Pattern-by-Treatment Interactions

Why allow treatment contrasts to vary by dropout pattern?

A pattern-mixture analysis describes outcome distributions separately for dropout patterns and then relates those distributions to treatment assignment. A natural question is whether the treatment contrast is the same in every pattern. If it is not, a model that forces one common contrast can obscure real differences in the fitted pattern-specific means. The converse warning is equally important: a difference between patterns does not, by itself, show that dropout caused a treatment response to change.

This tutorial formulates that question for one prespecified outcome visit. The earlier material on Treatment Contrasts Within Dropout Patterns defines a contrast within a pattern; here the focus is how to represent, compare, and test those contrasts together. Fixing the visit keeps the interaction question separate from how mean outcomes change over time.

Definition: Let \(D\) denote dropout pattern, with categories \(1,\ldots,K\), and let \(A\) be treatment assignment, coded \(0\) for control and \(1\) for active treatment. At a prespecified visit \(t\), define \(\mu_{da}(t)=E(Y_t\mid D=d,A=a)\). The pattern-specific treatment contrast is \(\Delta_d(t)=\mu_{d1}(t)-\mu_{d0}(t)\). A pattern-by-treatment interaction is present in the mean structure when these contrasts are not all equal across patterns.

This definition concerns a contrast conditional on the observed dropout-pattern category. For patterns in which \(Y_t\) is missing, \(\mu_{da}(t)\) is not simply an observed cell mean: it must be estimated under a model that supplies information about the missing outcome. Thus a numerical interaction may be a model-based contrast, not a direct comparison of fully observed outcomes. The model and its assumptions determine what can be learned.

A reference-cell model and its interpretation

Choose pattern 1 as the reference category. For the fixed visit \(t\), a saturated mean model for the \(2K\) pattern-by-treatment cells can be written with indicator variables \(I(D=d)\):

E(Y_t\mid D,A) = \beta_0+ \sum_{d=2}^{K}\beta_d I(D=d) +\beta_A A +\sum_{d=2}^{K}\gamma_d I(D=d)A.

The coefficients have direct meanings. In pattern 1, the control mean is \(\beta_0\), and the treatment mean is \(\beta_0+\beta_A\), so \(\Delta_1(t)=\beta_A\). In pattern \(d\ne1\), the control mean is \(\beta_0+\beta_d\) and the treatment mean is \(\beta_0+\beta_d+\beta_A+\gamma_d\). Therefore:

\Delta_d(t)=\beta_A+\gamma_d, \qquad \gamma_d=\Delta_d(t)-\Delta_1(t).

The interaction coefficient \(\gamma_d\) is not the treatment contrast in pattern \(d\); it is the difference between that contrast and the reference-pattern contrast. The omnibus null hypothesis of no pattern-by-treatment interaction is \(H_0:\gamma_2=\cdots=\gamma_K=0\). A nonzero coefficient identifies a difference from the reference pattern, while the omnibus test asks whether any of the contrasts differ.

Worked Example: Translating Cell Means Into Interaction Coefficients

Suppose hypothetical model-estimated means at a prespecified visit, measured in symptom-score points, are as follows. Assume these means come from a correctly specified pattern-specific model; the table alone does not establish how missing outcomes were identified.

Dropout pattern	Control mean	Treatment mean	Treatment contrast
1	10 points	12 points	2 points
2	8 points	11 points	3 points
3	6 points	10 points	4 points

The contrasts are treatment minus control: \(12-10=2\), \(11-8=3\), and \(10-6=4\) points. With pattern 1 as the reference, \(\beta_0=10\) points and \(\beta_A=2\) points. The pattern-2 control mean is \(10+\beta_2=8\), so \(\beta_2=-2\) points; the pattern-3 control mean gives \(\beta_3=6-10=-4\) points.

