Why Model Time Within Each Dropout Pattern?
In a pattern-mixture analysis, the outcome distribution is described separately within each missingness pattern, then related to the population through the pattern weights. For longitudinal outcomes, this means that the expected outcome can change over visit time differently across patterns. A single time trend imposed on everyone can conceal these differences; a pattern-specific trend makes them explicit.
The key limitation is that dropout determines which visits are observed within a pattern. A pattern that drops out after the first follow-up cannot, by itself, identify how its mean would have changed at later visits. A fitted curve may be evaluated beyond observed follow-up, but that value is an extrapolation requiring additional assumptions, not a pattern-specific empirical estimate.
For illustration, define pattern \(p=1\) as observed through visit 3, \(p=2\) as observed through visit 2 but not visit 3, and \(p=3\) as observed through visit 1 but not later visits. Let time \(t=0\) be baseline and let each unit of \(t\) represent one scheduled follow-up interval. These labels are only an example; a study should state its own pattern coding and time units.
Three Representations of Time
A linear model gives each pattern its own starting mean and rate of change:
Here, \(\beta_{0p}\) is the mean outcome in pattern \(p\) at the reference time \(t=0\), and \(\beta_{1p}\) is the expected change in outcome per time unit. The units of the slope are outcome units per time unit. This model assumes a constant mean change over time within that pattern.
A polynomial model allows curvature. A quadratic specification is
The coefficient \(\beta_{2p}\) describes curvature, not a constant increase per visit. The instantaneous rate of change is \(\beta_{1p}+2\beta_{2p}t\); for visits one unit apart, the model-implied change from \(t\) to \(t+1\) is \(\beta_{1p}+\beta_{2p}(2t+1)\). A cubic or higher-order polynomial adds terms, but extra flexibility also increases the number of coefficients to estimate.
A categorical-time model assigns a separate mean to each scheduled visit:
This saturated mean specification does not require equal changes between visits or a particular curve shape. Equivalently, with baseline as the reference, its visit effects are the differences \(\alpha_{pj}-\alpha_{p0}\). It uses more parameters than a linear trend, and a pattern needs observations at a visit to estimate its mean there from that pattern’s data.
In a combined regression representation, these models correspond to interactions between pattern indicators and the chosen time basis. For example, a pattern-specific linear model includes pattern-specific intercepts and slopes. This is closely related to the pattern-by-treatment interaction idea, with time rather than treatment distinguishing trajectories. The important practical point is that allowing pattern-specific coefficients avoids imposing equal time trends across dropout patterns.
Worked Longitudinal Comparison
Consider hypothetical fitted pattern-specific visit means for an outcome measured in points. The entries are available only at visits observed in that pattern. They are illustrative summaries, not data from a real study.
| Pattern | Observed visits | Fitted mean at \(t=0\) | Fitted mean at \(t=1\) | Fitted mean at \(t=2\) | Fitted mean at \(t=3\) |
|---|---|---|---|---|---|
| 1: observed through visit 3 | 0, 1, 2, 3 | 10 | 12 | 13 | 13 |
| 2: dropout after visit 2 | 0, 1, 2 | 10 | 11.5 | 12 | Not observed |
| 3: dropout after visit 1 | 0, 1 | 10 | 10.5 | Not observed | Not observed |
Worked Example: Calculating a Linear Trend for the Complete Pattern
For pattern 1, calculate the ordinary least-squares line through the four displayed means, using \(t=0,1,2,3\). This calculation treats each displayed visit mean equally; an individual-level analysis may weight information differently depending on its data and model.
The outcome values are \(10,12,13,13\). Their means are \(\bar{t}=(0+1+2+3)/4=1.5\) and \(\bar{x}=(10+12+13+13)/4=12\), where \(\bar{x}\) here denotes the mean of these four displayed outcome values. The least-squares slope is
The intercept is \(\hat{\beta}_{01}=\bar{x}-\hat{\beta}_{11}\bar{t}=12-(1)(1.5)=10.5\). Thus the fitted linear trend is \(\hat{\mu}_1(t)=10.5+t\), in points. It predicts 10.5, 11.5, 12.5, and 13.5 points at times 0 through 3. Those predictions summarize an average increase of 1 point per interval; they do not reproduce every displayed mean, and the model’s predicted value at baseline is not constrained to equal the observed mean of 10.
Worked Example: Describing Curvature With a Quadratic Trend
For pattern 1, the four displayed means are exactly represented by \(\mu_1(t)=10+2.5t-0.5t^2\). Check the predictions: at \(t=0\), \(10+2.5(0)-0.5(0)^2=10\); at \(t=1\), \(10+2.5-0.5=12\); at \(t=2\), \(10+5-2=13\); and at \(t=3\), \(10+7.5-4.5=13\).
The first-interval change is \(12-10=2\) points, the second is \(13-12=1\) point, and the third is \(13-13=0\) points. The quadratic coefficient is \(-0.5\) points per squared time unit, indicating a declining rate of increase over these visits. It does not mean the outcome itself decreases by half a point each interval.
For pattern 2, the three displayed means can be represented by the quadratic \(10+1.75t-0.25t^2\): the predictions are 10, \(10+1.75-0.25=11.5\), and \(10+3.5-1=12\). With exactly three visit means, a quadratic has three coefficients and can interpolate all three values. That exact fit offers no evidence that the underlying trajectory is truly quadratic; the model has no remaining visit-level information here to assess its shape.
