Joint Distributions of Intermediate and Final Outcomes

Why Model the Outcomes Jointly?

In a time-to-event MAMS design, an intermediate outcome may be available when the later event endpoint is not yet mature. The intermediate outcome can therefore inform which arms continue. But an arm’s intermediate results do not, by themselves, specify its later event results. To reason about that link, we need a model for the joint distribution of the intermediate outcome and the later endpoint.

This tutorial focuses on specifying that joint distribution and understanding the consequences of dependence for selection. It does not make a claim that the intermediate outcome is a valid substitute for the event endpoint. A joint model describes how outcomes vary together under its assumptions; it does not establish that changing the intermediate outcome would change the event endpoint.

Definition: Let \(Y\) denote an intermediate outcome observed before the final event readout, and let \(T\) denote the later event time. Their joint distribution is the distribution of the pair \((Y,T)\), including both marginal distributions and their dependence. For a prespecified horizon \(\tau\), define \(E=I(T>\tau)\), so \(E=1\) means event-free beyond \(\tau\). The joint distribution of \((Y,E)\) describes how the intermediate measurement and event-free status at that horizon occur together.

A selection rule often uses \(Y\), while the scientific endpoint is \(T\), or a summary of \(T\). If \(Y\) and the event endpoint are dependent, selecting arms using \(Y\) changes which event outcomes are likely to be seen among the selected arms. If they are independent, selection on \(Y\) carries no such information in the population. Both possibilities are assumptions to assess, not consequences of having measured the intermediate outcome.

Two Ways to Specify a Joint Model

One approach models an intermediate outcome and event status at a fixed horizon using a latent bivariate-normal distribution. It is convenient when the selection variable is continuous and the final endpoint is summarized as event-free status by \(\tau\). Let \(U\) and \(V\) be standardized latent variables with correlation \(\rho\), where \(U\) determines the intermediate outcome and \(V\) determines event-free status. For example:

\begin{pmatrix}U\\V\end{pmatrix} \sim N\!\left( \begin{pmatrix}0\\0\end{pmatrix}, \begin{pmatrix}1&\rho\\\rho&1\end{pmatrix} \right),\qquad Y=\mu_Y+\sigma_Y U,\qquad E=I(V>c).

Here \(c\) is chosen to give the desired marginal event-free probability: \(P(E=1)=1-\Phi(c)\). The model assumes a normal intermediate outcome, a probit threshold for event-free status at the chosen horizon, and a bivariate-normal latent pair. The parameter \(\rho\) describes latent dependence; it is not generally the ordinary correlation between \(Y\) and the binary variable \(E\).

A second approach models event time conditional on the intermediate outcome. One example is a proportional-hazards specification:

f_{Y,T}(y,t)=f_Y(y)f_{T\mid Y}(t\mid y),\qquad h(t\mid Y=y)=h_0(t)\exp(\gamma y).

The factorization is a probability identity; the hazard equation supplies a particular conditional model. Here \(h_0(t)\) is a baseline hazard and \(\gamma\) describes how the conditional hazard changes with \(Y\). For instance, if larger \(Y\) represents a more favorable response, a negative \(\gamma\) corresponds to a lower conditional event hazard as \(Y\) increases. This model requires a suitable hazard specification and, for ordinary right-censored likelihood calculations, censoring that is independent of the event time conditional on the modeled information. It describes conditional association, not a causal effect of \(Y\).

These formulations answer related but different modeling needs. A fixed-horizon model targets event-free status at a particular \(\tau\); a survival model represents event timing and can use information across follow-up. Neither is universally preferable. The choice should match the endpoint and data available at selection.

A Joint Probability Under a Latent-Normal Model

A simple special case makes the effect of dependence transparent. Suppose \(Y\) and the latent event-free variable \(V\) are standardized, both thresholds are zero, and \(E=I(V>0)\). Then \(Y>0\) is the intermediate criterion and \(E=1\) is event-free status. Under the bivariate-normal model:

P(Y>0,E=1)=\frac14+\frac{\arcsin(\rho)}{2\pi}.

This result follows from the probability of the upper-right quadrant of a centered bivariate normal distribution. At \(\rho=0\), the two variables are independent and the probability is \(1/4\). Differentiating the quadrant probability with respect to \(\rho\) gives \(1/(2\pi\sqrt{1-\rho^2})\); integrating from 0 to \(\rho\), and using the independent value \(1/4\), gives the displayed expression.

