Deriving the Joint Likelihood with Dropout Contributions

Why the Dropout Record Belongs in the Likelihood

In a shared-parameter model, a subject’s observed outcomes and dropout process are linked through latent shared effects. Once those effects are held fixed, the model specifies an outcome contribution and a dropout contribution. This tutorial derives how to combine them for one subject, without integrating over the shared effects.

Let $i$ index subjects, let $b_i$ denote subject $i$’s shared random effects, and let $Y_{i,\mathrm{obs}}$ denote the outcomes actually observed for that subject. Write $D_i$ for the dropout record, including the event time or observed follow-up time and whether dropout occurred. Let $\theta_y$ and $\theta_d$ denote the outcome-model and dropout-model parameters. The formulas below are conditional on a specified value of $b_i$.

Definition: A subject-level conditional joint likelihood contribution is the joint density or probability of that subject’s observed outcomes and dropout record, conditional on the shared effects. Under conditional independence of the outcome and dropout processes given $b_i$, it factors as $$ L_i(b_i;\theta_y,\theta_d) = p(Y_{i,\mathrm{obs}}\mid b_i;\theta_y)\, p(D_i\mid b_i;\theta_d). $$ The outcome factor is the joint density of the observed outcome values under the outcome model. The dropout factor is a density for an exact event time or a probability for an observed discrete-time record.

The factorization uses conditional independence given shared random effects: conditional on $b_i$, the outcome process and dropout process are independent. The observed-outcome density must still reflect the actual observations for that subject. If measurements after dropout are unavailable, their unobserved values are not inserted into $p(Y_{i,\mathrm{obs}}\mid b_i;\theta_y)$. Also, this factorization does not assert that outcomes and dropout are independent without conditioning on $b_i$; the shared effects can induce their association.

The contribution is conditional, not yet a marginal likelihood contribution. In particular, $L_i(b_i;\theta_y,\theta_d)$ is evaluated at a given $b_i$. Treating it as the likelihood based only on observed data without accounting for the latent effects would omit a necessary part of the model.

Continuous-Time Dropout and Right Censoring

Suppose dropout occurs at time $T_i$, and follow-up ends at time $C_i$ if dropout has not already occurred. The observed time is $U_i=\min(T_i,C_i)$, and let $\Delta_i=1$ indicate observed dropout by the end of follow-up and $\Delta_i=0$ indicate right censoring. Conditional on $b_i$, write $f_T(t\mid b_i)$ for the dropout-time density and $S_T(t\mid b_i)=P(T_i>t\mid b_i)$ for the dropout-free survival function.

Dropout contribution: For an observed event at $U_i$, the contribution is $f_T(U_i\mid b_i)$. For a subject right-censored at $U_i$, it is $S_T(U_i\mid b_i)$. Together, $$ p(D_i\mid b_i) = \{f_T(U_i\mid b_i)\}^{\Delta_i} \{S_T(U_i\mid b_i)\}^{1-\Delta_i}. $$ This expression assumes the censoring time is administrative or otherwise independent of the dropout time conditional on the modeled information. If the censoring mechanism is informative and is part of the analysis, it needs its own model and likelihood contribution.

An event-time contribution is a density, not a probability of dropout at one exact instant. A right-censoring contribution is a survival probability: the data establish that dropout did not occur before the observed follow-up ended, but do not specify a later event time. Consequently, the two cases have different units: a density has units inverse to time, whereas a survival probability is dimensionless.

When the dropout model is expressed through its hazard $h(t\mid b_i)$, define the cumulative hazard $H(t\mid b_i)=\int_0^t h(u\mid b_i)\,du$. The survival function is $S_T(t\mid b_i)=\exp\{-H(t\mid b_i)\}$, and the event-time density is $f_T(t\mid b_i)=h(t\mid b_i)S_T(t\mid b_i)$. Thus the dropout contribution can also be written

p(D_i\mid b_i) = \{h(U_i\mid b_i)\}^{\Delta_i} \exp\{-H(U_i\mid b_i)\}.

This form makes the event and censoring cases explicit. At an event time, the hazard supplies the event component and the exponential supplies survival up to that time. For a censored subject, $\Delta_i=0$, so only the survival through observed follow-up remains.

Justification: Since $S_T(t\mid b_i)=P(T_i>t\mid b_i)$, its instantaneous rate of decrease is $\frac{d}{dt}S_T(t\mid b_i)=-h(t\mid b_i)S_T(t\mid b_i)$, with $S_T(0\mid b_i)=1$. Dividing by $S_T(t\mid b_i)$ and integrating gives $\log S_T(t\mid b_i)=-\int_0^t h(u\mid b_i)\,du$. Therefore $S_T(t\mid b_i)=\exp\{-H(t\mid b_i)\}$. The event-time density is the hazard times the probability of remaining event-free to that time, so $f_T(t\mid b_i)=h(t\mid b_i)S_T(t\mid b_i)$.

