Why Construct a Joint Likelihood?
A joint dose model describes the probability of observing efficacy and toxicity together at a dose. Its likelihood is built from the outcomes actually recorded for the participants: a complete efficacy-toxicity pair contributes a joint probability, while a partially observed pair contributes a marginal probability. Writing these contributions explicitly is essential because multiplying the separate efficacy and toxicity probabilities would generally impose an independence assumption.
The earlier material on copula models, shared latent-variable models, independence, and ordinal joint outcomes provides several ways to specify joint probabilities. The likelihood construction begins after a model has been chosen: it translates the model and the observed records into a function of the unknown parameters.
Joint Outcome Categories and Participant Contributions
For participant \(i\), let \(d_i\) denote the assigned dose, \(z_i\) the recorded covariates, and \(Y_{Ei}\) and \(Y_{Ti}\) the efficacy and toxicity outcomes. For binary endpoints, code each as 0 or 1. Write \( \pi_{jk}(d_i,z_i;\theta)=P(Y_{Ei}=j,Y_{Ti}=k\mid d_i,z_i;\theta) \), where \(j,k\in\{0,1\}\) and \(\theta\) denotes the model parameters. The four probabilities must be nonnegative and sum to one at every dose and covariate profile.
If participants are conditionally independent given their doses and covariates, the likelihood for their outcome records is the product of their contributions:
This independence assumption concerns the outcomes across participants; it does not assert independence between efficacy and toxicity within a participant. The joint probabilities \(\pi_{jk}\) allow those two endpoints to be associated. If the study has additional dependence, such as clustering by site that is not represented in the model, the simple product may not be an appropriate sampling likelihood.
Dose and covariates enter each contribution at that participant’s observed values. In an adaptive trial, it is often appropriate to form an outcome likelihood conditional on the assigned doses and observed trial history when the allocation mechanism is known and does not involve the outcome-model parameters. If the allocation mechanism itself contains unknown parameters relevant to the analysis, its probability model may also need to be included.
A Categorical Model With Dose and Covariates
One direct specification treats the four binary outcome pairs as categories in a multinomial model. Use \((0,0)\) as the reference category and assign a linear predictor \(\eta_{jk}(d,z)\) to each of the other three categories. The resulting probabilities are
and
The linear predictors can include dose indicators, covariates, and specified interactions. For example, a term for a dose indicator allows the category probabilities to differ by dose, while a covariate term allows the joint outcome distribution to vary with a patient characteristic. The model is conditional on the included covariates; it does not make an unadjusted population claim unless that is the intended target.
The softmax form guarantees positive category probabilities that sum to one. It does not guarantee that each endpoint’s marginal probability follows a particular dose-response curve. If scientific requirements call for specific marginal dose-response shapes, a parameterization based on endpoint margins and an association structure may be more suitable. Whatever parameterization is chosen, the likelihood contribution is still the model probability of the observed joint category.
Worked Example: Likelihood From Complete Joint Categories
Suppose, hypothetically, six participants treated at one dose have the following complete binary outcomes: two with \((Y_E,Y_T)=(0,0)\), one with \((1,0)\), one with \((0,1)\), and two with \((1,1)\). At this dose, suppose the fitted joint model assigns probabilities \(0.50, 0.25, 0.15,\) and \(0.10\) to those four categories, respectively. These values are nonnegative and sum to \(1.00\), as required.
For the six individually recorded participants, the likelihood contribution is the product of their category probabilities:
This is the likelihood at the stated probability values; it is not the probability that the model or parameter values are true. If the data are represented only by category counts, the multinomial likelihood includes the combinatorial factor:
The factor counts the possible orderings of the six category labels. It does not depend on the unknown model parameters, so including or omitting it does not change likelihood-based estimates. It does matter when stating the probability of the grouped count outcome.
Covariates Enter Through Conditional Probabilities
A likelihood with covariates uses the same participant-level product, but evaluates each probability at that participant’s own \(d_i\) and \(z_i\). This is why covariate values must remain paired with their corresponding outcome records. A model that includes a covariate does not by itself justify a causal interpretation of that covariate’s coefficient.
Worked Example: A Dose and Covariate in a Joint Categorical Model
Consider a hypothetical model with dose 1 as the reference and \(I(d=2)\) equal to 1 at dose 2 and 0 otherwise. Let \(z\) be a binary patient covariate. Set the nonreference-category predictors to \( \eta_{10}=(\log 2)I(d=2) \), \( \eta_{01}=0 \), and \( \eta_{11}=(\log 1.5)z \). Suppose two independent participants both receive dose 2 and have \(z=1\). One has outcome \((1,0)\) and the other \((1,1)\).
For each participant, the denominator of the softmax probabilities is \(1+\exp(\log 2)+\exp(0)+\exp(\log 1.5)=1+2+1+1.5=5.5\). Thus the probabilities of \((0,0)\), \((1,0)\), \((0,1)\), and \((1,1)\) are, respectively, \(2/11\), \(4/11\), \(2/11\), and \(3/11\). The two observed outcomes therefore contribute
The dose and covariate values determine the probabilities used for both records. If either participant had a different dose or covariate value, their contribution would be evaluated using the corresponding predictors rather than reusing these probabilities.
Partially Observed Endpoints: Marginalize, Do Not Guess
When only part of a participant’s outcome pair is observed, the likelihood contribution is the probability of the observed information. This follows from the law of total probability: mutually exclusive possibilities for the unobserved endpoint are added, not selected or imputed as if known.
