Shared Effects in Discrete-Time Dropout Hazards

Why Model Dropout One Interval at a Time?

When follow-up is recorded at scheduled visits, the time of dropout is often known only up to an interval: a participant attended one visit but not the next. A discrete-time model represents that process through a sequence of conditional probabilities. This makes it possible to distinguish the chance of dropping out in a particular interval from the cumulative chance of having dropped out by a later visit.

In a shared-parameter model, the dropout process can depend on a latent subject effect that also enters the outcome model. Conditional on that effect, subjects with different latent outcome tendencies may have different dropout hazards. Marginally, the outcome and dropout can therefore be associated even if their processes are specified to be conditionally independent given the shared effect and observed history.

Definition: Let \(D_{it}=1\) mean that subject \(i\) first drops out during interval \(t\), and let \(R_{it}=1\) mean that the subject is still under observation and at risk of dropout at the start of interval \(t\). The discrete-time dropout hazard is \(h_{it}=P(D_{it}=1\mid R_{it}=1,\text{ information available for subject }i)\). It is a conditional probability for one interval, not the probability of dropout by that interval.

For clarity, suppose dropout is absorbing: after a subject drops out, that subject does not re-enter the risk set. Let intervals be indexed \(t=1,\ldots,T\), and let \(b_i\) denote a latent random effect also used in the outcome model. A person-period logistic model can be written as

\operatorname{logit}\{h_{it}(b_i)\} =\log\left\{\frac{h_{it}(b_i)}{1-h_{it}(b_i)}\right\} =\alpha_t+\boldsymbol{\gamma}^{\mathsf T}\mathbf{z}_{it}+\lambda b_i.

Here \(\alpha_t\) is an interval-specific baseline log-odds, \(\mathbf{z}_{it}\) contains observed predictors available for that interval, and \(\lambda\) is the coefficient linking the shared effect to dropout. The inverse-logit gives \(h_{it}(b_i)=\exp(\eta_{it})/\{1+\exp(\eta_{it})\}\), where \(\eta_{it}\) is the linear predictor. A positive \(\lambda\) means that larger values of \(b_i\) increase the conditional dropout odds, holding the other terms fixed. The substantive direction depends on how the outcome model defines \(b_i\); the sign alone does not say whether outcomes themselves are higher or lower.

Assumptions and Conditional Dropout Likelihood

The person-period representation requires a well-defined interval structure and an unambiguous risk set. Each subject contributes a record for every interval in which they are at risk. The model treats dropout as occurring in at most one interval and assumes that the recorded dropout status is correctly classified. If exact event times are recorded within intervals, collapsing them into intervals discards timing information.

For the standard Bernoulli specification, conditional on \(b_i\), the observed predictors and the relevant past history, the dropout indicator in each at-risk interval has probability \(h_{it}(b_i)\). With no re-entry, the conditional dropout likelihood contribution is

L_{D,i}(b_i) =\prod_{t:R_{it}=1} h_{it}(b_i)^{D_{it}}\{1-h_{it}(b_i)\}^{1-D_{it}}.

For a subject who drops out during interval \(k\), the contribution is the probability of remaining observed through intervals before \(k\), multiplied by the hazard in interval \(k\):

L_{D,i}(b_i) =\left\{\prod_{t=1}^{k-1}[1-h_{it}(b_i)]\right\}h_{ik}(b_i).

For a subject who is still observed at the end of interval \(K\), the contribution through \(K\) is instead \(\prod_{t=1}^{K}[1-h_{it}(b_i)]\). That expression describes right-censoring at the end of the observed follow-up, not a dropout event. The distinction matters: a hazard factor is included only for an interval in which dropout is observed.

Derivation: By the chain rule, the probability of first dropout in interval \(k\), conditional on \(b_i\), is \(P(D_{i1}=0,\ldots,D_{i,k-1}=0,D_{ik}=1\mid b_i)=\left\{\prod_{t=1}^{k-1}P(D_{it}=0\mid D_{i1}=\cdots=D_{i,t-1}=0,b_i)\right\}P(D_{ik}=1\mid D_{i1}=\cdots=D_{i,k-1}=0,b_i)\). Each factor averages over the histories available at that interval among subjects who have survived so far. Conditional on the history available at each step, the factors are \(1-h_{it}(b_i)\) before \(k\) and \(h_{ik}(b_i)\) at \(k\); if histories are marginalized, their sequential distributions must also be accounted for. This does not require the dropout indicators to be marginally independent.

