Current-Treatment Regression Versus Treatment-History Regression

Two Regression Questions, Two Treatment Comparisons

With treatment measured at two visits, an outcome model can include only the most recent treatment, $A_1$, or include both treatments, $A_0$ and $A_1$. These models do not ask the same regression question. The first compares outcomes across groups with different current treatment, allowing their earlier-treatment histories to differ as they do in the observed data. The second compares treatment histories while holding one visit’s treatment fixed.

That distinction matters when earlier treatment predicts later treatment and also affects the outcome. The current-treatment coefficient can then reflect both the current treatment contrast and differences in earlier treatment between the current-treatment groups. Including earlier treatment makes that comparison explicit, but does not by itself establish a causal interpretation.

Definition: Let $A_0$ and $A_1$ denote treatment at visits 0 and 1, respectively, and let $Y$ be the outcome measured after visit 1. A current-treatment regression includes $A_1$ but not $A_0$, for example $E[Y\mid A_1]=\alpha_0+\alpha_1 A_1$. An additive treatment-history regression includes both treatments, for example $E[Y\mid A_0,A_1]=\beta_0+\beta_1 A_0+\beta_2 A_1$. In the latter model, additivity means the conditional mean contains no $A_0A_1$ interaction.

For binary treatments, the history model’s coefficient $\beta_2$ is the modelled difference between $A_1=1$ and $A_1=0$ at a fixed value of $A_0$. The coefficient $\beta_1$ compares $A_0=1$ with $A_0=0$ at a fixed value of $A_1$. These are conditional comparisons. The current-only coefficient $\alpha_1$, by contrast, compares the two observed current-treatment groups, whose earlier-treatment distributions may differ.

Why the Current-Only Coefficient Can Differ

The following result makes the distinction precise. Assume $A_0,A_1$ are binary, $E[Y^2]<\infty$, $0<P(A_1=1)<1$, and the conditional mean follows the additive history model $E[Y\mid A_0,A_1]=\beta_0+\beta_1A_0+\beta_2A_1$. No causal interpretation is required for the result.

Result: The population least-squares coefficient of $A_1$ in a regression with an intercept and $A_1$ alone is $$ \alpha_1 = \beta_2+\beta_1\{E[A_0\mid A_1=1]-E[A_0\mid A_1=0]\}. $$ Thus, the current-only coefficient equals the history-model current-treatment coefficient plus a term reflecting how earlier treatment differs between the current-treatment groups.

Derivation. For a binary regressor with both values occurring, the population least-squares slope with an intercept is the difference in conditional outcome means. To verify this, write $p=P(A_1=1)$, $\mu_a=E[Y\mid A_1=a]$, and $\mu=E[Y]$. The slope is $\operatorname{Cov}(A_1,Y)/\operatorname{Var}(A_1)$. Since $\operatorname{Var}(A_1)=p(1-p)$ and $\operatorname{Cov}(A_1,Y)=p(1-p)(\mu_1-\mu_0)$, the slope is $\mu_1-\mu_0$. Under the stated conditional mean,

\begin{aligned} E[Y\mid A_1=a] &=\beta_0+\beta_1E[A_0\mid A_1=a]+\beta_2a,\\ \alpha_1 &=E[Y\mid A_1=1]-E[Y\mid A_1=0]\\ &=\beta_2+\beta_1\{E[A_0\mid A_1=1]-E[A_0\mid A_1=0]\}. \end{aligned}

The second term is the omitted-history contribution under this model. If earlier treatment has no association with the outcome given current treatment ($\beta_1=0$), or if the two current-treatment groups have the same mean earlier treatment, that term is zero. Otherwise, the current-only and history-model coefficients can differ. This identity is an algebraic result about regression coefficients; it does not say that either coefficient is a causal effect. $\square$

Worked Example: Earlier Treatment Increases the Current-Only Coefficient

Suppose $A_0$ is equally likely to be 0 or 1, and treatment probabilities at visit 1 are $P(A_1=1\mid A_0=1)=0.8$ and $P(A_1=1\mid A_0=0)=0.2$. Let the conditional outcome mean be $E[Y\mid A_0,A_1]=2A_0+3A_1$.

The joint probabilities of $(A_0,A_1)=(0,0),(0,1),(1,0),(1,1)$ are $0.5(0.8)=0.4$, $0.5(0.2)=0.1$, $0.5(0.2)=0.1$, and $0.5(0.8)=0.4$, respectively. In particular, $P(A_1=1)=P(A_1=0)=0.5$. Among those with $A_1=1$, the proportion with $A_0=1$ is $0.4/0.5=0.8$; among those with $A_1=0$, it is $0.1/0.5=0.2$.

The history model’s current-treatment coefficient is 3, and its earlier-treatment coefficient is 2. The current-only coefficient is $3+2(0.8-0.2)=4.2$. Directly, the mean outcome is $2(0.8)+3=4.6$ among $A_1=1$, and $2(0.2)=0.4$ among $A_1=0$; their difference is $4.6-0.4=4.2$. The current-only comparison is larger because the $A_1=1$ group also has more earlier treatment.

History Coefficients: Conditional Contrasts, Not Regime Effects

In an additive history model, each coefficient compares treatment values while fixing the other visit’s treatment. If the model is correctly specified, that is a precise statistical interpretation. It is not automatically a causal interpretation: causal conclusions require appropriate assumptions about treatment assignment and the outcome model. In particular, a contrast holding $A_1$ fixed while varying $A_0$ is not the same comparison as allowing later treatment to follow its natural course after changing earlier treatment.

