Gaussian Outcomes With Current-Outcome Dropout

Why Model Outcomes and Dropout Together?

A repeated-outcome model describes how outcomes vary over visits, but it does not by itself describe why some outcomes are unobserved. In a Diggle-Kenward selection model, the outcome distribution and the dropout mechanism are specified jointly. The selection component can depend on the outcome at the visit when dropout occurs, even when that current outcome is not observed. This makes the model a useful way to represent dropout related to an unobserved outcome value.

The focus here is the model’s structure: which parameters govern the mean trajectory, which govern within-subject covariance, and which govern selection. These roles are distinct, although they are estimated together from the observed-data likelihood.

Definition: For subject \(i\) at visit \(j\), let \(Y_{ij}\) be the outcome and let \(D_{ij}=1\) indicate dropout at visit \(j\), conditional on the subject having remained under observation through the preceding visit. Assume dropout is absorbing. Under the timing convention used here, the dropout decision at visit \(j\) is made before that visit’s outcome is measured: if \(D_{ij}=1\), \(Y_{ij}\) and later outcomes are unobserved; if the subject remains under observation through the visit-\(j\) measurement, \(Y_{ij}\) is observed. The latent \(Y_{ij}\) can nevertheless enter the dropout hazard.

Gaussian Outcomes and the Selection Hazard

For a subject with \(k\) planned visits, collect the outcomes in \(\mathbf{Y}_i=(Y_{i1},\ldots,Y_{ik})^\mathsf{T}\). Conditional on observed covariates, specify a multivariate normal outcome model:

\mathbf{Y}_i \mid \mathbf{x}_i \sim N\!\left(\boldsymbol{\mu}_i,\boldsymbol{\Sigma}_i\right), \qquad \boldsymbol{\mu}_i=\mathbf{X}_i\boldsymbol{\beta}.

Here \(\mathbf{X}_i\) is a design matrix for the mean trajectory, \(\boldsymbol{\beta}\) contains its regression coefficients, and \(\boldsymbol{\Sigma}_i\) is a positive-definite covariance matrix. The mean model may use visit indicators, a linear time trend, treatment, or other measured predictors as appropriate. The covariance model describes dependence and variability around that mean. For example, it might use a common variance and an autoregressive correlation structure.

Let \(h_{ij}\) denote the conditional probability of dropout at visit \(j\), given that the subject is still at risk for dropout at that visit. A logistic current-outcome hazard, optionally including observed covariates and previous outcomes, can be written as

\operatorname{logit}(h_{ij}) = \log\!\left(\frac{h_{ij}}{1-h_{ij}}\right) = \gamma_{0j}+\gamma_Y Y_{ij} +\boldsymbol{\gamma}_x^\mathsf{T}\mathbf{x}_i +\boldsymbol{\gamma}_{\mathrm{past}}^\mathsf{T}\mathbf{q}_{ij}.

The vector \(\mathbf{q}_{ij}\) represents included past-outcome terms; it can be empty in a current-outcome-only specification. Visit-specific intercepts \(\gamma_{0j}\) allow the baseline hazard to vary across visits. The coefficient \(\gamma_Y\) governs the association of the current outcome with dropout odds, conditional on the other terms. For an outcome measured in points, increasing \(Y_{ij}\) by one point multiplies those conditional odds by \(\exp(\gamma_Y)\). This is a model-based conditional association, not a causal effect.

Parameter roles: \(\boldsymbol{\beta}\) governs the outcome mean trajectory; parameters in \(\boldsymbol{\Sigma}_i\) govern outcome variability and within-subject covariance; and \(\boldsymbol{\gamma}\) governs the dropout hazard conditional on the model’s predictors. A nonzero \(\gamma_Y\) allows dropout probability to depend on the current outcome, including when that outcome is latent.

The main assumptions are that the Gaussian distribution adequately describes the conditional repeated outcomes, the selected mean and covariance structures are suitable, and the hazard correctly represents the dropout timing and its included predictors. The usual likelihood also treats subjects as independent conditional on modeled covariates. If dropout is possible for reasons not represented by the specified hazard, that part of the observation process is not captured by this model.

