Observed Histories at Sequential Decision Times

Why the Timing of a History Matters

In a longitudinal study, treatment is chosen more than once. At each decision, some information has already been observed and some has not. The distinction is fundamental: the information available immediately before a treatment decision is not the same as everything eventually recorded about a participant. A symptom measured after treatment begins, for example, cannot be part of the history available at the earlier decision, even if it is recorded on the same visit date.

Here the task is to describe the observed information in its correct temporal order. A pre-decision history is a record of covariates measured by that point and treatments assigned at earlier decisions. Defining it carefully makes the timing of a longitudinal data set explicit. By itself, this bookkeeping neither asserts a causal assumption nor identifies a causal effect.

Definition: Suppose treatment decisions occur at times $t=0,1,\ldots,K$. Let $A_t$ denote the treatment assigned at decision $t$, and let $L_t$ denote the covariate information measured before that decision. Write $\bar A_t=(A_0,\ldots,A_t)$ and $\bar L_t=(L_0,\ldots,L_t)$. The observed history immediately before decision $t$ is $$ H_t=(\bar L_t,\bar A_{t-1}). $$ For $t=0$, $\bar A_{-1}$ is empty, so $H_0=L_0$. In words, $H_t$ contains covariates measured through $t$ and treatments assigned strictly before $t$, but not $A_t$ or measurements made after $A_t$.

The notation $L_t$ describes what is known just before $A_t$, not everything measured during a calendar period labelled $t$. A common timeline is: measure $L_0$, assign $A_0$, measure $L_1$, assign $A_1$, and continue. In this ordering, $L_{t+1}$ may reflect what happened after $A_t$, but it is available before $A_{t+1}$. Thus $L_{t+1}$ belongs to $H_{t+1}$, not $H_t$.

The labels should follow actual measurement and decision times. If a study records several variables at a visit, their shared date does not settle whether they precede or follow the treatment choice. The protocol or data-collection sequence must establish their order. When order is genuinely unknown, the data do not justify treating a later measurement as pre-decision information.

Histories as Sequences and Information Sets

The vector notation $H_t$ is compact, but its contents can be checked by writing them out. For the first three decisions,

H_0=L_0,\qquad H_1=(L_0,A_0,L_1),\qquad H_2=(L_0,A_0,L_1,A_1,L_2).

This expansion reveals an alternating sequence: a covariate record available before a decision, the treatment assigned at that decision, then the next covariate record. The current treatment is excluded from the history at its own decision. Once it has been assigned, it becomes part of the information preceding the next decision.

For a more formal description, let $\mathcal H_t$ be the sigma-field generated by the recorded variables in $H_t$. A sigma-field represents the events that can be determined from those variables. The information immediately after decision $t$, but before the next covariate measurement, is $\mathcal H_t^+=\sigma(\mathcal H_t,A_t)$. After $L_{t+1}$ is measured, the next pre-decision information set is $\mathcal H_{t+1}$.

Result: Under the stated measurement order, $\mathcal H_t^+=\sigma(\mathcal H_t,A_t)$ and $\mathcal H_{t+1}=\sigma(\mathcal H_t,A_t,L_{t+1})$. In particular, the information set available before decision $t+1$ contains the earlier history, the treatment at $t$, and the intervening covariate measurement.

Proof. By definition, $\mathcal H_t$ is generated by $L_0,A_0,L_1,\ldots,A_{t-1},L_t$. Adding $A_t$ to its generators gives exactly the recorded information after decision $t$: $\sigma(\mathcal H_t,A_t)$. The next pre-decision history consists of the same variables together with $L_{t+1}$. Its generated sigma-field is therefore $\sigma(\mathcal H_t,A_t,L_{t+1})$, as claimed. This argument is about recorded information and its ordering; it does not impose any probabilistic independence condition. $\square$

Worked Example: A Two-Decision Treatment Sequence

A clinic measures baseline disease activity $L_0$, chooses an initial medication $A_0$, measures disease activity again at a follow-up visit $L_1$, and then chooses whether to increase the dose $A_1$. At the end of follow-up it records an outcome $Y$. Write the histories immediately before both treatment decisions and classify the variables.

Before the first decision, only baseline disease activity has been measured, so $H_0=L_0$. The initial medication $A_0$ is not part of $H_0$, because it is the treatment being chosen at that decision.

