Treatment Effect Estimation Under Shared-Parameter Dropout

What Treatment Effect Is Being Estimated?

In a longitudinal study with dropout, a treatment coefficient from the outcome submodel is not automatically the treatment effect for the population. It may describe a contrast conditional on latent effects, while the scientific target is often the contrast in mean outcomes for the original study population at a specified visit. These quantities can differ, especially when the outcome model is nonlinear or dropout changes which latent-effect values are represented among observed participants.

A shared-parameter model connects the longitudinal outcome and dropout processes through latent effects. Its joint likelihood uses both the observed outcomes and the dropout histories to estimate model parameters. To obtain a treatment contrast for the target population, predictions from the outcome submodel are then averaged over the target distribution of latent effects (and, when needed, covariates). This is a model-based estimate: it relies on the outcome model, dropout model, latent-effect distribution, and assumptions that identify the target.

Definition: Let \(A_i\) denote treatment assignment for participant \(i\), with \(A_i=1\) for treatment and \(A_i=0\) for control. Let \(Y_{ij}\) be the outcome at visit \(j\), \(R_{ij}=1\) if it is observed, and \(b_i\) the participant's shared latent effect or effects. A conditional treatment contrast compares outcome-model predictions at fixed \(b_i\) and covariates. A population-averaged treatment contrast compares predictions averaged over the target population distribution of \(b_i\) and covariates. For a mean difference at visit \(j\), write \(\Delta_j=E\{Y_{ij}(1)\}-E\{Y_{ij}(0)\}\), where the expectations refer to the specified target population.

The notation \(Y_{ij}(a)\) represents the outcome that would be observed at visit \(j\) under assignment \(a\), whether or not it was actually observed. A causal interpretation of \(\Delta_j\) requires a well-defined treatment, consistency, no interference, and treatment exchangeability and positivity for the target population. In a randomized trial, exchangeability is supported by random assignment. The shared-parameter model additionally makes assumptions about the outcome distribution, dropout process, latent-effect distribution, and their dependence. Without the causal assumptions, the contrast is an adjusted model-based comparison, not necessarily a causal effect.

How the Joint Fit Enters the Contrast

Let \(X_i\) denote observed covariates. A general outcome submodel specifies a conditional density \(f_Y\{Y_{ij}\mid A_i,X_i,b_i;\theta\}\), where \(\theta\) contains outcome parameters. A dropout model specifies the probability of the observed dropout history given treatment, covariates, and shared effects. For example, in discrete time it may specify a dropout hazard at each visit among participants still under follow-up. Denote the dropout-history contribution by \(f_R(R_i\mid A_i,X_i,b_i;\psi)\), with parameters \(\psi\).

For an observed history, the participant's likelihood contribution integrates out the latent effects. Schematically, if \(J_i\) is the last visit with an observed outcome, the contribution is

L_i(\theta,\psi) = \int \left\{\prod_{j:R_{ij}=1} f_Y(Y_{ij}\mid A_i,X_i,b;\theta)\right\} f_R(R_i\mid A_i,X_i,b;\psi) \,dF_b(b\mid X_i,A_i).

The form of \(f_R\) includes the appropriate dropout and non-dropout probabilities for the observed history. It must reflect the study's follow-up and censoring rules; administrative end of follow-up should not be treated as participant dropout. The integral means that the model does not simply discard participants after their last observed outcome. Their observed outcome history and dropout time both inform which latent effects are plausible, and thus contribute to estimation of the shared model.

The target contrast is calculated from the fitted outcome model, not by substituting a dropout hazard into the definition of the outcome. For a specified visit \(j\) and covariate value \(x\), the population prediction under assignment \(a\) is

\mu_j(a,x) = \int E(Y_{ij}\mid A_i=a,X_i=x,b;\theta)\,dF_b(b\mid x,a).

Average these predictions over the covariate distribution of the target population if the target is not a fixed covariate profile. Then contrast the treatment and control predictions on the chosen scale: difference, ratio, or another defined measure. The distribution used for averaging must match the target. In a randomized study with a common baseline population, it is generally inappropriate to average each arm over the latent-effect distribution among that arm's observed survivors.

Key distinction: The dropout model affects estimation of the shared model through the joint likelihood. The population treatment contrast is then obtained by averaging outcome predictions over the target distribution, including latent effects. It is not generally the contrast among observed survivors, whose latent-effect distribution may have been altered by dropout.

Linear and Nonlinear Outcome Models

For a linear outcome model with an identity link and mean-zero random effects, integration often has a simple form. For example, if

E(Y_{ij}\mid A_i,t_{ij},b_i) = \beta_0+\beta_A A_i+\beta_t t_{ij} +\beta_{At}A_i t_{ij}+b_{0i}+b_{1i}t_{ij},

and the target distribution of the random effects has \(E(b_{0i})=E(b_{1i})=0\) in both treatment arms, the population mean difference at time \(t\) is \(\beta_A+\beta_{At}t\). The shared dropout process can affect the fitted coefficients and their uncertainty, but the random effects cancel from this mean contrast under the stated assumptions.

