Tutorials › Pattern-Mixture and Selection Models › Marginal Treatment Comparisons From Pattern-Specific Fits

Home  /  Tutorials  /  Pattern-Mixture Models for Dropout

Pattern-Mixture Models for Dropout · Tutorial 223 of 499

Marginal Treatment Comparisons From Pattern-Specific Fits

Learn to combine pattern-specific fitted means and treatment-specific dropout-pattern proportions into a marginal treatment contrast, and interpret what that model-based comparison does—and does not—establish.

Advanced 11 min read

What You'll Learn

  • Define pattern-specific outcome means and treatment-arm pattern proportions.
  • Derive a marginal treatment mean using the law of total expectation.
  • Calculate a two-treatment contrast using treatment-specific pattern weights.
  • Separate within-pattern outcome differences from differences in pattern composition.
  • Diagnose incorrect weighting and clarify the assumptions behind model-based means.

From Pattern-Specific Fits to a Marginal Comparison

A fitted outcome mean within a dropout pattern answers a conditional question: what is the mean outcome for a treatment arm among people in that pattern? A treatment comparison is often intended to answer a population-level question instead: what is the mean outcome under each treatment in the whole treatment arm, and how far apart are those means? The bridge between the two is a weighted average. Each pattern-specific mean is weighted by the proportion of that treatment arm belonging to the pattern.

The weights generally differ between treatment arms. Dropout occurs after treatment assignment, so the distribution of dropout patterns can differ across arms. Using each arm’s own pattern proportions is essential when the target is the arm’s marginal mean under the observed treatment strategy. This calculation is a direct consequence of partitioning each arm into mutually exclusive patterns; it does not, by itself, make unobserved outcomes identifiable.

Definition: Let \(A\) denote treatment arm, with \(A=0\) for control and \(A=1\) for treatment. Let \(R\) denote dropout pattern, taking values \(r=1,\ldots,K\), and let \(Y\) be the endpoint of interest. For arm \(a\), define the pattern proportion \(\pi_{ar}=P(R=r\mid A=a)\) and the pattern-specific outcome mean \(\mu_{ar}=E(Y\mid A=a,R=r)\). The marginal mean for arm \(a\) is \(M_a=\sum_{r=1}^{K}\pi_{ar}\mu_{ar}\). The marginal treatment contrast, defined here as treatment minus control, is \(\Delta=M_1-M_0\).

The patterns must form an exhaustive, mutually exclusive classification of the individuals in each arm, so that \(\sum_r\pi_{ar}=1\). The endpoint and its scale must be the same for every pattern and both arms. For a continuous endpoint, for example, all \(\mu_{ar}\) must be means measured in the same units. If the scientific contrast is defined in the reverse direction, its sign is reversed; the weighting calculation itself is unchanged.

Why the Weighted Mean Is the Marginal Mean

Within arm \(a\), every individual has exactly one pattern. Decompose the mean by writing the indicator \(I(R=r)\), which equals one for pattern \(r\) and zero otherwise. Since the pattern indicators sum to one, the mean can be written as a sum of pattern-specific contributions:

$$ E(Y\mid A=a) =E\left[Y\sum_{r=1}^{K}I(R=r)\mid A=a\right] =\sum_{r=1}^{K}E\{YI(R=r)\mid A=a\}. $$

For any pattern \(r\), \(YI(R=r)\) is zero outside that pattern. Within the pattern, its conditional mean contribution is the pattern probability times the conditional outcome mean:

$$ E\{YI(R=r)\mid A=a\} =P(R=r\mid A=a)E(Y\mid A=a,R=r) =\pi_{ar}\mu_{ar}. $$

Summing these contributions gives \(M_a=\sum_r\pi_{ar}\mu_{ar}\). Applying the same result separately in the two arms gives the treatment contrast:

$$ \Delta =\sum_{r=1}^{K}\pi_{1r}\mu_{1r} -\sum_{r=1}^{K}\pi_{0r}\mu_{0r}. $$

In an analysis, the proportions and means are typically estimated. If \(\hat{\pi}_{ar}\) is the estimated proportion in pattern \(r\) in arm \(a\), and \(\hat{\mu}_{ar}\) is the fitted mean for that arm-pattern combination, then the plug-in estimate is \(\hat{M}_a=\sum_r\hat{\pi}_{ar}\hat{\mu}_{ar}\), and \(\hat{\Delta}=\hat{M}_1-\hat{M}_0\). The arithmetic is simple, but the interpretation depends on what the fitted means represent.

