Why One Regression Coefficient May Not Represent A Treatment Regime

Why the treatment history matters

With treatment at two visits, a person’s treatment history is the pair $(A_0,A_1)$, not a single treatment value. Here $A_0$ is treatment at the first visit and $A_1$ is treatment at the second. Even when both are binary, there are four possible histories: $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. A regression coefficient attached to one treatment variable summarizes a contrast defined by that model. It does not, just by being called a treatment coefficient, become the contrast between the complete histories $(1,1)$ and $(0,0)$.

The distinction is especially important when treatment effects can accrue at both visits, when treatment timing matters, or when an intermediate covariate is affected by earlier treatment. The quantity of causal interest may compare two treatment strategies across time, while a regression coefficient may compare people differing by one treatment value at a particular visit, conditional on other variables.

Definition: Let $Y(a_0,a_1)$ denote the outcome that would be observed if treatment were set to $a_0$ at visit 0 and $a_1$ at visit 1. The sustained-treatment versus never-treatment contrast on the mean-difference scale is $$ \Delta_{\mathrm{reg}}=E[Y(1,1)]-E[Y(0,0)]. $$ This is a contrast between two treatment histories. It is not defined as a regression coefficient.

The notation makes the target clear: both potential outcomes in the contrast specify treatment at both visits. To interpret this as a causal effect from observed data requires the usual conditions for longitudinal causal inference, including consistency, adequate treatment support for the histories being compared, and sequential exchangeability given the measured history. Those assumptions concern identification of the potential-outcome means; they do not make a regression coefficient equal to their contrast automatically.

A simple model shows the distinction

First suppose the potential-outcome mean follows an additive treatment-history model:

E[Y(a_0,a_1)]=\alpha+\beta_0a_0+\beta_1a_1.

Here $\beta_0$ and $\beta_1$ describe the mean change associated with setting treatment at the corresponding visit, with the other visit’s treatment held fixed in this model. The sustained-versus-never contrast follows by substituting the two histories into the mean model.

Result: Under the additive mean model above, $$ \Delta_{\mathrm{reg}}=\beta_0+\beta_1. $$ Thus, unless additional restrictions or a different coding are used, neither visit-specific coefficient alone is the sustained-versus-never contrast.

Derivation. Under $(1,1)$, the mean is $\alpha+\beta_0+\beta_1$. Under $(0,0)$, it is $\alpha$. Subtracting gives

E[Y(1,1)]-E[Y(0,0)] =(\alpha+\beta_0+\beta_1)-\alpha =\beta_0+\beta_1.

This derivation is a statement about the specified potential-outcome mean model. For the model parameters to have a causal interpretation from observed data, the identifying assumptions must also hold and the model must represent the relevant means adequately. The algebra alone does not establish those assumptions. Even under favorable identification, a coefficient for one visit remains a visit-specific parameter; the regime contrast combines the treatment changes specified by the two histories. $\square$

Worked Example: Two Different Visit Effects

Suppose the potential-outcome mean is $E[Y(a_0,a_1)]=20+2a_0+5a_1$. Calculate the mean under each history and the sustained-versus-never contrast.

For $(0,0)$, the mean is $20+2(0)+5(0)=20$. For $(1,0)$, it is $20+2(1)+5(0)=22$. For $(0,1)$, it is $20+2(0)+5(1)=25$. For $(1,1)$, it is $20+2(1)+5(1)=27$.

Therefore, $\Delta_{\mathrm{reg}}=27-20=7$. The first-visit coefficient is 2 and the second-visit coefficient is 5; neither coefficient alone equals 7. The contrast is their sum because the two regimes differ at both visits.

What one coefficient on cumulative treatment summarizes

A common simplification replaces the two treatment indicators with cumulative exposure, $C=A_0+A_1$, and fits a single-coefficient regression of $Y$ on $C$. That coefficient describes the linear association per one-unit increase in $C$ under the fitted model. It does not in general describe the contrast between the histories $(1,1)$ and $(0,0)$. A calculation under a particularly transparent treatment distribution makes this precise.

Result: Suppose $A_0$ and $A_1$ are independent Bernoulli random variables with probability $1/2$, and $Y=\alpha+\beta_0 A_0+\beta_1 A_1+\varepsilon$, where $E[\varepsilon\mid A_0,A_1]=0$. The population least-squares slope in the regression of $Y$ on an intercept and $C=A_0+A_1$ is $$ \frac{\beta_0+\beta_1}{2}. $$ The sustained-versus-never mean contrast is $\beta_0+\beta_1$, which is twice this slope.

