Treatment Probability Models With Clustered Assignments

Why the Assignment Unit Matters

When treatment is assigned to a clinic, school, worksite, or community, all members of that cluster may receive the same assignment. The data set can contain hundreds of people but only a much smaller number of independent assignment decisions. A treatment probability model must reflect that design: the probability of assignment is for a cluster, not a separate probability for each person who happens to share its assignment.

This distinction affects both prediction and uncertainty. Individual characteristics may help explain why a cluster was assigned treatment, but they do not turn one shared assignment into many independent assignments. A useful model therefore starts by identifying the assignment unit, then records which predictors belong to the cluster and which describe its members.

Definition: Let \(g=1,\ldots,G\) index clusters, \(i=1,\ldots,n_g\) index individuals within cluster \(g\), and \(t=1,\ldots,T\) index decision times. Let \(A_{gt}\) be a binary treatment assigned to cluster \(g\) at time \(t\). Under shared assignment, the individual treatment is \(A_{git}=A_{gt}\) for every member \(i\) of that cluster. A cluster-level treatment probability is a conditional probability for \(A_{gt}\), given specified cluster and treatment history information.

Write \(X_g\) for baseline cluster-level predictors, such as region or facility size, and \(L_{git}\) for individual-level covariates or histories. A model may use a cluster summary of the \(L_{git}\), such as their mean, or a prespecified set of member-level summaries. It may also use particular individual histories if the design or assignment process makes them relevant. The model must state what information is conditioned on; “cluster-level” describes the unit receiving treatment, not necessarily the level of every predictor.

For example, a clinic may be assigned a protocol partly according to its patient volume and partly according to the proportion of its patients with a risk factor. The protocol assignment is one clinic-level event. Patient volume and the risk-factor proportion are predictors of that event, not extra treatment assignments.

Specifying a Cluster-Level Treatment Model

For binary assignment, a common model uses the logistic link. Let \(H_{gt}\) denote the information used to predict assignment at time \(t\), including eligible prior cluster treatments and covariates defined for that decision. A simple specification is

\pi_{gt} = P(A_{gt}=1\mid H_{gt}) = \frac{\exp(\eta_{gt})}{1+\exp(\eta_{gt})}, \qquad \eta_{gt} = \beta_0+\beta_1^\top X_g+\beta_2^\top S_{gt}+\beta_3^\top A_{g,<t}.

Here \(S_{gt}\) is a vector of prespecified summaries of member-level information, and \(A_{g,<t}\) denotes the cluster’s prior treatment history. The conditional probability of the observed assignment is \(\pi_{gt}\) if \(A_{gt}=1\), and \(1-\pi_{gt}\) if \(A_{gt}=0\). A model can include nonlinear terms or interactions when substantively justified; the logistic form is an example of a probability model, not a guarantee that the chosen predictors are adequate.

The assignment likelihood is built from cluster-time decisions. If clusters are independent and the conditional model is correctly specified, its likelihood contribution is

L(\beta) = \prod_{g=1}^{G}\prod_{t=1}^{T} \pi_{gt}^{A_{gt}}(1-\pi_{gt})^{1-A_{gt}},

where each \(\pi_{gt}\) is conditional on the information specified for that decision. If the data have missing decisions or decision times at which a cluster is not eligible, the product includes only modeled, observed decisions. The treatment model must not count every cluster member as an independent Bernoulli assignment when assignment is shared.

Worked Example: One Assignment, Several Cluster Predictors

Suppose a clinic’s treatment assignment is modeled using an intercept, its standardized patient volume \(V_g\), and the proportion \(R_g\) of patients with a risk factor:

\operatorname{logit}(\pi_g)=-1+0.5V_g+2R_g.

For a clinic with \(V_g=1\) and \(R_g=0.25\), the linear predictor is

\eta_g=-1+0.5(1)+2(0.25)=0.

Therefore \(\pi_g=\exp(0)/(1+\exp(0))=1/2\). If this clinic has 40 patients, there are still 40 patient records but only one clinic assignment in this model. Repeating the same \(A_g\) 40 times and treating those repetitions as independent Bernoulli observations would incorrectly give the clinic’s single assignment 40-fold influence.