Worked Interval Calculations

Worked Example: Two Interval Hazards for One Subject

Hypothetical data and model. Suppose a subject is at risk in two intervals, has \(b_i=0.5\), and has no additional predictors. Let \(\alpha_1=-2.0\), \(\alpha_2=-1.5\), and \(\lambda=0.8\). The model assumptions here are that the subject remains at risk until dropout, the specified shared effect applies in both intervals, and each interval hazard follows the stated logistic model.

In interval 1, the linear predictor is \(-2.0+0.8(0.5)=-1.6\). Thus \(h_{i1}=\exp(-1.6)/\{1+\exp(-1.6)\}\approx0.1680\). In interval 2, if the subject remains at risk, the predictor is \(-1.5+0.8(0.5)=-1.1\), giving \(h_{i2}=\exp(-1.1)/\{1+\exp(-1.1)\}\approx0.2497\).

The probability of dropout in interval 2, conditional on being at risk at interval 1 and on \(b_i=0.5\), is \((1-h_{i1})h_{i2}\approx(0.8320)(0.2497)=0.2078\). The probability of remaining observed through both intervals is \((1-h_{i1})(1-h_{i2})\approx(0.8320)(0.7503)=0.6242\). The probability of dropout by the end of interval 2 is therefore \(1-(1-h_{i1})(1-h_{i2})\approx1-0.6242=0.3758\). These are different quantities: the interval-2 hazard is conditional on reaching interval 2, whereas the cumulative risk includes dropout in either interval.

More generally, conditional on a fixed value of \(b_i\), the probability of dropout by the end of interval \(K\) is one minus the probability of surviving all \(K\) intervals:

P(\text{dropout by }K\mid b_i) =1-\prod_{t=1}^{K}\{1-h_{it}(b_i)\}.

This is a conditional risk. To obtain a population-average risk under a random-effect distribution, average this conditional risk over that distribution. In general, one must not simply calculate a hazard at the mean random effect and treat it as the average hazard, because the inverse-logit and the product of survival probabilities are nonlinear.

Worked Example: Averaging Risk Over a Two-Point Shared-Effect Distribution

Hypothetical model. Let \(b_i\) take values \(+1\) and \(-1\), each with probability \(0.5\). Use two intervals, no additional predictors, \(\lambda=0.8\), \(\alpha_1=-2.0\), and \(\alpha_2=-1.5\). Thus the baseline terms and shared-effect coefficient are the same in both intervals; only \(b_i\) changes between the two conditional calculations.

For \(b_i=+1\), the predictors are \(-2.0+0.8=-1.2\) and \(-1.5+0.8=-0.7\). The hazards are \(h_{i1}\approx0.2315\) and \(h_{i2}\approx0.3318\). The conditional cumulative risk is

1-(1-0.2315)(1-0.3318) =1-(0.7685)(0.6682) \approx0.4865.

For \(b_i=-1\), the predictors are \(-2.0-0.8=-2.8\) and \(-1.5-0.8=-2.3\). The hazards are approximately \(0.0573\) and \(0.0911\), so the conditional cumulative risk is \(1-(0.9427)(0.9089)\approx0.1432\).

The population-average risk is the probability-weighted average of the two conditional risks: \(0.5(0.4865)+0.5(0.1432)\approx0.3149\). Using unrounded inverse-logit hazards gives the same result to four significant figures. Under this hypothetical distribution, the model-implied probability of dropout by the end of interval 2 is about \(0.3149\), or 31.49%. It is not the risk for a subject with \(b_i=0\); it averages over the stated distribution of latent effects.

Likelihood Contributions with Shared Outcomes

The dropout likelihood is one component of a joint model. Let \(Y_{it}\) denote an outcome observed in interval \(t\), and let \(f_Y(y_{it}\mid b_i,\text{history})\) denote its conditional probability mass or density under the outcome model. If the model specifies conditional factorization of the outcome and dropout processes given the shared effect and the observed history, a subject's conditional joint contribution is the product of the observed outcome contributions and the dropout contribution. The subject-level marginal likelihood then averages that product over the distribution of \(b_i\):

L_i=\int \left\{\prod_{t\in\mathcal{O}_i} f_Y(y_{it}\mid b_i,\text{history})\right\} L_{D,i}(b_i)\,f_b(b_i)\,db_i.

Here \(\mathcal{O}_i\) indexes intervals with observed outcomes and \(f_b\) is the specified random-effect density (or, for a discrete distribution, the corresponding probability mass function). The shared effect appears in both process models. Consequently, the outcome and dropout contributions must be multiplied conditional on \(b_i\) before integrating over \(b_i\); multiplying two separately averaged likelihoods would generally give a different result.