Worked Marginal Comparisons

Worked Example: Two Patterns and Two Treatment Arms

Hypothetical data. Suppose an endpoint is measured in points, with higher values indicating a better outcome. There are two patterns: complete follow-up and dropout before the endpoint. In the control arm, 80% of participants are in the complete-follow-up pattern and 20% are in the dropout pattern. The corresponding fitted means are 10 and 7 points. In the treatment arm, the proportions are 70% and 30%, and the fitted means are 12 and 8 points. Assume the fitted means represent the endpoint means for all members of their respective arm-pattern groups, including any unobserved endpoints predicted by the model.

The control marginal mean is

$$ \hat{M}_0=(0.80)(10)+(0.20)(7)=8.00+1.40=9.40\text{ points}. $$

The treatment marginal mean is

$$ \hat{M}_1=(0.70)(12)+(0.30)(8)=8.40+2.40=10.80\text{ points}. $$

Thus the estimated treatment-minus-control contrast is \(10.80-9.40=1.40\) points. Under these fitted pattern-specific means and pattern proportions, the treatment arm’s marginal mean is 1.40 points higher. This is a model-based comparison, not a claim that every participant benefits by 1.40 points.

Worked Example: Why Each Arm Needs Its Own Pattern Weights

Hypothetical data. Consider two patterns, \(R=1\) and \(R=2\), with endpoint means in points as follows: control means are 10 and 4, while treatment means are 12 and 7. Pattern proportions are \((0.80,0.20)\) in control and \((0.50,0.50)\) in treatment. Both sets of proportions sum to one.

Using the arm-specific proportions, the marginal means are

$$ \hat{M}_0=(0.80)(10)+(0.20)(4)=8.00+0.80=8.80, $$
$$ \hat{M}_1=(0.50)(12)+(0.50)(7)=6.00+3.50=9.50. $$

The marginal treatment contrast is \(9.50-8.80=0.70\) points. Although the fitted treatment mean is higher in both patterns, the treatment arm has a larger proportion in pattern 2, where the means are lower.

The contrast can be decomposed exactly, using control means as the reference for the pattern-composition component:

$$ \hat{M}_1-\hat{M}_0 =\sum_r\hat{\pi}_{1r}(\hat{\mu}_{1r}-\hat{\mu}_{0r}) +\sum_r(\hat{\pi}_{1r}-\hat{\pi}_{0r})\hat{\mu}_{0r}. $$

Here the within-pattern component, weighted by treatment proportions, is \((0.50)(12-10)+(0.50)(7-4)=1.00+1.50=2.50\) points. The composition component is \((0.50-0.80)(10)+(0.50-0.20)(4)=-3.00+1.20=-1.80\) points. Their sum is \(2.50-1.80=0.70\) points, matching the marginal contrast. This decomposition is descriptive and its allocation between components depends on the chosen reference means; the total contrast does not.

If, instead, both arms were weighted by the pooled proportions \((0.65,0.35)\), the means would be \(0.65(10)+0.35(4)=7.90\) for control and \(0.65(12)+0.35(7)=10.25\) for treatment, giving 2.35 points. That is a different, common-weight comparison, not the marginal contrast for the actual pattern composition of either arm.

Worked Example: Combining Three Pattern-Specific Fits

Hypothetical data. A trial defines three mutually exclusive patterns: endpoint observed at every visit, dropout after the first follow-up, and dropout before the first follow-up. The endpoint is a score in points. Control pattern proportions are \(0.60, 0.25, 0.15\), with fitted means \(20, 17, 14\). Treatment proportions are \(0.50, 0.30, 0.20\), with fitted means \(22, 18, 13\). Each arm’s proportions sum to one, and assume the fitted means target the endpoint for all people in each pattern.

For control,

$$ \hat{M}_0=(0.60)(20)+(0.25)(17)+(0.15)(14) =12.00+4.25+2.10=18.35\text{ points}. $$

For treatment,

$$ \hat{M}_1=(0.50)(22)+(0.30)(18)+(0.20)(13) =11.00+5.40+2.60=19.00\text{ points}. $$

Therefore, the treatment-minus-control marginal contrast is \(19.00-18.35=0.65\) points. The pattern-specific treatment differences are \(2\), \(1\), and \(-1\) points, respectively. The overall contrast is not their unweighted average, \((2+1-1)/3\); it reflects how common each pattern is in each treatment arm.

