From Dropout Histories to a Complete Analysis
A pattern-mixture analysis organizes a longitudinal data set according to each participant’s observed follow-up history. The analysis then describes outcomes within those patterns and, when the target is observed for every pattern, combines pattern-specific results using the patterns’ proportions in the trial. The crucial discipline is to keep the target clear: averaging observed outcomes does not, by itself, recover outcomes that were never measured.
The hypothetical trial below has visits at baseline, week 4, and week 8. For a transparent worked analysis, the numbers are for one randomized arm; in a two-arm trial, the same pattern-specific workflow can be carried out within each arm before making the intended comparison. Let \(R\) denote a participant’s observed visit history, and let \(Y_t\) denote the outcome at visit \(t\). The outcome is measured in mmHg. Participants are independent, and the normal models used below are working distributional models for the observed outcome vectors within each pattern.
Pattern-specific results are conditional descriptions, not automatically full-population results. For a visit at which only some patterns have observations, averaging those observed outcomes estimates a mean among participants observed at that visit. It is not the full-population mean unless additional assumptions justify that interpretation.
Worked Example: Coding Follow-Up Patterns
Suppose 60 hypothetical participants have the following histories: 36 provide all three measurements; 14 provide baseline and week 4 but not week 8; and 10 provide baseline only. Define patterns \(A\), \(B\), and \(C\) by these histories, respectively.
The sample pattern proportions are \(\hat p_A=36/60=0.6000\), \(\hat p_B=14/60=0.2333\), and \(\hat p_C=10/60=0.1667\). They sum to \(1.0000\). Each participant contributes to exactly one pattern, so these proportions partition the trial sample.
Baseline is observed in all three patterns. Week 4 is observed in patterns \(A\) and \(B\), and week 8 only in pattern \(A\). This availability map determines which visit-specific summaries can be calculated by weighting across all patterns, and which summaries are restricted to participants with observed outcomes.
Fitting the Observed Pattern-Specific Distributions
For a normal model, let the observed outcome vector for a participant in pattern \(r\) have distribution \(N(\boldsymbol{\mu}_r,\boldsymbol{\Sigma}_r)\), restricted to the visits actually observed in that pattern. The likelihood uses the observed components; it does not supply measurements at missed visits. With \(n_r\) independent vectors, the maximum-likelihood mean is their sample mean vector, and the maximum-likelihood covariance matrix divides the sum of cross-product deviations by \(n_r\), not \(n_r-1\). This distinction matters because the usual sample covariance matrix is unbiased under the model, while the likelihood estimate uses divisor \(n_r\).
In the hypothetical data, the pattern-specific sample means, sample standard deviations, and baseline–week 4 sample covariances are as follows. The covariance is included only for patterns \(A\) and \(B\), which have both measurements.
| Pattern | n | Baseline mean (SD) | Week 4 mean (SD) | Baseline–week 4 sample covariance | Week 8 mean (SD) |
|---|---|---|---|---|---|
| A: all visits | 36 | 120 (10) | 110 (12) | 90 | 98 (14) |
| B: through week 4 | 14 | 118 (11) | 115 (13) | 100 | Not observed |
| C: baseline only | 10 | 122 (9) | Not observed | Not estimable | Not observed |
Worked Example: Estimating a Pattern-Specific Covariance Matrix
For pattern \(A\), \(n_A=36\). The observed baseline and week 4 sample covariance matrix is \(\begin{pmatrix}100&90\\90&144\end{pmatrix}\). Under the bivariate normal likelihood, the maximum-likelihood covariance estimate is this matrix multiplied by \((n_A-1)/n_A=35/36\):
For example, \(144(35/36)=140\), and \(90(35/36)=87.50\). The determinant is \(97.2222(140)-(87.50)^2\approx 5{,}954.86\ \text{mmHg}^4\), which is positive; with a positive diagonal, this symmetric matrix is positive definite. That is a necessary check for a valid bivariate normal covariance estimate.
For pattern \(B\), the corresponding sample covariance matrix is \(\begin{pmatrix}121&100\\100&169\end{pmatrix}\). Multiplication by \(13/14\) gives \(\begin{pmatrix}112.3571&92.8571\\92.8571&156.9286\end{pmatrix}\ \text{mmHg}^2\). Its determinant is \(10449(13/14)^2\approx9{,}009.60\ \text{mmHg}^4\), also positive. Pattern \(C\) supplies a baseline distribution but cannot supply a baseline–week 4 covariance, because week 4 was not observed there.
These calculations establish valid estimated covariance matrices for the observed two-visit vectors; they do not establish that normality is adequate. Inspect the marginal distributions and model residuals within each pattern, paying particular attention to skewness, outliers, changing spread, and influential observations. Check that estimated covariance matrices are positive definite, and investigate near-singular matrices or implausibly large correlations. Small patterns can make diagnostics and estimates unstable even when the calculations are correct.
Combining Pattern-Specific Results Carefully
When a visit outcome is observed in every pattern, the empirical pattern-weighted mean equals the ordinary mean over all trial participants at that visit. To see why, let \(n_r\) and \(\bar y_{tr}\) be the size and observed mean for pattern \(r\). If every participant has \(Y_t\), then the total is \(\sum_r n_r\bar y_{tr}\), and division by \(n=\sum_r n_r\) gives the combined mean. The same weighting is a direct way to display how patterns contribute to the overall observed summary.
