Worked Three-Visit Gaussian Example

A Three-Visit Pattern-Mixture Calculation

A pattern-mixture model describes the outcome distribution separately for each missingness pattern, then combines those distributions using the proportions of people in the patterns. With three scheduled visits, this makes it possible to distinguish two different sources of information: observed outcomes within each pattern, and assumptions needed to describe outcomes after dropout. Keeping those sources separate is essential when calculating an overall mean.

Consider a hypothetical study with outcomes \(Y_1,Y_2,Y_3\) at three visits and monotone dropout: a participant who misses a visit does not return at a later visit. Let \(R=1\) denote dropout after visit 1, \(R=2\) dropout after visit 2, and \(R=3\) completion through visit 3. Thus, participants with \(R=r\) have outcomes observed through visit \(r\). This coding labels the patterns; it does not rank their outcomes or imply why dropout occurred.

For the Gaussian example, suppose the full outcome vector within pattern \(r\) follows \(Y\mid R=r\sim N_3(\mu_r,\Sigma_r)\), where \(Y=(Y_1,Y_2,Y_3)^\mathsf{T}\), \(\mu_r=(\mu_{r1},\mu_{r2},\mu_{r3})^\mathsf{T}\), and \(\Sigma_r\) is a positive-definite covariance matrix. Only the first \(r\) components are observed for pattern \(r\). The Gaussian specification describes the pattern-specific outcome distribution, but it does not by itself identify the unobserved components of \(\mu_r\) from observed data.

In the calculations below, observed components of each pattern mean are estimated by their pattern-specific sample means. For simplicity, suppose the covariance matrices are specified by a common positive-definite working covariance matrix; covariance estimation is not the target of this example. Any values assigned to mean components after dropout come from stated extrapolation restrictions, not from the unobserved outcomes. This distinction is substantive: the resulting population mean depends on those restrictions.

Worked Data and Pattern-Specific Means

The following hypothetical data contain 12 participants. The pattern counts are \(n_1=3\), \(n_2=4\), and \(n_3=5\). The observed outcome values are:

Weights and the Population Mean

For a sample representing the target population, the empirical pattern weights are the pattern counts divided by the total sample size. Here \(n=3+4+5=12\), so the weights are

They sum to one: \(0.2500+0.3333+0.4167=1.0000\). To calculate a population visit mean, a mean is needed for every pattern at that visit. At visit 1, all patterns have observed outcomes, so the empirical mixture mean can be calculated without an extrapolation for later outcomes.

Extrapolation Assumptions for Later Visits

At visits 2 and 3, not every pattern has observed outcomes. To show the full calculation, impose explicit mean restrictions for the missing components: in pattern \(R=1\), assume the mean increases by 1 outcome unit from visit 1 to visit 2 and by another 1 unit from visit 2 to visit 3; in pattern \(R=2\), assume the mean at visit 3 is 1 unit above the observed visit 2 mean. These are illustrative extrapolation assumptions, not findings established by the observed data. They give

The directly estimated components are the observed means calculated above. The components \(\hat{\mu}_{12}=11\), \(\hat{\mu}_{13}=12\), and \(\hat{\mu}_{23}=12\) are set by the stated restrictions. In an applied analysis, restrictions of this kind need substantive justification and should be examined for sensitivity.

Why the Mixture Formula Holds

The weighting step follows from conditioning on the mutually exclusive and exhaustive dropout patterns. For a visit \(j\), the law of total expectation gives

The patterns partition the population, so \(\sum_rP(R=r)=1\). The formula therefore averages the conditional means in proportion to the population sizes of their patterns. Replacing the population weights with \(n_r/n\), and the pattern means with estimates, gives the empirical plug-in estimate. This justification requires each pattern to be defined consistently and the weights to refer to the same target population as the means.

The formula does not say that observed participants in one pattern represent the missing outcomes in another. Instead, it says how to combine pattern-specific means once those means have been specified or estimated. When all components at a visit are observed within their patterns, the calculation uses observed data throughout. When some components are unobserved, the mixture formula remains valid, but the corresponding pattern means depend on modeling assumptions.

Sensitivity, Diagnostics, and Alternatives

The weights and observed means can be checked directly: counts must add to the total, weights must sum to one, and each displayed sample mean must use the observed values in its own pattern. For a Gaussian specification, the covariance matrix must be positive definite. Also examine whether fitted means and implied trajectories are plausible on the outcome scale. In very small patterns, sample means are imprecise and an unrestricted covariance matrix may be poorly estimated or not estimable; a parsimonious or externally specified covariance structure may be needed.

A useful sensitivity check changes an unobserved pattern mean while holding observed quantities and weights fixed. For example, increase the assumed visit 3 mean in pattern \(R=1\) from 12 to 16, leaving all other values unchanged. The visit 3 population mean changes by the pattern 1 weight times the four-unit increase: $$ \hat{\mu}_3^{\mathrm{new}}=12.8333+(0.2500)(16-12)=12.8333+1.0000=13.8333. $$ This shows exactly how much the population estimate depends on that extrapolation. It does not establish which assumed value is correct.

A common mistake is to calculate the visit 3 population mean using only the completers’ mean of 14, or to treat that value as the mean for everyone. That reasoning silently assumes that the other patterns have the same visit 3 mean as completers. The correction is to include every pattern with its weight and make explicit any assumptions used for unobserved pattern means. A second mistake is to call the imputed or restricted means observed-data estimates; label those components as assumption-dependent.

The main alternatives answer different questions. A complete-case mean uses only pattern \(R=3\) and does not, by itself, represent the full population. An available-case mean at a visit uses everyone observed at that visit, but its contributing pattern mix can differ across visits. A selection model instead represents the outcome distribution together with a model for missingness; it is a different factorization, not a justification for omitting pattern-specific assumptions. Pattern-mixture calculations make the pattern weights and the pattern-specific outcome assumptions visible, but cannot identify unobserved outcomes without restrictions or additional information.

Results are especially fragile when a dropout pattern is empty or very small, when pattern membership is incorrectly coded, or when extrapolated means are implausible. If a pattern has zero weight in the target population, its mean does not affect that population mixture, although it may still matter for other targets. If all patterns have positive weight but a visit-specific mean is unidentified in one or more patterns, the overall mean is not identified from the observed data alone. Report the observed estimates, the extrapolation assumptions, and sensitivity results separately.