Joint Factorization for Outcomes and Dropout Times

Why Factor the Joint Distribution?

In a repeated-outcome study, a participant’s observed outcomes and their dropout time are both part of the data-generating process. A model for outcomes alone does not describe how the observation pattern arises. A shared-parameter model represents both processes using a latent subject-specific effect: conditional on that effect, the outcome process and dropout process are modeled separately, and the effect links them in the joint distribution.

The factorization is useful because it makes the model’s components explicit. It also identifies what contributes to the observed data for a participant who drops out: the density of the outcomes actually observed, the probability of that dropout pattern, and the distribution of the latent effect, with unobserved outcomes integrated out.

Definition: Suppose outcomes are scheduled at visits \(1,\ldots,K\). Let \(Y_i=(Y_{i1},\ldots,Y_{iK})\) be subject \(i\)’s full outcome vector, \(b_i\) a shared latent effect, and \(S_i\) the first visit whose outcome is unobserved. Thus \(S_i=s\), for \(1\leq s\leq K\), means outcomes are observed at visits \(1,\ldots,s-1\) and missing from visit \(s\) onward. \(S_i=K+1\) means all \(K\) scheduled outcomes are observed.

The Joint Factorization

Write \(f_b(b_i)\) for the density or probability mass function of the shared effect, \(f_Y(y_i\mid b_i)\) for the conditional distribution of the full outcome vector, and \(g_s(b_i)=P(S_i=s\mid b_i)\) for the conditional probability of dropout pattern \(s\). Covariates and model parameters are suppressed in this notation; in an applied model, each component may depend on them.

The working factorization assumes that, given \(b_i\), the full outcome vector and dropout time are independent. In density notation, the joint distribution is

f(y_i,s_i,b_i)=f_b(b_i)\,f_Y(y_i\mid b_i)\,g_{s_i}(b_i).

This is a model assumption, not a consequence of sharing a latent effect. The shared effect can induce an association between outcomes and dropout after it is averaged over, even though the model separates the processes conditional on \(b_i\). The factorization also does not require outcomes at different visits to be conditionally independent: their dependence can be represented in the joint density \(f_Y(y_i\mid b_i)\).

A common way to specify \(g_s\) is with discrete dropout hazards. Let \(h_t(b_i)=P(S_i=t\mid S_i\geq t,b_i)\) for \(t=1,\ldots,K\). Among participants whose outcome at visit \(t\) would still be observed, \(h_t(b_i)\) is the probability that visit \(t\) is the first unobserved visit. The probability of a particular pattern is therefore

g_s(b_i)= \begin{cases} \displaystyle \left\{\prod_{t=1}^{s-1}[1-h_t(b_i)]\right\}h_s(b_i), & 1\leq s\leq K,\\[6pt] \displaystyle \prod_{t=1}^{K}[1-h_t(b_i)], & s=K+1. \end{cases}

The product before \(h_s\) records remaining observed through visits \(1,\ldots,s-1\); \(h_s\) records dropout before outcome \(s\). The final line is the probability of remaining observed through the end of scheduled follow-up. These probabilities sum to one: the dropout terms successively partition the probability of reaching each visit, with the final survival term accounting for those who reach the end.

Worked Example: A Dropout Pattern at a Fixed Shared Effect

Consider a hypothetical study with \(K=3\) visits. At a fixed value \(b\), suppose the dropout hazards are \(h_1(b)=0.10\), \(h_2(b)=0.20\), and \(h_3(b)=0.25\). A participant has outcomes observed at visits 1 and 2, so \(S=3\).

The probability of this pattern conditional on \(b\) is

$$ g_3(b)=[1-h_1(b)][1-h_2(b)]h_3(b) =(0.90)(0.80)(0.25)=0.180. $$

Suppose, additionally, that the conditional outcome density for the two observed values is \(f_Y(y_{i1},y_{i2}\mid b)=0.20\times0.15=0.030\). This product is appropriate here because, for this example, the specified conditional outcome model has a product density at the observed visits. At this fixed \(b\), the observed-data contribution before accounting for the distribution of \(b\) is \(0.030\times0.180=0.00540\). This is a density contribution, not a probability for the continuous outcome values.

From the Full Model to the Observed-Data Contribution

The observed record for a subject with \(S_i=s\) contains \(y_{i,\mathrm{obs}}=(y_{i1},\ldots,y_{i,s-1})\), along with the pattern \(s\). The remaining outcomes \(y_{i,\mathrm{mis}}=(y_{is},\ldots,y_{iK})\) are unobserved. To obtain the contribution for the actual record, integrate over those missing outcomes and over the unobserved shared effect.

Starting with the joint factorization, the observed-data contribution is

L_i = \int \int f_b(b_i)\, f_Y(y_{i,\mathrm{obs}},y_{i,\mathrm{mis}}\mid b_i)\, g_{s}(b_i)\, d y_{i,\mathrm{mis}}\,d b_i.

Because \(g_s(b_i)\) does not depend on the missing outcomes under this factorization, it can be taken outside the inner integral. Integrating the full outcome density over the missing coordinates gives the marginal density of the observed outcomes conditional on \(b_i\):

\int f_Y(y_{i,\mathrm{obs}},y_{i,\mathrm{mis}}\mid b_i)\, d y_{i,\mathrm{mis}} =f_{Y,\mathrm{obs}}(y_{i,\mathrm{obs}}\mid b_i).

Consequently,

L_i = \int f_b(b_i)\, f_{Y,\mathrm{obs}}(y_{i,\mathrm{obs}}\mid b_i)\, g_s(b_i)\,d b_i.

