Conditional Prediction of Outcomes After Dropout

Why Condition Predictions on Dropout?

After a participant drops out, their later outcomes are unobserved, but a longitudinal outcome model still defines a distribution for those outcomes. Under a Diggle–Kenward selection model, dropout may depend on the current outcome as well as on the history already observed. Consequently, the outcomes of participants who drop out need not follow the same conditional distribution as outcomes predicted from the outcome model alone. This tutorial derives that change for a Gaussian outcome model with a probit dropout hazard.

The prediction is a model-based distribution conditional on the observed history and dropout event. It is not a recovered measurement of a particular person’s missing outcome. Its form depends on both the outcome model and the specified selection equation.

Definition: Let $H$ denote the observed history immediately before a dropout decision at visit $t$. Let $W=(Y_t,Y_{t+1},\ldots,Y_T)^\top$ contain the current outcome and the subsequent outcomes that become unobserved. Suppose, conditional on $H$, $W\sim N(m,\Sigma)$, where $m$ and $\Sigma$ may depend on $H$. Let $D=1$ indicate dropout at visit $t$. A current-outcome probit selection equation is $$ P(D=1\mid W,H)=\Phi(a+\gamma Y_t), $$ where $a$ is a dropout linear-predictor term determined by the observed history and covariates, $\gamma$ is the current-outcome selection coefficient, and $\Phi$ is the standard normal distribution function.

The model assumes that the conditional outcome vector is multivariate Gaussian and that, given the current outcome and observed history, the dropout probability follows the stated probit equation. A latent normal error used to represent this probit event is independent of $W$ conditional on $H$. Dropout is monotone: after dropout at $t$, later outcomes are unobserved. The distribution derived below concerns the outcomes that would be defined by the outcome model, conditional on this dropout event.

For a vector formulation, let $e_1=(1,0,\ldots,0)^\top$, so $Y_t=e_1^\top W$, and set $b=\gamma e_1$. Then the hazard is $\Phi(a+b^\top W)$. Define $$ s=\sqrt{1+b^\top\Sigma b},\qquad q=\frac{a+b^\top m}{s},\qquad \lambda(q)=\frac{\phi(q)}{\Phi(q)},\qquad \delta(q)=\lambda(q)\{\lambda(q)+q\}, $$ where $\phi$ is the standard normal density. For finite $q$, $\Phi(q)>0$, so these quantities are defined.

The Distribution After Conditioning on Dropout

The key result gives the dropout probability and the first two moments of $W$ conditional on dropout. The conditional distribution is generally not multivariate normal: the probit hazard reweights the Gaussian outcome density. The moment formulas are nevertheless available in closed form.

Theorem: Under the stated Gaussian outcome and probit dropout model, $$ P(D=1\mid H)=\Phi(q), $$ and the conditional mean and covariance of the unobserved outcome vector are $$ E(W\mid H,D=1)=m+\frac{\Sigma b}{s}\lambda(q), $$ $$ \operatorname{Var}(W\mid H,D=1) =\Sigma-\frac{\Sigma b b^\top\Sigma}{s^2}\delta(q). $$ These expressions describe outcomes conditional on both the observed history and dropout; $m$ and $\Sigma$ alone describe outcomes conditional only on the history.

To justify the result, introduce an independent $E\sim N(0,1)$ and define $U=a+b^\top W+E$. The probit specification is equivalent to $D=1$ when $U>0$, because conditional on $W,H$, $P(U>0)=\Phi(a+b^\top W)$. Given $H$, $U$ is normal with mean $a+b^\top m$ and variance $s^2=1+b^\top\Sigma b$. Thus $P(U>0\mid H)=\Phi(q)$.

For the moments, Gaussian conditioning gives $$ E(W\mid U,H)=m+\frac{\Sigma b}{s^2}\{U-(a+b^\top m)\}, $$ and $$ \operatorname{Var}(W\mid U,H)=\Sigma-\frac{\Sigma b b^\top\Sigma}{s^2}. $$ For a standard normal $Z$, truncation to $Z>-q$ gives $E(Z\mid Z>-q)=\lambda(q)$ and $\operatorname{Var}(Z\mid Z>-q)=1-\delta(q)$. These identities follow by integrating the standard normal density over $(-q,\infty)$; in particular, $\int_{-q}^\infty z\phi(z)\,dz=\phi(q)$. Therefore, conditional on $U>0$, the mean of $U-(a+b^\top m)$ is $s\lambda(q)$, and its variance is $s^2\{1-\delta(q)\}$. Substitution into the conditional Gaussian moments yields the stated mean and covariance.

The mean shift is proportional to $\Sigma b$, not just to the current outcome’s variance. Thus selection on $Y_t$ can also shift predictions for later outcomes correlated with $Y_t$. The covariance update acts in the same direction: it reduces variance along the component of the outcome vector associated with the selection predictor.

