Why Regressions May Differ Across Dropout Patterns
A baseline covariate can help explain why people with the same dropout pattern have different outcome values. For example, baseline symptom severity may be associated with symptom scores at later visits among participants who complete follow-up, and that association need not match the association among participants who drop out earlier. A pattern-mixture regression allows these relationships to differ rather than imposing one common regression across all patterns.
The regression is conditional on dropout pattern: it describes the outcome distribution among people in a specified pattern, given their baseline covariates. It is not automatically a causal model of what dropout does, nor does fitting the observed outcomes in a pattern by itself reveal the outcomes that would have been measured after participants dropped out.
The covariates in \(X_i\) are measured at baseline. A component can be centered or scaled to make the intercept and slope easier to interpret, but the units must be stated. For a single covariate \(x\), the model is \(E(Y_{it}\mid R_i=r,x)=\alpha_{rt}+\beta_{rt}x\). The intercept is the fitted mean at \(x=0\); the slope is the fitted change in mean outcome for a one-unit increase in \(x\), holding other covariates fixed when they are included.
These are conditional associations within a pattern. Because dropout pattern is determined after baseline, conditioning on it can select participants with different characteristics. A pattern-specific slope therefore should not be interpreted as a causal effect of changing a baseline covariate.
Specification and What the Data Can Estimate
For each pattern and visit, specify the mean relationship that is scientifically plausible and supported by the data. A simple model uses a linear term for each baseline covariate. If the intended relationship is not plausibly linear, a linear slope may be a poor summary; the model should not be treated as correct merely because it is convenient. The current focus is the linear regression specification, not nonlinear effects.
The observed-data constraint is essential. In a monotone dropout design, participants in a pattern are observed only through certain visits. A regression for \(Y_{it}\) within pattern \(r\) can be directly estimated from that pattern’s observed outcomes at visit \(t\). If \(Y_{it}\) is missing for every member of the pattern, its regression coefficients cannot be estimated from that pattern’s observed outcomes alone. Assigning coefficients to that unobserved component is a modeling or sensitivity assumption, not a result supplied by the observed data.
At visits where the outcome is observed, ordinary least squares estimates the coefficients by minimizing the sum of squared residuals among participants in the relevant pattern. With one covariate and an intercept, the fitted slope and intercept are
The sums here include only participants in pattern \(r\) with \(Y_{it}\) observed. The denominator must be positive: if everyone in the pattern has the same covariate value, the data cannot estimate a slope for that covariate. With several covariates, the corresponding design matrix must have sufficient rank to distinguish the coefficients.
A linear conditional mean is the main assumption for interpreting these coefficients as the intended mean relationships. Independent participants and adequate covariate variation are also important for ordinary regression estimation and inference. Outcomes measured repeatedly on the same participant are not independent of one another; standard errors for joint or across-visit conclusions must respect that dependence. The covariance structure is a separate part of the longitudinal model and does not change the definition of the pattern-specific conditional mean.
Worked Example: Interpreting Separate Slopes
Suppose a hypothetical two-visit study uses baseline severity \(x\), measured in points above the study’s baseline mean. At visit 2, the fitted regressions are \(E(Y_{i2}\mid R_i=A,x)=50+2.0x\) for completers and \(E(Y_{i2}\mid R_i=B,x)=44+1.5x\) for an earlier-dropout pattern. Outcome scores are in points.
For pattern A, the intercept \(50\) is the fitted visit-2 mean for a completer whose baseline severity is at the baseline mean (\(x=0\)). Its slope, \(2.0\), means that within pattern A, a one-point higher baseline severity is associated with a fitted visit-2 mean 2.0 points higher.
For pattern B, the corresponding fitted mean at \(x=0\) is 44 points, and the within-pattern slope is 1.5 outcome points per one-point increase in baseline severity. These are pattern-specific associations, not evidence that dropout caused a 6-point difference between patterns or that severity has different causal effects.
