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Shared-Parameter Analysis of Relapse and Attrition

Learn to specify and interpret a shared-parameter model for repeated relapse outcomes and attrition, and to assess how its dropout linkage affects the estimated treatment contrast.

Advanced 10 min read

What You'll Learn

  • Specify a random-effects outcome model and a dropout hazard that share a latent severity parameter.
  • Factor the observed-data likelihood under explicit conditional-independence assumptions.
  • Interpret how observed outcomes and dropout jointly update information about latent severity.
  • Calculate a treatment contrast and its Wald interval under a fitted model.
  • Diagnose model misfit and assess sensitivity to the assumed outcome–dropout linkage.

Why Link Relapse Outcomes and Attrition?

In an addiction trial, participants with worsening substance use may be more likely to stop attending visits or submitting reports. An analysis of observed outcomes alone can then give an incomplete picture: the chance that an outcome is observed may carry information about the outcome itself. A shared-parameter model represents one possible link by giving each participant a latent severity effect that influences both repeated substance-use outcomes and dropout.

This is a model-based way to analyze incomplete longitudinal data, not a guarantee that the missing outcomes have been recovered from the observed data. Its conclusions depend on the assumed outcome and dropout models, including the way the latent effect connects them. The model therefore needs both a careful likelihood and an assessment of how the treatment contrast changes under plausible alternatives for that connection.

Model and Assumptions

Definition: Let \(A_i\) denote participant \(i\)'s treatment assignment, with \(A_i=1\) for the trial treatment and \(A_i=0\) for control. Let \(Y_{it}\) be the substance-use severity score at visit \(t\), where larger values mean more severe use. Let \(b_i\) be a participant-specific latent severity effect. A shared-parameter model specifies both the repeated-outcome distribution and the dropout process conditional on \(b_i\), and integrates over the distribution of \(b_i\) to obtain the observed-data likelihood.

For illustration, consider the following outcome model, in which \(t\) is the visit number:

$$ Y_{it}\mid b_i,A_i \sim N\left(\beta_0+\beta_A A_i+\beta_t t+\beta_{At}A_i t+b_i,\ \sigma^2\right), \qquad b_i\sim N(0,\sigma_b^2). $$

The treatment contrast in mean severity at visit \(t\) is \(\beta_A+\beta_{At}t\): a negative contrast favors treatment because higher scores indicate greater severity. The random effect \(b_i\) shifts a participant's expected severity across visits. Its population mean is fixed at zero, so the treatment-specific population mean is obtained by averaging over \(b_i\).

Assume monotone dropout: after leaving follow-up, a participant has no later scheduled outcomes. At each visit \(t<T\), among participants still followed, let \(h_{it}\) be the probability of dropping out immediately after that visit. One possible dropout model is:

$$ \operatorname{logit}(h_{it})=\gamma_0+\gamma_A A_i+\gamma_t t+\alpha b_i, \qquad \operatorname{logit}(u)=\log\left(\frac{u}{1-u}\right). $$

Here \(\alpha\) is the shared-parameter association. If \(\alpha>0\), higher latent severity is associated with a higher dropout hazard, conditional on treatment and visit. This does not establish that severity causes dropout. Rather, it describes a joint probability model for the observed outcomes and follow-up process.

The likelihood below requires an important additional assumption: the repeated outcomes are conditionally independent given \(b_i\), treatment, and the covariates in the model. The conditional normal distributions alone do not imply this independence. If residual correlation remains after conditioning on \(b_i\), the outcome model must instead specify an appropriate joint conditional distribution. We also assume that, conditional on \(b_i\), treatment, visit, and included covariates, the dropout process is independent of the outcome residuals. The displayed hazard does not additionally depend on a participant's most recent observed score; that alternative can be modeled explicitly if clinically justified.

Observed-Data Likelihood

Let \(m_i\) be the last visit at which participant \(i\) has an observed outcome. If the participant drops out after visit \(m_i<T\), the dropout transition at \(m_i\) is observed. If the participant completes follow-up, the at-risk transitions are observed through \(T-1\). Write \(d_{it}=1\) for dropout immediately after visit \(t\) and \(d_{it}=0\) for continued follow-up. Let \(K_i=m_i\) for a participant who drops out before \(T\), and \(K_i=T-1\) for a completer. Conditional on \(b_i\), the participant's contribution is:

$$ L_i(b_i)= \left\{\prod_{t=1}^{m_i} f_Y(y_{it}\mid b_i,A_i)\right\} \left\{\prod_{t=1}^{K_i} h_{it}^{d_{it}}(1-h_{it})^{1-d_{it}}\right\}. $$

The first product includes only observed outcomes. The second includes the dropout or continued-follow-up transition at each visit when the participant was at risk. The marginal likelihood contribution is:

$$ L_i=\int L_i(b)\,\phi(b;0,\sigma_b^2)\,db, \qquad L=\prod_i L_i, $$

where \(\phi(b;0,\sigma_b^2)\) is the normal density for the random effect. The integral, rather than a likelihood evaluated at \(b_i=0\), accounts for uncertainty about each participant's latent severity. In practice it is evaluated numerically, for example by quadrature or simulation-based methods.

