Why a Pattern Indicator Is Not a Pattern-Specific Model
In a pattern-mixture analysis, the distribution of outcomes is described conditional on dropout pattern. That conditioning is substantive: it asks how the observed trajectory differs among people with different patterns of outcome availability. A common shortcut is to put a dropout-pattern indicator into one pooled regression and assume that this has specified a separate trajectory model for every pattern. Usually it has not. The terms included in the regression determine which features may differ and which are forced to be shared.
Let \(D_i=g\) indicate that individual \(i\) belongs to dropout pattern \(g\), and let \(Y_{it}\) denote the outcome at visit \(t\), when that outcome is observed. A pattern-specific description concerns the conditional distribution of the outcomes available under pattern \(g\). It can include a pattern-specific mean trajectory, but a distribution also involves features such as variance, dependence between visits, and outcome type. The distinction matters even when the scientific aim is only to compare conditional means: a pooled regression can impose a mean structure that is too restrictive.
What an Additive Pattern Term Assumes
For a numeric time variable \(t\), consider a pooled linear regression with two patterns, \(A\) and \(B\):
The indicator \(I(D_i=B)\) equals 1 for pattern \(B\) and 0 otherwise. The model permits the patterns to have different intercepts, but it gives them the same time slope, \(\beta_t\). Equivalently, the fitted difference between the pattern \(B\) and pattern \(A\) means is \(\beta_D\) at every time. These are restrictions, not consequences of having included a pattern variable.
The restriction follows directly by subtracting the two conditional means:
Because this difference does not depend on \(t\), the model cannot represent pattern-specific slopes. Adding more covariates does not automatically fix this issue. Unless the model includes the relevant interactions, it generally imposes common covariate effects as well as common time effects.
Worked Example: A Shared Slope Compromises the Pattern Means
Suppose hypothetical observed cell means are as follows. Pattern \(A\) is observed at visits \(t=0,1,2\), and pattern \(B\) at \(t=0,1\). Assume equal numbers of observations in each displayed cell. Consider fitting an ordinary least-squares regression with a pattern main effect and a common linear time slope.
| Pattern | Visit \(t\) | Observed mean outcome |
|---|---|---|
| A | 0 | 10 |
| A | 1 | 12 |
| A | 2 | 14 |
| B | 0 | 10 |
| B | 1 | 11 |
Within pattern \(A\), the mean time is 1 and the mean outcome is 12. Its contribution to the numerator of the common-slope calculation, per equally sized cell, is \( (-1)(10-12)+(0)(12-12)+(1)(14-12)=4 \); the corresponding denominator is \( (-1)^2+0^2+1^2=2 \). For pattern \(B\), the mean time is 0.5 and mean outcome is 10.5. The numerator contribution is \( (-0.5)(10-10.5)+(0.5)(11-10.5)=0.5 \), and the denominator contribution is \( (-0.5)^2+(0.5)^2=0.5 \). Thus the pooled common slope is
The fitted pattern-specific intercepts are the group means minus the common slope times each group’s mean time. They are \(12-1.8(1)=10.2\) for pattern \(A\), and \(10.5-1.8(0.5)=9.6\) for pattern \(B\). The fitted lines are therefore \(10.2+1.8t\) and \(9.6+1.8t\). At \(t=1\), the model predicts 12 for pattern \(A\) and 11.4 for pattern \(B\), compared with observed means of 12 and 11. The model has compromised between the two different observed slopes: 2 for \(A\) and 1 for \(B\). A pattern main effect did not preserve both pattern-specific trajectories.
Interactions Allow Different Mean Trajectories
To allow two patterns to have different linear slopes, add a pattern-by-time interaction:
For pattern \(A\), the fitted intercept and slope are \(\beta_0\) and \(\beta_t\). For pattern \(B\), they are \(\beta_0+\beta_D\) and \(\beta_t+\beta_{Dt}\). The difference in means is now \(\beta_D+\beta_{Dt}t\), which can change over time. With more than two patterns, pattern-specific time terms or a full set of pattern-by-time interactions can provide the corresponding flexibility. Interactions with other covariates are needed when their effects are also meant to differ by pattern.
Worked Example: Recovering Two Different Mean Lines
Use the hypothetical cell means in the previous table. The pattern \(A\) means lie on \(10+2t\), and the pattern \(B\) means lie on \(10+1t\). Under the interaction model, these lines correspond to
For pattern \(A\), the mean is \(10+2t\), giving 10, 12, and 14 at the three observed visits. For pattern \(B\), it is \(10+0+(2-1)t=10+t\), giving 10 and 11 at its two observed visits. In this example the interaction represents both sets of observed means, unlike the additive model’s common slope.
This flexibility has a boundary. Pattern \(B\) has no observed outcome at \(t=2\) in the example, so its mean at that visit is not an observed pattern-specific mean. Continuing the fitted line would be a model-based extrapolation, not a direct estimate supported by pattern \(B\)'s data at \(t=2\). Interactions let coefficients vary; they do not create observations where a pattern has none.
Mean Interactions Do Not Specify the Whole Distribution
Even a model with a separate mean at every pattern and visit does not necessarily specify separate trajectory distributions. A Gaussian regression, for example, commonly adds a residual-variance assumption. If the fitted model uses one residual variance for everyone, it assumes equal residual variances across patterns unless it explicitly allows otherwise. A model for repeated outcomes also needs a dependence structure if it is to describe their joint distribution. Equal means do not imply equal variances or equal within-person correlations.
