Separate-Time and Pooled Treatment Models

Why the Meaning of “Pooled” Matters

At each follow-up time, a binary treatment model estimates the probability of treatment given the history available before that treatment decision. Fitting one regression per time and fitting one regression to records from all times are not automatically two ways of estimating the same model. The answer depends on whether the pooled regression gives each time its own parameters or requires parameters to be shared across times.

This distinction matters when estimated treatment probabilities will be used to construct inverse probability weights. A model that forces a common relationship across times can smooth estimates, but it can also miss real changes in treatment assignment. Conversely, fitting separate models allows time-specific relationships but uses fewer observations per fit. These are modeling choices, not merely different data layouts.

Definition: Let $A_{it}\in\{0,1\}$ be the treatment for individual $i$ at time $t$, and let $H_{it}$ denote the observed covariate and treatment history available before that decision. A separate-time logistic model specifies $$ \operatorname{logit} P(A_{it}=1\mid H_{it})=x_{it}^{\mathsf T}\beta_t, $$ where $x_{it}$ is the chosen design vector and $\beta_t$ is allowed to vary with time. A pooled model stacks contributions from multiple times into one likelihood. It is unrestricted by time if it retains a separate parameter vector $\beta_t$ for each time; it is constrained if some or all parameters must be shared across times.

Here $\operatorname{logit}(p)=\log\{p/(1-p)\}$, and its inverse is $\operatorname{expit}(z)=1/\{1+\exp(-z)\}$. The design vector is specified by the analyst; it need not contain the same covariates at every time. For the exact equivalence result below, however, the separate and pooled formulations must use the same observed treatment records, design vectors, offsets if any, and logistic likelihood contributions.

Likelihood Equivalence With Separate Parameter Blocks

For a set of observed treatment decisions at time $t$, write $p_{it}(\beta_t)=\operatorname{expit}(x_{it}^{\mathsf T}\beta_t)$. The Bernoulli log-likelihood contribution is $$ \ell_t(\beta_t)=\sum_i \left[A_{it}\log p_{it}(\beta_t)+(1-A_{it})\log\{1-p_{it}(\beta_t)\}\right]. $$ The separate-time analysis maximizes each $\ell_t$ over its own parameter vector. A pooled model with distinct parameter blocks has log-likelihood $\ell_{\mathrm{pool}}(\beta_1,\ldots,\beta_K)=\sum_{t=1}^K\ell_t(\beta_t)$. Thus, in this formulation, “pooling” combines the likelihood contributions but does not impose common coefficients.

Theorem: Suppose the pooled log-likelihood is exactly the sum of the separate-time log-likelihoods, and each time-specific likelihood has a finite maximizer. Then every tuple formed by choosing a separate-time maximizer at each time is a pooled maximizer. Conversely, each pooled maximizer has a time-specific maximizing parameter in every block. The coefficient vectors need not be unique. If they are nonunique, fitted probabilities on the observed design rows are nevertheless the same across maximizers.

Maximize each contribution.
For every $t$, choose $\hat\beta_t$ such that $\ell_t(\hat\beta_t)\geq \ell_t(\beta_t)$ for every allowable $\beta_t$.

Add the inequalities.
For any candidate tuple $(\beta_1,\ldots,\beta_K)$, summing the time-specific inequalities gives $\sum_t\ell_t(\hat\beta_t)\geq\sum_t\ell_t(\beta_t)$. Thus the tuple of separate maximizers is a pooled maximizer.

Check the converse.
If a pooled maximizer had a block $\beta_s$ that did not maximize $\ell_s$, replacing just that block by a maximizer of $\ell_s$ would increase the sum. That contradicts pooled maximality. Hence every block of a pooled maximizer is a separate-time maximizer.

This proves a statement about the set of maximizers; it does not, by itself, prove that there is only one maximizing coefficient vector. Logistic log-likelihood is strictly concave in the vector of observed linear predictors, but it may not be strictly concave in $\beta_t$ when the design matrix is rank deficient. For example, duplicate columns permit different coefficient vectors with the same linear predictors. In that case the coefficient estimates are not unique, while the fitted probabilities on the observed rows are unchanged. If each time-specific maximizer is unique, the coefficient vectors from the pooled and separate fits are identical. The same fitted-probability formula and observed design rows are also essential to the probability-equivalence claim. $\square$

A finite maximizer is a substantive condition: complete or quasi-complete separation can prevent ordinary logistic maximum likelihood from having a finite estimate. The likelihood factorization still identifies the same supremum under the unrestricted parameter blocks, but one must not describe a nonexistent finite maximizer as an estimate.