The interactions are the differences in contrasts: \(\gamma_2=3-2=1\) point and \(\gamma_3=4-2=2\) points. For example, the fitted treatment mean in pattern 2 is \(10-2+2+1=11\) points. The treatment contrast increases by 1 point from pattern 1 to pattern 2, and by 2 points from pattern 1 to pattern 3. These descriptions are conditional on dropout pattern; they are not, without additional assumptions, causal effects in subgroups defined before treatment.

Testing whether the contrasts vary

Let \(\hat{\boldsymbol{\gamma}}=(\hat{\gamma}_2,\ldots,\hat{\gamma}_K)^\mathsf{T}\), and let \(\widehat{\mathrm{Var}}(\hat{\boldsymbol{\gamma}})\) be its estimated covariance matrix. A Wald test of the omnibus interaction null uses:

W= \hat{\boldsymbol{\gamma}}^\mathsf{T} \left\{\widehat{\mathrm{Var}}(\hat{\boldsymbol{\gamma}})\right\}^{-1} \hat{\boldsymbol{\gamma}}.

Under \(H_0\), a large-sample reference distribution is chi-squared with \(K-1\) degrees of freedom, provided the estimator is approximately normal and the covariance estimate is appropriate. In a finite-sample linear model, the reference distribution and degrees of freedom should instead follow the specified estimation procedure. This test concerns the collection of differences in contrasts; it is not a test that any one treatment contrast is zero.

Worked Example: An Omnibus Wald Test

Suppose a hypothetical fitted model has three dropout patterns and estimates \(\hat{\gamma}_2=1.0\) and \(\hat{\gamma}_3=2.0\) symptom-score points. Assume the estimated covariance matrix is \(\begin{pmatrix}0.25&0.10\\0.10&0.36\end{pmatrix}\) points squared, and that the large-sample Wald approximation is suitable. Test \(H_0:\gamma_2=\gamma_3=0\).

The determinant of the covariance matrix is \(0.25(0.36)-0.10^2=0.08\). Its inverse is \(\begin{pmatrix}4.50&-1.25\\-1.25&3.125\end{pmatrix}\) per point squared. Substituting the estimate vector gives:

W= \begin{pmatrix}1&2\end{pmatrix} \begin{pmatrix}4.50&-1.25\\-1.25&3.125\end{pmatrix} \begin{pmatrix}1\\2\end{pmatrix} =4.50-5.00+12.50=12.00.

There are \(K-1=2\) restrictions, so the reference distribution is \(\chi^2_2\). The upper-tail p-value is \(P(\chi^2_2\ge12.00)=0.00248\). At the 0.05 significance level, reject the null of equal treatment contrasts across patterns. Under the model and approximation assumptions, these data provide evidence that at least one pattern-specific contrast differs from the reference-pattern contrast. The test does not identify which pair differs, nor does it establish a causal explanation for the variation.

Focused comparisons and what they add

After an omnibus test, a focused comparison can describe a particular difference. For patterns 3 and 2, the difference in treatment contrasts is \(\Delta_3(t)-\Delta_2(t)=\gamma_3-\gamma_2\). Its estimated variance must account for covariance between the two estimated interaction coefficients. Ignoring that covariance can give an incorrect standard error.

Worked Example: Comparing Two Nonreference Patterns

Use the estimates and covariance matrix from the preceding hypothetical example. The estimated difference between pattern 3 and pattern 2 treatment contrasts is \(2.0-1.0=1.0\) point. Its variance and standard error are:

\widehat{\mathrm{Var}}(\hat{\gamma}_3-\hat{\gamma}_2) =0.36+0.25-2(0.10)=0.41, \qquad SE=\sqrt{0.41}=0.6403\text{ points}.

Using a normal critical value of 1.96, the 95% confidence interval is \(1.0\pm1.96(0.6403)\), or \((-0.255,\,2.255)\) points. Under the model and normal approximation, the interval describes plausible values for the difference in pattern-specific treatment contrasts; it includes zero. This does not contradict the omnibus result: the omnibus test combines evidence across two interaction dimensions, while this interval addresses only one contrast. If several pairwise comparisons are examined, multiplicity should be considered when making confirmatory claims.

