Pattern Mixture Model

Pattern Mixture Model in Trial-Based Cost-Effectiveness Analysis

A pattern mixture model treats an incomplete outcome, such as a trial participant's total QALYs or NHS costs, as a weighted blend of distributions, one for each group of people who share the same pattern of observed and missing values. The weights are the shares of the sample in each pattern, and the distribution for people whose values are missing has to be supplied by an explicit assumption, because the data say nothing about it. That feature has made the model a favoured way of asking whether a trial result survives the possibility that people who stopped returning questionnaires were in worse health than their recorded data would predict, a situation described as missing not at random (MNAR). In trial-based economic evaluation the assumption is usually one number, an offset or a percentage by which missing values differ from those predicted under missing at random (MAR), and the analysis is repeated across plausible values of it. The other main framework is the selection model, and causes of gaps are treated on the missing data page.

Factorising the outcome by missing data pattern

Every model for incomplete data has to describe how the outcome and the indicator recording whether it was seen behave together. Little's 1993 paper in the Journal of the American Statistical Association set out two ways of splitting that joint distribution. A selection model specifies the outcome distribution for everyone and then the probability of being missing given the outcome, while a pattern mixture model specifies the outcome distribution among respondents and among nonrespondents, together with the marginal probability of each pattern.

$$f(Y, R \mid X) = f(Y \mid R, X)\, P(R \mid X)$$

where $Y$ is the outcome (for example QALYs over 12 months), $R$ is the missing data pattern ($R = 1$ if $Y$ is observed and $R = 0$ if not, in the simplest case), $X$ holds fully observed covariates such as treatment arm and baseline utility, $f(Y \mid R, X)$ is the outcome distribution within a pattern and $P(R \mid X)$ is the probability of that pattern. A selection model writes the same joint distribution as $f(Y \mid X)\, P(R \mid Y, X)$. Summing over patterns gives the distribution the analysis needs.

$$f(Y \mid X) = \sum_{r} P(R = r \mid X)\, f(Y \mid R = r, X)$$

Here the sum runs over every pattern $r$ in the data, so the population distribution is a mixture of the pattern-specific distributions. In longitudinal trials the patterns are often defined by the time of dropout, so completers and people who left after 3 or 6 months form separate strata. Little and Wang described the approach as stratifying incomplete data by the pattern of missing values and formulating distinct models within each stratum. Molenberghs and colleagues showed in 1998 that the taxonomy of missing completely at random (MCAR), MAR and MNAR, developed mainly within selection modelling, also applies to pattern mixture models.

Why the missing pattern needs an identifying assumption

Little called pattern mixture models chronically underidentified. With one incomplete outcome there are no data on its distribution in the stratum where it is missing, so the parameters for that stratum have to be fixed by restrictions or prior information. His complete-case restrictions tie them to their counterparts among complete cases, and other restrictions tie them to parameters in other missing value patterns.

The 2010 National Research Council (NRC) panel report on missing data in clinical trials gives the simplest case. With $\mu_1$ the mean among respondents, $\mu_0$ the mean among nonrespondents and $\pi$ the proportion responding, the full-data mean is a weighted average of the two, but the data hold no information about $\mu_0$. MAR here amounts to setting $\mu_0$ equal to $\mu_1$, an assumption the panel stresses is itself untestable.

Delta and scaling adjustments to imputed values

The NRC panel makes the link between the two strata explicit with a sensitivity parameter. Each value is a different unverifiable assumption.

$$\mu = \pi \mu_1 + (1-\pi)(\mu_1 + \Delta)$$

where $\mu$ is the full-data mean in one trial arm, $\pi$ is the proportion of that arm with an observed outcome, $\mu_1$ is the mean among those observed and $\Delta$ is the nonrespondents' mean minus the respondents' mean. A value of zero for $\Delta$ corresponds to MAR, and the panel suggests estimating $\mu$ over a range of values including zero, with a separate value allowed in each arm. For binary outcomes it suggests the log odds scale, which keeps the implied mean between 0 and 1.