Worked Example: One Observed Endpoint in a Binary Pair
Use the hypothetical joint probabilities from the first example: \(\pi_{00}=0.50\), \(\pi_{10}=0.25\), \(\pi_{01}=0.15\), and \(\pi_{11}=0.10\). Suppose one participant’s efficacy is observed as 1 but toxicity is unavailable. Under an ignorable observation process, sum over the two possible toxicity values:
For a different participant with toxicity observed as 1 but efficacy unavailable, the contribution is \( \pi_{01}+\pi_{11}=0.15+0.10=0.25 \). If these two records are conditionally independent, their combined contribution is \(0.35\times0.25=0.0875\). A record with neither endpoint observed contributes \(1\) to an outcome-only likelihood under an ignorable observation process, because the probabilities of all possible joint outcomes sum to one. That does not mean the missing record provides information about the outcome-model parameters.
Justification and Assumptions for Marginal Contributions
The marginal contribution can be derived by considering all complete outcomes compatible with the observed record. If efficacy is observed as \(j\), the two complete binary pairs \((j,0)\) and \((j,1)\) are disjoint and together represent exactly the event \(Y_E=j\). Therefore,
This sum is valid as an outcome likelihood when the missingness mechanism can be ignored for inference on \(\theta\), under the model’s assumptions. A common sufficient setup is that the observation process is missing at random given observed information and that its parameters are distinct from the outcome-model parameters. If the probability that toxicity is recorded depends on its unobserved value even after conditioning on observed information, the observed-data likelihood generally needs a model for the observation process as well. Simply summing the outcome probabilities would then omit relevant information or make an unsupported assumption.
The same principle applies to categorical endpoints with more than two levels. Sum over every unobserved category compatible with the observed record. For example, if efficacy has categories \(0,1,2\) and toxicity has categories \(0,1\), observing efficacy category 2 alone contributes the sum of the two joint probabilities for \((2,0)\) and \((2,1)\). For time-to-event or other noncategorical outcomes, the corresponding operation is integration over unobserved outcome values rather than a finite sum.
Likelihood Checks, Alternatives, and Failure Modes
Before using a joint likelihood, check that the outcome categories are defined consistently, the probabilities sum to one at every relevant dose and covariate profile, and each recorded contribution matches the observation. These basic checks catch many coding and specification errors. Additional checks concern the assumptions behind the likelihood and whether the model can represent the data adequately.
- Do not substitute marginal probabilities for joint probabilities. For a complete pair \((1,1)\), the contribution is \(\pi_{11}\), not \(P(Y_E=1)P(Y_T=1)\), unless conditional independence is part of the model.
- Do not treat an unobserved endpoint as a particular category. If its value is unknown, sum or integrate over compatible values under the missingness assumptions; do not silently code it as 0.
- Check sparse categories and covariate support. If some joint categories are rare or certain dose-covariate combinations have little or no data, model parameters may be weakly identified or estimates may be unstable.
- Use log likelihoods for computation when appropriate. Products of many small probabilities can underflow numerically. The equivalent log likelihood is a sum of log contributions, \( \ell(\theta)=\sum_i\log L_i(\theta) \), provided each contribution is positive.
- Inspect model fit and dependence assumptions. Poor fit to observed joint category frequencies can indicate that the chosen dose, covariate, or association structure is inadequate. Agreement with the separate endpoint margins alone does not establish that the joint distribution is adequate.
A full joint likelihood uses the probability of each observed pair, preserving the modeled dependence. Multiplying separate efficacy and toxicity likelihoods is a different approach: it is appropriate only under a suitable conditional independence model, or as an explicitly chosen approximation. A composite likelihood can combine marginal or lower-dimensional pieces when a full joint model is difficult to construct, but it is not generally the same likelihood and its uncertainty calculations may require adjustment. Bayesian analysis uses the same likelihood contribution structure and combines the likelihood with a prior; the prior changes the inferential calculation, not the meaning of the observed-data contribution.
A likelihood is not a guarantee that the model is true. It quantifies how well parameter values assign probability to the observed records under the stated model and assumptions. Dose-specific probabilities may be extrapolated to poorly observed doses by the model, so estimates there can be driven strongly by its functional form. In dose-finding work, that uncertainty matters: a precisely computed likelihood under a misspecified joint model can still misrepresent benefit and toxicity together.
Check Your Understanding
Work through these questions, then compare your reasoning with the solutions.
- A complete pair has joint probability \(\pi_{01}=0.12\). What is its likelihood contribution, and when could a product of marginal probabilities replace it?
- If efficacy is observed as 0 and toxicity is unobserved, write the contribution using the four binary joint probabilities.
- At a dose, suppose \(\pi_{00}=0.40\), \(\pi_{10}=0.20\), \(\pi_{01}=0.25\), and \(\pi_{11}=0.15\). Find the contribution when only toxicity equal to 1 is observed.
- Why is the outcome-only contribution for a record with neither endpoint observed equal to 1 under an ignorable observation process, and when might that treatment be inadequate?
Solutions. 1. The contribution is \(0.12\). A product of marginal probabilities is justified only if the model imposes the relevant conditional independence. 2. It is \(\pi_{00}+\pi_{01}\), summing over both toxicity categories. 3. It is \(\pi_{01}+\pi_{11}=0.25+0.15=0.40\). 4. Summing all four joint probabilities gives \(1\), so the record supplies no outcome information under that likelihood. If recording depends on unobserved outcomes, or the observation process is otherwise informative, the observation mechanism may need to be modeled.