For a hypothetical participant with \(x=3\), the fitted means are $$ \widehat{Y}_{2,A}=50+2.0(3)=56\text{ points}, \qquad \widehat{Y}_{2,B}=44+1.5(3)=48.5\text{ points}. $$ The arithmetic gives two predictions for the same covariate value under two different conditional pattern models. The comparison is meaningful as a comparison of fitted pattern-specific means; it is not a treatment contrast.
Fitted Means With Several Baseline Covariates
For a covariate vector, a fitted mean is obtained by substituting the subject’s covariate values into the regression for the specified pattern and visit. Each coefficient multiplies its corresponding covariate, and the terms are added to the intercept. Centering and scaling should be applied consistently: if a model uses age in decades above 50, a participant aged 60 has a covariate value of 1, not 60.
Worked Example: Substituting a Covariate Vector
In a hypothetical dropout pattern \(r=C\), the fitted mean at visit 2 is \(E(Y_{i2}\mid R_i=C,s_i,a_i)=62+1.8s_i-0.35a_i\). Here \(s_i\) is baseline symptom severity in points above its mean, and \(a_i\) is age in decades above 50. The outcome is also measured in points. For a participant with \(s_i=4\) and age 60, \(a_i=1\).
Substitute both values: $$ \widehat{Y}_{2,C}=62+1.8(4)-0.35(1) =62+7.2-0.35 =68.85\text{ points}. $$
Holding age fixed, one additional point of baseline severity is associated with a fitted mean 1.8 outcome points higher within pattern C. Holding severity fixed, being one decade older corresponds to a fitted mean 0.35 outcome points lower. These interpretations describe the specified conditional mean model; they do not establish causal effects.
Short Derivation: Why Centering Changes the Intercept
Centering a covariate changes the reference value for the intercept but does not change fitted values when the model includes an intercept. For a single covariate, write \(z=x-c\), where \(c\) is a fixed centering constant. Starting with the regression \(\alpha+\beta x\), substitute \(x=z+c\):
Thus the centered model has intercept \(\alpha+\beta c\) and the same slope \(\beta\). At \(x=c\), the centered value is \(z=0\), so its intercept is the fitted mean at \(x=c\). This can make the intercept correspond to a typical participant rather than an arbitrary zero. The transformation must be done within the model’s stated covariate scale; it does not make an unsupported prediction reliable.
Worked Example: Estimating a Pattern-Specific Line From Data
Consider hypothetical visit-1 observations for three participants in pattern A and three in pattern B. The baseline covariate \(x\) is measured in points above its mean, and all six outcomes below are observed at visit 1.
| Pattern | Baseline \(x\) | Visit-1 outcome \(Y\) (points) |
|---|---|---|
| A | 0 | 10 |
| A | 1 | 12 |
| A | 2 | 14 |
| B | 0 | 8 |
| B | 1 | 9 |
| B | 2 | 10 |
For pattern A, \(\bar{x}_A=1\) and \(\bar{Y}_{1A}=12\). The slope numerator is \((0-1)(10-12)+(1-1)(12-12)+(2-1)(14-12)=2+0+2=4\); the denominator is \((0-1)^2+(1-1)^2+(2-1)^2=2\). Therefore \(\hat{\beta}_{A1}=4/2=2\), and \(\hat{\alpha}_{A1}=12-2(1)=10\). The fitted line is \(\widehat{Y}_{1,A}=10+2x\).
For pattern B, \(\bar{x}_B=1\) and \(\bar{Y}_{1B}=9\). The numerator is \((0-1)(8-9)+(1-1)(9-9)+(2-1)(10-9)=1+0+1=2\); the denominator is again 2. Thus \(\hat{\beta}_{B1}=2/2=1\), and \(\hat{\alpha}_{B1}=9-1(1)=8\). The fitted line is \(\widehat{Y}_{1,B}=8+x\).