For a nonlinear link, the conditional contrast usually does not equal the population-averaged contrast. In a logistic outcome model, for instance, averaging \(\operatorname{logit}^{-1}(\eta+b)\) over \(b\) is not the same as applying the inverse-logit to the average linear predictor. Thus, exponentiating a conditional treatment coefficient gives a conditional odds ratio; it does not in general give a population-averaged risk ratio or risk difference.

Worked Example: A Time-Specific Contrast in a Linear Model

Suppose a hypothetical longitudinal study fits the linear outcome model above, with time measured in weeks. The joint outcome-dropout fit gives \(\hat{\beta}_A=1.20\) outcome points and \(\hat{\beta}_{At}=0.35\) points per week. The fitted random intercept and slope have mean zero in both arms. Estimate the population mean difference at week 4.

Under the identity-link model and the stated mean-zero assumption, integrating over the random effects gives \(\Delta(4)=\beta_A+4\beta_{At}\). Substituting the fitted values,

\widehat{\Delta}(4) = 1.20+4(0.35) = 2.60\text{ points}.

For illustration, suppose the estimated variances and covariance of \(\hat{\beta}_A\) and \(\hat{\beta}_{At}\) are \(0.16\), \(0.0100\), and \(-0.0125\), respectively. The estimated variance of the contrast is

\widehat{\operatorname{Var}}\{\widehat{\Delta}(4)\} = 0.16+4^2(0.0100)+2(4)(-0.0125) = 0.220, \qquad SE= \sqrt{0.220}=0.4690.

A Wald 95% confidence interval is \(2.60\pm1.96(0.4690)\), or approximately \(1.681\) to \(3.519\) points. Under the fitted model and its assumptions, the treatment group is estimated to have a mean outcome 2.60 points higher at week 4; the interval describes uncertainty in that estimated contrast. This calculation assumes the displayed covariance matrix pertains to the fitted parameters and uses the specified linear contrast. In an actual analysis, uncertainty from all fitted parameters and any standardization over covariates must also be included.

Worked Example: Integrating a Binary Outcome Model

Consider a hypothetical binary outcome at one visit. Suppose a fitted logistic model has two equally weighted latent-effect values, \(b=-1\) and \(b=1\), for the target population. The control linear predictors at those values are \(-1.50\) and \(0.50\). The treatment coefficient is \(0.80\), so the treatment predictors are \(-0.70\) and \(1.30\). Calculate the population-averaged risks, risk difference, and risk ratio. These values are illustrative; the two-point distribution is used to make the integration visible.

The inverse-logit is \(\operatorname{logit}^{-1}(u)=1/(1+e^{-u})\). Average the predicted risks over the two equally weighted latent-effect values. Using full precision in the calculations,

\begin{aligned} \hat p_0 &=\tfrac12\{\operatorname{logit}^{-1}(-1.50) +\operatorname{logit}^{-1}(0.50)\}\\ &=\tfrac12(0.1824255+0.6224593) =0.4024424,\\ \hat p_1 &=\tfrac12\{\operatorname{logit}^{-1}(-0.70) +\operatorname{logit}^{-1}(1.30)\}\\ &=\tfrac12(0.3318122+0.7858350) =0.5588236. \end{aligned}

The risk difference is \(0.5588236-0.4024424=0.1563812\), which rounds to \(0.1564\), or 15.64 percentage points. The risk ratio is \(0.5588236/0.4024424=1.3886\), which rounds to \(1.389\). These are population-averaged estimates under the specified latent-effect distribution. The conditional odds ratio from the treatment coefficient is \(e^{0.80}=2.226\); it is a different measure on a different scale and is not the risk ratio. Rounding the risks before calculating the contrasts can change the last reported digits, so retain full precision until the final results.

Why Survivor Comparisons Can Differ

Dropout can make the latent-effect distribution among observed participants different from its distribution in the original target population. A comparison of observed means therefore targets a survivor-conditioned quantity, not necessarily the population contrast at that visit. The following calculation separates those quantities in a small hypothetical example with known model values.

Worked Example: Population Contrast Versus Observed-Survivor Contrast

Suppose half the population has \(b=-1\) and half has \(b=1\). At the target visit, the outcome is \(Y=10+2A+b\). Thus the treatment effect is 2 outcome units at either value of \(b\). Suppose the probabilities of remaining observed to that visit are 0.8 and 0.4 for control participants with \(b=-1\) and \(b=1\), and 0.5 and 0.2 for treated participants with \(b=-1\) and \(b=1\), respectively.

