Conditional Dropout Odds and Marginal Risks

From Conditional Dropout to Marginal Risk

A dropout model can describe how the chance of dropping out changes with a participant’s current outcome. Its fitted probabilities are conditional on that outcome. A marginal dropout risk answers a different question: among participants at risk of dropout, what proportion is expected to drop out when their outcomes vary according to a specified distribution? Converting between these quantities requires averaging probabilities over the outcome distribution, not simply substituting a typical outcome.

This distinction matters in longitudinal selection models because the current outcome may be unobserved for participants who drop out at that visit. The outcome distribution used for averaging must therefore be specified by the model and the target population, rather than automatically taken to be the distribution among participants who remain observed.

Definition: At visit \(t\), let \(R_t=1\) indicate that a participant is at risk of dropout, let \(D_t=1\) indicate dropout at that visit, let \(Y_t\) be the current outcome, and let \(H_{t-1}\) denote the history available before the dropout decision. The conditional dropout risk at outcome \(y\) and history \(h\) is \(P(D_t=1\mid R_t=1,Y_t=y,H_{t-1}=h)\). The marginal dropout risk at history \(h\) averages this conditional probability over the distribution of \(Y_t\) among the target participants at risk, conditional on that history.

For a logistic current-outcome model, write the conditional risk as

q_t(y,h)=P(D_t=1\mid R_t=1,Y_t=y,H_{t-1}=h) =\frac{\exp\{\eta_t(h)+\gamma_t y\}}{1+\exp\{\eta_t(h)+\gamma_t y\}}.

Here \(\eta_t(h)\) is the history-specific intercept and \(\gamma_t\) is the change in conditional log odds per one-unit increase in the outcome, under this model. The model assumes that, conditional on the specified history and current outcome, the dropout probability follows this logistic form. Its marginal risk is

r_t(h)=\int q_t(y,h) f_t(y\mid R_t=1,H_{t-1}=h)\,dy.

The density \(f_t\) is the modelled outcome distribution for the target risk set. For a discrete outcome, the integral is replaced by a sum over its possible values. If the target is a population that averages over histories as well, the history-specific risks must also be averaged over the target distribution of \(H_{t-1}\).

Why Averaging Is Necessary

The logistic function is nonlinear. Consequently, in general, the mean of the conditional probabilities is not the conditional probability evaluated at the mean outcome:

E\{q_t(Y_t,h)\mid R_t=1,H_{t-1}=h\} \ne q_t\{E(Y_t\mid R_t=1,H_{t-1}=h),h\}.

The right-hand side is a plug-in probability for a participant whose outcome equals the mean. It is not, in general, the expected dropout risk across participants with a spread of outcomes. Nor is the marginal risk obtained by averaging conditional odds and converting that average to a probability. The risk is the average of the conditional probabilities themselves.

A short Taylor expansion explains one way outcome variability can matter. Let \(g(u)=1/(1+\exp(-u))\), and let \(\mu_t(h)\) and \(v_t(h)\) be the mean and variance of the at-risk outcome distribution at history \(h\). Expanding \(g\{\eta_t(h)+\gamma_tY_t\}\) around \(\eta_t(h)+\gamma_t\mu_t(h)\), then taking expectations, gives the second-order approximation

r_t(h)\approx g\{\eta_t(h)+\gamma_t\mu_t(h)\} +\frac{\gamma_t^2v_t(h)}{2} g''\{\eta_t(h)+\gamma_t\mu_t(h)\}, \qquad g''(u)=g(u)\{1-g(u)\}\{1-2g(u)\}.

The first-order term involving \(Y_t-\mu_t(h)\) has expectation zero; the second-order term depends on the outcome variance and the curvature of the logistic function. This is an approximation, not a substitute for integration when an accurate marginal risk is needed. It also shows why a common shift in the outcome mean is not the only distributional feature that can affect the risk.

Worked Example: Averaging Over a Three-Point Outcome Distribution

Suppose a hypothetical visit has conditional dropout risk \(q(y)=\operatorname{logit}^{-1}(-2+y)\), where \(\operatorname{logit}^{-1}(u)=1/(1+\exp(-u))\). Among participants at risk, suppose \(Y_t\) takes values 0, 1, and 2 with probabilities 0.20, 0.50, and 0.30. These probabilities describe the assumed outcome distribution for this risk set.

