Worked Analysis with Binary Repeated Outcomes

Why Joint Modeling Matters

A binary outcome observed repeatedly can be related to dropout even when the dropout process is not directly determined by any one recorded response. For example, an unobserved tendency toward worsening health might both lower the chance of a favorable response and increase the chance of leaving follow-up. A shared-random-effects model represents that connection with a latent subject-specific quantity that enters both the outcome model and the dropout hazard.

The model does not recover outcomes after dropout from the data alone. Rather, it specifies a joint probability model for the observed outcomes and the observation process. Estimation and predictions therefore depend on the stated assumptions about the random effect, the outcome process, and dropout.

Model and Timing Convention

Suppose there are \(T\) scheduled visits. Let \(Y_{it}\) be the binary outcome for subject \(i\) at visit \(t\), observed through that subject’s last observed visit \(m_i\). Dropout is monotone: once a subject drops out, later outcomes are not observed. Define dropout in interval \(t\) as occurring after visit \(t\) and before visit \(t+1\). Thus hazards are defined for \(t=1,\ldots,T-1\); a subject last seen at visit \(m_i<T\) drops out in interval \(m_i\), while a subject observed through visit \(T\) has no dropout event in the modeled intervals.

Definition: A shared-random-effects model links the repeated outcomes and dropout by including the same subject-specific random effect \(b_i\) in both conditional models. In the specification below, \(b_i\sim N(0,\tau^2)\), and, conditional on \(b_i\) and covariates, each visit outcome is Bernoulli while dropout follows a discrete-time hazard.

For a treatment or baseline covariate \(A_i\), one simple specification is

\begin{aligned} Y_{it}\mid b_i,A_i &\sim \operatorname{Bernoulli}(p_{it}),& \operatorname{logit}(p_{it})&=\alpha_t+\beta A_i+b_i,\\ D_{it}\mid b_i,A_i,\text{at risk at }t &\sim \operatorname{Bernoulli}(h_{it}),& \operatorname{logit}(h_{it})&=\gamma_t+\xi A_i+\lambda b_i, \quad t=1,\ldots,T-1. \end{aligned}

Here \(D_{it}=1\) denotes dropout in interval \(t\), conditional on being observed through visit \(t\). Visit-specific intercepts allow outcome probabilities and dropout hazards to change over time. The coefficient \(\lambda\) describes how the shared latent effect is associated with dropout: for a one-unit increase in \(b_i\), the conditional dropout odds are multiplied by \(e^\lambda\), holding \(A_i\) and interval fixed. The outcome coefficient \(\beta\) is a conditional log-odds contrast for \(A_i\), holding the shared effect and visit fixed.

This specification assumes conditional independence of the outcome sequence and dropout process given \(b_i\) and \(A_i\), with the outcome observations conditionally independent across visits in this simple version. It also assumes the chosen normal distribution for \(b_i\), the logit links, and the hazard’s dependence on \(b_i\) are adequate. In particular, the hazard has no additional direct dependence on the most recent observed outcome. These are modeling assumptions, not consequences of the data structure.

Observed-Data Likelihood

Write \(f(b)\) for the normal density of \(b_i\). For a fixed value \(b\), the outcome contribution for a subject observed through visit \(m_i\) is the product of the Bernoulli probabilities for their observed responses. The dropout contribution is the probability of surviving each interval in which the subject remained under follow-up, multiplied by the event hazard if dropout occurred.

L_i(b)= \left\{\prod_{t=1}^{m_i}p_{it}(b)^{Y_{it}}\bigl[1-p_{it}(b)\bigr]^{1-Y_{it}}\right\} \begin{cases} \displaystyle\left\{\prod_{t=1}^{m_i-1}[1-h_{it}(b)]\right\}h_{i,m_i}(b),&m_i<T,\\[6pt] \displaystyle\prod_{t=1}^{T-1}[1-h_{it}(b)],&m_i=T. \end{cases}

The subject’s observed-data likelihood contribution integrates over the unobserved random effect:

L_i=\int L_i(b)f(b)\,db.

The likelihood for independent subjects is \(\prod_i L_i\). The integral is generally evaluated numerically, for example by adaptive quadrature, and the resulting log likelihood is maximized over the model parameters. The integration must be over the joint contribution: the same \(b\) affects both outcomes and dropout.

Why the Shared Effect Is Integrated Jointly

Conditional on a particular \(b\), the outcome and dropout factors multiply under the stated conditional-independence assumption. But after \(b\) is integrated out, the two observed processes are generally associated. An outcome history that makes some values of \(b\) more plausible also changes the implied dropout probabilities.

Condition on the latent effect.
For each possible \(b\), calculate the probability of the observed outcomes and the observed retention or dropout pattern.