Justification: Conditional on \(b_i\), the assumed independence of the repeated outcome residuals gives the product of observed outcome densities. Conditional independence of the dropout process and those residuals gives the separate product of transition probabilities. Multiplying these conditional contributions gives \(L_i(b_i)\). Since \(b_i\) is unobserved, the law of total probability averages that joint contribution over its specified distribution, producing \(L_i\). If repeated outcomes are not conditionally independent, their product must be replaced by their specified joint observed-outcome density.

Worked Example: A Conditional Likelihood Contribution

Worked Example: A Conditional Likelihood Contribution

Suppose, hypothetically, that a participant is assigned to treatment (\(A_i=1\)), has latent effect \(b_i=0.5\), and reports a severity score of \(3.2\) at visit 1 before dropping out. Use \(\beta_0=2.0\), \(\beta_A=-0.4\), \(\beta_t=0.2\), \(\beta_{At}=-0.1\), \(\sigma=1\), and dropout parameters \(\gamma_0=-2.0\), \(\gamma_A=0.3\), \(\gamma_t=0.2\), \(\alpha=0.8\). Assume the stated conditional normal outcome model and logistic hazard apply.

The visit-1 conditional mean is:

$$ \mu_{i1}=2.0-0.4(1)+0.2(1)-0.1(1)(1)+0.5=2.2. $$

The outcome density at \(y_{i1}=3.2\) is therefore:

$$ f_Y(3.2\mid b_i=0.5,A_i=1) =\frac{1}{\sqrt{2\pi}}\exp\left[-\frac{(3.2-2.2)^2}{2}\right] \approx 0.2420. $$

The dropout log-odds after visit 1 are \(-2.0+0.3+0.2+0.8(0.5)=-1.1\). Thus:

$$ h_{i1}=\frac{\exp(-1.1)}{1+\exp(-1.1)}\approx 0.2497. $$

Because the participant drops out after that visit, the conditional contribution for the observed outcome and dropout transition is \(0.2420\times0.2497\approx0.06043\). This is a contribution conditional on \(b_i=0.5\), not the participant's marginal likelihood. To obtain the latter, the contribution must be integrated over the random-effects distribution. The assumed conditional independence of the outcome residual and dropout transition is what permits this product.

How the Shared Effect Informs Severity

A dropout event can contribute information about \(b_i\) through the hazard model. In a model with \(\alpha>0\), dropout makes higher values of \(b_i\) more plausible, all else equal. The observed outcomes also inform \(b_i\). The two sources of information are combined in the likelihood; neither should be treated as a direct observation of the latent effect.

Worked Example: Updating Latent Severity With Dropout

For a transparent calculation, suppose a hypothetical model gives \(b_i\) a two-point distribution: \(P(b_i=1)=P(b_i=-1)=0.5\). This is an illustrative alternative to the normal random-effects distribution above. At one visit, let \(Y_i\mid b_i\sim N(2+b_i,1)\), and let the probability of dropout immediately after the visit be \(\operatorname{logit}^{-1}(-2+b_i)\). A participant has observed \(Y_i=3\) and then drops out.

For \(b_i=1\), the outcome mean is 3, so its density at 3 is \(0.3989\), and the dropout probability is \(\operatorname{logit}^{-1}(-1)=0.2689\). For \(b_i=-1\), the outcome mean is 1, so its density at 3 is \(0.0540\), and the dropout probability is \(\operatorname{logit}^{-1}(-3)=0.04743\). The unnormalized posterior weights are:

$$ w_+=0.5(0.3989)(0.2689)\approx0.05365,\qquad w_-=0.5(0.0540)(0.04743)\approx0.001280. $$

Normalizing gives:

$$ P(b_i=1\mid Y_i=3,\text{ dropout}) =\frac{0.05365}{0.05365+0.001280} \approx0.9767. $$

The high-severity effect was already more plausible from the outcome alone: the corresponding posterior probability is \(0.3989/(0.3989+0.0540)\approx0.8808\). Dropout raises it further under the specified model. This result depends on the assumed prior distribution and hazard; it is not a model-free conclusion about the participant.

Estimating and Interpreting the Treatment Contrast

Parameters are typically estimated by maximizing the integrated likelihood, with uncertainty calculated from the fitted model. A treatment contrast is then reported for a clinically relevant visit, along with its standard error and an interval. Under the linear mean model above, the contrast at visit \(t\) is \(\beta_A+\beta_{At}t\). Because the random effect has the same mean in both arms, it cancels from this marginal mean contrast. The dropout linkage can nevertheless change the fitted coefficients: the likelihood uses the dropout pattern to help estimate the distribution of latent severity and the outcome parameters.

