Binary Outcomes With Pattern-Specific Logistic Models

Why model binary outcomes by dropout pattern?

Repeated outcomes are sometimes binary: a participant may meet a response criterion at each visit, experience an event by each visit, or answer “yes” to a repeated question. If some participants stop attending, their observed response histories differ in length. A pattern-mixture analysis describes the outcome distribution separately within dropout patterns. For binary responses, logistic regression gives a convenient way to describe how the probability of a response varies with visit, treatment, or baseline covariates within each pattern.

The interpretation requires care. A treatment coefficient within a dropout pattern compares response odds between treatment groups among participants in that pattern, conditional on the other model covariates. It is not automatically a causal treatment effect, and it does not by itself predict responses that were never observed after dropout.

Definition: Let $Y_{ij}$ be the binary outcome for participant $i$ at visit $j$, coded 1 for the event of interest and 0 otherwise. Let $G_i=g$ denote participant $i$'s dropout pattern, and let $m_g$ be the last visit observed for pattern $g$ under monotone dropout. A pattern-specific logistic mean model specifies $$ \operatorname{logit}\{P(Y_{ij}=1\mid G_i=g,A_i,X_i)\} =\log\left\{\frac{P(Y_{ij}=1\mid G_i=g,A_i,X_i)} {1-P(Y_{ij}=1\mid G_i=g,A_i,X_i)}\right\} =\eta_{ijg}, $$ where $A_i$ is treatment, $X_i$ denotes baseline covariates, and $\eta_{ijg}$ is a specified linear predictor. The model describes response probabilities at visits observed in pattern $g$, unless additional assumptions are made to extend it beyond those visits.

Specifying pattern-specific effects

A straightforward specification gives each pattern its own intercept and treatment and time effects. For example, with a numeric visit index $t_j$ and one baseline covariate $X_i$,

\operatorname{logit}\{P(Y_{ij}=1\mid G_i=g,A_i,X_i)\} =\beta_{0g}+\beta_{tg}t_j+\beta_{Ag}A_i+\beta_{Xg}X_i.

Here $\beta_{Ag}$ is the conditional treatment log odds ratio in pattern $g$, for the same visit and covariate value. Exponentiating it gives the treatment odds ratio, $\exp(\beta_{Ag})$. Similarly, if time is entered linearly and measured in visits, $\exp(\beta_{tg})$ is the conditional odds ratio for a one-visit increase, holding treatment and covariates fixed. These interpretations depend on the model's logit-linear form; they are not assertions that the probability changes by a constant amount.

The coefficients can be allowed to differ by pattern, or some can be constrained to be common. A pooled model with pattern indicators and pattern-by-treatment or pattern-by-time interactions is an equivalent way to express these choices. For example, a common treatment effect imposes $\beta_{Ag}=\beta_A$ for every pattern; allowing treatment interactions relaxes that restriction. A common coefficient is a modeling constraint, not a fact established merely by pooling the data.

The binary-response likelihood contribution for one observed response is Bernoulli:

P(Y_{ij}=y_{ij}\mid G_i=g,A_i,X_i) =p_{ijg}^{\,y_{ij}}(1-p_{ijg})^{1-y_{ij}}, \qquad p_{ijg}=P(Y_{ij}=1\mid G_i=g,A_i,X_i).

Multiplying these contributions across visits gives a likelihood for a participant's observed responses only under conditional independence of responses given the model variables. Repeated outcomes within a person are often correlated, so a product of Bernoulli terms should not be treated as a correct joint likelihood without that assumption. A logistic mean model specifies visit-specific probabilities, but it does not alone specify the dependence among a participant's repeated responses. Inference must use an approach whose variance and dependence assumptions suit the analysis; a working-independence estimating procedure with participant-clustered standard errors, for example, has a different interpretation from a fully specified independent-response likelihood.

Worked Example: Treatment Odds Ratios in Two Patterns

Suppose a hypothetical study has complete-follow-up pattern $C$ and a pattern $D$ in which participants are observed through visit 2 and then drop out. At visit 2, fitted pattern-specific models (with treatment $A=0$ or 1) are

\operatorname{logit}(p_{ijC})=-1.2+0.7A_i, \qquad \operatorname{logit}(p_{ijD})=-0.5+0.8A_i.