Multiply within each latent-effect value.
Combine the outcome and dropout factors at that same \(b\), as required by the joint model.

Average over the random-effect distribution.
Integrate the product against \(f(b)\). In general, this is not equal to the product of separately integrated outcome and dropout factors.

This order of operations is the key likelihood calculation. If one integrated the two factors separately and multiplied the results, one would effectively treat their latent-effect variation as independent, contrary to the shared-effect model.

Worked Example: A Dropout Likelihood Contribution

Consider a hypothetical three-visit study. A subject receives \(A_i=0\), has \(Y_{i1}=Y_{i2}=1\), and drops out after visit 2, so \(m_i=2<T=3\). For a transparent hand calculation, approximate the random-effect distribution by two equally weighted quadrature nodes, \(b=-1\) and \(b=1\). Use outcome intercepts \(\alpha_1=\alpha_2=0\), and dropout intercepts \(\gamma_1=-2\), \(\gamma_2=-1\), with \(\lambda=1\). This two-node calculation illustrates integration; it is not a claim that the random effect is actually discrete.

At \(b=-1\), both observed outcome probabilities are \(\operatorname{logit}^{-1}(-1)=0.2689\). The interval-1 dropout hazard is \(\operatorname{logit}^{-1}(-3)=0.04743\), and the interval-2 hazard is \(\operatorname{logit}^{-1}(-2)=0.1192\). The conditional likelihood contribution is therefore

(0.2689)^2(1-0.04743)(0.1192)\approx 0.00821.

At \(b=1\), the outcome probabilities are \(0.7311\), the interval-1 hazard is \(\operatorname{logit}^{-1}(-1)=0.2689\), and the interval-2 hazard is \(\operatorname{logit}^{-1}(0)=0.5000\). Thus the contribution at this node is

(0.7311)^2(1-0.2689)(0.5000)\approx 0.1954.

Averaging these conditional contributions using the two equal weights gives

L_i\approx 0.5(0.00821)+0.5(0.1954)=0.1018.

The event hazard used here is \(h_{i2}\), which is defined because there are three scheduled visits and dropout occurred in interval 2, after visit 2 and before visit 3. The likelihood also includes survival through interval 1.

Predicting Dropout Risk From an Observed History

For a subject observed through visit \(t<T\), a useful prediction is the probability of dropout in the next interval, conditional on the subject’s observed outcomes and continued follow-up through visit \(t\). The observed history updates the distribution of \(b\): random-effect values that make the observed outcomes and survival through earlier intervals more likely receive more weight.

P(D_{it}=1\mid \text{observed history through }t,\text{ at risk at }t) = \frac{\int h_{it}(b)W_i(b)f(b)\,db} {\int W_i(b)f(b)\,db},

where \(W_i(b)\) is the probability of the subject’s observed outcomes through visit \(t\), multiplied by the probabilities of surviving intervals \(1,\ldots,t-1\). The current interval hazard is not included in \(W_i(b)\), because the subject is being treated as still at risk at the start of that interval. For numerical integration, this ratio is calculated using the same quadrature nodes in numerator and denominator.

Worked Example: Conditional Dropout Risk After Visit 2

Use the two quadrature nodes and model parameters from the previous example. Suppose a subject has \(Y_{i1}=Y_{i2}=1\) and is observed through visit 2, so they have survived interval 1 and are at risk in interval 2. At \(b=-1\), the weight before normalizing is

0.5(0.2689)^2(1-0.04743)\approx 0.03445.

At \(b=1\), it is

0.5(0.7311)^2(1-0.2689)\approx 0.1954.

The total is approximately \(0.2298\), so the normalized weights are about \(0.1499\) for \(b=-1\) and \(0.8501\) for \(b=1\). The predicted interval-2 dropout risk is the weighted average of the two interval-2 hazards:

0.1499(0.1192)+0.8501(0.5000)\approx 0.4429.

Under this illustrative model and quadrature approximation, the predicted risk of dropout between visits 2 and 3 is about 44.3% for a subject with this observed history who is still under follow-up at visit 2. This is a model-based conditional prediction, not an observed frequency or a guarantee about an individual.

Worked Example: Marginal Risk Across the Random-Effect Distribution

With the same two-node approximation, calculate the probability of dropout before visit 3 for a subject before observing any outcomes. At \(b=-1\), the probability of remaining through both modeled intervals is

(1-0.04743)(1-0.1192)\approx 0.8390.

At \(b=1\), it is

(1-0.2689)(1-0.5000)\approx 0.3655.

