Restricting Treatment Models to Eligible Decision Times

Why Eligibility Defines the Treatment Model

A treatment model describes assignment at a time when treatment could actually be assigned. In a longitudinal study, that opportunity may vary over time and across individuals. A person may already have received an absorbing treatment, may no longer meet a protocol criterion, or may be outside the study’s decision process. At such a time, an observed absence of treatment is not necessarily an observed decision to withhold treatment.

The distinction determines which records belong in the treatment-model risk set. Including ineligible records as untreated can make the model estimate a mixture of two processes: whether a decision was possible and, conditional on a decision being possible, which treatment was assigned. The treatment assignment model should address the second process.

Definition: Let $R_{it}$ be an eligibility indicator for individual $i$ at decision time $t$, determined just before the treatment decision: $R_{it}=1$ means the individual is eligible to receive treatment at that decision, and $R_{it}=0$ means no treatment decision is available. The treatment-model risk set at time $t$ is $\mathcal{R}_t=\{i:R_{it}=1\}$. Let $A_{it}$ denote the treatment assignment only when $R_{it}=1$. Let $H_{it}$ denote the information available just before the decision, including the recorded history relevant to eligibility and assignment.

For binary treatment, the assignment probability of interest is $P(A_{it}=1\mid H_{it},R_{it}=1)$, or the corresponding probability for the treatment level under study. The condition $R_{it}=1$ is not an optional adjustment: it specifies the population of decision opportunities for which the assignment is defined. When $R_{it}=0$, $A_{it}$ is undefined under this setup, rather than observed to equal zero.

Eligibility should be specified from the study protocol and the timing of information. It must be determined before the assignment being modeled. If a treatment itself changes later eligibility, that later risk set may depend on prior treatment; this does not make later ineligible records untreated decisions. It means that the later assignment model applies to those who remain eligible. Eligibility is also distinct from missingness: a person eligible for a decision whose assignment was not recorded has missing treatment data, not structural ineligibility.

Fitting the Assignment Model on the Risk Set

At each time, the response-model data should contain eligible decision opportunities. For a binary treatment, a correctly specified conditional model can be written as

P(A_{it}=1\mid H_{it},R_{it}=1)=g^{-1}\{\eta_t(H_{it};\theta_t)\},\qquad i\in\mathcal{R}_t,

where $g^{-1}$ is the chosen inverse link and $\eta_t$ is the model’s linear predictor or other specified function. The link may vary; the risk-set restriction does not. A likelihood based on independent eligible decision observations has contributions only from those observations. In a longitudinal model, the contribution at time $t$ is present only if $R_{it}=1$.

If individuals are sampled independently and each person’s treatment history is modeled sequentially, a person-level assignment likelihood can be factored by the chain rule when each factor conditions on all preceding assignments for that person. If a model treats multiple individuals in a cluster as conditionally independent, that is an additional assumption, not a consequence of defining each person’s risk set. The conditioning information and dependence structure must support the factorization being used.

Result: For a fixed cluster $c$, order its eligible decision opportunities as $k=1,\ldots,m_c$, and let $Z_{ck}$ be the assignment at opportunity $k$. Write $G_{ck}$ for all information being conditioned on, including the assignments at the preceding opportunities in this ordering. Then $$ P(Z_{c1}=z_{c1},\ldots,Z_{cm_c}=z_{cm_c}\mid \text{pre-decision information}) =\prod_{k=1}^{m_c}P(Z_{ck}=z_{ck}\mid G_{ck}). $$ This is the probability chain rule. A product of individual factors that condition only on each person’s own history is justified only if those factors also correctly describe the joint assignment law—for example, under conditional independence across individuals given the modeled histories. Otherwise the joint model must account for the dependence.

To verify the result, apply the chain rule successively: the joint probability is the probability of the first assignment multiplied by the probability of the second given the first, then by the probability of the third given the first two, and so on. Each conditioning set $G_{ck}$ must contain those preceding assignments, as well as the other specified pre-decision information. If it omits a preceding assignment that predicts the current one, the proposed factor need not equal the chain-rule factor. Thus the risk-set restriction and the dependence assumptions are separate requirements.

Worked Example: Excluding Structurally Ineligible Records

At one decision time, 30 people are observed. Ten are not eligible under the protocol; no treatment decision is defined for them. Among the 20 eligible people, 6 receive treatment and 14 do not. Suppose the assignment model has only an intercept.