Worked Example: An Exponential Dropout Time

Consider a hypothetical subject with shared effect fixed at $b_i=0$. Suppose the observed outcome is $Y_i=50$ points and, conditional on $b_i=0$, $Y_i\sim N(50,\sigma^2)$, where $\sigma=1/(0.60\sqrt{2\pi})\approx0.6649$ points. The outcome density at 50 is $$ p(Y_i=50\mid b_i=0) = \frac{1}{\sigma\sqrt{2\pi}} = 0.60\ \text{point}^{-1}. $$ Assume an exponential dropout time with constant hazard $\lambda=0.20$ per month, and suppose dropout is observed at $t=3$ months. The event-time density is $$ f_T(3\mid b_i=0) = \lambda e^{-\lambda t} = 0.20e^{-(0.20)(3)} = 0.20e^{-0.60} \approx 0.1097623\ \text{month}^{-1}. $$

Conditional independence gives the subject-level joint contribution: $$ L_i(b_i=0) = 0.60(0.1097623\ldots) = 0.0658574\ldots \approx 0.06586\ \text{point}^{-1}\text{month}^{-1}. $$ The displayed product uses the unrounded event-time density before rounding the result. If the event-time density is instead rounded first to $0.1098$, the displayed multiplication is $0.60(0.1098)=0.06588$. The outcome factor is a density at the observed outcome, and the dropout factor is a density at the observed event time; the product is not itself a probability of this exact pair of values.

For comparison, if this subject had instead been known to remain free of dropout through 3 months and then been right-censored, the dropout contribution would be $S_T(3\mid b_i=0)=e^{-0.60}\approx0.5488$, not the event density. The joint contribution in that case would be $0.60(0.5488)\approx0.3293\ \text{point}^{-1}$.

Worked Example: A Time-Varying Hazard

For another hypothetical subject, suppose $Y_i\mid b_i=0\sim N(0,1)$ and the observed outcome is $Y_i=0$. Its outcome density is $1/\sqrt{2\pi}\approx0.3989423$ per outcome unit. Let the dropout hazard be $h(t\mid b_i=0)=0.10+0.05t$ per month, where $t$ is measured in months and the slope has units month$^{-2}$. Dropout is observed at $t=2$ months.

The cumulative hazard and hazard at the event time are $$ H(2\mid b_i=0) = \int_0^2(0.10+0.05u)\,du = 0.10(2)+0.025(2^2) = 0.30, \qquad h(2\mid b_i=0)=0.10+0.05(2)=0.20\ \text{month}^{-1}. $$ Hence the event-time density is $$ f_T(2\mid b_i=0) = 0.20e^{-0.30} \approx0.1481636\ \text{month}^{-1}. $$ The conditional joint contribution is $$ L_i(b_i=0) = 0.3989423(0.1481636) \approx0.05911, $$ in inverse outcome units times inverse months. The hazard at 2 months alone is not the event-time density: the survival factor $e^{-0.30}$ is also required.

Discrete-Time Dropout Contributions

If dropout status is observed only at scheduled intervals, use conditional interval probabilities rather than a continuous-time density. Let $q_{ij}(b_i)$ be the probability of dropout during interval $j$, conditional on $b_i$ and on being at risk at the start of that interval. The probability of remaining under observation through interval $j$, conditional on being at risk at its start, is $1-q_{ij}(b_i)$.

Discrete-time contribution: If dropout occurs in interval $k$, the dropout contribution is $$ p(D_i\mid b_i) = \left\{\prod_{j=1}^{k-1}[1-q_{ij}(b_i)]\right\}q_{ik}(b_i). $$ If the subject is observed through interval $m$ without dropout, the contribution is $$ p(D_i\mid b_i) = \prod_{j=1}^{m}[1-q_{ij}(b_i)]. $$ These expressions assume that each interval probability is conditional on still being at risk at that interval’s start.

Worked Example: Dropout in the Third Interval

Suppose a hypothetical subject has observed outcomes $Y_{i,\mathrm{obs}}$ whose joint density, evaluated under the outcome model conditional on $b_i=0$, is $0.40$ per outcome unit. The subject has interval dropout probabilities $q_{i1}=0.10$, $q_{i2}=0.20$, and $q_{i3}=0.30$, conditional on reaching each interval. Dropout is recorded during interval 3.

The subject must first remain observed through intervals 1 and 2, then drop out in interval 3. Thus $$ p(D_i\mid b_i=0) = (1-0.10)(1-0.20)(0.30) = (0.90)(0.80)(0.30) = 0.216. $$ The joint contribution is $$ L_i(b_i=0) = 0.40(0.216) = 0.0864\ \text{per outcome unit}. $$ If the subject instead remained observed through all three intervals, the dropout-record contribution would be $(0.90)(0.80)(0.70)=0.504$, and the joint contribution would be $0.40(0.504)=0.2016$ per outcome unit. In discrete time the dropout part is a probability, not an event-time density.