Worked Example: Separate Fits and an Unrestricted Pooled Fit

Suppose there are two times, and each model has only an intercept. At time 1, 2 of 5 individuals receive treatment; at time 2, 3 of 5 receive treatment. For a group with $y$ treated observations out of $m$, the log-likelihood for its treatment probability $p$, ignoring a constant, is $y\log p+(m-y)\log(1-p)$. Its maximum occurs at $\hat p=y/m$.

At time 1, $\hat p_1=2/5=0.4$, so the separate intercept estimate is $\hat\beta_1=\operatorname{logit}(0.4)=\log(2/3)$, approximately $-0.4055$. At time 2, $\hat p_2=3/5=0.6$, giving $\hat\beta_2=\operatorname{logit}(0.6)=\log(3/2)$, approximately $0.4055$.

The unrestricted pooled fit has two intercept parameters, one for each time, and log-likelihood $\ell_1(\beta_1)+\ell_2(\beta_2)$. Its maximizing probabilities are therefore $0.4$ at time 1 and $0.6$ at time 2, just as in the separate fits. Pooling has not averaged these probabilities; it has merely written the same two likelihood contributions in one objective.

What Changes When Parameters Are Shared?

A constrained pooled regression does something different: it maximizes the sum of the time-specific contributions over a restricted parameter space. For example, a shared-intercept model requires $p_{it}=p$ at every time when there are no covariates. It cannot reproduce different time-specific probabilities. More generally, a shared-coefficient model forces the treatment probability to follow the same specified relationship with the design variables at each time.

Worked Example: A Shared Intercept Changes the Probabilities

Use the two-time data from the preceding example, but now require a single intercept and therefore one common treatment probability $p$. There are $2+3=5$ treated decisions among $5+5=10$ observations. The pooled log-likelihood, apart from a constant, is $$ \ell(p)=5\log p+5\log(1-p). $$

Differentiating gives $\ell'(p)=5/p-5/(1-p)$. Setting this equal to zero yields $5(1-p)=5p$, so $\hat p=1/2$. The second derivative is $\ell''(p)=-5/p^2-5/(1-p)^2<0$ for $0<p<1$, confirming a maximum. Thus the constrained pooled fit predicts $0.5$ at both times, whereas the separate fits predict $0.4$ and $0.6$. The difference comes from the common-parameter restriction, not from pooling records as such.

A general criterion makes the distinction precise. If the separate-time maximizers produce fitted probabilities that can all be represented by one parameter value satisfying the pooled model’s restrictions, then that value is also a maximizer of the constrained pooled likelihood: it already attains the unrestricted maximum. If no allowed shared parameter can reproduce those separate fitted probabilities, the constrained model cannot attain that unrestricted maximum and must give different fitted probabilities on at least some observed rows.

Worked Example: A Shared Slope Can Hide Time Differences

Now let $X$ be a binary covariate and fit an intercept and slope at each time. At time 1, suppose there are four observations in each $X$ group: one treated when $X=0$, and three treated when $X=1$. The fitted probabilities are $1/4$ and $3/4$. The intercept and slope are $$ \hat\beta_{10}=\operatorname{logit}(1/4)=-\log 3,\qquad \hat\beta_{11}=\operatorname{logit}(3/4)-\operatorname{logit}(1/4)=\log 9. $$

At time 2, suppose there are again four observations in each $X$ group, with two treated in each group. The fitted probabilities are $1/2$ and $1/2$, giving $\hat\beta_{20}=0$ and $\hat\beta_{21}=0$. Separate-time fits therefore show a different covariate relationship across times.