Derivation. For a regression with an intercept, the population slope is $\operatorname{Cov}(C,Y)/\operatorname{Var}(C)$, provided $\operatorname{Var}(C)>0$. Independence and the Bernoulli$(1/2)$ distributions give $\operatorname{Var}(A_0)=\operatorname{Var}(A_1)=1/4$ and $\operatorname{Cov}(A_0,A_1)=0$. The conditional mean-zero assumption makes $\operatorname{Cov}(C,\varepsilon)=0$. Consequently,

\begin{aligned} \operatorname{Var}(C)&=\operatorname{Var}(A_0+A_1)=\frac14+\frac14=\frac12,\\ \operatorname{Cov}(C,Y) &=\operatorname{Cov}(A_0+A_1,\beta_0A_0+\beta_1A_1+\varepsilon)\\ &=\frac{\beta_0}{4}+\frac{\beta_1}{4}. \end{aligned}

Dividing the covariance by the variance gives $(\beta_0+\beta_1)/2$. By the additive mean model, the $(1,1)$ versus $(0,0)$ contrast is $\beta_0+\beta_1$. This proves the stated relationship under these assumptions; it is not a universal formula for every distribution of treatment histories or every regression specification. $\square$

Worked Example: A Cumulative-Exposure Slope

Use the model $Y=10+2A_0+5A_1+\varepsilon$ from the preceding result, with all four treatment histories equally likely and $E[\varepsilon\mid A_0,A_1]=0$. The possible values of $C=A_0+A_1$ are 0, 1, and 2. Their corresponding mean outcomes are 10 for $C=0$, an average of 13.5 for $C=1$, and 17 for $C=2$.

The two histories with $C=1$ have means 12 and 15, so their average is $(12+15)/2=13.5$. To verify the regression slope directly, $E[C]=1$, $E[Y]=13.5$, $E[CY]=(0+1(12)+1(15)+2(17))/4=15.25$, and $E[C^2]=(0+1+1+4)/4=1.5$. Hence $\operatorname{Cov}(C,Y)=15.25-1(13.5)=1.75$ and $\operatorname{Var}(C)=1.5-1^2=0.5$. The slope is $1.75/0.5=3.5$.

The sustained-versus-never contrast is $17-10=7$. The one-unit cumulative-exposure slope is 3.5, not 7. It summarizes a one-unit increase in $C$ in this treatment distribution; the regime contrast changes $C$ by two units and also specifies the full histories being compared.

Adjustment can change which comparison is being made

In longitudinal settings, a later covariate $L_1$ may be affected by $A_0$, predict $A_1$, and predict $Y$. Adjusting for $L_1$ can be necessary for a particular conditional treatment comparison, but that does not turn the coefficient on $A_1$ into the total effect of setting both treatments. In particular, holding $L_1$ fixed in a regression comparison is not the same intervention as allowing $L_1$ to take the value it would naturally take under each treatment regime.

Worked Example: Feedback and the Adjusted Later-Treatment Coefficient

Let $U$ have mean zero and positive variance, and suppose the intermediate covariate follows $L_1=2A_0+U$. Suppose the outcome model is $Y=A_0+4A_1+3L_1+\varepsilon$, with $E[\varepsilon\mid A_0,A_1,L_1]=0$. Assume treatment histories have sufficient support and that the regression model is correctly specified. The later treatment may be assigned based on $L_1$; the conditional mean model still specifies an adjusted association of 4 for $A_1$, holding $A_0$ and $L_1$ fixed.

Under the intervention $(0,0)$, $L_1=U$, so the mean outcome is $0+0+3E[U]=0$. Under $(1,1)$, $L_1=2+U$, so the mean outcome is $1+4+3(2+E[U])=11$. The sustained-versus-never contrast is therefore 11.

In the conditional outcome equation, the coefficient on $A_1$ is 4. It compares $A_1=1$ with $A_1=0$ at the same $A_0$ and $L_1$; it does not include the change in $L_1$ induced by setting $A_0=1$. The total regime contrast includes that pathway: its components are the direct $A_0$ contribution 1, the $A_1$ contribution 4, and the induced $L_1$ contribution $3(2)=6$, summing to $1+4+6=11$.