These models assume a linear treatment term on the log-odds scale and no other covariates or treatment-by-visit interaction at this visit. For pattern $C$, the treatment odds ratio is $\exp(0.7)=2.014$. For pattern $D$, it is $\exp(0.8)=2.226$. Thus, within each pattern, the fitted treatment-group response odds are respectively 2.014 and 2.226 times the control-group odds.

For pattern $C$, the fitted control and treatment probabilities are

p_{0C}=\frac{1}{1+\exp(1.2)}=0.2314752, \qquad p_{1C}=\frac{1}{1+\exp(0.5)}=0.3775407.

The model-based probability difference is $0.3775407-0.2314752=0.1460655$, or approximately 0.1461 (14.61 percentage points). This difference is not the odds ratio: the odds ratio is constant under this model, while the probability difference depends on the baseline probability.

For pattern $D$, $p_{0D}=1/\{1+\exp(0.5)\}=0.3775407$ and $p_{1D}=1/\{1+\exp(-0.3)\}=0.5744425$. The difference is $0.5744425-0.3775407=0.1969018$, or approximately 19.69 percentage points. The two pattern-specific odds ratios describe conditional associations among the respective patterns; they do not establish that treatment caused the difference or that outcomes after dropout follow these probabilities.

Deriving and reading an odds ratio

The odds-ratio interpretation follows directly from the logistic model. At fixed pattern, visit, and covariate values, suppose the linear predictor is $\eta_0$ for $A=0$ and $\eta_0+\beta_{Ag}$ for $A=1$. Since the odds corresponding to a logit $\eta$ equal $\exp(\eta)$, the ratio of treatment to control odds is

\frac{\exp(\eta_0+\beta_{Ag})}{\exp(\eta_0)} =\exp(\beta_{Ag}).

The baseline terms cancel, which is why the conditional odds ratio does not depend on the covariate values in this no-interaction model. If treatment interacts with time, the treatment log odds ratio includes the relevant time term and can differ by visit. If treatment interacts with a covariate, it can differ by covariate value. In either case, report the contrast at stated values rather than interpreting a single coefficient as universal.

A one-unit time effect is likewise tied to its coding. If $t_j$ is coded as 0, 1, 2, a coefficient applies to a one-visit increase. A contrast spanning $k$ visits has odds ratio $\exp(k\beta_{tg})$, provided the linear time effect is appropriate. The probability at a given predictor value is recovered by the inverse-logit transformation $p=\exp(\eta)/\{1+\exp(\eta)\}$.

Worked Example: Interpreting Pattern-Specific Time Slopes

In a hypothetical analysis, visit is coded $t=0,1,2$. At fixed treatment and baseline covariates, suppose pattern $C$ has time coefficient $0.35$, while pattern $D$ has time coefficient $-0.10$. Assume a linear time effect on the logit scale is adequate across these visits and both patterns have observations at the visits being compared.

For pattern $C$, the fitted odds ratio for a one-visit increase is $\exp(0.35)=1.419$. For a two-visit increase, from $t=0$ to $t=2$, it is $\exp(2\times0.35)=\exp(0.70)=2.014$. In this model, the response odds at visit 2 are estimated to be 2.014 times those at visit 0, conditional on treatment and covariates.

For pattern $D$, the one-visit odds ratio is $\exp(-0.10)=0.9048$, and the two-visit odds ratio is $\exp(-0.20)=0.8187$. The ratio of the one-visit odds ratios for pattern $C$ versus pattern $D$ is $\exp(0.35-(-0.10))=\exp(0.45)=1.568$. This comparison describes different fitted time trends on the conditional odds scale; uncertainty in the estimates would be needed to judge how precisely that difference is estimated.

The calculation relies on the stipulated linear logit-time models and on comparing visits represented in both patterns. It should not be used to claim a pattern $D$ response probability at visits after dropout, where that pattern supplies no observed outcomes, without an explicit extrapolation assumption.