Assumptions, Interpretation, and Checks

The weighted identity is exact for well-defined conditional means and pattern proportions. Its use as an estimate requires additional care. Pattern proportions must refer to the target treatment arm and population, and the fitted means must correspond to those same arm-pattern groups. If an endpoint is missing for some members of a pattern, their outcomes are not revealed by the weighting formula. The model must supply fitted means that represent the target outcomes in those groups; the validity of those predictions depends on the model and its assumptions about unobserved outcomes.

For a causal interpretation of the arm contrast, treatment assignment must support the intended comparison—for example, in a randomized trial, the assigned strategies should be well-defined, consistency should hold, and there should be no interference between participants. In addition, missing endpoint values must be handled under defensible assumptions or an appropriate design. Randomization alone does not recover outcomes that were never observed. Without these conditions, the result should be described as a model-based marginal comparison between arms, not automatically as a causal treatment effect.

Several checks catch common implementation errors:

  • Verify that pattern categories are mutually exclusive and exhaustive and that proportions sum to one within each arm, allowing for rounding.
  • Confirm that each weight is calculated within the same arm as its corresponding fitted mean. Do not use the control pattern distribution to weight the treatment means when the target is each arm’s own marginal mean.
  • Use pattern-specific means for the same endpoint, population, and measurement scale. A fitted coefficient or a prediction at one covariate value is not automatically a pattern-level mean.
  • Check that a pattern with a positive estimated proportion has a defined fitted mean. An empty arm-pattern cell cannot provide an empirical pattern-specific mean without additional modeling.
  • Keep the contrast direction explicit. With treatment minus control, a positive result means a higher treatment-arm mean on the stated outcome scale.

A useful distinction is between a marginal comparison and a conditional one. Comparing \(\mu_{1r}\) with \(\mu_{0r}\) within each pattern describes treatment-arm differences conditional on pattern membership. It is not the same estimand as comparing \(M_1\) with \(M_0\), because the marginal comparison also incorporates each arm’s pattern composition. Treating the dropout pattern as if it were an ordinary baseline subgroup can therefore obscure the target: pattern membership occurs after treatment assignment and its distribution may itself differ by arm.

This calculation also differs from forcing a common pattern distribution across arms. Common weights answer a standardized comparison for a chosen shared pattern distribution; they can be useful for a different target, but they do not reproduce the two arm-specific population means defined here. If covariates are included in pattern-specific regressions, the fitted quantities must still be converted into means for the intended target population before combining them. The weighting formula does not determine that covariate averaging step.

Common mistake: “The treatment effect is the average of the pattern-specific treatment differences, so I will give every pattern equal weight.” This is wrong unless equal pattern weights are actually the intended target. The marginal contrast is the treatment-arm weighted sum of its pattern-specific means minus the corresponding control-arm weighted sum. The weights are the arm-specific pattern proportions, not automatically equal weights or pooled proportions.

Pattern-weighted summaries matter because they put fitted stratum-level results back onto an overall outcome scale. They also make clear which part of the comparison comes from fitted outcomes within patterns and which part is associated with different pattern distributions. The resulting estimate remains dependent on the quality of the pattern-specific fits, especially where dropout leaves outcomes unobserved; careful weighting cannot repair a misspecified or unsupported mean model.

Check Your Understanding

Answer each question, then compare your work with the solutions below.

  1. In control, two pattern proportions are \(0.75\) and \(0.25\), with means 8 and 4 points. In treatment, they are \(0.60\) and \(0.40\), with means 9 and 5 points. Calculate both marginal means and the treatment-minus-control contrast.
  2. Why is an unweighted average of the pattern-specific treatment differences generally not the marginal treatment contrast?
  3. Suppose one arm’s estimated pattern proportions sum to \(0.92\). Name one check to make before computing a weighted mean.
  4. Does randomization alone justify treating model-predicted means for missing outcomes as known true means? Explain briefly.

Solutions. (1) Control: \(0.75(8)+0.25(4)=6+1=7\) points. Treatment: \(0.60(9)+0.40(5)=5.4+2=7.4\) points. The contrast is \(7.4-7=0.4\) points. (2) Patterns need not be equally common, and their proportions may differ between arms; the marginal contrast uses each arm’s proportions. (3) Check for an omitted or misclassified pattern, an incorrect denominator, or a transcription error; only rounding would justify a small discrepancy. (4) No. Randomization supports the treatment comparison but does not reveal unobserved outcomes; predictions for missing outcomes require defensible modeling assumptions.

Previous

Next

Back to Tutorials
Clinical Biostats