The conditional risks at the three outcome values are

q(0)=\operatorname{logit}^{-1}(-2)=0.1192,\quad q(1)=\operatorname{logit}^{-1}(-1)=0.2689,\quad q(2)=\operatorname{logit}^{-1}(0)=0.5000.

Weight each probability by the corresponding conditional risk:

$$ r=(0.20)(0.1192)+(0.50)(0.2689)+(0.30)(0.5000) =0.02384+0.13445+0.15000 =0.3083. $$

The marginal dropout risk is therefore about 30.8% for this specified at-risk population. Its mean outcome is \(0(0.20)+1(0.50)+2(0.30)=1.10\). Substituting that mean into the conditional model instead gives \(\operatorname{logit}^{-1}(-2+1.10)=0.2891\), not 0.3083. The difference arises because the conditional probability is nonlinear in the outcome.

Continuous Outcomes and Numerical Integration

For a continuous outcome, the same definition applies, but a probability density replaces the discrete weights. A closed-form expression for the logistic model averaged over a normally distributed outcome is generally unavailable. Numerical integration or simulation can nevertheless calculate the marginal risk. The outcome distribution and the conditional dropout model must both be specified; a numerical method cannot correct a misspecified distribution.

Worked Example: A Logistic Model With a Normal Outcome

For a hypothetical risk set, suppose \(Y_t\sim N(0,1)\), where the second argument is the variance, and suppose \(q(y)=\operatorname{logit}^{-1}(-1+0.8y)\). Thus the outcome mean is 0 and its standard deviation is 1, in the outcome’s measurement units. The marginal risk is

r=\int_{-\infty}^{\infty} \operatorname{logit}^{-1}(-1+0.8y)\, \frac{1}{\sqrt{2\pi}}\exp(-y^2/2)\,dy.

For example, a five-node Gauss–Hermite approximation for a standard normal variable uses nodes \(0,\pm1.356,\pm2.857\), with respective weights \(0.533333\), \(0.222076\) for each of \(\pm1.356\), and \(0.011257\) for each of \(\pm2.857\). Evaluating the conditional risks gives approximately \(0.2689\) at 0, \(0.5212\) and \(0.1106\) at \(1.356\) and \(-1.356\), and \(0.7834\) and \(0.0361\) at \(2.857\) and \(-2.857\). Therefore,

\begin{aligned} r &\approx (0.533333)(0.2689) +(0.222076)(0.5212+0.1106)\\ &\quad +(0.011257)(0.7834+0.0361)\\ &\approx 0.1434+0.1403+0.0092 =0.2929. \end{aligned}

The estimated marginal risk is about 29.3%. Plugging the mean outcome, 0, into the conditional model gives \(q(0)=0.2689\), which is lower than the integrated risk. The quadrature result is an approximation; in practice, one can increase the number of nodes or compare with simulation to check numerical stability.

Marginalizing Across Groups and Histories

The target population matters even if the conditional dropout model is held fixed. Consider groups with different outcome distributions. The group-specific marginal risks must first be calculated using each group’s own at-risk outcome distribution. A population risk is then a weighted average of those risks, using the group proportions in the target population. If the conditional intercept also varies by history or group, use the appropriate intercept for each component before averaging.

Worked Example: The Same Conditional Model, Different Marginal Risks

Suppose a binary outcome \(Y_t\) has conditional dropout risk \(q(y)=\operatorname{logit}^{-1}(-1.5+1.2y)\). In hypothetical Group A, 25% of those at risk have \(Y_t=1\); in Group B, 75% do. The conditional risks are

q(0)=\operatorname{logit}^{-1}(-1.5)=0.1824,\qquad q(1)=\operatorname{logit}^{-1}(-0.3)=0.4256.