Conditional Histories and Marginal Probabilities

For repeated assignments, conditioning must be explicit. Let \(C_g\) denote a specified covariate history for cluster \(g\), and let \(\mathbf A_g=(A_{g1},\ldots,A_{gT})\). By the chain rule, for any treatment sequence \(\mathbf a_g\) with positive conditional probability,

P(\mathbf A_g=\mathbf a_g\mid C_g) = \prod_{t=1}^{T} P(A_{gt}=a_{gt}\mid A_{g1}=a_{g1},\ldots,A_{g,t-1}=a_{g,t-1},C_g).

The factors in this identity condition on the same \(C_g\), as well as the prior treatments in the sequence. If covariates are fixed baseline predictors, this is a direct way to express the conditional treatment-history probability. If \(C_g\) includes time-varying covariates, the identity remains true when each factor is conditioned on the entire specified \(C_g\). That factor may then condition on covariates observed after the decision time, so it need not be the sequential probability model used for a causal analysis.

In a sequential treatment model, the probability at time \(t\) is often defined using only information available by that decision, such as prior treatment and covariate history through \(t\). The product of those sequential probabilities is a treatment-mechanism factorization under the specified sequential data-generating model. It should not automatically be described as \(P(\mathbf A_g\mid\text{the entire realized time-varying covariate path})\): conditioning on future covariates can change the conditional treatment probabilities. State the conditioning set that the model actually uses.

An unconditional treatment-history probability is obtained by averaging the conditional probability over the covariate distribution, not by evaluating the conditional factors at an arbitrary covariate value. In general,

P(\mathbf A_g=\mathbf a_g) = \int P(\mathbf A_g=\mathbf a_g\mid C_g=c)\,dF_{C_g}(c).

Worked Example: Conditional Products and Marginalization

Let \(X_g\) be a binary baseline cluster characteristic with \(P(X_g=0)=P(X_g=1)=1/2\). Suppose

P(A_{g1}=1\mid X_g=0)=0.2,\qquad P(A_{g1}=1\mid X_g=1)=0.8,

and

P(A_{g2}=1\mid A_{g1}=1,X_g=0)=0.25,\qquad P(A_{g2}=1\mid A_{g1}=1,X_g=1)=0.75.

For the treatment history \((1,1)\), the conditional probabilities are \(0.2(0.25)=0.05\) when \(X_g=0\) and \(0.8(0.75)=0.60\) when \(X_g=1\). Thus the unconditional history probability is

P(A_{g1}=1,A_{g2}=1) = \frac12(0.05)+\frac12(0.60)=0.325.

A tempting but incorrect calculation is to average each displayed conditional factor separately and multiply: \([ (0.2+0.8)/2 ][(0.25+0.75)/2]=0.25\). The second factor must account for the distribution of \(X_g\) among clusters with \(A_{g1}=1\). Indeed, \(P(A_{g1}=1)=0.5\), and Bayes’ rule gives \(P(X_g=1\mid A_{g1}=1)=0.8\). Therefore

P(A_{g2}=1\mid A_{g1}=1) =(0.2)(0.25)+(0.8)(0.75)=0.65, \qquad (0.5)(0.65)=0.325.

The conditional product followed by integration gives the correct marginal probability; multiplying separately averaged factors does not.

What the Cluster-Level Likelihood Counts

The likelihood also clarifies why a large number of individual records does not substitute for a large number of assignment units. For a single decision per cluster, write \(W_g\) for the model’s predictor vector, including an intercept, and \(\pi_g=\operatorname{expit}(\beta^\top W_g)\). The log-likelihood is

\ell(\beta) = \sum_{g=1}^{G} \left\{ A_g\log(\pi_g)+(1-A_g)\log(1-\pi_g) \right\}.

Result: Under independent clusters and a correctly specified conditional Bernoulli model, the expected score contribution from each cluster is zero. This statement concerns the model’s assignment unit; it does not require independence among individuals within a cluster.

To verify the result, differentiate the cluster contribution. Since \(\partial\pi_g/\partial\beta=\pi_g(1-\pi_g)W_g\), its score is

U_g(\beta) = \frac{\partial\ell_g(\beta)}{\partial\beta} = W_g(A_g-\pi_g).

Conditional on \(W_g\), correct specification means \(E[A_g\mid W_g]=\pi_g\). Hence

E[U_g(\beta)\mid W_g] = W_g\{E[A_g\mid W_g]-\pi_g\} = 0, \qquad E[U_g(\beta)]=0. \quad\square

This derivation also identifies the relevant amount of information: the score sums over clusters, not over patients duplicated under a shared assignment. For repeated decisions, the score contributions are indexed by the modeled cluster-time decisions, with their specified histories and dependence accounted for. Standard errors must likewise respect the independent sampling or assignment units; if clusters are independent, cluster-level contributions provide the natural basis for variance estimation.