Leurent and colleagues, in a 2018 PharmacoEconomics tutorial on trial-based cost-effectiveness analysis, set out a procedure that needs only standard software. Missing values are first filled by multiple imputation under MAR, each imputed value is then shifted by an offset $\delta$ or multiplied by a scale factor $c$, and each modified dataset is analysed as usual, with results combined by Rubin's rules. With $c = 0.9$, a participant who dropped out is assumed to have, on average, 10% lower quality of life than a similar participant who stayed. The tutorial notes that an additive offset is common for outcomes on an interpretable scale anchored at 0 and 1, such as EQ-5D utilities, and that a multiplicative change may be more intuitive for costs, where a fixed sum depends on context. Faria and colleagues had earlier presented the same shift of MAR-imputed costs and QALYs as a practical approximation to the pattern mixture model.

Choosing the shift and reporting the scenarios

Because the shift cannot be estimated from the data, its value has to come from outside the dataset. Leurent and colleagues describe formal elicitation, in which trial investigators, clinicians or patients give their beliefs about how the missing data differ, and the elicited distribution can be carried into the analysis by drawing a different shift for each imputed dataset. A tipping point analysis instead moves the parameter until the conclusion changes, and then asks experts whether that value is plausible, an application of threshold analysis. An intermediate route agrees a most likely scenario and several extreme ones with those involved in the trial, for example at a steering committee meeting. The tutorial favours prespecifying the choice before results are known and letting the parameter differ by arm, since the missing data mechanism need not be the same in both.

For reporting, the tutorial proposes a table of incremental cost, incremental QALYs, incremental net monetary benefit and probability of cost-effectiveness by scenario, alongside overlaid cost-effectiveness acceptability curves.

Worked example with differential dropout in a two-arm trial

All numbers are illustrative. A trial randomises 200 participants to each arm and measures QALYs over 12 months. In the intervention arm 160 complete follow-up ($\pi = 0.80$) with an observed mean of 0.720 QALYs, and in the control arm 180 complete ($\pi = 0.90$) with a mean of 0.700. The 40 and 20 dropouts had poorer baseline utility, so multiple imputation under MAR predicts means for them of 0.640 and 0.630. Costs are treated as MAR, with an incremental cost of GBP 100, and the threshold is an illustrative GBP 20,000 per QALY.

1. MAR analysis. The intervention arm mean is 0.80 × 0.720 + 0.20 × 0.640 = 0.704 and the control mean is 0.90 × 0.700 + 0.10 × 0.630 = 0.693. Incremental QALYs are 0.011, and incremental net monetary benefit is 20,000 × 0.011 = 220 less 100, or GBP 120.

2. Complete-case comparison. Completers alone give 0.720 less 0.700, or 0.020 incremental QALYs, and a net benefit of 20,000 × 0.020 = 400 less 100, or GBP 300. This assumes the dropouts resemble the completers, ignoring their poorer baseline health.

3. Same shift in both arms. Suppose dropouts in both arms are 0.05 QALYs worse than the MAR prediction. The intervention mean falls by 0.20 × 0.05 = 0.010 to 0.694 and the control mean by 0.10 × 0.05 = 0.005 to 0.688. Incremental QALYs fall to 0.006, and net benefit to 20,000 × 0.006 = 120 less 100, or GBP 20. The equal shift does not cancel, because twice the share of the intervention arm is imputed.

4. Shift in the intervention arm only. With the shift applied to intervention dropouts alone, the intervention mean is 0.694 and the control mean stays at 0.693. Incremental QALYs are 0.001, and net benefit is 20,000 × 0.001 = 20 less 100, or minus GBP 80, so the conclusion reverses.

5. Tipping points. Net benefit is zero when incremental QALYs equal 100 / 20,000 = 0.005, a fall of 0.006 from the MAR value. A shift in the intervention arm alone moves incremental QALYs by 0.20 times the shift, so the tipping point is a downward shift of 0.006 / 0.20 = 0.03 QALYs. An equal shift in both arms moves them by 0.20 less 0.10, or 0.10 times the shift, so the tipping point is a downward shift of 0.006 / 0.10 = 0.06 QALYs.

6. Interpretation. The intervention is cost-effective under MAR, but a shortfall of 0.03 QALYs among its 40 dropouts, about 5% of their MAR-predicted value of 0.640, would remove the advantage. Whether that is plausible is a clinical judgement about who left the trial and why. A real analysis would also report each scenario's confidence interval after Rubin's rules.