The current-only coefficient also has a precise interpretation: for binary $A_1$, it is the observed difference in mean outcomes between the $A_1=1$ and $A_1=0$ groups. The two groups can contain different mixtures of earlier-treatment histories. Consequently, the coefficient does not generally isolate the current treatment contrast from that history difference.

Worked Example: Earlier Treatment Reverses the Current-Only Contrast

Let $P(A_0=1)=P(A_0=0)=0.5$, with $P(A_1=1\mid A_0=1)=0.2$ and $P(A_1=1\mid A_0=0)=0.8$. Suppose $E[Y\mid A_0,A_1]=2A_0+A_1$.

Here $P(A_1=1)=0.5$. The $A_1=1$ group contains $A_0=1$ with probability $0.1/0.5=0.2$; the $A_1=0$ group contains $A_0=1$ with probability $0.4/0.5=0.8$. The history model’s current-treatment coefficient is $1$, while its earlier-treatment coefficient is $2$.

The current-only coefficient is $1+2(0.2-0.8)=-0.2$. Checking by group means gives $E[Y\mid A_1=1]=2(0.2)+1=1.4$ and $E[Y\mid A_1=0]=2(0.8)=1.6$, also yielding $1.4-1.6=-0.2$. Thus the observed current-treatment comparison is negative even though the current-treatment coefficient in the stated history model is positive. This is a regression comparison, not evidence that the causal effect of current treatment is negative.

Model Form, Diagnostics, and Limitations

The additive history model imposes a restriction: the current-treatment contrast is the same at both values of earlier treatment. That restriction can be inappropriate. For example, if the conditional mean contains an interaction term $\beta_3A_0A_1$, the current-treatment contrast is $\beta_2$ when $A_0=0$ and $\beta_2+\beta_3$ when $A_0=1$. A single additive coefficient cannot represent both contrasts unless $\beta_3=0$. A current-only model is more restrictive still: it does not distinguish the earlier-treatment histories at all.

Before interpreting fitted coefficients, examine the four observed history groups $(A_0,A_1)$. Very small or empty groups make within-history comparisons imprecise or impossible. Strong association between $A_0$ and $A_1$ can also make the history-model coefficients unstable, even when both are estimable. Check whether outcome patterns across the history groups are compatible with the chosen additive form; a systematic difference in current-treatment contrasts across $A_0$ values signals that a common contrast may not fit well.

Neither regression repairs confounding merely by including a treatment variable. A current-only regression does not account for earlier treatment; a history regression accounts for it only as represented in the model and does not automatically address other causes of treatment and outcome. Nor does adding treatment history make a coefficient a comparison of complete treatment regimes. The models are useful for describing which groups are compared and for diagnosing how treatment history changes that comparison, not as stand-alone guarantees of causal identification.

State the fitted variables.
Identify whether the regression includes $A_1$ alone or both $A_0$ and $A_1$.

Name the comparison.
For a current-only model, describe the two observed current-treatment groups. For a history model, specify which other visit’s treatment is held fixed.

Check support and form.
Inspect the treatment-history groups and assess whether an additive conditional mean is plausible before treating a single coefficient as a common contrast.

Common pitfall: “The regression includes $A_1$, so its coefficient is the effect of treatment at visit 1.” This overlooks that a current-only regression compares groups that may differ in $A_0$. The correction is to state the coefficient’s actual comparison: an observed current-treatment group difference for the current-only model, or a conditional model contrast for the history model. Neither description alone supplies the assumptions needed for a causal claim.

Worked Example: Reading a Treatment-History Model

Suppose the fitted conditional mean is $E[Y\mid A_0,A_1]=5+4A_0+2A_1$, with binary treatments. At $A_0=0$, the modelled mean is 5 for $A_1=0$ and 7 for $A_1=1$, a difference of 2. At $A_0=1$, it is 9 for $A_1=0$ and 11 for $A_1=1$, also a difference of 2. Thus the $A_1$ coefficient encodes a common current-treatment contrast under the additive model.

For the earlier-treatment contrast, at $A_1=0$ the means are 5 and 9, a difference of 4; at $A_1=1$ they are 7 and 11, again a difference of 4. The $A_0$ coefficient therefore encodes a common earlier-treatment contrast under this model. These calculations interpret the fitted conditional mean; causal interpretations require additional justification.

Check Your Understanding

Answer the questions, then compare with the solutions below.

In a current-only regression with binary $A_1$, what difference does the population slope represent?
Suppose $\beta_1=3$, $\beta_2=2$, $E[A_0\mid A_1=1]=0.7$, and $E[A_0\mid A_1=0]=0.4$. Find the current-only coefficient under the additive history model.
What does $\beta_2$ represent in an additive history regression, and what assumption makes that contrast common across earlier-treatment groups?
Why does including $A_0$ in a regression not, by itself, prove that the coefficient of $A_1$ is a causal effect?

Solutions. (1) It is $E[Y\mid A_1=1]-E[Y\mid A_1=0]$, the observed difference in mean outcomes between current-treatment groups. (2) The coefficient is $2+3(0.7-0.4)=2.9$. (3) It is the modelled difference between $A_1=1$ and $A_1=0$ holding $A_0$ fixed; the additive conditional-mean specification, with no $A_0A_1$ interaction, makes this contrast common across values of $A_0$. (4) Other causes of treatment and outcome may remain unaccounted for, and the model’s conditional contrast need not equal a causal contrast.

Current-Treatment Regression Versus Treatment-History Regression

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