Worked Example: Reading Categorical-Time Contrasts
For pattern 1, a categorical-time model takes the four fitted means directly: \(\alpha_{10}=10\), \(\alpha_{11}=12\), \(\alpha_{12}=13\), and \(\alpha_{13}=13\) points. Relative to baseline, the visit contrasts are \(12-10=2\), \(13-10=3\), and \(13-10=3\) points at visits 1, 2, and 3. The contrast between visits 3 and 2 is \(13-13=0\) points.
These contrasts describe the pattern-specific mean trajectory without imposing linearity or curvature. Pattern 2 supplies means through visit 2, so its observed categorical contrasts are \(11.5-10=1.5\) points at visit 1 and \(12-10=2\) points at visit 2. A categorical mean for pattern 2 at visit 3 cannot be calculated from this table: that visit is unobserved in that pattern.
Why the Polynomial Formula Has Changing Slopes
The linear model has a constant change because subtracting its mean at \(t\) from its mean at \(t+1\) leaves the same slope. For a quadratic model, the same subtraction gives a time-dependent change:
Thus \(\beta_{2p}\) changes the increment from one interval to the next by \(2\beta_{2p}\). For pattern 1 in the example, \(\beta_{1p}=2.5\) and \(\beta_{2p}=-0.5\). The implied increments are \(2.5-0.5=2\), \(2.5-1.5=1\), and \(2.5-2.5=0\) points, matching the displayed means.
Choosing a Time Structure and Checking Its Limits
| Time representation | What it assumes or allows | Main caution |
|---|---|---|
| Linear | Constant mean change per time unit within each pattern | Can miss curvature or changing visit-to-visit increments |
| Polynomial | Smooth curvature represented by powers of time | Higher-order terms use more parameters and can behave poorly outside observed times |
| Categorical | A separate mean at each observed scheduled visit | Uses more parameters and does not estimate a missing pattern-visit mean from that pattern’s data |
The models answer related but different questions. Linear time is parsimonious and easily summarized by a rate. A polynomial can represent a smooth bend while retaining a compact formula. Categorical time is flexible for the scheduled visits but does not assert a smooth path between them. Selection among them should consider the visit schedule, scientific plausibility, pattern sizes, residual behavior, and the intended interpretation, not just which model gives the most convenient curve.
A useful diagnostic is to compare observed pattern-specific means with fitted means at each observed visit. Persistent, ordered discrepancies—for example, repeated underprediction at early visits and overprediction later—suggest the time structure may be inadequate. Also inspect residuals against time and check whether fitted changes are plausible on the outcome’s scale. A high-order polynomial may fit observed visits closely yet imply implausible rises or falls between, or beyond, those visits.
Parameter counts matter. A linear trend requires two mean coefficients per pattern; a quadratic requires three; a categorical model requires one mean for each observed visit in that pattern. Small or sparsely observed patterns may not support even these modest structures. If a pattern has only one observed visit, its mean at that visit can be described, but its within-pattern time slope cannot be determined from those data. If it has two visits, a line through the two pattern means can be calculated, but the data provide no within-pattern check of linearity.
The assumption behind a fitted pattern-specific mean is that its chosen time basis adequately describes the conditional mean at the observed visits, and that the available data in that pattern support its coefficients. If inference uses a Gaussian pattern-specific model, distributional and covariance assumptions are also relevant; the mean-structure choice alone does not establish them. Pattern weights are needed to form population-level summaries, but a pattern-specific curve is not itself a population-average trajectory.
A further distinction is essential: comparing two patterns at a visit where both have observed data is a comparison of their conditional means. It is not automatically a contrast for the full population or a causal effect. In particular, a predicted mean for a visit after a pattern’s dropout point depends on the model’s extrapolation and assumptions about unobserved outcomes. It should not be presented as if that value were directly observed in the pattern.
Common Mistake
More generally, a fitted curve should not be judged only by its visual smoothness. Check whether coefficients are estimable in each pattern, whether the model fits observed visits, and whether the reported comparisons stay within the patterns’ shared observed time support. These checks reveal when a flexible time representation is describing data and when it is mainly imposing structure.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A pattern-specific linear model has intercept 8 points and slope 0.6 points per month. What mean does it predict after 3 months, and what does the slope mean?
- A quadratic mean is \(9+2t-0.4t^2\). Calculate the mean at \(t=2\) and the model-implied change from \(t=1\) to \(t=2\).
- A dropout pattern has observed outcomes at baseline and one follow-up visit only. Can its own data identify a quadratic time trend? Explain.
- Why can a categorical-time model be useful even when it uses more coefficients than a linear model?
Solutions. (1) The prediction is \(8+0.6(3)=9.8\) points. The slope means the modeled conditional mean increases by 0.6 points per month within that pattern. (2) At \(t=2\), the mean is \(9+2(2)-0.4(2^2)=9+4-1.6=11.4\). The change from 1 to 2 is \(11.4-[9+2(1)-0.4(1^2)]=11.4-10.6=0.8\) points. (3) No. A quadratic has three coefficients, while two observed time means provide at most two independent equations; the curvature is not identified without additional constraints or information. (4) It permits a separate mean at each observed visit and therefore avoids imposing a constant slope or polynomial shape. Its flexibility costs parameters and does not supply means at visits missing within that pattern.