Worked Example: Joint Contribution for Dropout in the Second Interval

Hypothetical data and conditional models. A subject has \(b_i=0\), is observed in two intervals, records binary outcomes \(Y_{i1}=1\) and \(Y_{i2}=0\), and drops out at the end of interval 2. Suppose the outcome model gives \(P(Y_{i1}=1\mid b_i=0)=0.8\) and \(P(Y_{i2}=1\mid b_i=0)=0.6\). Suppose the dropout hazards at \(b_i=0\) are \(h_{i1}=0.2\) and \(h_{i2}=0.3\). We assume the interval-2 outcome is recorded before the end-of-interval dropout assessment, so both outcomes are observed, and we use the stated conditional factorization given \(b_i\).

The outcome contribution is \(0.8(1-0.6)=0.8(0.4)=0.32\). The dropout contribution is survival through interval 1 followed by dropout in interval 2: \((1-0.2)(0.3)=0.8(0.3)=0.24\). Thus the conditional joint contribution at \(b_i=0\) is \(0.32(0.24)=0.0768\).

If this subject were instead known to remain observed through the end of interval 2, the dropout contribution would be \((1-0.2)(1-0.3)=0.8(0.7)=0.56\), not \(0.24\). The likelihood contribution changes because the observed dropout pattern changes. In a random-effects model, \(0.0768\) is a conditional contribution at the specified \(b_i\), not yet a marginal contribution; marginalization requires averaging the joint contribution over the model for \(b_i\).

Diagnostics, Alternatives, and Limitations

The person-period layout makes the risk-set structure inspectable. Before interpreting coefficients, check that each subject contributes only intervals for which they are at risk, that a dropout event is recorded once, and that later intervals are not coded as additional dropout events. Check the time scale and interval boundaries as well: a hazard can change substantially with interval width, and a period-specific intercept is not automatically comparable across differently defined intervals.

Potential warning signs include nearly empty intervals, separation (for example, no observed dropouts in an interval for a predictor group), extreme fitted hazards near zero or one, and unexplained patterns in observed dropout rates across intervals. Also compare predicted and observed interval-specific dropout proportions within relevant covariate or outcome-history groups. Such comparisons can reveal gross misfit, although they cannot verify the latent-effect distribution or the assumed link between outcome and dropout.

The model is useful when event time is naturally recorded in discrete periods and interval-specific hazards are the target. A continuous-time hazard model instead represents risk over continuous time and may be more appropriate when exact event times are available; it is not obtained merely by relabeling these interval probabilities. Another alternative is a model without a shared latent effect, which does not capture the particular latent association represented here. The shared-effect model has its own limitations: results depend on the chosen outcome model, random-effect distribution, link, and conditional-independence specification. The likelihood calculations do not by themselves establish that these assumptions are correct or that the shared effect is identified by the available data.

Common pitfall: A student may call \(h_{it}\) “the probability of dropout by visit \(t\)” and add the hazards across intervals. That is incorrect because each hazard is conditional on still being at risk, and the subject must survive earlier intervals to reach a later one. Use \(1-\prod_{t=1}^{K}\{1-h_{it}\}\) for cumulative risk conditional on the shared effect, then average that risk over the random-effect distribution if a population-average risk is wanted.

Check Your Understanding

Answer each question using the definitions and formulas above. Solutions are provided below.

A subject has hazards \(0.10\) and \(0.20\) in two intervals. What is the probability of dropout in interval 2, conditional on being at risk in interval 1?
For those same hazards, calculate the cumulative dropout risk by the end of interval 2.
A subject drops out in interval 3 after surviving intervals 1 and 2. Write the conditional dropout likelihood contribution.
In a shared-effect model, what does a positive \(\lambda\) imply about dropout odds as \(b_i\) increases, holding other terms fixed?
Why must outcome and dropout likelihood contributions be combined conditional on \(b_i\) before averaging over \(b_i\)?

Solutions. (1) The subject must survive interval 1 and then drop out in interval 2: \((1-0.10)(0.20)=0.18\). (2) The risk is \(1-(1-0.10)(1-0.20)=1-(0.90)(0.80)=0.28\). (3) The contribution is \([1-h_{i1}(b_i)][1-h_{i2}(b_i)]h_{i3}(b_i)\). (4) Since the log-odds include \(\lambda b_i\), a positive \(\lambda\) makes the conditional dropout odds increase with \(b_i\), with other terms fixed. (5) Both process contributions depend on the same latent effect; integrating their product preserves that shared dependence, whereas multiplying separately averaged contributions generally does not.