Worked Example: Separating Trajectory, Covariance, and Selection

Suppose a hypothetical study plans three measurements of a biomarker, in points, at times \(t=0,1,2\). Let the mean be \(\mu_{ij}=50+2t_j\). Suppose the residual standard deviation is \(4\) points and the within-subject correlation between visits \(j\) and \(\ell\) is \(0.5^{|j-\ell|}\). Let the dropout hazard be \(\operatorname{logit}(h_{ij})=-3+0.04Y_{ij}\), with no past-outcome or covariate terms.

Trajectory. The mean vector is \((50,52,54)^\mathsf{T}\) points. The intercept \(50\) is the modeled mean at time zero, and the slope \(2\) is the modeled increase per time unit. These are outcome-model parameters.

Covariance. The marginal variance at each visit is \(4^2=16\) points squared. The lag-one covariance is \(16(0.5)=8\), and the lag-two covariance is \(16(0.5^2)=4\). Thus

\boldsymbol{\Sigma}_i = \begin{pmatrix} 16&8&4\\ 8&16&8\\ 4&8&16 \end{pmatrix} \quad\text{points}^2.

Selection. At a current value of \(50\) points, the log odds are \(-3+0.04(50)=-1\), so the hazard is \(1/(1+\exp(1))\approx0.269\). At \(55\) points, the log odds are \(-3+0.04(55)=-0.8\), giving \(1/(1+\exp(0.8))\approx0.310\). The current-outcome coefficient is \(0.04\) per point, so a one-point increase multiplies conditional dropout odds by \(\exp(0.04)\approx1.041\). It does not change the outcome mean or covariance parameters.

Observed-Data Likelihood and Visit Timing

The joint model combines the Gaussian outcome density with the dropout hazards. If a subject completes all \(k\) measurements, then the observed-data contribution includes the Gaussian density of all outcomes and survival through each dropout decision. If the subject drops out at visit \(j\), the outcomes through visit \(j-1\) are observed, while \(Y_{ij}\) is latent under the timing convention above. Its value must be integrated over, rather than treated as observed or replaced by a fixed value.

L_i = \begin{cases} f_{\boldsymbol{\beta},\boldsymbol{\Sigma}_i} (y_{i1},\ldots,y_{ik}) \displaystyle\prod_{\ell=1}^{k} \{1-h_{i\ell}(y_{i\ell})\}, & \text{if the subject completes all } k \text{ visits},\\[1.2em] \displaystyle \int f_{\boldsymbol{\beta},\boldsymbol{\Sigma}_i} (y_{i1},\ldots,y_{i,j-1},y_{ij}) \left\{\prod_{\ell=1}^{j-1}(1-h_{i\ell}(y_{i\ell}))\right\} h_{ij}(y_{ij})\,dy_{ij}, & \text{if dropout occurs at visit } j. \end{cases}

In the dropout expression, \(y_{i1},\ldots,y_{i,j-1}\) are the observed outcomes. The density shown is the Gaussian marginal density for outcomes through visit \(j\); later outcomes have been integrated out of the Gaussian model. Earlier survival factors appear once each, and the hazard at visit \(j\) describes dropout there. There are no later hazard factors for this subject because dropout is absorbing. This factorization follows from multiplying the outcome density by the probabilities of the observed dropout history, then integrating over the unobserved current outcome.

Worked Example: Contribution for Dropout at Visit Two

Consider a hypothetical three-visit study using the Gaussian model from the previous example. One subject has \(y_{i1}=50\) points and drops out at visit 2, so \(Y_{i2}\) is unobserved. The mean of \(Y_{i2}\) conditional on \(Y_{i1}=50\) is \(52+0.5(50-50)=52\) points, and its conditional variance is \(16(1-0.5^2)=12\) points squared. Thus \(Y_{i2}\mid Y_{i1}=50\sim N(52,12)\).

At visit 1, the hazard evaluated at \(50\) is \(h_{i1}(50)=\operatorname{logit}^{-1}(-1)\approx0.269\), so the probability of remaining under observation through that decision is approximately \(0.731\). The dropout-at-visit-2 contribution is therefore

L_i = f_{Y_{i1}}(50)(0.731) \int_{-\infty}^{\infty} f_{Y_{i2}\mid Y_{i1}}(y\mid 50) \operatorname{logit}^{-1}(-3+0.04y)\,dy.