The eligible-data likelihood for the treatment probability $p$ is proportional to $p^6(1-p)^{14}$. Its log-likelihood is $6\log(p)+14\log(1-p)$, and its derivative is $6/p-14/(1-p)$. Setting the derivative to zero gives $6(1-p)=14p$, so $\hat p=6/20=0.30$.

If the ten ineligible people are incorrectly coded as untreated, the fitted intercept-only probability becomes $6/30=0.20$. That number estimates the fraction treated among all 30 records, including people for whom no assignment was possible. It is not the probability of treatment conditional on an eligible decision. The ten records should not enter this assignment likelihood as treatment zeros.

Worked Example: A Later Risk Set Depends on Earlier History

Suppose 100 people are eligible at time 1. Forty receive treatment. Under the protocol, those treated people are not eligible for the time-2 decision. Of the 60 untreated people, 15 are no longer eligible at time 2 for a separate, recorded protocol reason. The remaining 45 are eligible, and 9 receive treatment at time 2.

At time 1, the observed assignment fraction is $40/100=0.40$. At time 2, the eligible denominator is $60-15=45$, so the observed treatment fraction is $9/45=0.20$. The 40 previously treated people and the 15 people no longer eligible do not contribute time-2 assignment outcomes. In particular, $9/100=0.09$ is not the time-2 conditional assignment fraction: it uses people outside the time-2 risk set in its denominator.

A time-2 model can use the history available before that decision, including the recorded reason for eligibility where relevant. Its predictions concern the eligible population represented by its risk-set observations. They do not, without additional assumptions, describe what assignment would have been for those who were ineligible.

What the Risk-Set Likelihood Targets

The reason for restricting the model can be expressed through the expected log-likelihood. Consider a single eligible decision opportunity with history $H=h$, binary assignment, and true conditional probability $p_0(h)=P(A=1\mid H=h,R=1)$. A proposed model assigns probability $q(h)$, with $0<q(h)<1$. Its conditional expected log-likelihood is

p_0(h)\log q(h)+\{1-p_0(h)\}\log\{1-q(h)\}.

This expectation is over assignments among eligible opportunities at that history. Ineligible observations have no assignment outcome in this expression and cannot supply evidence about $p_0(h)$.

Result: For any eligible history $h$ with $0<p_0(h)<1$, the conditional expected Bernoulli log-likelihood is uniquely maximized at $q(h)=p_0(h)$.

For proof, subtract the expected log-likelihood at $q(h)$ from its value at $p_0(h)$. The difference is

p_0(h)\log\frac{p_0(h)}{q(h)} +\{1-p_0(h)\}\log\frac{1-p_0(h)}{1-q(h)}.

This is the Kullback–Leibler divergence between Bernoulli distributions with probabilities $p_0(h)$ and $q(h)$, and it is nonnegative, with equality only when the two distributions agree. Hence the expected log-likelihood is maximized uniquely at $q(h)=p_0(h)$. This pointwise result assumes the candidate probabilities are in $(0,1)$; boundary cases can be handled by the corresponding limiting log-likelihood. In a parametric model, the same argument applies to the model’s expected log-likelihood, but the true conditional probability is attainable only if the model is correctly specified. $\square$

In practice, a model is usually fitted across histories rather than separately at every $h$. This result explains the target when the model is correctly specified; it does not guarantee correct specification, adequate data, or causal identification. Positivity is also relevant: for treatment strategies being considered, the needed assignment probabilities must be positive within the eligible histories that occur.

Worked Example: A Covariate-Specific Risk Set

Among eligible decision opportunities, suppose a binary covariate $X$ has two strata. For $X=0$, 30 people are eligible and 6 receive treatment; for $X=1$, 20 are eligible and 12 receive treatment. The stratum-specific empirical conditional probabilities are

\hat p(0)=\frac{6}{30}=0.20,\qquad \hat p(1)=\frac{12}{20}=0.60.

For an intercept-and-$X$ logistic model that fits these two strata exactly, the intercept is $\log\{0.20/0.80\}=\log(0.25)$, and the coefficient of $X$ is $\log\{0.60/0.40\}-\log\{0.20/0.80\}=\log(1.5)-\log(0.25)=\log(6)$. Thus the fitted probabilities are 0.20 for $X=0$ and 0.60 for $X=1$. These calculations use eligible observations only; records with $R=0$ have no assignment outcome to add to either stratum.

If one of the strata had no eligible observations, its conditional assignment probability could not be learned empirically from that stratum. If a stratum had eligible observations but only one treatment level, the empirical probability would be on the boundary. Such cases call for careful assessment of sparse support and model assumptions, not for filling the denominator with ineligible records.