A pooled model with one shared intercept and one shared slope has, across both times, 3 treated of 8 observations when $X=0$, and 5 treated of 8 when $X=1$. With a binary $X$, an intercept and slope can represent the two group probabilities freely. The pooled estimates are consequently $\hat p_0=3/8$ and $\hat p_1=5/8$. They give $$ \hat\beta_0=\operatorname{logit}(3/8)=\log(3/5),\qquad \hat\beta_1=\operatorname{logit}(5/8)-\operatorname{logit}(3/8)=\log(25/9). $$

The calculations check directly: $\operatorname{expit}\{\log(3/5)\}=3/8$, and adding the slope gives $\log(3/5)+\log(25/9)=\log(5/3)$, whose expit is $5/8$. The shared-slope pooled fit estimates a compromise across times; it does not equal either time-specific relationship.

Worked Example: When a Shared Model Can Agree With Separate Fits

Suppose that at each of two times and within each of two $X$ groups, there are four observations, with one treated at $X=0$ and three treated at $X=1$. Each separate fit estimates probabilities $1/4$ and $3/4$, and hence intercept $-\log 3$ and slope $\log 9$.

In the pooled data, the $X=0$ group has 2 treated out of 8 and the $X=1$ group has 6 treated out of 8. The pooled probabilities are still $1/4$ and $3/4$. The corresponding shared intercept and slope are therefore $-\log 3$ and $\log 9$, identical to the separate estimates. Here the separate maximizers satisfy the pooled model’s shared-parameter restriction, so both approaches yield the same fitted probabilities.

Comparison, Diagnostics, and Limitations

Approach	Parameter structure	What it guarantees	Main limitation
Separate-time regressions	A distinct parameter vector is estimated at every time.	Allows the fitted relationship to vary freely by time, within each specified model.	Each fit uses only that time’s treatment decisions, so estimates may be unstable when data are sparse.
Unrestricted pooled likelihood	All time-specific parameter blocks are included in one likelihood.	With identical rows and likelihood contributions, has the same maximizing fitted probabilities as the separate fits.	It does not, by itself, borrow information across times or reduce the number of parameters.
Constrained pooled regression	Some parameters are shared across times.	Uses a lower-dimensional model when the shared restrictions are appropriate.	Can conceal time-specific differences and yield misspecified treatment probabilities.

Several checks help identify problems. Compare treatment counts and fitted probabilities by time, and, where covariates are included, examine whether calibration or residual patterns differ across times or covariate groups. A pooled model that looks acceptable overall can still systematically overpredict in one time period and underpredict in another. Check as well that the separate and pooled fits use the same eligible observations and the same pre-treatment predictors; otherwise, differences cannot be attributed to pooling restrictions alone.

Sparse data and separation can make separate estimates extreme or non-finite. Pooling with shared parameters may stabilize estimates, but stability does not establish that the restrictions are correct. At the other extreme, an unrestricted pooled fit has the same information and fitted values as separate fits, so changing its layout alone cannot solve sparse-data problems. Finally, likelihood contributions from repeated decisions do not make a person’s records independent in the marginal sense. The conditional treatment likelihood and the approach used to quantify uncertainty are distinct considerations.

Common pitfall: “The data were pooled, so the fitted probabilities must be pooled averages.” This is false. A pooled likelihood with separate parameter blocks reproduces the separate fits, while a shared-parameter model may average across time as a consequence of its restrictions. Also, showing that separate maximizers form a pooled maximizer does not prove that the coefficient maximizer is unique. State the parameter structure and distinguish coefficient uniqueness from fitted-probability equality.

Check Your Understanding

Answer each question, then check the solutions below.

Two time-specific likelihoods are added in one pooled objective, but each has its own parameter vector. What condition makes this equivalent to separate fitting?
At two times, 4 of 10 and 6 of 10 treatment decisions are positive. What are the separate intercept-only fitted probabilities, and what is the shared-intercept pooled probability?
Why can coefficient estimates differ across maximizing solutions while fitted probabilities on the observed rows remain identical?
If separate-time fitted probabilities can be represented by one shared-parameter model, what follows about the constrained pooled maximum?

Solutions. (1) The pooled objective must be exactly the sum of the same separate-time likelihood contributions, with the same rows, design variables, and parameter spaces kept separate by time. (2) The separate probabilities are $4/10=0.4$ and $6/10=0.6$; the shared probability is $(4+6)/(10+10)=0.5$. (3) A rank-deficient design can give distinct coefficient vectors with the same linear predictors on observed rows; the logistic likelihood then gives the same fitted probabilities. (4) The shared parameter value reproduces the unrestricted separate maxima, so it also maximizes the constrained pooled likelihood. $\square$