Estimation, checks, and failure modes

Before interpreting estimates, check that the data support the chosen model within each pattern. A pattern may have few participants, few events, or no events in a treatment-by-visit cell. Logistic regression can then produce imprecise estimates or complete or quasi-complete separation, in which a predictor or combination of predictors nearly or perfectly distinguishes 1s from 0s. Symptoms include very large coefficient estimates or standard errors, convergence warnings, and fitted probabilities extremely close to 0 or 1. Simplifying the model, combining defensible categories, or using an appropriate penalized method may help, but each changes the analysis and should be reported.

Also check whether a linear trend in visit is plausible on the logit scale, whether important treatment-by-time or covariate interactions are needed, and whether fitted probabilities are compatible with the observed event frequencies within pattern and visit. Examine pattern-specific sample sizes and event counts rather than relying only on a pooled summary: a pooled fit can conceal poor support in a particular pattern. If repeated responses are correlated, assess whether the chosen estimation and variance method accounts for within-participant dependence. A good-fitting mean model does not itself validate an assumed joint distribution.

Pattern-specific models have a useful flexibility-versus-stability tradeoff. Separate coefficients avoid forcing identical treatment or time associations across patterns, but use fewer observations per coefficient and can be unstable. A pooled model with selected common effects can improve precision if the constraints are reasonable; interactions allow heterogeneity where supported. Neither choice should be made solely because one produces a smaller standard error.

Important limitation: A logistic model fitted to observed responses within a dropout pattern describes those responses under its stated covariate and functional-form assumptions. It does not reveal the unobserved post-dropout binary outcomes. Predicting those outcomes requires additional assumptions beyond the observed pattern-specific fits. Further, because dropout pattern is defined by follow-up behavior, conditional comparisons within pattern need not retain the interpretation of an overall randomized treatment comparison.

Common mistake

A common mistake is to say, “The treatment coefficient is 0.8, so treatment raises the response probability by 80 percentage points in this dropout pattern.” The coefficient is on the log-odds scale, not the probability scale. Its exponent, $\exp(0.8)=2.226$, is an odds ratio. The probability change depends on the starting odds and must be calculated from the fitted probabilities, as in the first worked example. Another mistake is to extend a fitted pattern-specific curve beyond the last observed visit as though the extrapolated values were data-based estimates. Such an extension is an added modeling assumption, not a consequence of fitting the observed outcomes.

Key takeaway: Specify the response probability on the logit scale separately by dropout pattern or encode pattern differences through interactions. Interpret exponentiated coefficients as conditional odds ratios at clearly stated covariate and visit values, check data support and dependence assumptions, and distinguish observed-response modeling from extrapolation after dropout.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

A pattern-specific treatment coefficient is $-0.4$. What is its treatment odds ratio, and how should it be described?
A pattern has time coefficient $0.2$ with time coded in visits. What is the model-implied odds ratio for a three-visit increase?
For a fitted logit of $-1$, calculate the response probability. If a treatment coefficient of $0.8$ applies, calculate the treated probability and probability difference.
Why does fitting a logistic mean model to visits observed in a dropout pattern not by itself determine outcomes after that pattern's last visit?

Solutions. (1) $\exp(-0.4)=0.6703$. Conditional on the modeled covariates and visit, the treatment-group odds are estimated to be 0.6703 times the control-group odds in that pattern. (2) $\exp(3\times0.2)=\exp(0.6)=1.822$. (3) The control probability is $1/\{1+\exp(1)\}=0.2689414$. The treated logit is $-1+0.8=-0.2$, giving probability $1/\{1+\exp(0.2)\}=0.4501660$. The model-based difference is $0.4501660-0.2689414=0.1812246$, approximately 0.1812 (18.12 percentage points); the odds ratio is $\exp(0.8)=2.226$. (4) There are no observed responses at those later visits in that pattern. A prediction there requires extrapolating the model and adding assumptions about the unobserved outcomes; the observed-data fit alone cannot verify them.

Binary Outcomes With Pattern-Specific Logistic Models

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