missing data by splitting the sample according to the pattern of missingness, for instance those who completed the study against those who dropped out at various points, and analysing each group under its own assumptions before combining them. This lets the outcomes of people with different missing-data patterns be modelled differently, which suits data missing not at random, where dropouts may differ systematically from completers. Making explicit assumptions about each pattern is its approach. Stratifying by how data are missing is its method. Little and Rubin (2002) describe this.
Source: Little & Rubin 2002
How does a pattern-mixture model work?
A pattern-mixture model works by dividing the sample into groups defined by their pattern of missing data, specifying a model for the outcome within each pattern, and then averaging over the patterns to obtain overall estimates. Because the outcome for those with missing data cannot be identified from their own data alone, assumptions are made linking the patterns. So a pattern-mixture model works by modelling the outcome conditional on the missingness pattern and combining across patterns, with identifying assumptions bridging the observed and missing data, which makes the assumptions about how missingness relates to outcomes explicit rather than implicit, distinguishing it from approaches that assume missing at random.
Source: Little 1993
When is a pattern-mixture model used?
A pattern-mixture model is used when data may be missing not at random and the analyst wishes to make explicit, transparent assumptions about how those with missing data differ from those with complete data, often in sensitivity analyses. So a pattern-mixture model is used to handle missing not at random data and to examine the sensitivity of conclusions to assumptions about the missingness, which is valuable because the missing not at random possibility cannot be dismissed and standard methods assuming missing at random may be biased, making the pattern-mixture approach a way to explore how results depend on the assumed differences between missingness patterns.
Source: Little 1993
How does a pattern-mixture model relate to missing not at random data?
A pattern-mixture model relates to missing not at random data by providing a framework to model it: it groups individuals by missingness pattern and makes explicit assumptions about the outcomes in patterns with missing data, which can encode a dependence of missingness on the unobserved values. So a pattern-mixture model is well suited to missing not at random situations, since it allows the analyst to specify how those who drop out or have missing data differ, rather than assuming missing at random, and this makes it a common tool for sensitivity analyses that probe the impact of missing not at random assumptions on the conclusions.
Source: Little 1993
How does a pattern-mixture model differ from a selection model?
A pattern-mixture model and a selection model are two frameworks for missing not at random data that factor the joint distribution of the outcome and the missingness differently: the pattern-mixture model conditions the outcome on the missingness pattern, while the selection model models the missingness as depending on the outcome. So the two differ in how they decompose the relationship between outcomes and missingness, with the pattern-mixture model stratifying by pattern and the selection model modelling the probability of missingness given the values, and both make untestable assumptions, so they are used, often together, in sensitivity analyses to examine missing not at random possibilities from complementary angles.
Source: Little 1993
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 22 Dec 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-ES-SA-154
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