Assumptions, diagnostics, and interpretation limits

The algebraic definition of the interaction requires only means and a contrast scale, but valid estimates and tests require more. The mean model must represent the relevant pattern-by-treatment cells, and the covariance estimate must reflect the dependence and variance structure of the fitted data. For observed outcomes analyzed with ordinary independent-cell regression, this typically entails independent participants and a suitable error model. If variances differ across cells, a method using a suitable robust or model-based covariance estimate may be needed. If some outcomes at \(t\) are missing within a pattern, estimates for those outcomes also rely on the assumptions and structure used to model the missing data. A fully unobserved cell cannot be recovered from the interaction formula alone.

Check the number of participants and outcome availability in each pattern-by-treatment cell, the scale and coding of treatment and pattern, and whether the fitted cell means reproduce the intended contrasts. Sparse or empty cells can make estimates unstable or nonidentifiable. Large standard errors, sensitivity to modest modeling changes, or a covariance matrix that is nearly singular are warnings that the interaction is weakly supported. For observed outcomes, inspect residual behavior and variance differences in relation to the chosen model; such checks cannot verify assumptions about outcomes that were never observed.

A key interpretation limit is that dropout pattern is generally determined after treatment assignment and may be related to outcomes. Conditioning on it can create selected groups whose composition differs across arms. Consequently, \(\Delta_d(t)\) is a pattern-conditional model contrast, not automatically a causal effect for a well-defined population. A significant interaction is evidence of modeled variation in conditional contrasts, not evidence that dropout itself modifies treatment response or that the same contrasts would apply under a different dropout process.

The interaction model is more flexible than a common-treatment-contrast model, which constrains every \(\gamma_d\) to zero and can be more precise if that constraint is appropriate. Testing coefficients one at a time is less suitable for the overall question because it does not jointly test all interaction restrictions. Comparing fitted cell means is useful for interpretation, but uncertainty must still be calculated from their joint covariance. The Wald test is convenient; its approximation may be poor with small or sparse samples, so the inferential method should match the fitted model and available sample size.

Common mistake: “The interaction is significant, so treatment has a different causal effect for people who would belong to each dropout pattern.” The test only establishes evidence against equality of the model-based contrasts conditional on observed pattern, under its assumptions. Because pattern is post-treatment and missing outcomes may require extrapolation, it does not alone establish causal effect modification.

Key takeaway: With a reference pattern, each interaction coefficient is the difference between that pattern’s treatment contrast and the reference pattern’s contrast. The omnibus test evaluates all such differences jointly; interpretation must retain the model-based and conditional nature of the comparison.

Check Your Understanding

Answer each question, then compare with the solutions below.

In a model with pattern 1 as reference, what does \(\gamma_3= -1.5\) mean, in context?
There are four dropout patterns. How many restrictions are tested by the omnibus no-interaction null?
Why can an empty treatment-by-pattern cell prevent an interaction contrast from being estimated directly from observed cell means?
A Wald test has \(W=5.99\) with 2 degrees of freedom. At \(\alpha=0.05\), what is the conclusion, approximately?

Solutions. 1. The treatment-minus-control contrast in pattern 3 is 1.5 outcome units lower than the corresponding contrast in pattern 1; it is not the pattern-3 contrast itself. 2. There are \(K-1=3\) restrictions, one for each nonreference pattern. 3. A cell mean needs outcome information for that treatment-pattern combination; without observations, the cell contrast is not determined by the observed means alone and would require additional modeling assumptions. 4. The upper-tail probability for \(\chi^2_2\) at 5.99 is approximately 0.050, just below 0.05 using the usual 5.991 critical value only if the statistic exceeds it; since 5.99 is slightly less than 5.991, do not reject at the exact 0.05 cutoff (the rounded p-value is about 0.050).