Worked Example: A Two-Visit Prediction After Dropout

Suppose a hypothetical biomarker is measured at a dropout visit and one later visit. Conditional on the observed history $H$, let $W=(Y_t,Y_{t+1})^\top$ have mean $m=(0,0)^\top$ points and covariance $$ \Sigma=\begin{pmatrix}4&2\\2&3\end{pmatrix} $$ in squared points. The determinant is $4(3)-2(2)=8>0$, and the diagonal entries are positive, so this covariance matrix is positive definite. Assume $P(D=1\mid W,H)=\Phi(-1+0.5Y_t)$. The coefficient is in inverse points, and dropout is more likely at higher current outcomes.

Here $a=-1$, $b=(0.5,0)^\top$, and $$ \Sigma b=\begin{pmatrix}2\\1\end{pmatrix},\qquad b^\top\Sigma b=1,\qquad s=\sqrt{2},\qquad q=-\frac{1}{\sqrt{2}}\approx-0.7071. $$ The model-implied dropout probability conditional on $H$ is $\Phi(q)\approx0.2398$. Using $\phi(q)\approx0.3107$, $\lambda(q)=\phi(q)/\Phi(q)\approx1.296$, and $\delta(q)=\lambda(q)\{\lambda(q)+q\}\approx0.7631$.

The mean shift is $$ \frac{\Sigma b}{s}\lambda(q) =\begin{pmatrix}2\\1\end{pmatrix}\frac{1.296}{\sqrt{2}} \approx\begin{pmatrix}1.833\\0.9164\end{pmatrix} $$ points. Thus the conditional predicted means after dropout are approximately $1.833$ points for $Y_t$ and $0.9164$ points for $Y_{t+1}$, rather than zero.

For the covariance, the rank-one matrix $\Sigma b b^\top\Sigma$ is $(\Sigma b)(\Sigma b)^\top=\begin{pmatrix}4&2\\2&1\end{pmatrix}$. Hence $$ \operatorname{Var}(W\mid H,D=1) =\begin{pmatrix}4&2\\2&3\end{pmatrix} -\frac{0.7631}{2}\begin{pmatrix}4&2\\2&1\end{pmatrix} \approx\begin{pmatrix}2.474&1.237\\1.237&2.618\end{pmatrix} $$ in squared points. For example, the current-outcome variance is $4-2(0.7631)\approx2.474$. These are conditional moments under the specified model, not values observed after dropout.

Using the Formula With an Observed History

In practice, $m$ and $\Sigma$ are conditional quantities. First use the outcome model to obtain the distribution of the unobserved outcome vector given the participant’s observed history and covariates. Then use the dropout equation to update that distribution. If the observed history enters the hazard, its contribution is held fixed when calculating $a$; the current unobserved outcome remains in the selection term.

Worked Example: History-Specific Prediction With No Mean Shift in Dropout Probability

For a hypothetical participant whose observed history yields $m=(5,6)^\top$ points and the same covariance matrix $\Sigma$ as above, suppose the history-specific hazard is $P(D=1\mid W,H)=\Phi(-2.5+0.5Y_t)$. This could arise from a model with a history term; once $H$ is fixed, its contribution is included in $a=-2.5$. The outcome covariance is positive definite as checked in the preceding example, and the latent probit error is assumed independent of the outcomes.

Now $\Sigma b=(2,1)^\top$, $s=\sqrt{2}$, and $a+b^\top m=-2.5+0.5(5)=0$, so $q=0$. Therefore $P(D=1\mid H)=0.5$, $\lambda(0)=\phi(0)/\Phi(0)\approx0.7979$, and $\delta(0)=\lambda(0)^2\approx0.6366$.

The conditional mean is $$ m+\frac{\Sigma b}{s}\lambda(0) =\begin{pmatrix}5\\6\end{pmatrix} +\begin{pmatrix}2\\1\end{pmatrix}\frac{0.7979}{\sqrt{2}} \approx\begin{pmatrix}6.128\\6.564\end{pmatrix} $$ points. The conditional variances are $4-2(0.6366)\approx2.727$ and $3-0.5(0.6366)\approx2.682$ squared points; the conditional covariance is $2-0.6366\approx1.363$ squared points. Although the dropout probability is one half for this history, dropout still changes the outcome distribution: the probability is evaluated at the history-specific mean, while the selection event also reweights individual outcomes around that mean.

Worked Example: Negative Selection on the Current Outcome

Keep the history-specific outcome distribution $m=(5,6)^\top$ and $\Sigma=\begin{pmatrix}4&2\\2&3\end{pmatrix}$, but suppose the hazard is $P(D=1\mid W,H)=\Phi(2.5-0.5Y_t)$. This is a different hypothetical model in which lower current outcomes make dropout more likely. The covariance remains positive definite, and the same independent-error probit assumption is used.