This example also identifies a frequent interpretive trap: a coefficient can be correctly estimated for the conditional regression model and still answer a different question from the regime contrast. Whether adjustment is appropriate depends on the estimand and causal structure. The numerical example is deliberately simple; in an actual study, identifying the regime means requires adequate measurement and the relevant longitudinal causal assumptions, not merely a regression fit.

Comparing the quantities and checking the analysis

Quantity	What it compares	Why it may differ from the regime contrast
Coefficient for $A_1$ adjusted for $A_0,L_1$	A conditional later-treatment comparison at fixed modeled covariates	It does not by itself include earlier treatment effects or changes in $L_1$
Coefficient for $C=A_0+A_1$	A modeled change per one-unit increase in cumulative exposure	It collapses histories with different timing and depends on the model and treatment distribution
$E[Y(1,1)]-E[Y(0,0)]$	Two specified complete treatment histories	It is a regime-level target, not a single conditional regression parameter by definition

Regression remains useful. A history-specific outcome model can be used to estimate means for specified histories, and a target-aligned analysis can compare the resulting regime means. The key is to calculate the target contrast rather than assume that a convenient coefficient is that contrast. A marginal structural model is another standard framework for causal questions with time-varying treatment and confounding; the present point is about defining the target and interpreting coefficients, not about deriving that framework.

Useful diagnostics begin with the coding and the question. Check whether the model includes separate visit indicators or only a cumulative summary; determine which histories are combined by that coding; and state explicitly what is held fixed in an adjusted coefficient. Then compare the coefficient’s units and contrast with the intended regime: a one-visit change, a one-unit change in cumulative exposure, and a change from never to sustained treatment are not interchangeable. Sparse or absent observations for a history also raise a support problem: a fitted coefficient cannot supply information that the data do not contain without relying on additional modeling assumptions.

Common mistake: “The adjusted coefficient on treatment is 4, so sustained treatment changes the mean outcome by 4.” This reasoning silently treats a coefficient for one modeled treatment comparison as if it changed treatment at both visits and included all consequences of the earlier treatment. The correction is to write the two histories, calculate or estimate $E[Y(1,1)]$ and $E[Y(0,0)]$, and subtract those means. Under the additive model the result is $\beta_0+\beta_1$; with an evolving covariate, the regime contrast may additionally include changes in that covariate’s distribution.

Key takeaway: A regression coefficient is tied to its model’s treatment coding and conditioning set. A treatment-regime contrast is tied to two complete treatment histories. Equality between them requires a specific model and contrast calculation; it should never be inferred from the word “adjusted.”

Check Your Understanding

Answer each question, then compare with the brief solutions below.

If $E[Y(a_0,a_1)]=8+3a_0+2a_1$, what is the mean contrast for $(1,1)$ versus $(0,0)$?
Under the balanced treatment-history assumptions in the cumulative-exposure result, what is the population slope for $C=A_0+A_1$ when $\beta_0=6$ and $\beta_1=2$?
Why can an adjusted coefficient for $A_1$ fail to represent the effect of the $(1,1)$ history versus $(0,0)$?
If a regression includes only $C=A_0+A_1$, which two distinct histories are combined at $C=1$? Why does that matter?

Solutions. (1) The means are $8+3+2=13$ and $8$, so the contrast is $13-8=5$. (2) The slope is $(6+2)/2=4$, while the regime contrast is $6+2=8$. (3) The adjusted coefficient compares the later treatment at fixed modeled covariates; it need not include the earlier treatment’s effect or changes in intermediate covariates. (4) The histories are $(1,0)$ and $(0,1)$. They have the same cumulative exposure but different timing, so a single cumulative-exposure value does not retain which visit was treated.

Quantity	What it compares	Why it may differ from the regime contrast
Coefficient for \(A_1\) adjusted for \(A_0,L_1\)	A conditional later-treatment comparison at fixed modeled covariates	It does not by itself include earlier treatment effects or changes in \(L_1\)
Coefficient for \(C=A_0+A_1\)	A modeled change per one-unit increase in cumulative exposure	It collapses histories with different timing and depends on the model and treatment distribution
\(E[Y(1,1)]-E[Y(0,0)]\)	Two specified complete treatment histories	It is a regime-level target, not a single conditional regression parameter by definition