Building Covariate Histories for Denominator Models

Why the Covariate History Must Be Built One Decision at a Time

A sequential denominator model estimates the probability of the treatment actually received at each decision, conditional on the information relevant to that decision. Its covariate predictors therefore cannot be chosen from an undated list of variables. Each variable must be attached to a measurement time, and the history for decision $t$ must contain exactly the covariates observed by that decision. If an available covariate is omitted, the model may fail to represent the conditional treatment probability; if a future covariate is included, the model uses information unavailable when treatment was assigned.

Use the convention that $L_0$ is baseline covariate information observed before follow-up, and $L_t$ is the covariate vector measured before treatment decision $A_t$, for $t=1,\ldots,K$. Thus $L_2$ is available for decision 2 when it is measured before $A_2$; it is not a post-decision measurement under this convention. The earlier material on longitudinal treatment weights establishes the role of sequential treatment probabilities. Here the focus is the covariate component of the information used to model them.

Definition: The observed covariate history available before decision $t$ is $$ \bar L_t=(L_0,L_1,\ldots,L_t). $$ The denominator probability at decision $t$ conditions on this history and on the past treatment history: $$ P(A_t=a_t\mid \bar A_{t-1},\bar L_t), $$ where $\bar A_{t-1}=(A_1,\ldots,A_{t-1})$, with an empty past-treatment history at $t=1$. A covariate predictor for this denominator must be constructed from $\bar L_t$, not from measurements made after $A_t$.

This definition is about availability, not whether every component will ultimately be included in a fitted model. The full history is the set of measured covariate information that could be relevant. A chosen model may use all of it, or a defensible summary, but a summary entails a restriction that should be made explicit. The denominator probability is also conditional on past treatment; the construction and coding of those treatment predictors is a separate issue.

From a History to Model Predictors

For binary treatment, a sequential denominator model commonly represents the conditional probability with a logistic regression. In a separate model for decision $t$, the covariate part of the linear predictor can be written generically as a function of the entries in $\bar L_t$:

\operatorname{logit} P(A_t=1\mid \bar A_{t-1},\bar L_t) = \text{terms for past treatment} + f_t(L_0,L_1,\ldots,L_t).

The function $f_t$ specifies how the available covariates enter: for example, as separate numeric or indicator predictors. This tutorial does not prescribe a functional form. The essential construction rule is that $f_t$ can use $L_0,\ldots,L_t$, but not $L_{t+1}$ or later measurements. If treatment probabilities are modeled in pooled long format, the same rule applies to each decision row: the row’s covariate predictors must correspond to the history available at that row’s decision.

Worked Example: Listing Histories Across Three Decisions

Suppose an observational study follows participants through three treatment decisions. Age and disease severity are recorded at baseline as $L_0$. A biomarker is measured before each decision and is denoted $L_1,L_2,L_3$, respectively. The histories are:

Decision 1: $\bar L_1=(L_0,L_1)$. The biomarker measured before $A_1$ is included.

Decision 2: $\bar L_2=(L_0,L_1,L_2)$. The measurement $L_2$, taken before $A_2$, is available and belongs in this history.

Decision 3: $\bar L_3=(L_0,L_1,L_2,L_3)$. All three pre-decision biomarker measurements and baseline information are available.

A model for decision 2 that uses only $(L_0,L_1)$ is not using the full observed history under this schedule: it leaves out $L_2$. Conversely, including $L_3$ in the decision-2 model would use a measurement obtained after $A_2$. Both errors are prevented by writing out the history before selecting predictors.

Worked Example: Calculating a Decision-Specific Probability

At decision 2, let $L_0$ indicate high baseline severity, $L_1$ indicate an elevated biomarker before decision 1, and $L_2$ indicate an elevated biomarker before decision 2. Consider a fitted model whose covariate contribution is

\operatorname{logit} P(A_2=1\mid \bar A_1,\bar L_2) = -1+0.4L_0+0.7L_1+0.8L_2,

with the past-treatment terms held fixed for this calculation. For a participant with $L_0=1,L_1=0,L_2=1$, the linear predictor is

$$ -1+(0.4)(1)+(0.7)(0)+(0.8)(1)=0.2. $$

The corresponding probability is

\frac{\exp(0.2)}{1+\exp(0.2)} = \frac{1.2214}{2.2214} \approx 0.550.

If $L_2$ were mistakenly left out, the linear predictor would be $-1+0.4=-0.6$, giving $\exp(-0.6)/(1+\exp(-0.6))\approx0.354$. These are different fitted probabilities because the model assigns a coefficient to $L_2$. The calculation does not establish that the coefficient or model is correct; it shows why the history must first be indexed consistently with measurement time.

When Can a Reduced History Be Used?

A model need not include every recorded value as a separate term, but replacing the full history with a summary is not automatically harmless. Let $S_t=g_t(\bar L_t)$ be a summary computed from the available covariate history. It can replace that history in the conditional treatment probability only if the omitted distinctions do not carry additional information about treatment, given the summary and past treatment.

Theorem: Suppose $S_t=g_t(\bar L_t)$ is a measurable function of the observed history. If $$ A_t\ \perp\!\!\!\perp\ \bar L_t\mid(\bar A_{t-1},S_t), $$ then $$ P(A_t=a\mid\bar A_{t-1},\bar L_t) = P(A_t=a\mid\bar A_{t-1},S_t) $$ for each treatment value $a$, almost surely. Thus the summary retains the treatment-assignment information in the full history, conditional on past treatment.