The full-population control mean is \(\{9+11\}/2=10\), and the full-population treatment mean is \(\{11+13\}/2=12\). The population contrast is therefore \(12-10=2\) outcome units. Among observed controls, the two outcome values have weights proportional to their survival probabilities, giving

\bar Y_{0,R=1} = \frac{0.8(9)+0.4(11)}{0.8+0.4} = \frac{11.6}{1.2} = 9.667.

Among observed treated participants, the corresponding survivor mean is

\bar Y_{1,R=1} = \frac{0.5(11)+0.2(13)}{0.5+0.2} = \frac{8.1}{0.7} = 11.57.

The observed-survivor difference is \(11.57-9.667=1.905\) units (using unrounded means, \(1.9048\)), not the full-population contrast of 2. The difference arises because survival selects different mixtures of latent effects in the two arms. In a real analysis these quantities are not known from the dropout probabilities alone; the joint model estimates them under its assumptions.

Inference, Diagnostics, and Alternatives

For a linear contrast of fitted parameters, a Wald standard error can be computed from the fitted covariance matrix. For nonlinear contrasts or marginal predictions that involve integration, use a delta method based on the full fitted-parameter covariance matrix, or a suitable bootstrap that refits the joint model in each resample. A confidence interval should reflect uncertainty in the outcome parameters and, where relevant, the dropout parameters and latent-effect distribution. Reporting only the conditional treatment coefficient's standard error may understate or misrepresent uncertainty in a marginal contrast.

Before interpreting an estimate, check the following:

Target and scale: State the visit, population, and contrast scale. A conditional log-odds coefficient, marginal risk difference, and survivor mean difference are not interchangeable.
Dropout history: Check that the modeled hazard or dropout probability matches the actual observation schedule, including the distinction between dropout and administrative censoring.
Fit and integration: Review convergence, numerical integration accuracy, and sensitivity to quadrature settings. Unstable estimates or large changes with integration settings undermine precision.
Overlap and support: Look for treatment-covariate or latent-effect regions that are weakly represented in observed histories. Population predictions in such regions rely heavily on model extrapolation.
Model adequacy: Check outcome and dropout fit, and assess sensitivity to plausible latent-effect distributions and link functions. A well-converged fit is not proof that these assumptions are correct.

A shared-parameter analysis is not the only way to address dropout. Inverse-probability weighting uses estimated observation probabilities to reweight observed outcomes and requires adequate positivity and a correctly specified observation model. Multiple imputation specifies a distribution for missing outcomes and combines completed-data analyses; its validity depends on the imputation assumptions. A pattern-mixture approach directly models outcome distributions by missingness pattern and requires identifying restrictions for unobserved outcomes. These approaches target related questions but make different modeling commitments. The standard material on Overlaps Missing Data provides broader missing-data context; the focus here is how contrasts are constructed from a shared-parameter fit.

A central limitation is that outcomes after dropout are unobserved. The observed data alone generally cannot verify the assumed relationship between latent effects, outcomes, and dropout in those unobserved periods. Shared parameters create a structured extrapolation, not direct information about every missing outcome. Consequently, report the estimand, outcome and dropout specifications, latent-effect assumptions, estimation method, uncertainty, and sensitivity of conclusions to plausible alternatives. A precise estimate from one joint model does not remove uncertainty about its untestable assumptions.

Common Mistake

Pitfall: “The fitted treatment coefficient is the population treatment effect, so I can report it without further calculation.” This reasoning is valid only for particular outcome models and estimands. In a linear identity-link model, a coefficient or linear combination may equal the marginal mean contrast under stated mean-zero assumptions. In a nonlinear model, averaging conditional predictions generally changes the contrast. The correction is to define the target first, predict under each treatment assignment, integrate over the target distribution of latent effects and covariates, and then calculate the contrast on the intended scale.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A logistic outcome model has treatment coefficient \(\beta_A\). Does \(e^{\beta_A}\) generally equal the population-averaged risk ratio? Explain.
In a linear random-intercept model with treatment effect \(\beta_A\) and mean-zero random intercepts in both arms, what does integrating over the random intercept do to the mean difference?
For the binary worked example, which values should be used when calculating the risk difference and risk ratio: the unrounded integrated risks or their rounded display values?
Why can the treatment contrast among observed survivors differ from the contrast for the original study population?
Name two assumptions or diagnostics that should be considered before interpreting a shared-parameter treatment contrast.

Solutions. (1) No. \(e^{\beta_A}\) is the conditional odds ratio under the logistic model; integration over latent effects produces population-averaged risks, from which a risk ratio must be calculated. (2) The random intercept averages to zero in each arm, so it cancels from the marginal mean difference, which is \(\beta_A\), provided the stated assumptions hold. (3) Use the unrounded integrated risks for arithmetic, then round the final contrast. (4) Dropout can change the latent-effect mixture among observed participants, so their conditional outcome means need not represent the original population. (5) Examples include correct specification of the dropout and outcome models, a plausible latent-effect distribution, adequate treatment and covariate overlap, and reliable convergence and numerical integration.