Pattern-Mixture Model

Definition

A pattern-mixture model (PMM) handles nonignorable, missing-not-at-random (MNAR) dropout in longitudinal studies by stratifying the observed-data distribution by the dropout pattern — the number of complete observations each subject provides — and then analyzing the joint model of outcome and pattern without specifying a separate dropout mechanism. In contrast to selection models (Heckman 1976; Diggle-Kenward 1994), which model the dropout probability conditional on the outcome, PMMs treat the pattern as a between-subjects factor and estimate pattern-specific trajectory parameters.

Key Ideas

• Pattern indicator: Let $d_i \in \{1, 2, \ldots, T\}$ denote the last observed time for subject $i$ (dropout pattern). The full data distribution factorizes as $f(\mathbf{y}_i, d_i) = f(\mathbf{y}_i | d_i) \cdot P(d_i)$, whereas a selection model factorizes as $f(d_i | \mathbf{y}_i) \cdot f(\mathbf{y}_i)$.
• Pattern-as-covariate: PMM within a mixed-effects framework enters $d_i$ as a between-subjects dummy covariate in $\mathbf{W}_i\boldsymbol{\alpha}$; the fixed-effects vector $\boldsymbol{\alpha}$ then includes pattern-specific intercepts and slopes.
• No dropout model required: Unlike selection models, PMM does not require specifying or estimating $P(d_i | \mathbf{y}_i)$, avoiding the need for exclusion restrictions or auxiliary predictors of dropout.
• Identification under MNAR: PMMs identify the within-pattern parameters from completers in each stratum; extrapolation to unobserved time points requires assumptions (e.g., parallel trends across patterns).
• MAR as special case: If pattern effects are zero ($\boldsymbol{\alpha}_\text{pattern} = \mathbf{0}$), the PMM reduces to a missing at random (MAR) model; a likelihood-ratio test of pattern effects diagnoses potential MNAR.

How It Works

1. Define patterns: Identify the distinct dropout times $d \in \{d_1, \ldots, d_K\}$ observed in the data; group subjects accordingly.
2. Augment the design matrix: Add $K-1$ dummy variables for patterns (or pattern × time interactions) to $\mathbf{W}_i$.
3. Fit mixed-effects regression model (MRM): Estimate the augmented mixed-effects model by marginal maximum likelihood estimator (MMLE); the pattern dummies absorb systematic differences in mean trajectory across dropout patterns.
4. Interpret pattern effects: Significant pattern × time interaction indicates that the trajectory shape differs between completers and early dropouts — a signal of potential nonignorable missing data.
5. Sensitivity: Compare inferences (treatment effects, slopes) with and without pattern terms; robustness across specifications strengthens conclusions.

Why It Matters

• Practical for MNAR: Avoids the strong functional-form assumptions of selection models (Heckman 1976) and joint random-effects models (Diggle-Kenward 1994).
• Transparent: Pattern effects are directly interpretable as stratum-specific mean trajectories, not latent selection processes.
• Nested test for MAR: A $\chi^2$ test of all pattern coefficients = 0 provides a formal diagnostic for ignorable dropout within the MRM framework.
• Applicable to binary/ordinal outcomes: PMM can be combined with Random Effects Probit or ordinal mixed-effects models (Hedeker-Gibbons 1994) in the same augmented-covariate approach.

Open Questions

• Extrapolation assumptions: Pattern-specific models are identified only within the observed range of each pattern; marginalizing over patterns to get overall treatment effects requires untestable assumptions about missing trajectories (Little 1993, 1994).
• Many patterns: With $T$ time points, up to $T$ dropout patterns may appear; sparse patterns require pooling (e.g., early vs. late dropout) or regularization.
• Selection vs. PMM: Neither model is falsifiable from observed data alone; both should be run as sensitivity analyses (Little-Rubin 1987; Little 1995).

Related

• Mixed-Effects Regression
• Random Effects Probit
• Selection Model
• Generalized Estimating Equations
• Donald Hedeker
• Robert D. Gibbons
• Gibbons-Hedeker (2000) Applications of Mixed-Effects Models in Biostatistics