Proof. Conditional independence states that the conditional distribution of $A_t$ given $(\bar A_{t-1},S_t,\bar L_t)$ equals its conditional distribution given $(\bar A_{t-1},S_t)$, almost surely. Because $S_t$ is determined by $\bar L_t$, conditioning on $(\bar A_{t-1},\bar L_t)$ also determines $S_t$. Hence

P(A_t=a\mid\bar A_{t-1},\bar L_t) = P(A_t=a\mid\bar A_{t-1},S_t,\bar L_t) = P(A_t=a\mid\bar A_{t-1},S_t),

which is the claimed equality. The condition is substantive: it does not follow merely from choosing a convenient summary. $\square$

Worked Example: A Summary That Loses History Information

Suppose $L_1,L_2$ are binary and the true treatment probability at a decision is

P(A=1\mid L_1,L_2) = \operatorname{expit}(-1+L_1+2L_2), \qquad \operatorname{expit}(x)=\frac{\exp(x)}{1+\exp(x)}.

Consider using only the count $S=L_1+L_2$. Two histories with $S=1$ are $(L_1,L_2)=(1,0)$ and $(0,1)$. For the first, the logit is $-1+1+0=0$, so the probability is $0.5$. For the second, the logit is $-1+0+2=1$, so the probability is $\exp(1)/(1+\exp(1))\approx0.731$.

The same summary value therefore corresponds to different treatment probabilities. In this example, conditioning on $S$ alone cannot reproduce the full-history probability for both patterns. A summary can be useful for reducing model complexity, but it must preserve distinctions relevant to treatment assignment or be treated as an approximation.

Omission and Marginalization

The effect of omitting a covariate can be stated precisely. If $X$ is part of the history and $Z$ denotes the remaining conditioning information, then the treatment probability given $Z$ averages the probability given both $Z$ and $X$ over the conditional distribution of $X$:

Result: For binary treatment, $$ P(A_t=1\mid Z) = \sum_x P(A_t=1\mid Z,X=x)P(X=x\mid Z) $$ when $X$ is discrete. For continuous $X$, the sum is replaced by the corresponding conditional integral.

Derivation. For any fixed value of $Z$, partition the event $\{A_t=1\}$ according to the possible values of $X$. The law of total probability conditional on $Z$ gives

P(A_t=1\mid Z) = \sum_x P(A_t=1,X=x\mid Z) = \sum_x P(A_t=1\mid X=x,Z)P(X=x\mid Z).

The same argument applies with an integral for continuous $X$. Thus omitting $X$ does not generally leave the probability unchanged; it changes the target conditional probability into a mixture. That mixture may be an appropriate model target in some settings, but it is not the full-history denominator probability when the latter is required.

Diagnostics, Alternatives, and Limitations

History construction should be checked before fitting. A practical audit is to list, for each decision, every covariate’s measurement time and verify that it is included only in histories for decisions at or after its recorded availability. Then inspect the actual predictor columns in the analysis data: a decision-2 row should not accidentally receive a decision-3 measurement, and an available decision-2 covariate should not be dropped because of an off-by-one index.

After fitting, useful warning signs include implausible predicted probabilities, large changes when a substantively relevant history component is added, and very sparse combinations of history values. These checks can reveal coding or model instability, but they cannot prove that the history captures all causes of treatment assignment. The model also depends on accurate measurement, adequate functional form, and sufficient observed support for the treatment probabilities being estimated.

A current-covariate-only model is simpler than a full-history model, but it discards earlier measurements unless they are represented through a justified summary. A model with a carefully chosen summary can reduce dimensionality, but the theorem above shows the condition required for it to preserve the conditional probability. Separate-time and pooled treatment models are alternative ways to organize estimation; whichever is used, the covariate history supplied for each decision must obey the same availability rule. These choices do not eliminate the risks of omitted variables, measurement error, or sparse history patterns.

A common mistake is to treat “the latest covariate” as the entire history, or to assume a variable belongs to the next decision simply because its subscript is $t$. Under the convention here, $L_t$ is measured before $A_t$, so it belongs to $\bar L_t$. The correction is to define the measurement schedule first, then write $\bar L_t=(L_0,\ldots,L_t)$ and construct predictors from that list. If a study instead measures a variable after $A_t$, its data definition and index must reflect that different schedule.

Check Your Understanding

Answer each question, then compare with the solutions below.

If $L_0$ is baseline and $L_t$ is measured before $A_t$, what is the covariate history for decision 3?
A covariate $W_2$ is measured after $A_2$ but before $A_3$. Should it enter the decision-2 or decision-3 history?
Why can two histories with the same summary value still imply different treatment probabilities?
What does a model estimate after a discrete covariate $X$ is omitted from the conditioning set?

Solutions. (1) $\bar L_3=(L_0,L_1,L_2,L_3)$. (2) It is unavailable for decision 2 and available for decision 3, so it belongs in the decision-3 history under this measurement schedule. (3) The summary may combine distinct histories that have different conditional treatment probabilities; the worked example gives probabilities $0.5$ and approximately $0.731$ for two patterns with the same sum. (4) It estimates a conditional probability averaged over the distribution of $X$ given the remaining conditioning information, by the law of total probability.