Worked Example: Dependence Changes the Chance of Meeting Both Criteria

Suppose the hypothetical latent-normal model has \(\rho=0.60\), with \(Y>0\) defined as meeting the intermediate criterion and \(E=1\) defined as event-free at the chosen horizon. Each criterion has marginal probability 0.50. Calculate the probability of meeting both.

Model check. The proposed correlation is admissible because \(-1<0.60<1\). The standardized normal margins and zero thresholds imply \(P(Y>0)=P(E=1)=0.50\).

Calculation.

P(Y>0,E=1) =\frac14+\frac{\arcsin(0.60)}{2\pi} =\frac14+\frac{0.6435}{6.2832} =0.3524.

Under independence, the probability would be \(0.50\times0.50=0.2500\). With positive latent dependence it is 0.3524, so meeting the intermediate criterion is associated with a greater chance of being event-free at the horizon in this model. This is a population probability under the stated model, not a guarantee for an individual or evidence that the intermediate outcome causes event-free survival.

From an Intermediate Selection Rule to Event Outcomes

If an arm continues only when its intermediate outcome exceeds a threshold, the relevant joint-model quantity includes probabilities conditional on passing that threshold. The latent-normal example gives a direct calculation when both thresholds are zero.

Worked Example: Event-Free Probability Among Those Passing a Threshold

Use the same hypothetical model, with \(\rho=0.60\), and suppose an arm is considered promising when \(Y>0\). Find the event-free probability at the horizon conditional on passing this criterion.

The probability of passing is \(P(Y>0)=0.50\). The joint probability of passing and being event-free, calculated above, is 0.3524. Therefore:

P(E=1\mid Y>0) =\frac{P(E=1,Y>0)}{P(Y>0)} =\frac{0.3524}{0.50} =0.7048.

The model gives a 0.7048 event-free probability among individuals with \(Y>0\), compared with the marginal probability \(P(E=1)=0.50\). This conditional probability illustrates why dependence matters for predictions made from an intermediate criterion. In an arm-selection design, however, a rule usually acts on an arm-level summary, such as an estimated response rate or mean; the individual-level conditional probability is not automatically the probability that an arm will be selected or its final treatment effect.

A survival-time formulation can instead use the intermediate outcome directly as a covariate in a conditional event-time distribution. The next example uses a particularly simple version, with a constant hazard within each response group. Its purpose is to show how a joint distribution can be assembled from a marginal distribution for the intermediate outcome and conditional event-time distributions.

Worked Example: A Binary Response and Conditional Event-Free Survival

Suppose, hypothetically, that an early response \(R\) has probability \(P(R=1)=0.60\). From a prespecified landmark, assume constant event hazards of 0.04 per month for responders and 0.10 per month for nonresponders. Assume event times follow exponential distributions within these groups, and censoring is independent of event time within each group. Calculate the probability of being event-free for 12 months and the joint probability of response and 12-month event-free status.

Under a constant hazard \(\lambda\), the survival probability through time \(t\) is \(S(t)=\exp(-\lambda t)\). Thus:

P(T>12\mid R=1)=\exp(-0.04\times12)=\exp(-0.48)=0.6188,

P(T>12\mid R=0)=\exp(-0.10\times12)=\exp(-1.20)=0.3010.

The joint probability of response and event-free status is \(P(R=1,T>12)=P(R=1)P(T>12\mid R=1)=0.60\times0.6188=0.3713\). For nonresponse it is \(P(R=0,T>12)=0.40\times0.3010=0.1204\). Adding the two disjoint cases gives:

$$ P(T>12)=0.3713+0.1204=0.4917. $$

In this model, response and 12-month event-free status are dependent: the conditional event-free probabilities differ by response group. The exponential and constant-hazard assumptions are restrictive; if the hazard changes over time, the survival probability must instead be obtained from the specified time-varying hazard. The landmark also matters: the event-time origin here is the landmark, and the calculation concerns individuals eligible to be assessed at that point.

How Dependence Can Change Arm Selection

An arm-selection rule may rank treatments by an intermediate measure, while the goal is to retain arms with favorable later event outcomes. If the joint distributions differ across arms, the ranking by the intermediate measure need not match the ranking by event-free probability. The following hypothetical probabilities make that distinction explicit.