Worked Example: Treatment Contrast Under Alternative Linkage Assumptions

Suppose hypothetical model fits for severity at week 12 produce the following treatment contrasts and standard errors. These are illustrative fitted outputs, not estimates from an actual trial. Assume higher scores indicate worse severity, and use a Wald interval with multiplier 1.96.

Fixed linkage value \(\alpha\)Estimated week-12 contrastStandard error
0-0.700.34
0.5-0.820.36
1.0-1.050.41

For the primary illustrative fit with \(\alpha=0.5\), the estimated treatment-minus-control contrast is \(-0.82\) severity-score points at week 12. Its Wald statistic is \(-0.82/0.36=-2.278\). The two-sided standard normal p-value is approximately \(0.0227\), and the 95% Wald confidence interval is:

$$ -0.82\pm1.96(0.36)=-0.82\pm0.7056=(-1.526,\,-0.114). $$

Under this fitted model, the treatment group is estimated to have 0.82 points lower severity at week 12 than the control group. The interval is a model-based range of values compatible with the data and assumptions at the stated confidence level; it is not a probability statement that the fixed parameter lies in this particular interval. The p-value describes the extremeness of a statistic under the null contrast and the model, not the probability that the null is true.

The three estimates differ as the assumed linkage changes: \(-0.70\), \(-0.82\), and \(-1.05\) points. This illustrates why the linkage is a substantive modeling choice. The table alone does not establish which \(\alpha\) is correct; conclusions should be discussed across clinically plausible values, with uncertainty and model adequacy considered.

Diagnostics, Alternatives, and Limitations

The shared-parameter model is useful when a latent participant characteristic plausibly helps explain both repeated severity and attrition. It makes that explanation explicit and yields a joint likelihood. It is not uniquely identified merely because the likelihood can be fitted: participants' unobserved outcomes after dropout are not directly observed, so the assumed distribution and linkage matter. In particular, the size of \(\alpha\) is often weakly informed by the observed data and can be sensitive to the chosen random-effects distribution and hazard form.

Useful checks include comparing observed and model-predicted dropout proportions by treatment and visit; checking outcome residual patterns among observed visits; and examining whether the fitted model reproduces observed outcome summaries and dropout patterns together. Inspect convergence, numerical integration accuracy, and the estimated random-effects distribution. Implausibly large parameter estimates, unstable estimates across starting values, or a treatment contrast that changes sharply with small modeling changes are warnings, not reasons to select the most favorable result.

The model should also be compared conceptually with alternatives. A selection model specifies the outcome distribution together with a model for missingness conditional on the outcome; a pattern-mixture model specifies outcome distributions conditional on the missingness pattern. The shared-parameter model instead links the outcome and dropout processes through latent effects. These approaches express different assumptions about unobserved outcomes, and observed-data fit alone may not distinguish them. The earlier material on sensitivity analyses provides relevant context for reporting how assumptions affect conclusions.

A shared latent effect can be too simple if dropout also depends on recent observed relapse, adverse events, access to care, or other factors not represented in \(b_i\). Conversely, the model can impose excessive dependence if the residual-independence assumptions are false. Include justified covariates, specify residual dependence when needed, and report the model and sensitivity choices clearly. Even a well-fitting model does not turn an association between latent severity and dropout into a causal mechanism.

Common Pitfall

Common mistake: Multiplying the observed outcome densities and dropout probabilities after stating only the marginal conditional normal distribution of each repeated outcome. Those marginal distributions do not specify the joint distribution of repeated outcomes. Correction: explicitly assume conditional independence of outcomes given the shared effect and covariates, as done here, or specify a joint conditional outcome distribution and use its appropriate observed-data density. Independence between the outcome process and dropout process conditional on \(b_i\) does not itself imply independence among repeated outcomes.

Check Your Understanding

Answer each question, then compare with the solutions below.

  1. In the displayed model, what does a positive value of \(\alpha\) mean, and what does it not establish?
  2. Why is it necessary to integrate over \(b_i\) in the participant likelihood?
  3. Suppose the estimated treatment-minus-control contrast is \(0.4\) severity points and higher scores mean worse outcomes. Which group has the lower estimated severity?
  4. What assumption justifies multiplying the repeated outcome densities in the conditional likelihood?

Solutions. (1) Positive \(\alpha\) means higher latent severity is associated with higher dropout odds, conditional on included variables; it does not establish that severity causes dropout. (2) \(b_i\) is unobserved, so its uncertainty must be averaged over its specified distribution to form the marginal likelihood. (3) The positive contrast means the treatment group is estimated to have 0.4 points higher severity, so control has the lower estimated severity. (4) The model must assume that repeated outcomes are conditionally independent given \(b_i\), treatment, and covariates, or specify another joint conditional outcome distribution whose observed-data density is used.

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