Here $b=(-0.5,0)^\top$, $\Sigma b=(-2,-1)^\top$, $s=\sqrt{2}$, and $a+b^\top m=2.5-0.5(5)=0$. Thus $q=0$, $P(D=1\mid H)=0.5$, and $\lambda(0)\approx0.7979$, $\delta(0)\approx0.6366$. The mean becomes $$ \begin{pmatrix}5\\6\end{pmatrix} +\begin{pmatrix}-2\\-1\end{pmatrix}\frac{0.7979}{\sqrt{2}} \approx\begin{pmatrix}3.872\\5.436\end{pmatrix} $$ points. The covariance is unchanged from the preceding example because it depends on $b$ through $\Sigma b b^\top\Sigma$, which is the same for $b$ and $-b$. The example separates two ideas: the sign of $\gamma$ determines the direction of the mean shift, while, at $q=0$, reversing that sign does not change the covariance update.

Diagnostics, Alternatives, and Limits

The closed-form result is useful only if its assumptions are a defensible description of the analysis. Check that the outcome model gives a plausible conditional mean and covariance for the future outcome vector, and that the probit hazard uses the correct temporal ordering: observed history is available before dropout, whereas $Y_t$ is the current outcome whose value may be unobserved. Check the direction and scale of $\gamma$, and inspect the implied dropout probabilities across plausible current-outcome values and histories. Implausibly near-zero or near-one probabilities, or a selection coefficient whose units do not match the outcome scale, can signal a specification or coding problem.

Also check numerical behavior when $q$ is far into the negative tail. In that region $\Phi(q)$ can be very small, so calculating $\phi(q)/\Phi(q)$ directly may be unstable even though the conditional distribution is mathematically defined. Stable normal-tail calculations are preferable. Finally, estimated parameters are uncertain: formulas evaluated at fitted values give plug-in predictions, not intervals that automatically account for estimation uncertainty.

A model that ignores dropout would use $m$ and $\Sigma$ directly, which is appropriate for this conditional prediction only when the selection mechanism does not reweight outcomes (for example, $\gamma=0$ in this specification). A pattern-mixture formulation instead describes outcome distributions by missingness pattern; it parameterizes the problem from a different direction and does not, by itself, supply this probit-based update. The selection model makes the link through the specified dropout equation, so predictions depend on that assumption.

The result is limited to a Gaussian conditional outcome vector and a probit selection equation linear in the current outcome with history fixed. It does not establish that the dropout mechanism is correct or identify its parameters from the observed data without further assumptions. With non-Gaussian outcomes or a different link, the same moment formulas need not apply. Moreover, the conditional law is generally skewed rather than Gaussian, so replacing it with a normal distribution having these moments may give inaccurate tail probabilities or interval predictions.

Common pitfall: It is wrong to predict future outcomes from $W\mid H$ and then argue that the dropout event is irrelevant because dropout occurred after the observed history. In this model dropout depends on the current outcome, so conditioning on $D=1$ reweights the outcomes: the mean changes by $\Sigma b\,\lambda(q)/s$. The correction is to condition jointly on $H$ and $D=1$, using the selection-adjusted distribution.

Key takeaway: Under Gaussian outcomes and a current-outcome probit hazard, dropout changes predictions for both the current unobserved outcome and correlated future outcomes. The change is determined by the outcome covariance with the selection predictor and by the history-specific standardized hazard $q$.

Check Your Understanding

Answer each question, then compare with the solutions below.

If $\gamma=0$, what happens to the conditional mean and covariance of $W$ after dropout?
Why can a dropout hazard depending only on $Y_t$ shift the predicted mean of $Y_{t+1}$?
For $q=0$, state the dropout probability and the values of $\lambda(q)$ and $\delta(q)$, to four significant figures.
In the negative-selection example, why does the mean shift point downward while the covariance update is the same as in the positive-selection example?
Give one reason the displayed moment formulas may be inadequate for predicting a tail probability after dropout.

Solutions. (1) Then $b=0$, $s=1$, and $\Sigma b=0$, so the conditional mean remains $m$ and covariance remains $\Sigma$. (2) If $Y_{t+1}$ is correlated with $Y_t$, the corresponding entry of $\Sigma b$ is nonzero; selection on $Y_t$ therefore shifts its conditional mean too. (3) The dropout probability is $\Phi(0)=0.5$; $\lambda(0)=\phi(0)/0.5\approx0.7979$, and $\delta(0)=\lambda(0)^2\approx0.6366$. (4) Reversing the sign of $b$ reverses $\Sigma b$, hence the mean shift, but leaves $\Sigma b b^\top\Sigma$ unchanged, hence the covariance update. (5) The selected distribution is generally skewed rather than normal; its mean and covariance do not determine its tail probabilities as they would for a normal distribution.

Conditional Prediction of Outcomes After Dropout

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