Putting the Subject Contribution Together

The outcome and dropout pieces are assembled using the same conditional factorization for either time scale. The outcome factor may itself be a joint density over several measurements; it should not be split into a product of measurement-specific densities unless the outcome model justifies that product. For example, correlated outcome errors generally require a joint density that accounts for their dependence.

Specify the observed outcomes.
Identify which values were recorded before dropout and evaluate their joint density under the outcome model conditional on $b_i$.

Encode the dropout record.
For continuous time, distinguish an observed event time from right censoring. For discrete time, identify the event interval or the last interval observed without dropout.

Evaluate the dropout contribution.
Use an event density or survival probability in continuous time, or the appropriate product of interval probabilities in discrete time.

Multiply conditionally.
Under conditional independence given $b_i$, multiply the observed-outcome density by the dropout-record contribution, keeping the conditioning and units clear.

For independent subjects conditional on their shared effects, the conditional likelihood across subjects is the product of their subject-level contributions. This statement describes the likelihood conditional on those effects. Removing the conditioning requires handling the distribution of the latent effects; that is a different operation from deriving the subject-level factors.

Diagnostics, Alternatives, and Limitations

Several checks catch errors before the contributions are combined. A continuous-time hazard should be nonnegative, its cumulative hazard should be nondecreasing, and the resulting survival probability should lie between zero and one. A discrete-time dropout probability must lie between zero and one, and the probability of an event in interval $k$ must include survival through every earlier interval. Verify that event times are treated as densities and censored times as survival probabilities. Check units as well: multiplying an outcome density by an event-time density produces a joint density with both sets of inverse units.

A frequent mistake is to use $h(t\mid b_i)$ by itself as the event-time contribution. The hazard is an instantaneous rate conditional on remaining at risk, not the unconditional event-time density. The correction is $f_T(t\mid b_i)=h(t\mid b_i)S_T(t\mid b_i)$. Another error is to give a censored subject a density at the censoring time; censoring tells us the event time exceeds that time, so its contribution is survival through follow-up.

The likelihood construction also depends on what the recorded timing means. If dropout is known only to have occurred within a time interval but an exact event time was not observed, substituting the interval midpoint into a continuous-time density is generally not correct. The relevant event contribution is the probability of an event in that interval, obtained from the survival probabilities at its endpoints. Likewise, an outcome observation recorded at a visit and an event time recorded between visits need to be aligned with the model’s specified observation schedule.

The shared-parameter likelihood here is distinct from using only an outcome likelihood that discards dropout records, or from specifying a pattern-mixture factorization by outcome-observation pattern. Those are alternative modeling formulations, with their own assumptions and interpretations. The current contribution relies specifically on the chosen shared effects and the stated conditional independence. A good fit to observed outcomes or dropout times alone does not establish that this assumption, or the unobserved-outcome model, is correct.

Finally, these expressions do not resolve whether the latent-effect distribution, hazard form, or outcome model is appropriate, nor do they establish that the parameters are identifiable from the observed data. They provide the subject-level factors once those models have been specified. If dropout is interval-censored, censoring is informative, or the process has additional observation mechanisms, the likelihood must reflect those features rather than force them into the basic event-time or discrete-time formula.

Key takeaway: Conditional on shared effects, the subject contribution is the observed-outcome density multiplied by the correct dropout-record contribution. Use an event-time density for an observed continuous-time dropout, a survival probability for right censoring, and conditional interval probabilities for discrete-time dropout.

Check Your Understanding

Work through the questions, then compare your reasoning with the solutions.

A subject’s event-time hazard is $0.15$ per week at an observed dropout time of 4 weeks, and the survival probability to 4 weeks is $0.70$. What is the event-time density at 4 weeks?
A subject is right-censored after 5 months. What dropout contribution is used, and why is it not the hazard at 5 months?
In discrete time, the dropout probabilities are $0.05$, $0.10$, and $0.20$. The subject drops out during interval 3. Compute the dropout-record contribution.
Suppose the observed-outcome density is $0.25$ per outcome unit and the dropout-record contribution is $0.40$. Under conditional independence, what is the joint contribution and its units?

Solutions. (1) The event-time density is hazard times survival: $0.15(0.70)=0.105$ per week. (2) Use $S_T(5\mid b_i)$, the probability of remaining free of dropout through 5 months conditional on the shared effects. Right censoring establishes survival through follow-up, not an event at 5 months, so it contributes a survival probability rather than a hazard or event density. (3) The subject survives intervals 1 and 2, then drops out in interval 3: $(1-0.05)(1-0.10)(0.20)=(0.95)(0.90)(0.20)=0.171$. (4) The joint contribution is $0.25(0.40)=0.10$ per outcome unit; the dropout factor is a probability and therefore adds no inverse-time unit.