Evidence from the Ten Top Tips weight-loss trial

The Leurent tutorial applied the method to the Ten Top Tips (10TT) trial of a weight-loss intervention for obese adults in UK general practice. Of 537 participants, 313 (58%) completed the 2-year follow-up, and only 31% had complete quality of life and cost data. Under MAR, incremental net monetary benefit at GBP 20,000 per QALY was minus GBP 49, with a 48% probability of cost-effectiveness. Reducing imputed quality of life by 5% in the intervention arm alone moved net benefit to minus GBP 713 and the probability to 19%, while the same reduction in the control arm alone gave GBP 550 and 75%. Equal shifts in both arms kept the probability between 41% and 48%, and allowing missing costs to be up to 10% higher had only a marginal effect.

Pattern mixture models beside other missing data methods

Selection models express departures from MAR as a relation between the chance of being missing and the unobserved value. Leurent and colleagues note that their typical sensitivity parameter, a log odds ratio for how a unit change in the outcome affects the chance of being observed, is harder to elicit and communicate than a mean difference, and the NRC panel warns that selection models are highly sensitive to parametric assumptions about the full-data distribution. Leurent and colleagues treat shared parameter models, which link outcome and missingness through a latent variable, and reference-based imputation, which borrows the distribution of another trial arm, as special cases of the two main frameworks.

The European Medicines Agency guideline on missing data in confirmatory trials lists pattern mixture, selection and shared parameter models as approaches for investigating different MNAR scenarios. The NRC panel explains that last observation carried forward, sometimes mistakenly thought valid under MCAR or MAR, in general makes an MNAR assumption and attaches no uncertainty to the filled-in value, whereas a pattern mixture analysis states its departure from MAR and varies it.

Limits of pattern mixture sensitivity analysis

The shift parameter is an assumption, not an estimate. Leurent and colleagues stress that it is not derived from the data, and the NRC panel, describing one such model, notes that its fit to the observed data is unchanged across values of the sensitivity parameters, so goodness of fit cannot choose between scenarios.

Equal shifts in both arms are not automatically neutral. The tutorial notes that they tend to cancel when the missing data pattern is broadly similar across arms, and the worked example leaves a residual effect when one arm has twice the dropout rate of the other. The number of parameters also grows quickly, since separate shifts for costs and quality of life in each arm already give four, and binary and survival outcomes raise extra problems of model compatibility and elicitation.

Uptake has been low. A review by Leurent, Gomes and Carpenter of 52 trial-based cost-effectiveness analyses found that only 2 studies (4%) ran sensitivity analyses allowing for MNAR data, and both modified multiply imputed values with a simplified pattern mixture approach. The tutorial lists carrying such results into wider decision models as further research, so a model built on trial estimates can inherit a missing data assumption that its own sensitivity analysis never varies.

Sources

• Committee for Medicinal Products for Human Use. Guideline on missing data in confirmatory clinical trials. EMA/CPMP/EWP/1776/99 Rev. 1. European Medicines Agency; adopted June 2010, effective 1 January 2011. https://www.ema.europa.eu/en/documents/scientific-guideline/guideline-missing-data-confirmatory-clinical-trials_en.pdf
• Faria R, Gomes M, Epstein D, White IR. A guide to handling missing data in cost-effectiveness analysis conducted within randomised controlled trials. PharmacoEconomics. 2014;32(12):1157-1170. https://doi.org/10.1007/s40273-014-0193-3
• Leurent B, Gomes M, Carpenter JR. Missing data in trial-based cost-effectiveness analysis: an incomplete journey. Health Economics. 2018;27(6):1024-1040. https://doi.org/10.1002/hec.3654
• Leurent B, Gomes M, Faria R, Morris S, Grieve R, Carpenter JR. Sensitivity analysis for not-at-random missing data in trial-based cost-effectiveness analysis: a tutorial. PharmacoEconomics. 2018;36(8):889-901. https://doi.org/10.1007/s40273-018-0650-5
• Little RJA. Pattern-mixture models for multivariate incomplete data. Journal of the American Statistical Association. 1993;88(421):125-134. https://doi.org/10.1080/01621459.1993.10594302
• Little RJA, Wang Y. Pattern-mixture models for multivariate incomplete data with covariates. Biometrics. 1996;52(1):98-111. https://doi.org/10.2307/2533148
• Molenberghs G, Michiels B, Kenward MG, Diggle PJ. Monotone missing data and pattern-mixture models. Statistica Neerlandica. 1998;52(2):153-161. https://doi.org/10.1111/1467-9574.00075
• National Research Council. The Prevention and Treatment of Missing Data in Clinical Trials. Washington, DC: National Academies Press; 2010. Chapters 4 and 5. https://doi.org/10.17226/12955