Dependence, Diagnostics, and Alternatives

The risk-set definition is made at the decision-opportunity level, but the likelihood may involve dependence between opportunities. For one individual followed over time, the chain rule uses that individual’s preceding assignments when they belong in the conditional history. Across individuals in a cluster, shared clinicians, households, or group-level decisions may induce dependence. A product of person-level conditional probabilities is not automatically a valid joint likelihood merely because every person is eligible.

Worked Example: Eligible Individuals With Correlated Assignments

Two eligible people have fixed modeled histories. Their assignments are perfectly correlated: with probability $0.5$, both receive treatment, and with probability $0.5$, neither does. Each person’s marginal treatment probability conditional on their fixed history is $0.5$. Yet the true joint probability of observing $(1,1)$ is $0.5$, not $0.5\times0.5=0.25$.

If the two individual probabilities are multiplied as though assignments were conditionally independent, the resulting joint likelihood is wrong. In the ordering person 1 then person 2, the chain-rule factors are $P(A_1=1)=0.5$ and $P(A_2=1\mid A_1=1)=1$, whose product is $0.5$, the correct joint probability. The second factor must include the preceding assignment here. Alternatively, use a joint model for the cluster. Eligibility restriction alone does not resolve dependence.

Useful diagnostics focus on whether the risk set has been constructed and modeled as intended:

Audit the decision rule. For every time, confirm that $R_{it}$ is determined before assignment and matches the protocol. Review a sample of eligible and ineligible records against the operational definition.
Check denominators by time and history. Report eligible counts and counts for each treatment level. Unexpectedly large drops, empty strata, or treatment levels absent among eligible people can reveal coding errors or weak support.
Separate ineligibility from missingness. Confirm that eligible but unrecorded assignments are not silently recoded as ineligible or untreated.
Assess dependence at the level of assignment. If multiple people share assignment mechanisms, check whether the fitted likelihood accounts for dependence. A variance adjustment alone does not repair a misspecified joint assignment probability.
Inspect predictions where decisions occur. Evaluate calibration and model fit among eligible observations, by time and important histories. Predictions for ineligible records are not a validation target for the assignment model.

A tempting alternative is to retain every person-time row and code $A_{it}=0$ whenever $R_{it}=0$. That approach is appropriate only if “no treatment” is genuinely the outcome of an eligible decision. When ineligible records are structural non-decisions, the approach changes the modeled quantity. Restricting to eligible observations is the direct alternative; if the scientific or operational question instead concerns eligibility and assignment jointly, that requires a model that explicitly represents both processes.

The restriction has limits. It estimates assignment probabilities for eligible decision opportunities and does not identify assignment probabilities for histories where no one is eligible. Sparse risk sets can make fitted probabilities unstable, and a correct risk-set definition does not guarantee a correct functional form or adequate positivity. Nor does a treatment model by itself establish the assumptions needed for causal interpretation. Its role here is narrower: to represent assignment conditional on the information available when an eligible decision is made.

Key takeaway: At time $t$, estimate assignment probabilities using observations with $R_{it}=1$, and treat $R_{it}=0$ as no decision rather than as an untreated assignment. When assignments may be dependent, ensure that the likelihood conditions on the preceding assignments needed for a valid factorization, or specify a suitable joint model.

Check Your Understanding

Answer each question, then compare with the solutions below.

At a decision time, 80 people are eligible and 24 receive treatment. Another 20 people are ineligible. What is the eligible-sample treatment fraction, and why is the fraction $24/100$ different?
An eligible person’s treatment was not recorded. Should that record be coded as $R=0$? Explain.
State the condition needed to multiply individual assignment probabilities into a joint likelihood when assignments in a cluster may be dependent.
Can a time-specific assignment probability be learned directly for a history that never occurs among eligible individuals? Why or why not?

Solutions. (1) The eligible-sample fraction is $24/80=0.30$. The fraction $24/100=0.24$ includes 20 people for whom no assignment was possible, so it is not the conditional fraction among eligible decisions. (2) No. The person is eligible, so $R=1$; the assignment is missing and should be handled as missing data rather than structural ineligibility or an untreated decision. (3) The factors must be the correct sequential conditional probabilities, conditioning on all preceding assignments needed to describe the joint law; conditional independence given modeled histories is one sufficient alternative. (4) No. With no eligible observations at that history, the data contain no assignment outcomes there, so the probability is not empirically identified without additional modeling assumptions.