Worked Example: The Intermediate Ranking and Event Ranking Differ

Suppose two hypothetical arms are assessed using a binary intermediate response \(R\) and event-free status \(E\) at a fixed later horizon. A rule selects the arm with the higher response probability. Assume the following joint probabilities:

Arm	\(P(R=1)\)	\(P(E=1\mid R=1)\)	\(P(E=1\mid R=0)\)
A	0.60	0.75	0.60
B	0.70	0.60	0.60

The response rule selects Arm B because \(0.70>0.60\). To compare event-free probabilities, apply the law of total probability separately in each arm:

P(E=1\mid A)=0.60(0.75)+0.40(0.60)=0.45+0.24=0.69,

P(E=1\mid B)=0.70(0.60)+0.30(0.60)=0.42+0.18=0.60.

Thus the intermediate rule selects B, even though the specified event-free probability is higher for A: 0.69 versus 0.60. In A, response and event-free status are associated; in B, the event-free probability is the same in both response groups. This example shows that an intermediate-only ranking does not determine the later endpoint ranking. It does not show which rule is best, nor does it account for sampling uncertainty in estimated arm-level probabilities.

Assumptions, Diagnostics, and Alternatives

A joint model should be checked as a model for both outcomes, not just for the intermediate variable or the event endpoint separately. The marginal distributions can each look reasonable while their dependence is misspecified. Useful checks and sensitivity analyses include:

Check each margin. Compare the assumed distribution for \(Y\) with its observed scale and shape. For event times, examine whether the chosen hazard or survival form is plausible over follow-up.
Check joint behavior. Compare observed joint frequencies or conditional event summaries across intermediate-outcome groups with model-based quantities. With continuous \(Y\), inspect whether conditional event patterns are adequately represented across its range.
Assess dependence sensitivity. Recalculate relevant arm-level predictions under plausible dependence values, including independence when scientifically credible. A selection conclusion that changes under modest changes in dependence is fragile.
Check timing and missingness. The intermediate outcome must be defined at the intended selection time. Delayed assessment, events before assessment, or informative missingness can change which individuals contribute and undermine a simple joint model.
Respect the scope of the model. A latent correlation, a conditional hazard coefficient, or an observed association is not a validation of the intermediate outcome as a surrogate endpoint.

The main alternative to a dependent joint model is an independence assumption, which sets the dependence to zero. It simplifies calculations but should be used only when defensible for the outcomes and population at hand. A latent-normal threshold model is compact for continuous intermediate outcomes and fixed-horizon binary status, but depends on its latent-normal and threshold assumptions. A conditional survival model retains event-time information and permits covariate-dependent hazards, but its conclusions depend on the hazard specification and censoring assumptions. More flexible joint models may be appropriate when these structures fit poorly, at the cost of additional parameters and stronger data requirements.

This discussion treats one arm’s paired outcomes at a time. In a multi-arm trial, observations sharing a control group can also be statistically dependent across treatment comparisons. That is a separate source of dependence and should not be confused with the within-person association between an intermediate outcome and a later event endpoint.

Common Pitfall

Common mistake: “The intermediate outcome is associated with event-free survival, so selecting the arm with the best intermediate result must select the arm with the best event outcome.”

Correction: An association within individuals does not guarantee that arms have the same association structure, nor that their intermediate and event rankings agree. State the joint model, calculate the event quantity relevant to the decision, and examine sensitivity to dependence assumptions. The model describes a relationship; it does not establish a causal mechanism or a surrogate-endpoint claim.

Check Your Understanding

Answer each question, then compare your reasoning with the brief solutions.

In the latent-normal threshold model, what does \(\rho\) describe, and why is it not generally the observed correlation between \(Y\) and \(E\)?
If the two zero-threshold variables are independent and each has probability 0.50 of exceeding its threshold, what is the probability that both criteria are met?
Under a conditional survival model, what does a negative \(\gamma\) mean when larger \(Y\) is a more favorable response?
Using the two-arm example, which arm does the response rule select, and what are the two event-free probabilities?

Solutions. (1) \(\rho\) is the correlation between the standardized latent variables; thresholding one variable to form binary \(E\) changes the observed dependence. (2) Independence gives \(0.50\times0.50=0.25\). (3) The specified conditional hazard decreases as \(Y\) increases, since \(\exp(\gamma Y)\) decreases with \(Y\) when \(\gamma<0\). (4) The response rule selects B, with response probability 0.70; the event-free probabilities are 0.69 for A and 0.60 for B.