Before the second decision, the clinic has observed baseline activity, the initial medication, and follow-up activity. Thus $H_1=(L_0,A_0,L_1)$. The follow-up measurement $L_1$ can be part of this history because it occurs before the dose-increase decision $A_1$. It cannot be moved into $H_0$, because it was not measured before $A_0$.

The outcome $Y$ is recorded after the treatment decisions in this example. It is not part of either pre-decision history. Whether $Y$ is binary or continuous does not change its temporal position.

A Chain-Rule Factorization of the Observed Data

A correctly ordered history also provides a convenient way to write the observed-data distribution. This is a probability chain rule, not a causal identification formula. Suppose $Y$ is recorded after the last treatment $A_K$, and the observed sequence is $L_0,A_0,L_1,A_1,\ldots,L_K,A_K,Y$. For discrete variables, let $p$ denote probability masses; for variables with densities, let it denote the corresponding densities. More generally, the same statement can be written with conditional probability distributions. In that form, assume regular conditional distributions exist, as they do on standard Borel state spaces.

Result: The observed joint distribution factors in chronological order as $$ p(l_0,\bar a_K,\bar l_K,y) = p(l_0) \left\{\prod_{t=0}^{K}p(a_t\mid h_t)\right\} \left\{\prod_{t=0}^{K-1}p(l_{t+1}\mid h_t,a_t)\right\} p(y\mid h_K,a_K). $$ Here each conditional distribution is an observed-data conditional distribution. The factorization alone makes no claim about interventions or causal effects.

Derivation. Apply the chain rule to the sequence in the order it is observed. Its first factor is $p(l_0)$. The next variable is $A_0$, whose preceding variables are exactly $H_0=L_0$, giving $p(a_0\mid h_0)$. Next comes $L_1$; its preceding variables are $H_0,A_0$, giving $p(l_1\mid h_0,a_0)$. At a general decision $t$, the preceding variables before $A_t$ are precisely $H_t$, so its factor is $p(a_t\mid h_t)$. If $t<K$, the next variable $L_{t+1}$ is conditioned on $H_t,A_t$. The final variable $Y$ follows $H_K,A_K$, giving $p(y\mid h_K,a_K)$. Multiplying these successive conditional factors is the chain rule and yields the displayed expression. No factor has been removed using independence. $\square$

Worked Example: Checking a Path Probability

Consider two treatment decisions, $A_0$ and $A_1$, with binary $L_0,L_1,Y$. For the observed path $L_0=1,A_0=1,L_1=1,A_1=1,Y=1$, suppose the relevant conditional probabilities are $P(L_0=1)=0.40$, $P(A_0=1\mid L_0=1)=0.75$, $P(L_1=1\mid L_0=1,A_0=1)=0.60$, $P(A_1=1\mid L_0=1,A_0=1,L_1=1)=0.50$, and $P(Y=1\mid L_0=1,A_0=1,L_1=1,A_1=1)=0.80$. Find the probability of this complete observed path.

The first decision history is $H_0=L_0$; the second is $H_1=(L_0,A_0,L_1)$. Applying the chain rule in that order gives

\begin{aligned} &P(L_0=1,A_0=1,L_1=1,A_1=1,Y=1)\\ &=0.40(0.75)(0.60)(0.50)(0.80)\\ &=0.30(0.60)(0.50)(0.80)\\ &=0.18(0.50)(0.80)\\ &=0.09(0.80)=0.072. \end{aligned}

The result is $0.072$. Each factor is conditional on the variables that precede it in the observed sequence. This calculation does not mean that setting either treatment to 1 would produce this probability for the outcome; it describes the probability of an observed path under the data-generating distribution.

Worked Example: Sorting Variables by Their Measurement Time

In a study, a participant reports smoking status before a treatment decision at month 0. Treatment $A_0$ is then assigned. At month 2, the study measures a symptom score $S_1$; at month 3 it records an adverse event $E_1$; at month 4, before the next decision, it measures symptom score $S_2$ and assigns treatment $A_1$. The primary outcome is recorded at month 8. Decide which recorded values belong to $H_0$ and $H_1$.

Before $A_0$, the measured smoking status is the only listed covariate, so $H_0$ contains smoking status. Neither $S_1$ nor $E_1$ belongs in this history: both are measured after the first treatment decision.

Before $A_1$, the observed information includes smoking status, $A_0$, $S_1$, $E_1$, and $S_2$, provided the adverse event is recorded before the month-4 decision. Therefore $H_1$ contains all five of these components. The outcome at month 8 is not in $H_1$, and neither is $A_1$, because both occur after the second pre-decision history is defined.

