Likelihood for Pattern-Specific Normal Models

Why the Observed Subvector Matters

In a pattern-mixture analysis, outcomes are described conditionally on dropout pattern. For a three-visit study with monotone dropout, a participant may contribute all three outcomes, only the first two, or only the first. The outcome-density contribution to the likelihood must match the visits actually observed: a participant observed at visits 1 and 2 contributes a bivariate density, not a three-dimensional density with an invented value at visit 3.

The observed-data likelihood combines two pieces: the probability of each dropout pattern and the outcome density conditional on that pattern. This factorization is a joint-probability identity; it does not assert that dropout is independent of outcomes. The normal-model assumption is made for the conditional outcome distributions used in the likelihood.

The pattern-specific model need not impose the same mean or covariance across patterns. The notation \(\mu_{g,O_g}\) means retain the entries of \(\mu_g\) for observed visits; \(\Sigma_{g,O_gO_g}\) is the corresponding principal submatrix. For example, if a pattern observes visits 1 and 2, only the first two means and the \(2\times2\) covariance block enter that pattern’s outcome density.

Likelihood and Its Components

Assume subjects are independent, their pattern labels are observed, and the conditional observed subvectors follow the specified normal distributions. Let \(n_g\) be the number of subjects in pattern \(g\), with \(\sum_g n_g=N\). When pattern probabilities and conditional outcome parameters are unrestricted across patterns, the likelihood for the labelled observations is

Here \(\phi_q(y;\mu,\Sigma)\) is the \(q\)-dimensional normal density,

The first product is the pattern-probability contribution. The second is the product of conditional outcome densities. Their separation follows from \(P(G_i=g,Y_{i,O_g}=y)=P(G_i=g)f(y\mid G_i=g)\). It does not imply that pattern and outcome are independent: the conditional density is explicitly allowed to vary with \(g\).

If the data are summarized as a pattern-count vector alone, its multinomial likelihood includes the coefficient \(N!/\prod_g n_g!\). For the labelled observations, the pattern contribution is \(\prod_g\pi_g^{n_g}\), without that coefficient. The coefficient does not involve model parameters, so it does not change maximum-likelihood estimates. If mean or covariance parameters are constrained to be shared across patterns, the displayed factorization still describes the likelihood, but the outcome parameters no longer maximize independently pattern by pattern.

Estimating a Pattern-Specific Normal Distribution

For a particular pattern, temporarily write its \(n_g\) observed subvectors as \(y_1,\ldots,y_{n_g}\), each of dimension \(q_g\), and write its observed mean and covariance parameters as \(\mu\) and \(\Sigma\). Its log likelihood, omitting no parameter-dependent terms, is

For fixed positive-definite \(\Sigma\), differentiating with respect to \(\mu\) gives the maximizer \(\bar y_g=n_g^{-1}\sum_i y_i\). Define the residual scatter matrix \(S_g=\sum_i(y_i-\bar y_g)(y_i-\bar y_g)^\mathsf{T}\). The next result requires \(S_g\) to be positive definite; merely having at least one observation in the pattern is not enough.

The covariance claim can be justified directly. At \(\mu=\bar y_g\), put \(A=S_g/n_g\), which is positive definite under the stated condition. Up to constants, maximizing the log likelihood over \(\Sigma\) is equivalent to minimizing \(\log|\Sigma|+\operatorname{tr}(\Sigma^{-1}A)\). Let \(B=A^{-1/2}\Sigma A^{-1/2}\), with positive eigenvalues \(\lambda_1,\ldots,\lambda_{q_g}\). Then

For every \(\lambda>0\), \(\log\lambda+1/\lambda\geq1\), with equality only at \(\lambda=1\): its derivative is \((\lambda-1)/\lambda^2\), so its minimum occurs at 1. The expression is therefore minimized only when every eigenvalue is 1, or \(B=I\), which gives \(\Sigma=A=S_g/n_g\). This also establishes why the estimate is a maximum when the scatter is positive definite.

Observed Marginals, Not Imputed Full Vectors

A pattern’s likelihood contribution depends on its observed marginal distribution. It does not use the conditional distribution of its missing outcomes given the observed ones. Consequently, an observed-data likelihood may estimate means and covariances for observed visits in a pattern without identifying that pattern’s mean or covariance for visits never observed in that pattern. Such extrapolation requires additional structure or assumptions.

Pattern Counts and Diagnostics

When pattern probabilities are unrestricted and the outcome parameters are variation-independent of them, maximizing the pattern-count part gives \(\hat\pi_g=n_g/N\). For a pattern with no participants, this estimate is on the boundary at zero, and its conditional outcome distribution has no data contribution. The count estimate does not supply information about unobserved outcomes within that pattern.

Before interpreting fitted normal parameters, check that each pattern is coded consistently, the subvector has the intended visits and order, and every covariance matrix used in a density is positive definite. For a separately fitted \(q_g\)-dimensional covariance, \(n_g>q_g\) is necessary for positive-definite residual scatter, but not sufficient: the observed vectors must also span all \(q_g\) dimensions around their mean. A singular or nearly singular scatter matrix signals that the unrestricted covariance estimate does not exist or is numerically unstable.

A common mistake is to report \(S_g/n_g\) as the MLE whenever \(n_g>0\), then treat a singular result as an ordinary estimate with a minor caveat. That is incorrect. If \(S_g\) is singular, some residual direction has zero variation. Covariance eigenvalues in that direction can approach zero without increasing the residual quadratic penalty, while the normal-density determinant term makes the likelihood grow without bound. Thus there is no finite maximum over positive-definite covariance matrices. A constrained or regularized covariance model may be useful, but it changes the estimation problem and must be stated explicitly.

This likelihood also differs from complete-case analysis: it uses each pattern’s observed marginal density, rather than discarding everyone with incomplete outcomes. It differs from a shared-parameter normal model when pattern-specific distributions are left unrestricted; imposing common means or covariances couples the contributions and changes the estimates. Weighted estimating equations are another approach to pattern-specific means, but they are not this likelihood and do not make its normal-density assumptions disappear.

The main limitation is identification. The observed-data likelihood describes the distribution of what was observed within each recorded pattern, together with the pattern frequencies. It does not, without additional restrictions, identify the distributions of outcomes missing within those patterns. A well-computed likelihood is not by itself evidence that a normal model fits well, nor does it settle how pattern-specific distributions should be combined for a population-level target.