Worked Example: Baseline Population Summary
Baseline is measured for all 60 participants. The pattern-weighted baseline mean is
This is the observed sample mean at baseline, not an extrapolation. A pooled sample variance can also be calculated by adding within-pattern and between-pattern sums of squares. The within-pattern sum is \(35(100)+13(121)+9(81)=5802\ \text{mmHg}^2\). The between-pattern sum is \(36(120-119.8667)^2+14(118-119.8667)^2+10(122-119.8667)^2\approx94.9333\ \text{mmHg}^2\). Therefore the pooled sample variance and standard deviation are
The denominator is \(59\), because this is the sample variance of 60 observed baseline values. The decomposition accounts for both variation within patterns and differences between pattern means.
Worked Example: Week 4 Available-Case Summary
At week 4, outcomes are observed for 36 participants in pattern \(A\) and 14 in pattern \(B\), but not for the 10 in pattern \(C\). Among the 50 observed participants, the mean is
This is an available-case mean. Its observed-pattern weights are \(36/50=0.72\) and \(14/50=0.28\), not the full-trial proportions \(36/60\) and \(14/60\). The sample variance among these 50 observed values includes within- and between-pattern sums of squares:
The degrees-of-freedom denominator is \(50-1=49\), and the within-pattern terms use \(n_r-1\) times each pattern’s sample variance. This variance describes the observed week 4 values; it is not a variance estimate for all 60 participants unless further assumptions support that target.
At week 8, the mean of 98 mmHg and standard deviation of 14 mmHg describe the 36 completers in pattern \(A\). They do not, on their own, estimate the week 8 mean for all 60 participants. The 24 week 8 outcomes in patterns \(B\) and \(C\) are missing, so a full-trial week 8 summary requires additional assumptions or information about those outcomes. Pattern weights alone cannot fill in the missing pattern-specific distributions.
Diagnostics, Alternatives, and Reporting
An end-to-end analysis should connect the data structure, fitted models, diagnostics, and the estimand reported. A practical sequence is:
Use observed histories to assign each participant once; confirm the pattern counts sum to the analyzed sample size.
State the distributional model and ensure each pattern’s likelihood uses only outcomes observed for that pattern.
Inspect pattern-specific distributions and residuals, verify covariance validity, and flag sparse patterns or unstable estimates.
Weight across all patterns only when the target is observed in all of them; otherwise label the result as available-case or pattern-specific.
Identify the participants represented by each estimate and state when missing outcomes require assumptions beyond the observed data.
Pattern-stratified fits make differences in observed outcome distributions visible and provide a direct basis for pattern-specific reporting. They are not the only possible analysis. Complete-case analysis uses only participants with complete follow-up, but can be inefficient and may not represent the full trial. Available-case summaries use all observed outcomes at a visit, but their composition can change across visits. A pooled longitudinal model can borrow information across observations under its model assumptions, while selection models represent the relationship between outcomes and observation through a different factorization. These alternatives answer related questions, but they do not make the same assumptions or necessarily target the same summary.
| Approach | What it summarizes | Key caution |
|---|---|---|
| Pattern-specific fit | Observed outcome distributions within each follow-up history | Small patterns can yield imprecise or unstable fits |
| Available-case summary | Participants observed at a particular visit | Not automatically a full-trial summary when some patterns are absent |
| Complete-case analysis | Participants observed at all specified visits | Excludes incomplete histories and can change the represented group |
| Pooled longitudinal model or selection model | A model-based summary under its own specification and assumptions | Do not treat model choice as a substitute for stating the target and assumptions |
The value of the end-to-end approach is not that pattern-mixture modeling automatically solves missing outcomes. It makes the observed histories, conditional distributions, and weighting explicit, so readers can distinguish what the data directly summarize from what depends on further modeling. Reporting should include pattern definitions and counts, the model and its diagnostics, units, pattern-specific estimates, the weights used for any combined estimate, and a clear label for available-case summaries.
Check Your Understanding
Use the hypothetical trial above. Suggested solutions follow each question.
- Why do the three pattern proportions sum to one? Solution: Each participant is assigned to exactly one of the mutually exclusive observed histories, so the counts sum to 60 and \(36/60+14/60+10/60=1\).
- What is the maximum-likelihood baseline variance estimate for pattern \(A\), given its sample standard deviation of 10 mmHg and \(n_A=36\)? Solution: The sample variance is \(100\ \text{mmHg}^2\). The normal likelihood estimate is \((35/36)100=97.2222\ \text{mmHg}^2\).
- Why is 111.4 mmHg not automatically the full-trial week 4 mean? Solution: It averages the 50 observed week 4 outcomes in patterns \(A\) and \(B\); pattern \(C\)'s 10 week 4 outcomes are unobserved. The result is an available-case mean, not a mean over all 60 measured outcomes.
- What should be reported for week 8 if no assumptions are made about missing outcomes in patterns \(B\) and \(C\)? Solution: Report the pattern-\(A\) completer summary, such as its mean of 98 mmHg, and identify it as applying to the 36 completers. Do not label it the full-trial week 8 mean.
- What is the pooled baseline sample standard deviation across all 60 participants? Solution: The combined sum of squares is \(5802+94.9333=5896.9333\ \text{mmHg}^2\). Dividing by 59 gives \(99.9480\ \text{mmHg}^2\); its square root is approximately \(9.9974\ \text{mmHg}\).