Worked Example: Equal Means, Different Pattern Variances
Suppose, hypothetically, that a single visit is observed in both patterns and that the conditional outcome distributions are \(Y\mid D=A\sim N(10,1)\) and \(Y\mid D=B\sim N(10,16)\). The second argument of \(N\) is the variance. A pooled regression with a pattern main effect estimates equal pattern means in this example: the pattern coefficient is 0. But if it also assumes a common Gaussian residual variance, it cannot represent both pattern distributions.
With equal numbers of observations in the two patterns and these means, the pooled common-variance normal likelihood has variance equal to the average of the two within-pattern variances:
That common value is neither pattern’s variance. A pattern-specific normal model could instead use variance 1 for \(A\) and 16 for \(B\). Thus, a pattern indicator can correctly represent a mean contrast while the pooled distributional model remains wrong for both patterns' variability. For multiple visits, analogous restrictions can arise through a shared covariance structure.
A useful way to express the distinction is that a conditional mean model specifies \(E(Y_{it}\mid D_i=g,t)\), not the entire conditional law of the outcome. A full pattern-specific distributional model must also state an appropriate outcome family and any pattern-specific dispersion or dependence features required by the analysis. This is especially important for binary, count, or otherwise non-Gaussian outcomes, where the chosen link and distribution determine what a coefficient means and what distribution is being modeled.
Derivation: The Restrictions Are Visible in the Equations
The comparison between the additive and interaction models can be established without estimation details. In the additive model, the conditional mean for pattern \(A\) is \(\beta_0+\beta_t t\), and for \(B\) it is \(\beta_0+\beta_D+\beta_t t\). Differentiating each linear mean with respect to time gives the same slope:
With the interaction term, the derivatives are \(\beta_t\) for \(A\) and \(\beta_t+\beta_{Dt}\) for \(B\). The slopes can differ by exactly \(\beta_{Dt}\). This shows both what the interaction adds and what it does not add: it relaxes a restriction on conditional means, but says nothing by itself about pattern-specific variances or outcome distributions.
More generally, separate regressions for each pattern can be written as distinct conditional mean functions \(m_g(t,x)\), where \(x\) represents other predictors. A pooled regression matches those mean functions only if its terms and functional form allow the required pattern-specific coefficients. A fully interacted model can represent separate coefficients for a chosen basis of time and covariates, provided the data support estimation; it still needs distributional assumptions beyond the mean.
Model Choice, Diagnostics, and Failure Modes
Three common strategies answer different modeling needs. A pooled model with pattern main effects is parsimonious but assumes common slopes for predictors not interacted with pattern. A pooled model with relevant interactions allows some or all conditional mean coefficients to vary while retaining a shared parameterization. Separate pattern-specific fits allow parameters to be estimated independently, but can be unstable when patterns contain few observations. None is automatically best: the choice should match the intended conditional distributions and the information available within each pattern.
| Approach | What it can represent | Important limitation |
|---|---|---|
| Pooled regression with pattern main effects | Pattern-specific intercepts, with common slopes for other terms | Can hide different pattern-specific time or covariate effects |
| Pooled regression with pattern interactions | Different means for the pattern-specific terms that are interacted | Uninteracted terms and shared distributional parameters remain constrained |
| Separate pattern-specific fits | Distinct fitted parameters within each pattern | Small or sparse patterns can yield unstable estimates |
Diagnostics should examine the model within patterns rather than relying only on an overall fit summary. Plot observed outcomes or pattern-specific summaries against time; look for slopes or curvature that differ by pattern. Inspect residual spread within each pattern and, for repeated outcomes, whether the assumed dependence structure is plausible. Check how many observations each pattern contributes at each visit and whether the proposed form can be estimated from that support. Large standard errors, unstable estimates after small modeling changes, and predictions driven mainly by extrapolation are warning signs, not evidence that the patterns truly share a trajectory.
A common mistake is to reason, “Dropout pattern is in the regression, so the model is already pattern-specific.” The correction is to inspect the equation term by term. A pattern main effect allows different intercepts; without interactions, it usually forces common slopes. Even a fully interacted mean model is not automatically a separate model for each pattern’s outcome distribution. State the mean, variance, outcome-family, and dependence assumptions explicitly, and distinguish fitted values from quantities unsupported by a pattern’s observed visits.
Check Your Understanding
Answer each question using the model structure, not just the presence of a pattern indicator. Brief solutions follow.
- In a model with a pattern main effect and a common time term, what restriction is imposed on the two patterns’ linear mean trajectories?
- If the fitted mean is \(\beta_0+\beta_D I(D=B)+\beta_t t+\beta_{Dt}I(D=B)t\), what is the time slope in pattern \(B\)?
- Can a pattern-by-time interaction alone allow the patterns to have different residual variances? Explain.
- Suppose pattern \(A\) has a mean of 5 at \(t=0\) and 9 at \(t=2\). What is its linear slope? If pattern \(B\)'s mean changes from 5 to 7 over the same interval, what interaction coefficient represents the slope difference when \(A\) is the reference pattern?
- Why should a pattern-specific prediction at a visit with no observations in that pattern be described cautiously?
Solutions. (1) The slopes are shared, so the difference between pattern means is constant over time. (2) The slope is \(\beta_t+\beta_{Dt}\). (3) No. It relaxes a restriction on the mean; a variance model must separately allow pattern-specific residual variances. (4) Pattern \(A\)'s slope is \((9-5)/(2-0)=2\) outcome units per time unit. Pattern \(B\)'s slope is \((7-5)/2=1\), so the slope difference \(B-A\), represented by \(\beta_{Dt}\), is \(1-2=-1\) outcome units per time unit. (5) The prediction is extrapolated from the model rather than estimated from outcomes observed for that pattern at that visit; its credibility depends on the modeling assumptions.