Why Joint Symptom Outcomes Need a Joint Model
A hypothetical lupus trial records fatigue and pain scores at a scheduled follow-up, with larger scores indicating worse symptoms. Some participants have both scores, some have only one, and some have neither. A treatment comparison based only on participants with both scores can describe that observed subgroup, but it does not by itself describe the trial population. A pattern-mixture analysis makes the response pattern explicit and combines pattern-specific outcome distributions using the proportions in each pattern.
The two symptoms also matter jointly. Separate fatigue and pain means do not reveal how often participants have low scores on both outcomes, or how fatigue and pain vary together. The joint distribution therefore involves both pattern-specific means and the dependence between outcomes within each pattern.
Use four patterns: \(B\), both symptoms observed; \(F\), fatigue observed but pain missing; \(P\), pain observed but fatigue missing; and \(N\), neither observed. In this example, the pattern proportions are \(0.60, 0.15, 0.10,\) and \(0.15\), respectively, in each arm. Equal pattern proportions are a simplifying feature of the hypothetical data, not a requirement of the method.
The pattern-specific means below include observed means where a symptom is recorded and assumed means where it is missing. These missing-outcome means are explicit modeling assumptions, not measured facts. The table summarizes only the means; a complete joint model also specifies within-pattern variation and dependence.
| Pattern | Weight in each arm | New treatment: fatigue mean | New treatment: pain mean | Control: fatigue mean | Control: pain mean |
|---|---|---|---|---|---|
| \(B\): both observed | 0.60 | 3.50 | 3.50 | 4.00 | 4.00 |
| \(F\): fatigue only | 0.15 | 5.00 | 5.00 assumed | 6.00 | 6.00 assumed |
| \(P\): pain only | 0.10 | 4.75 assumed | 4.25 | 5.00 assumed | 5.00 |
| \(N\): neither observed | 0.15 | 6.50 assumed | 6.50 assumed | 7.00 assumed | 7.00 assumed |
From Pattern-Specific Means to Population Contrasts
For each arm and symptom, the pattern-mixture marginal mean is the weighted sum of its pattern-specific means. This calculation targets the mean under the specified full-data model for the randomized arm; it is not the mean among complete cases. The pattern weights should be estimated from the observed response patterns, while the missing-outcome distributions need defensible assumptions. For this worked calculation, treat the displayed weights and means as fixed hypothetical inputs.
Worked Example: Weighting the Fatigue and Pain Means
For the new-treatment arm, the fatigue mean is $$ 0.60(3.50)+0.15(5.00)+0.10(4.75)+0.15(6.50) =2.10+0.75+0.475+0.975=4.30. $$ The corresponding control fatigue mean is $$ 0.60(4.00)+0.15(6.00)+0.10(5.00)+0.15(7.00) =2.40+0.90+0.50+1.05=4.85. $$ Thus the treatment-minus-control fatigue contrast is \(4.30-4.85=-0.550\) score points.
For pain, the new-treatment mean is $$ 0.60(3.50)+0.15(5.00)+0.10(4.25)+0.15(6.50) =2.10+0.75+0.425+0.975=4.25. $$ The control pain mean is $$ 0.60(4.00)+0.15(6.00)+0.10(5.00)+0.15(7.00) =2.40+0.90+0.50+1.05=4.85. $$ The treatment-minus-control pain contrast is \(4.25-4.85=-0.600\) score points. Because lower scores indicate fewer symptoms, both point estimates favor the new treatment under the displayed assumptions: by 0.550 points for fatigue and 0.600 points for pain.
These are model-based point estimates, not evidence by themselves that the treatment effect is statistically distinguishable from zero. The calculation also does not establish that the assumed means for missing outcomes are correct.
A Sensitivity Shift for the Neither-Observed Pattern
A useful sensitivity analysis asks what happens if the assumed outcomes in pattern \(N\) are worse under new treatment and better under control than the initial assumptions suggest. Let \(\delta\) be a nonnegative shift in score points. In this illustration, add \(\delta\) to each pattern-\(N\) symptom mean in the new-treatment arm and subtract \(\delta\) from each pattern-\(N\) symptom mean in the control arm. All other pattern-specific means and weights are held fixed. This is a chosen sensitivity model, not a consequence of the observed data.
Because pattern \(N\) has weight \(0.15\) in each arm, shifting its mean changes the treatment-minus-control contrast by \(0.15\delta-(-0.15\delta)=0.30\delta\). Therefore the fatigue contrast becomes \(-0.550+0.30\delta\), while the pain contrast becomes \(-0.600+0.30\delta\). The distinct starting contrasts must be retained: the two symptoms do not have the same baseline contrast.
Worked Example: Finding the Symptom-Specific Tipping Points
A tipping point here is the value of \(\delta\) at which a model-based treatment-minus-control point contrast reaches zero. For fatigue, solve $$ -0.550+0.30\delta=0 \quad\Longrightarrow\quad \delta=\frac{0.550}{0.30}=1.833\text{ score points}. $$ For pain, solve $$ -0.600+0.30\delta=0 \quad\Longrightarrow\quad \delta=\frac{0.600}{0.30}=2.000\text{ score points}. $$ Thus, as \(\delta\) increases from zero, the fatigue point contrast reaches zero first, at about 1.833 points; the pain point contrast reaches zero at 2.000 points. At \(\delta=2\), fatigue has contrast \(-0.550+0.600=0.050\) points, and pain has contrast \(-0.600+0.600=0\) points. Under this sensitivity scenario, the point estimates no longer favor treatment for fatigue at \(\delta=2\), while the pain contrast is exactly zero.