This example also shows why the word “covariate” does not by itself determine whether a measurement belongs in a particular history. A symptom score can be a later history component, while a symptom score measured after the final decision cannot be a pre-decision component for that decision. The ordering, not the variable’s name, settles the question.

Common Errors, Diagnostics, and Limits

A history is sometimes mistakenly treated as a list of every measured variable in the study. That makes the time ordering disappear. A practical diagnostic is to take each variable in a proposed $H_t$ and ask: “Was this value recorded before $A_t$ was assigned?” If not, it does not belong in the pre-decision history at $t$. A visit-level data table that does not make this order clear deserves scrutiny before it is used for longitudinal analysis.

Wrong reasoning: “The month-2 symptom score is recorded in the same study period as the initial treatment, so it belongs in the baseline history.” Correction: If treatment $A_0$ was assigned before the month-2 measurement, that score is post-$A_0$ and cannot be part of $H_0$.
Wrong reasoning: “A variable is called a confounder, so it must be included in every history.” Correction: A history is defined first by what has actually been measured before the decision. Whether a variable has a particular causal role is a separate question.
Wrong reasoning: “Everything measured at the same visit is either all before or all after treatment.” Correction: Check the protocol’s sequence. Some measurements may precede a decision and others follow it, even if they share a visit label.

A long history can be written in two equivalent ways: as the compact tuple $H_t=(\bar L_t,\bar A_{t-1})$, or as its expanded chronological list. The tuple is useful in formulas; the expanded list is often easier for checking a data set. In a wide data set, repeated variables may appear in separate columns such as $L_0,L_1,L_2$. In a long data set, they may appear as repeated rows indexed by visit. Neither layout changes which values were available at a decision.

The chain-rule factorization is another useful check: the conditional distribution for $A_t$ should condition on $H_t$, and the distribution for the next measurement $L_{t+1}$ should condition on the history and $A_t$. If a purported observed-data factorization conditions a treatment on a covariate measured only afterward, its chronology is inconsistent. Conversely, seeing a factorization in this order does not establish that the recorded covariates are sufficient for causal analysis.

There are important limits to what a correctly written history accomplishes. It does not fix measurement error, determine whether all relevant information was recorded, or resolve an ambiguous temporal order. Nor does it establish consistency, exchangeability, positivity, or a causal interpretation of any conditional distribution. Those are distinct questions. The history is the observed information available at the decision; assumptions about its relationship to potential outcomes must be stated separately.

Check Your Understanding

For each question, identify the relevant history or explain the timing. Solutions follow the questions.

At decision $t=2$, which treatment variables belong in $H_2$, and does $A_2$ belong?
Measurements occur in this order: $L_0,A_0,L_1,A_1,L_2,A_2,Y$. Write $H_2$ and state why $Y$ is excluded.
Suppose $P(L_0=0)=0.6$, $P(A_0=1\mid L_0=0)=0.5$, and $P(L_1=1\mid L_0=0,A_0=1)=0.4$. What is the joint probability of the path $L_0=0,A_0=1,L_1=1$?
A symptom is measured after $A_1$ but before $A_2$. Is it part of $H_1$, $H_2$, both, or neither? Explain.

Solutions. (1) $H_2=(L_0,L_1,L_2,A_0,A_1)$, with the tuple understood to retain the ordered components; it includes treatments strictly before decision 2, not $A_2$. (2) $H_2=(L_0,A_0,L_1,A_1,L_2)$. The outcome is recorded after the treatment sequence and is not available before $A_2$. (3) By the chain rule, the path probability is $0.6(0.5)(0.4)=0.12$. (4) It belongs in $H_2$ if it is measured before decision $A_2$, but not in $H_1$, which is defined before $A_1$. Its exact placement depends on the stated measurement order, as given.

Observed Histories at Sequential Decision Times

What You'll Learn

Why the Timing of a History Matters

Histories as Sequences and Information Sets

Worked Example: A Two-Decision Treatment Sequence

A Chain-Rule Factorization of the Observed Data

Worked Example: Checking a Path Probability

Worked Example: Sorting Variables by Their Measurement Time

Worked Example: Sorting Variables by Their Measurement Time

Common Errors, Diagnostics, and Limits

Check Your Understanding

Previous

Next