Long-Format Data for Pooled Treatment Models

Why Long Format Matters

A pooled treatment model uses records from multiple treatment occasions in one estimation data set. To represent those occasions explicitly, the data are arranged in long format: a person who contributes several observed treatment decisions has several rows, one for each decision. The rows preserve which person and which decision time each treatment value belongs to.

Reshaping is a data-organization step, not a causal adjustment by itself. It does not determine which past variables belong in a treatment model, correct confounding, or solve informative loss to follow-up. Its job is to give each modeled treatment decision a clearly indexed record, with the variables associated with that decision attached to the correct row.

For binary treatment, \(A_{it}\) is coded 0 or 1. The time field identifies the decision occasion; it is not a substitute for the person identifier. The history field \(H_{it}\) is a placeholder for the information used to describe the treatment probability at time \(t\). Its exact construction and the coding of past treatments require separate decisions. The essential timing rule here is that information called pre-decision must be recorded before \(A_{it}\) is assigned.

From Repeated Records to Person-Period Rows

A wide data set commonly has one row per person and separate columns for repeated measurements, such as \(A_{i0}, A_{i1}, A_{i2}\). Reshaping converts the treatment columns into rows while retaining the person identifier and matching each period’s information to its own treatment. A row is included only for a decision that was observed and eligible under the study’s definition. A missed or ineligible decision is not silently recoded as \(A_{it}=0\).

A useful data audit is to verify that each included \((i,t)\) pair occurs exactly once, that the observed times are in the intended order, and that all fields on a row refer to the same decision. If the design schedules \(K\) decisions for everyone with complete follow-up, there should be \(nK\) rows. With incomplete follow-up, the row count is \(\sum_i\sum_t R_{it}\), not necessarily \(nK\).

What the Pooled Model Receives

For each included row, a pooled model represents the conditional probability of treatment at that decision. Write

Here \(\theta\) denotes the model parameter, and \(T_{it}=t\) records the decision time. The notation states the role of the row variables without prescribing how the history is constructed or how time is encoded. A pooled model estimates its treatment-probability relationship from the stacked rows; the model specification determines which columns and functional forms it uses.

For a binary treatment, the contribution of one observed decision to the conditional likelihood is

If \(A_{it}=1\), this is \(p_{it}(\theta)\); if \(A_{it}=0\), it is \(1-p_{it}(\theta)\). This compact expression makes clear why the treatment column must contain the treatment assigned at that row’s time, rather than a person-level summary or a treatment from another period.

Likelihood and Score on the Stacked Data

The distinction between dependence within a person and dependence between people is essential. Within a person, the chain rule expresses a joint treatment-history probability as a product of successive conditional probabilities. Across people, multiplying person-level likelihood contributions requires independent people, conditional on any modeled covariates, or an explicit joint model for their dependence.

To verify the result, first fix one person \(i\). The chain rule gives the probability of that person’s observed sequence as a product of the conditional probabilities at each observed decision, with the conditioning information appropriate to that sequence. Under the stated model these factors are \(q_{it}(\theta)\), so the person-level contribution is \(\prod_{t:R_{it}=1}q_{it}(\theta)\). Conditional independence across people makes the joint probability the product of these person-level contributions. Substituting their products and using \(\log\prod x_j=\sum\log x_j\) gives the displayed log-likelihood. No step treats repeated rows from one person as independent unconditional observations. \(\square\)

For a common logistic pooled model, let \(X_{it}\) be the row’s design vector and set \(p_{it}(\theta)=\operatorname{expit}(X_{it}^{\mathsf T}\theta)\), where \(\operatorname{expit}(u)=1/(1+\exp(-u))\). The log-likelihood contribution is

Differentiating this expression gives \(X_{it}\{A_{it}-p_{it}(\theta)\}\). Therefore the pooled score is

This follows by differentiating each row contribution and summing the derivatives of the log-likelihood. It shows the computational role of long format: every observed treatment decision contributes a residual, \(A_{it}-p_{it}(\theta)\), multiplied by that row’s design vector. The derivation does not make rows from the same person independent.

Diagnostics, Alternatives, and Limits

Before fitting a pooled model, inspect the stacked records rather than relying only on the original wide file. Useful checks include:

• Duplicate decisions: confirm that an \((i,t)\) key appears no more than once unless repeated records at the same decision are intentional and modeled.
• Time alignment: compare the time index on each row with the original treatment column and verify that linked measurements belong to the same occasion.
• Impossible sequences: check whether rows occur after a person was no longer eligible under the study design, or whether scheduled decisions are missing without an explained reason.
• Missingness versus no treatment: distinguish an observed \(A_{it}=0\) from an absent or unobserved decision. Coding missing treatment as zero changes the treatment record.
• Unequal row counts: tabulate rows per person and by time. Unexpected patterns can reveal accidental row loss, duplicated records, or follow-up differences that need substantive attention.

Long format is the natural alternative to fitting a separate model at each time when the goal is to pool decision records; separate-time models instead estimate time-specific models. The earlier material on separate-time and pooled treatment models distinguishes those modeling strategies. Either strategy still requires an appropriately defined treatment record and assumptions suited to the study. Wide format can remain convenient for data collection or descriptive summaries, but it does not directly supply one observation per treatment decision to a pooled row-based model.

A common mistake is to say, “There are six rows, so the sample size for the likelihood is six independent observations.” The rows are six treatment decisions, but multiple decisions may belong to the same person. The within-person likelihood uses conditional factors, while inference must respect the dependence structure and the independent sampling units. If people are independent, person-level contributions can be multiplied as above; standard errors should still account for repeated observations within person rather than treating all rows as independent clusters. If people share clusters or otherwise depend on one another, that dependence must also be addressed. Reshaping alone supplies no such correction.

Finally, a correctly organized table cannot repair a poorly defined observation process. If treatment decisions are absent because of dropout, illness, or another process related to treatment or covariates, simply omitting those rows can change the target distribution. Nor does a long table establish that the chosen history is sufficient for confounding control, that positivity holds, or that the probability model is adequate. Those are separate modeling and causal assumptions, not consequences of the file layout.