Gaussian Pattern-Specific Models

Why Specify a Distribution Within Each Dropout Pattern?

In a longitudinal study, people who stop providing outcomes at different visits form different dropout patterns. A pattern-mixture model represents the population by the pattern probabilities and the outcome distributions conditional on pattern. For continuous outcomes, a multivariate normal distribution is one possible specification for each pattern. It provides a compact description of the trajectory’s mean at each visit and the variability and dependence among visits.

The Gaussian assumption is about the distribution of the full outcome trajectory within a pattern, including visits that are missing for that pattern. This distinction matters: observed outcomes can inform the distribution’s observed margins, but they do not automatically reveal the unobserved parts of a dropout group’s trajectory. A normal model makes those parts mathematically explicit; it does not make them empirically identified.

The factorization into a pattern probability and a conditional outcome distribution is a representation of the joint distribution of outcomes and pattern. By itself, it does not assert that dropout is missing at random, nor does it determine the unobserved outcomes. The course’s earlier material on pattern-mixture factorization and observed-data contributions gives the general framework; here the focus is the Gaussian specification and its implications in a three-visit design.

What the Mean Vector and Covariance Matrix Say

For pattern \(g\), the component \(\mu_{gj}\) is the mean of visit \(j\)’s outcome among people in pattern \(g\). Its units are the outcome’s units. For example, if outcomes are a symptom score, \(\mu_{g2}\) is the average visit-2 score in pattern \(g\), including the model-defined value at visit 2 even if that visit is unobserved for some patterns.

The diagonal entry \(\sigma_{gjj}\) is the variance at visit \(j\) within pattern \(g\), in squared outcome units. An off-diagonal entry \(\sigma_{gjk}\) is the covariance between visits \(j\) and \(k\), also in squared outcome units. Positive covariance indicates that deviations above the pattern-specific mean at one visit tend to accompany deviations above the mean at the other; negative covariance indicates an opposite tendency. Covariance is scale-dependent, so the corresponding correlation, \(\sigma_{gjk}/\sqrt{\sigma_{gjj}\sigma_{gkk}}\), is often easier to compare across visit pairs.

A valid covariance matrix must be symmetric and positive definite. In particular, variances must be positive, correlations must lie between \(-1\) and \(1\), and every linear combination of the trajectory must have positive variance unless a singular model is explicitly intended. Positive definiteness is not merely a computational convenience: it is required for an ordinary nonsingular multivariate normal density.

Observed Data and the Gaussian Likelihood

Under monotone dropout, a participant in pattern \(g\) contributes the observed prefix \((Y_1,\ldots,Y_g)\). A marginal of a multivariate normal distribution is also multivariate normal. Consequently, the observed prefix in pattern \(g\) has mean vector \(\boldsymbol{\mu}_{g,1:g}\), the first \(g\) entries of \(\boldsymbol{\mu}_g\), and covariance matrix \(\boldsymbol{\Sigma}_{g,1:g;1:g}\), the leading \(g\times g\) block of \(\boldsymbol{\Sigma}_g\).

This gives the observed-data contribution for one participant with observed values \(\mathbf y_{1:g}\): $$ P(G=g)\,f(\mathbf y_{1:g}\mid G=g) =\pi_g\,\phi_g\!\left(\mathbf y_{1:g}; \boldsymbol{\mu}_{g,1:g},\boldsymbol{\Sigma}_{g,1:g;1:g}\right), $$ where \(\phi_g(\cdot;\boldsymbol{\mu},\boldsymbol{\Sigma})\) is the \(g\)-variate normal density. The omitted outcome components are integrated out; the contribution is not the density of a completed three-entry vector whose missing entries have been observed.

How Many Parameters Are There, and Which Are Identified?

An unrestricted three-entry Gaussian mean vector has three parameters. A symmetric \(3\times3\) covariance matrix has three variances and three distinct covariances, hence six parameters. Therefore each pattern’s full-trajectory distribution has nine parameters. Across three patterns there are \(3(9)=27\) outcome-distribution parameters. The three pattern probabilities add two free parameters because they sum to one, for a total of 29 nominal parameters.

The observed data do not identify all 29 parameters without additional restrictions or assumptions. In pattern 3, all three outcomes are observed, so its observed Gaussian distribution contains three mean and six covariance parameters. In pattern 2, only the two-dimensional observed marginal is directly informed: two means and three covariance entries, or five parameters. In pattern 1, only the univariate visit-1 marginal is directly informed: one mean and one variance, or two parameters. The pattern frequencies inform the two free pattern probabilities, provided the patterns occur in the data.

The remainder is concrete. For pattern 2, the observed data inform \(\mu_{21},\mu_{22}\) and the covariance entries for visits 1 and 2. They do not inform \(\mu_{23}\), \(\sigma_{213}\), \(\sigma_{223}\), or \(\sigma_{233}\): the visit-3 mean, its covariances with earlier visits, and its variance. For pattern 1, only \(\mu_{11}\) and \(\sigma_{111}\) are informed; the other seven entries of its full Gaussian distribution involve unobserved visits. The likelihood contribution depends on the observed leading block and not on these missing-visit entries. Positive-definiteness requirements constrain possible values, but do not select a unique completion.

Assumptions, Diagnostics, and Alternatives

A Gaussian pattern-specific analysis requires careful interpretation of its assumptions. The outcomes must be suitably modeled as continuous, and within each pattern the full trajectory is assumed to follow a multivariate normal distribution with a valid covariance matrix. The pattern definition and the visit ordering must correctly represent the monotone dropout process. For direct estimation of a pattern’s observed marginal, there must be participants in that pattern and enough observed data to support the parameters being estimated. Estimating a full covariance matrix from a small complete-pattern sample can be unstable even when the mathematical model is valid.

Diagnostics should examine observed outcomes within each pattern, not pretend that missing outcomes are available for checking. Useful warning signs include strong skewness or heavy tails, extreme observations that dominate estimated covariances, patterns with very small sample sizes, and sample covariance matrices that are nearly singular. Compare observed marginal distributions with the fitted Gaussian margins, and inspect whether fitted variances and correlations are plausible. Such checks can reveal lack of fit in observed portions; they cannot validate a model’s unobserved pattern-specific means or covariances.

The main alternative within a Gaussian formulation is to impose a shared outcome distribution or structured relationships across patterns, reducing the number of free parameters. Such constraints can make more parameters estimable, but they change the assumptions and may obscure genuine differences between patterns. Selection models use a different factorization, modeling outcomes and missingness through a selection mechanism. Neither representation is automatically correct: conclusions depend on the substantive assumptions and on how unobserved outcomes are handled.

The practical limitation is therefore not that Gaussian pattern-specific models are unusable, but that their full-data description can be more extensive than the observed evidence. A model can fit all observed margins well while implying very different outcomes after dropout. Report which parameters are estimated from observed data and which are fixed or constrained by assumptions; do not describe the latter as directly learned from the dropout group’s unseen outcomes.