Averaging retention over the two equally weighted nodes gives approximately \(0.5(0.8390)+0.5(0.3655)=0.6023\). Thus the marginal probability of dropout before visit 3 is \(1-0.6023=0.3977\), or about 39.8%. This differs from the 44.3% conditional risk above because the observed history shifts the weights toward the larger-\(b\) node, which has higher hazards under this model.

Interpreting a Fitted Model

Suppose a hypothetical fitted model estimates \(\beta=0.60\) for a binary treatment indicator in the outcome model. Holding visit and \(b_i\) fixed, the treatment-group odds of a positive outcome are multiplied by \(e^{0.60}\approx1.82\) relative to the comparison group. This is a conditional odds ratio, not a risk ratio or a marginal treatment effect. A marginal contrast averages over the random-effect distribution and may differ.

Likewise, if \(\lambda=0.70\), then a one-unit increase in \(b_i\) multiplies the conditional dropout odds by \(e^{0.70}\approx2.01\), holding treatment and interval fixed. Since \(b_i\) is latent and its scale is tied to the assumed distribution, this coefficient is a feature of the specified model rather than a directly observed subject-level effect.

Parameter estimates are obtained from the integrated likelihood, not by first estimating each subject’s random effect as if it were known. Numerical integration and optimization should be checked using the convergence and model-checking principles relevant to shared-parameter fits.

Assumptions, Diagnostics, and Alternatives

The model is most interpretable when its assumptions are made explicit. In this specification, visit times are scheduled and fixed, dropout is monotone, and the observed outcomes precede or coincide with the last recorded visit. The conditional outcome and hazard models have the stated forms, and the same random effect is distributed according to the assumed normal density. Conditional on that effect and the covariates, no further dependence between the modeled outcomes and dropout remains.

Signs that the model may be inadequate include poor calibration of observed dropout frequencies across visits or covariate groups, systematic mismatch in the distribution of binary response patterns, unstable estimates of \(\lambda\) or \(\tau^2\), and material changes in predictions under plausible alternative random-effect distributions or link functions. A useful check is to compare observed pattern frequencies and outcome histories with simulations from the fitted joint model. Such checks can reveal lack of fit, but agreement with observed data cannot verify assumptions about outcomes after dropout.

The shared-effect model is not the only approach. An outcome-only model that ignores dropout is simpler but does not represent the dependence induced by a latent factor associated with both processes. A selection model specifies outcome distributions conditional on missingness-related quantities; a pattern-mixture model specifies outcome distributions by missingness pattern. These formulations organize assumptions differently. The shared-random-effects model is especially useful when a subject-level latent tendency is a scientifically meaningful way to connect repeated responses and dropout, but its predictions depend on that latent structure and distribution.

Common Pitfalls and Limitations

Using a hazard outside the defined follow-up intervals. With \(T\) visits and hazards only for intervals \(1,\ldots,T-1\), dropout after the final visit is not an event in the model. A subject last observed at \(T\) contributes survival through interval \(T-1\), not a hazard \(h_{iT}\).
Multiplying separately integrated contributions. Integrating the outcome and dropout factors independently discards their common dependence on \(b_i\). Multiply them conditional on \(b_i\), then integrate.
Calling a conditional coefficient marginal. The log-odds coefficient \(\beta\) conditions on \(b_i\). It is not automatically a population-averaged treatment contrast.
Treating a fitted association as proof of mechanism. A nonzero \(\lambda\) describes association through the assumed latent effect; it does not establish that the latent effect is the true cause of dropout.

The central limitation is that the outcomes after dropout are unobserved. The joint likelihood uses observed outcomes and dropout patterns to estimate model parameters, but the distribution of missing outcomes remains dependent on unverifiable modeling assumptions. Prediction and sensitivity conclusions should be reported accordingly.

Check Your Understanding

Answer each question, then compare with the solutions below.

With \(T=4\), a subject last observed at visit 3 and then lost has which dropout hazard in their likelihood contribution?
Why must the outcome and dropout factors be multiplied before integrating over \(b_i\)?
In the model above, what does \(e^\lambda\) represent?
If a subject is observed through visit 2 and is at risk in interval 2, should the prediction weights include survival through interval 2?

Solutions. (1) The event is dropout in interval 3, so the contribution includes \(h_{i3}\), as well as survival through intervals 1 and 2. This hazard is defined because \(3<T=4\). (2) Both processes depend on the same \(b_i\), so their joint conditional likelihood is a product at each fixed \(b_i\); integrating that product preserves their shared latent dependence. (3) It is the multiplicative change in conditional dropout odds for a one-unit increase in \(b_i\), holding covariates and interval fixed. (4) No. The subject is at risk at the start of interval 2, so the weights include observed outcomes through visit 2 and survival through interval 1; the interval-2 hazard is averaged over those weights to calculate risk in interval 2.