For discrete outcomes, the integral over missing outcomes is replaced by a sum. If \(s=K+1\), no outcomes are missing and \(f_{Y,\mathrm{obs}}\) is the density of the full vector. If \(s=1\), no outcomes are observed; the outcome-density contribution is one, leaving the probability of the pattern averaged over \(b_i\). In either case, the pattern itself remains part of the observed record.

Key distinction: At a specified value \(b_i\), the conditional contribution is \(f_{Y,\mathrm{obs}}(y_{i,\mathrm{obs}}\mid b_i)g_s(b_i)\). The subject’s marginal observed-data contribution averages this product over \(f_b(b_i)\). In general, the average of a product is not the product of averages.

Worked Example: Complete Follow-Up and the Survival Pattern

In a hypothetical three-visit study, suppose the hazards at a fixed \(b\) are \(0.10\), \(0.20\), and \(0.25\). A participant completes all visits, so \(S=4=K+1\). The conditional probability of this pattern is

$$ g_4(b)=(1-0.10)(1-0.20)(1-0.25) =(0.90)(0.80)(0.75)=0.540. $$

Suppose the joint conditional density of the participant’s three observed outcomes is \(f_Y(y_{i1},y_{i2},y_{i3}\mid b)=0.12\). Because all outcomes are observed, there is no missing-outcome integral to perform at this fixed \(b\). The conditional observed-data contribution is

0.12\times0.540=0.0648.

This calculation uses the joint outcome density as given; it does not assume the three outcomes are independent. The value \(0.0648\) is still conditional on \(b\). The marginal contribution would require averaging it, with the appropriate \(b\)-specific outcome density and pattern probability, over the distribution of \(b\).

Worked Example: Integrating Over a Discrete Shared Effect

Suppose, hypothetically, that \(b\) has two possible values, \(b_a\) and \(b_b\), each with probability \(0.50\). For a particular observed outcome record and dropout pattern, suppose the conditional observed-outcome densities are \(0.20\) at \(b_a\) and \(0.10\) at \(b_b\), while the conditional pattern probabilities are \(0.30\) and \(0.60\), respectively.

The marginal contribution is the weighted sum of the products:

$$ L_i=(0.50)(0.20)(0.30)+(0.50)(0.10)(0.60) =0.030+0.030=0.060. $$

A tempting alternative is to multiply the separately averaged quantities. That gives \([(0.50)(0.20)+(0.50)(0.10)]\times[(0.50)(0.30)+(0.50)(0.60)]=(0.15)(0.45)=0.0675\), which is not the required integral. The same latent effect indexes both components, so their product must be formed at each \(b\) before averaging.

Assumptions, Diagnostics, and Alternatives

The factorization requires a well-defined observation pattern and a specified distribution for the shared effect. More substantively, it assumes the conditional separation represented by \(f_Y(y\mid b)g_s(b)\). If dropout can depend on the observed outcome history even after conditioning on \(b\), the dropout component may instead need to be written \(P(S=s\mid b,y_{\mathrm{obs}})\); the stated factorization would then be inadequate unless that dependence is represented in the model.

Useful checks include verifying that all fitted hazards lie between zero and one, that the implied pattern probabilities sum to one, and that predicted dropout frequencies are reasonably compatible with observed frequencies across visits. Outcome-model diagnostics should also examine the observed outcomes by visit and dropout pattern. Such checks can reveal problems with the specified components, but fit to observed records cannot establish that the assumptions governing unobserved outcomes are correct.

The shared-parameter factorization differs in emphasis from pattern-mixture models, which organize outcome distributions by observation pattern, and selection models, which organize the joint distribution through outcome and observation components in a different order. These are alternative factorizations, not automatic checks on one another. A shared-effect model is only as credible as its chosen latent-effect distribution and conditional outcome and dropout specifications. In particular, the distribution of outcomes after dropout is informed through the model, not directly observed for participants who have dropped out.

A common mistake is to multiply the observed-outcome density by a dropout probability evaluated at an average latent effect, or to omit the dropout-pattern probability because the outcomes are the main scientific interest. The correction is to use the joint model: multiply the components at each value of \(b_i\), include the probability of the observed pattern, and then integrate over \(b_i\). Another mistake is to treat missing future outcomes as zero. They must instead be marginalized from the outcome distribution; zero is a value, not a generic representation of unobserved data.

Record the pattern.
For each subject, identify \(S_i\) and which scheduled outcomes are observed.

Write the conditional components.
Specify the outcome density and the probability of the observed dropout pattern at the same value of \(b_i\).

Marginalize.
Integrate or sum over missing outcomes, then integrate the product against the shared-effect distribution.

Key takeaway: The observed-data contribution is an average over shared effects of the conditional density for observed outcomes multiplied by the conditional probability of the subject’s dropout pattern. Missing outcomes are integrated out, not replaced by values.

Check Your Understanding

Answer each question, then compare with the solutions below.

With \(K=3\), what pattern does \(S=2\) represent?
If \(h_1=0.10\), \(h_2=0.20\), and \(h_3=0.25\), what is the probability of remaining observed through all three visits?
Why must the outcome-density and pattern-probability product be averaged over \(b\), rather than generally multiplying their separate averages?
Under the factorization in this tutorial, what happens to unobserved future outcomes in the observed-data contribution?

Solutions. (1) Visit 1 is observed; visits 2 and 3 are unobserved. (2) \((0.90)(0.80)(0.75)=0.540\). (3) Both components depend on the same \(b\), so their product at each \(b\) is integrated; the average of a product need not equal the product of averages. (4) Their values are integrated out of the full outcome density, yielding the marginal density of the observed outcomes conditional on \(b\).