Substituting the conditional normal density gives

L_i = f_{Y_{i1}}(50)(0.731) \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi(12)}} \exp\!\left[-\frac{(y-52)^2}{2(12)}\right] \frac{1}{1+\exp\{3-0.04y\}}\,dy.

The integral averages the visit-2 dropout probability over the conditional distribution of the unobserved visit-2 outcome. For example, at \(y=52\), the hazard is \(\operatorname{logit}^{-1}(-3+0.04(52))=\operatorname{logit}^{-1}(-0.92)\approx0.285\). This value is the hazard at that particular latent outcome, not the integral’s average. The integral is evaluated as part of fitting the observed-data likelihood; it cannot generally be replaced by the hazard at the conditional mean.

Model Checks, Comparisons, and Limitations

The Gaussian outcome specification and the dropout hazard should be checked as separate components as well as jointly. Useful checks include plotting observed outcome distributions and trajectories by visit, examining whether residual variability or correlation changes over time, and checking whether the chosen mean structure leaves systematic patterns. For the hazard, check whether the selected functional form and visit effects are plausible given the observed dropout pattern and measured predictors. Such checks can expose poor fit, but observed data cannot directly reveal the distribution of outcomes that were unobserved at dropout.

The main alternative is to model dropout as independent of the unobserved current outcome after conditioning on observed information, as in an appropriate missing-at-random analysis. A current-outcome selection model instead makes the dependence explicit through \(\gamma_Y\). It is not a data-free correction: conclusions can depend on the chosen hazard link, covariates, and assumptions about the latent outcome distribution. In particular, observed data alone may not identify all aspects of a nonignorable selection mechanism. The selection specification should therefore be justified and, where relevant, assessed through sensitivity analyses.

A common mistake is to treat “dropout at visit \(j\)” as proof that \(Y_{ij}\) was measured before dropout. Under the timing used here, dropout occurs before the visit-\(j\) measurement, so the current outcome is missing and enters the likelihood through integration. If a study’s operational timing instead measures the outcome before recording dropout, the observation and hazard ordering must be specified accordingly; silently mixing the two conventions gives the wrong likelihood.

Another mistake is to interpret \(\gamma_Y\) as a coefficient in the mean trajectory. It belongs to the conditional dropout model: in the logistic specification, \(\exp(\gamma_Y)\) is the multiplicative change in conditional dropout odds for a one-unit increase in current outcome, holding the other hazard predictors fixed. It is neither a change in mean outcome nor evidence of a causal effect.

Specify the outcome model.
Choose the Gaussian mean structure and a valid covariance model for the planned repeated measurements.

Specify the dropout timing and hazard.
State when dropout occurs relative to measurement, then define the conditional hazard and its current-outcome term.

Form the observed-data likelihood.
Multiply the outcome density by the dropout-history probabilities and integrate over outcomes unobserved because of dropout.

Assess assumptions and sensitivity.
Check outcome and hazard fit where observable, and recognize that latent-outcome selection assumptions are not fully checked by observed data alone.

Check Your Understanding

Answer each question using the timing convention and model definitions above. Brief solutions follow the questions.

In a logistic hazard with \(\gamma_Y=0.03\) per point, what is the multiplier on conditional dropout odds for a 5-point increase in current outcome?
A subject drops out at visit 3 after observed visits 1 and 2. Which outcome values must be integrated over in the likelihood contribution under this tutorial’s timing convention?
Which parameter component governs within-subject covariance: \(\boldsymbol{\beta}\), \(\boldsymbol{\Sigma}_i\), or \(\boldsymbol{\gamma}\)?
Why is the dropout hazard evaluated at the conditional mean of a missing current outcome generally not a replacement for integrating over that outcome?

Solutions. (1) The multiplier is \(\exp(5\times0.03)=\exp(0.15)\approx1.162\); conditional dropout odds are multiplied by about 1.162, holding other predictors fixed. (2) \(Y_{i3}\) is integrated over. Later outcomes are also unobserved, but their Gaussian values have already been marginalized out when using the density through visit 3; there are no later dropout hazards after absorbing dropout. (3) \(\boldsymbol{\Sigma}_i\) governs covariance. (4) The hazard is a nonlinear function of the outcome, so in general the average of the hazard over the conditional outcome distribution differs from the hazard evaluated at the conditional mean.

Gaussian Outcomes With Current-Outcome Dropout

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