The pattern-\(N\) contribution also illustrates why the direction and weight of a shift matter. At \(\delta=2\), the assumed pattern-\(N\) means are \(6.50+2=8.50\) for new-treatment participants and \(7.00-2=5.00\) for controls. Their weighted difference is $$ 0.15(8.50)-0.15(5.00)=1.275-0.750=0.525. $$ At \(\delta=0\), the same pattern contributed \(0.15(6.50)-0.15(7.00)=-0.075\). The change in its contribution is \(0.525-(-0.075)=0.600=0.30(2)\), consistent with the contrast formula.
Why the Joint Distribution Changes Joint Conclusions
The mean calculations above do not specify how fatigue and pain co-vary. A bivariate model might represent the two scores within each arm and pattern by a joint distribution with a mean vector and covariance matrix. For example, a bivariate normal specification requires a positive-semidefinite covariance matrix: both variances must be nonnegative and the absolute covariance cannot exceed the product of the standard deviations. The bivariate normal form is an assumption to check, not a feature automatically guaranteed by using a pattern-mixture model.
The importance of dependence can be seen without assuming normal scores. Consider the event that a participant has both a favorable fatigue score and a favorable pain score, defined here as each score being at or below a prespecified threshold. Suppose, hypothetically, that within pattern \(B\) the probability of a favorable fatigue score is \(0.60\) and the probability of a favorable pain score is also \(0.60\). Those marginal probabilities alone do not determine the probability of both.
Worked Example: Same Margins, Different Joint Probability
Under conditional independence in pattern \(B\), the probability of both favorable scores is \(0.60(0.60)=0.36\). Under a different joint distribution with probability \(0.50\) for both favorable, the two discordant cells can each have probability \(0.10\), and the probability of neither favorable is \(0.30\). The margins remain \(0.50+0.10=0.60\) for each symptom, but the joint probability is now \(0.50\), rather than \(0.36\).
For a population illustration, suppose the probability of both favorable outcomes is \(0.20\) in the remaining patterns combined. With pattern \(B\) weight \(0.60\), the overall probability is \(0.60(0.36)+0.40(0.20)=0.216+0.080=0.296\) under independence. Using the alternative pattern-\(B\) joint probability gives \(0.60(0.50)+0.40(0.20)=0.300+0.080=0.380\). These are different population probabilities despite identical symptom-specific margins in pattern \(B\). The example's probabilities are illustrative and are not calculated from the score means in the earlier table.
Assumptions, Diagnostics, and Alternatives
A defensible analysis needs more than arithmetic. The outcome scales and assessment time must be defined consistently across arms and patterns; treatment assignment and the intended population-level comparison must be clear. The response pattern must be recorded correctly, and its estimated frequency must represent the analysis population. Any distribution assigned to an unobserved symptom within a pattern is an assumption. In particular, the observed outcomes in a pattern do not generally identify the missing symptom's mean, variance, or association with the other symptom.
Before interpreting results, examine pattern counts by arm. Very small or empty patterns make pattern-specific estimates unstable or unavailable, and an assumed pattern weight of zero cannot support an estimate for that stratum. Check observed score ranges, distributions, and within-pattern associations; assess whether the chosen joint distribution is plausible for bounded or skewed symptom scales. For a bivariate normal model, inspect whether its implied tails and covariance are reasonable and confirm that each fitted covariance matrix is valid. Compare conclusions across plausible shifts and dependence assumptions, and report which inputs are observed and which are assumed.
The pattern-mixture approach is useful because it exposes the assumptions attached to each missingness pattern. It does not make missing outcomes empirically known, and a sensitivity result is not proof that one assumption is right. A selection model instead represents outcome and response jointly through a model for response conditional on outcomes. A complete-case analysis uses only participants with both outcomes observed and changes the target to that selected group unless additional conditions justify broader interpretation. These approaches answer related questions through different modeling choices; none can recover information absent from the data without assumptions.
Check Your Understanding
Use the definitions and hypothetical values in this tutorial. The solutions are included so that the arithmetic and interpretation can be checked.
- With the table's pattern weights and control pain means, calculate the weighted control pain mean.
- Under the stated sensitivity shift, what is the pain contrast at \(\delta=1\), and which direction does that point estimate favor?
- Why can two models with the same marginal probability of favorable fatigue and favorable pain have different probabilities of both being favorable?
- Name one diagnostic for the pattern-specific joint model and explain what a problem with it would imply.
Solutions. (1) \(0.60(4.00)+0.15(6.00)+0.10(5.00)+0.15(7.00)=2.40+0.90+0.50+1.05=4.85\) score points. (2) The pain contrast is \(-0.600+0.30(1)=-0.300\) score points; because lower scores are better, this point estimate favors the new treatment. (3) Marginal probabilities do not specify dependence: the joint probability depends on how often favorable fatigue and favorable pain occur in the same participants. (4) For example, check whether the estimated within-pattern covariance matrix is positive semidefinite; an invalid matrix cannot represent a valid joint distribution and signals a problem with the fitted model or estimates.