Worked Example: Individual Histories Inform a Shared Decision

A district assigns one program to each school. Before assignment, school \(g\) has 20 pupils, of whom 6 had a prior high-risk indicator. Let \(R_g=6/20=0.30\), and suppose a fitted model is \(\operatorname{logit}(\pi_g)=-1+2R_g\). Then

\eta_g=-1+2(0.30)=-0.40, \qquad \pi_g=\frac{\exp(-0.40)}{1+\exp(-0.40)} \approx \frac{0.6703}{1.6703}\approx 0.401.

The 20 pupils’ histories contribute to the summary \(R_g\), but the predicted probability \(0.401\) is the probability of the school’s assignment. Under shared assignment, each pupil’s treatment is the same school-level value. A model that instead treats the 20 pupil records as 20 separate treatment outcomes would answer a different question and understate uncertainty if it ignored within-school dependence.

Diagnostics, Alternatives, and Limitations

Assessment should focus on both the assignment mechanism and the scale of the model. Check that predictors are measured before the relevant decision, that cluster summaries are defined consistently, and that predicted probabilities are not concentrated near zero or one without substantive explanation. Compare predicted and observed assignment frequencies across important cluster types. For repeated decisions, inspect each decision time and relevant history strata rather than relying only on a single overall fit statistic.

A common diagnostic warning is that a model appears to have an enormous sample size because there are many individual rows, while treatment varies only across a modest number of clusters. The effective information for an assignment-level model is governed largely by the number and distribution of clusters and assignments. Sparse treatment categories or near-separation among cluster predictors can produce unstable estimates and extreme predicted probabilities.

The main alternative is to model genuinely individual-level assignments when treatment can vary among individuals. That requires an individual assignment probability and an appropriate account of dependence within clusters; it is not a relabeling of a shared cluster assignment. When assignment is shared, modeling the cluster outcome directly is the coherent approach. Flexible prediction methods may also be used, but they must still respect the assignment unit and the conditioning information. A more flexible algorithm cannot repair a mistaken unit of analysis or unavailable predictors.

Finally, a well-fitted treatment probability model is not proof that all relevant confounding has been measured, nor does it resolve positivity problems. Its probabilities are conditional on the variables included and on the model’s assumptions. Keep the interpretation narrow: it estimates the specified conditional assignment mechanism.

Common pitfall: “There are 500 individuals, so the assignment model has 500 independent treatment observations.” If treatment was assigned once to each of 10 clinics, there are 10 clinic-level assignments, not 500 independent ones. Use clinic-level treatment probabilities and account for the cluster structure; individual histories may enter as predictors or summaries, but do not multiply the number of assignments by the number of members.

Check Your Understanding

Answer each question, then compare with the solutions below.

A treatment is assigned once to each nursing home and shared by all residents. What is the assignment unit, and how many assignment observations are there for 12 homes?
Give one cluster-level predictor and one way individual histories could enter a cluster-level model.
Why is multiplying two separately averaged conditional probabilities generally not a valid way to obtain an unconditional treatment-history probability?
For one cluster, \(P(A_1=1\mid X=0)=0.3\), \(P(A_1=1\mid X=1)=0.7\), and \(P(A_2=1\mid A_1=1,X)=0.2\) for \(X=0\) and \(0.6\) for \(X=1\). If \(P(X=0)=P(X=1)=1/2\), calculate \(P(A_1=1,A_2=1)\).
What does a very small fitted assignment probability warn you to examine?

Solutions. (1) The nursing home is the assignment unit, so there are 12 assignment observations. (2) Region is a cluster-level predictor; the proportion of residents with a specified prior history is a summary of individual histories. (3) The distribution of covariates among clusters with the earlier treatment can differ from the overall covariate distribution, so the later conditional factor must be averaged with the appropriate conditional distribution. (4) The conditional history probabilities are \(0.3(0.2)=0.06\) for \(X=0\) and \(0.7(0.6)=0.42\) for \(X=1\). Averaging gives \(0.5(0.06)+0.5(0.42)=0.24\). (5) Examine sparse treatment patterns, near-separation, predictor coding, model fit, and whether the relevant covariate strata have adequate overlap; very small probabilities can signal instability or a positivity concern.