At \(x=1.5\), the fitted means are \(10+2(1.5)=13\) points in pattern A and \(8+1.5=9.5\) points in pattern B. Each line is estimated from the observations in its own pattern. These tiny, perfectly linear hypothetical samples illustrate the calculation only; in practice, three observations provide little basis for assessing model fit or obtaining stable inference.
Diagnostics, Comparisons, and Limits
Pattern-specific fitting can reveal different baseline-outcome relationships, but it uses fewer participants per regression than a common model. Inspect pattern-specific sample sizes, covariate ranges, residual plots, and influential observations. A slope based on a narrow range of baseline values may be unstable. Predictions far outside that range are extrapolations, even when the fitted line produces a precise-looking number. If the estimated design matrix is rank-deficient, some coefficients cannot be separately estimated.
Check that the outcome is available at the visit being modeled for the participants used to fit that pattern’s regression. If the pattern has no observed outcomes at that visit, do not describe a coefficient as empirically estimated there. It may be specified as an assumption for an unobserved component, but the distinction should be explicit. Also check that covariates are coded in the units used in the interpretation and that fitted values are plausible on the outcome’s scale.
| Approach | What it allows | Main caution |
|---|---|---|
| Separate regression within each pattern | Intercepts and baseline-covariate slopes can differ by pattern. | More coefficients must be supported by each pattern’s data; sparse patterns can give unstable estimates. |
| One common regression across patterns | Uses a shared covariate relationship and can be more parsimonious. | Imposes common coefficients; it may not represent pattern-specific associations. |
| Pattern-specific intercepts without covariate slopes | Allows pattern means to differ without modeling within-pattern covariate gradients. | Does not adjust or describe how baseline covariates relate to outcomes within a pattern. |
The pattern-mixture formulation makes pattern membership part of the conditional description. It does not, by itself, identify missing outcomes after dropout. For a pattern’s unobserved visits, results depend on additional assumptions about those outcome distributions. Nor do separate regressions automatically yield a population-average regression: a population mean combines pattern-specific means using pattern weights, and those means can depend on covariates. The earlier material on pattern weights and marginal means provides that mixture perspective.
A common mistake is to fit a regression using all observed records and then claim its slope describes every dropout pattern. That pooled slope generally imposes or averages relationships across patterns; it is not the separate slope for each pattern. The correction is to define the target pattern and visit first, fit or specify that pattern’s regression, and report clearly whether the outcome component is observed or assumption-based. A second mistake is to interpret a pattern-specific slope causally. The correction is to call it a conditional association unless a separate causal design and assumptions justify a causal interpretation.
The practical value of these regressions is disciplined description: they permit baseline covariates to explain variation within each pattern, produce fitted conditional means for specified subjects, and make differences in covariate relationships explicit. Their limitations are equally important: patterns can be small, observed covariate support can be weak, and unobserved outcomes cannot be recovered solely by calculating a regression line.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- A pattern-specific model is \(\widehat{Y}=30+0.8x\), where \(x\) is baseline score in points above its mean and \(Y\) is measured in points. Interpret the slope and calculate the fitted mean for \(x=5\).
- Why can a regression for a visit after every member of a dropout pattern has left follow-up not be estimated directly from that pattern’s observed outcomes?
- A covariate is centered as \(z=x-10\), and the original model is \(20+3x\). Write the equivalent model using \(z\), and state its fitted mean at \(x=10\).
- Give one reason a prediction at a baseline covariate value far outside a pattern’s observed range is questionable.
Solutions. (1) Within the specified pattern, each one-point increase in baseline score is associated with a fitted mean 0.8 outcome points higher. At \(x=5\), the fitted mean is \(30+0.8(5)=34\) points. (2) There are no observed outcomes at that visit in the pattern to supply information about the regression coefficients; a value there requires additional modeling assumptions. (3) Since \(x=z+10\), \(20+3x=20+3(z+10)=50+3z\); at \(x=10\), \(z=0\) and the fitted mean is 50. (4) The prediction extrapolates beyond observed covariate support, where the assumed linear relationship may not hold and the data cannot check it.