For Group A, the marginal risk is

$$ r_A=(0.75)(0.1824)+(0.25)(0.4256) =0.1368+0.1064=0.2432. $$

For Group B, it is

$$ r_B=(0.25)(0.1824)+(0.75)(0.4256) =0.0456+0.3192=0.3648. $$

Thus the marginal dropout risks are 24.3% and 36.5%, respectively, despite using the same conditional model. The difference results from the groups’ different distributions of the current outcome. If the target population were 40% Group A and 60% Group B, its marginal risk would be \((0.40)(0.2432)+(0.60)(0.3648)=0.3162\), or 31.6%. This calculation assumes those group proportions describe the target risk set.

Assumptions, Checks, and Alternatives

The averaging formula is exact for a correctly specified conditional risk and the stated outcome distribution. Its use in practice rests on several substantive and computational requirements:

Correct risk-set definition. The distribution being averaged must correspond to participants at risk at the dropout decision, conditional on the histories included in the target. Using a distribution from a different visit or population changes the estimand.
Compatible outcome and dropout models. The outcome distribution must represent the current outcome for the risk set, including outcomes that may be unobserved when dropout occurs. If the observed outcomes of continuing participants are used as a substitute without justification, the resulting risk may be biased.
Appropriate conditional model. The logistic form, its included history terms, and its dependence on the current outcome must be reasonable over the outcome range used in integration. Extrapolating a linear predictor far beyond observed outcomes can make the marginal calculation sensitive to unsupported model assumptions.
Accurate integration. For numerical integration, check that increasing quadrature nodes or simulation draws does not materially change the estimate. Large changes indicate inadequate numerical approximation or a difficult tail contribution.

Useful diagnostics include comparing the assumed outcome distribution with observed outcome data where available, checking its fit across histories and groups, and inspecting the range of simulated or quadrature outcomes against the range supported by data. These checks cannot verify the distribution of outcomes that are systematically unobserved after dropout. That part of the model is an assumption and should be examined through sensitivity analyses when it materially affects conclusions.

The main alternative to explicit integration is the plug-in calculation \(q\{E(Y_t)\}\). It is simple, but generally answers a different question and ignores outcome variation. A model that does not depend on \(Y_t\) has a constant conditional risk within the specified history, so averaging over \(Y_t\) leaves that risk unchanged. That simplification is valid only when the conditional model actually omits current-outcome dependence or its coefficient is zero. A different link function may change the shape of the conditional probability curve and the details of integration, but it does not remove the need to average conditional probabilities over the target outcome distribution.

Common pitfall: “The mean outcome is 1.10, so the marginal dropout risk is the logistic-model risk at 1.10.” This is generally wrong because averaging a nonlinear probability is not the same as evaluating it at an average outcome. Calculate the weighted average or integral of conditional probabilities using the specified at-risk outcome distribution.

The marginal risk is a population summary, not an individual-level conditional probability. It depends on both the conditional dropout model and the outcome distribution over which it is averaged. Reporting only the conditional model does not determine a unique marginal risk unless the target population and its outcome distribution are also specified.

Check Your Understanding

Use the stated assumptions for each question. Brief solutions follow.

A binary outcome has conditional dropout risks 0.10 when \(Y_t=0\) and 0.40 when \(Y_t=1\). If 30% of those at risk have \(Y_t=1\), calculate the marginal risk.
For a nonlinear logistic dropout model, explain why \(q\{E(Y_t)\}\) need not equal \(E\{q(Y_t)\}\).
Two groups have the same conditional dropout model but different distributions of \(Y_t\). Can their marginal dropout risks differ? State why.
Name the outcome distribution needed to calculate a history-specific marginal dropout risk, and identify one reason the observed completers’ outcome distribution might not be an adequate substitute.

Solutions. (1) The risk is \((0.70)(0.10)+(0.30)(0.40)=0.07+0.12=0.19\), or 19%. (2) A logistic probability is nonlinear in \(Y_t\); averaging its values generally differs from evaluating it at the mean. The conditional probabilities must be averaged over the outcome distribution. (3) Yes. The marginal risk weights the conditional risks by the group-specific outcome probabilities, so different outcome distributions can produce different weighted averages. (4) Use the distribution of the current outcome among the target participants at risk, conditional on the specified history. Completers may not represent that distribution if dropout depends on the current outcome, because those who remain observed can have a different outcome composition.