Why Regress Outcomes on Exposure?
When one person's treatment can affect another person's outcome, an outcome regression may include both the individual's own treatment and a summary of peers' treatment. This gives a compact way to describe how the outcome varies across exposure conditions, and to calculate contrasts implied by a specified model. But a regression coefficient is not automatically a causal effect: its interpretation depends on the exposure definition, the assignment design, the model form, and the target population.
Let \(Z_i\) be individual \(i\)'s own treatment, and let \(G_i\) be the mapped peer exposure assigned to that individual. An exposure mapping summarizes the aspects of other people's assignments treated as relevant for \(i\)'s outcome. For example, \(G_i\) might be the proportion of \(i\)'s network neighbors who receive treatment. Write \(Y_i(z,g)\) for the potential outcome under own treatment \(z\) and mapped peer exposure \(g\). This notation presumes that assignments summarized by the same \((z,g)\) are treated as equivalent for the outcome question at hand.
An Additive Regression for Two Binary Exposures
A basic model for a continuous outcome and binary own treatment and peer exposure is
Here \(Z_i\) and \(G_i\) each take values 0 or 1. The model says that the conditional mean outcome is additive in the two exposures: changing own treatment from 0 to 1 changes the fitted mean by \(\beta_Z\) at either peer-exposure level, while changing peer exposure from 0 to 1 changes it by \(\beta_G\) at either own-treatment level. These are statements about the model's fitted means. Causal interpretation additionally requires that the exposure conditions identify the potential outcomes of interest, that relevant conditions have positive probability under the design, and that the mean model is adequate for the target contrasts.
For ordinary least squares (OLS), the coefficients are chosen to minimize the sum of squared differences between observed outcomes and fitted values. If the outcome is measured in points, the coefficients for binary exposures are in points. Other outcome types may require another regression model; then coefficients need not be differences in the outcome's original units.
Worked Example: Fitting an Additive Model to Four Exposure Means
Suppose a hypothetical randomized network study has the following mean outcomes, measured in symptom-score points. Each of the four exposure conditions has the same number of observed individuals.
| Own treatment \(Z\) | Peer exposure \(G\) | Mean outcome (points) |
|---|---|---|
| 0 | 0 | 10 |
| 1 | 0 | 13 |
| 0 | 1 | 12 |
| 1 | 1 | 16 |
Fit the additive model \(E(Y\mid Z,G)=\beta_0+\beta_Z Z+\beta_G G\). Because the four conditions have equal sample sizes, the least-squares estimate of the intercept is the grand mean minus the fitted contributions from the two balanced binary predictors. The grand mean is
The mean outcome for \(Z=1\) is \((13+16)/2=14.5\), and for \(Z=0\) it is \((10+12)/2=11\). Thus \(\hat{\beta}_Z=14.5-11=3.5\) points. Similarly, the means for \(G=1\) and \(G=0\) are 14 and 11.5, so \(\hat{\beta}_G=2.5\) points. The intercept is \(12.75-(3.5/2)-(2.5/2)=9.75\) points. The fitted model is therefore
For example, its fitted mean at \((Z,G)=(1,1)\) is \(9.75+3.5+2.5=15.75\) points. The observed mean in that condition was 16 points. The four fitted means are 9.75, 13.25, 12.25, and 15.75 points, so the model does not pass exactly through all four observed means.
The coefficients imply a model-based own-treatment contrast of 3.5 points at either value of \(G\), and a model-based peer-exposure contrast of 2.5 points at either value of \(Z\). The observed differences, however, are 3 and 4 points for own treatment at \(G=0\) and \(G=1\), respectively; for peer exposure they are 2 and 3 points at \(Z=0\) and \(Z=1\). The additive model averages these differences rather than reproducing each one. Equal cell sizes justify the simple coefficient arithmetic, but do not establish that the additive restrictions are appropriate.
Interpreting Model-Based Contrasts
A model-based contrast is obtained by predicting outcomes under two exposure conditions and subtracting the predictions. For a model with a covariate \(C_i\),
the fitted contrast for changing peer exposure from \(g_0\) to \(g_1\), holding own treatment and covariates fixed, is \(\hat{\beta}_G(g_1-g_0)\). Likewise, the fitted contrast for changing own treatment from 0 to 1 is \(\hat{\beta}_Z\). These expressions follow from subtraction of the model's predictions. Their causal interpretation is not supplied by the subtraction itself.
If the desired estimand is an average contrast for a target population, calculate predicted outcomes under each condition for each target individual, then average the predictions and take their difference. For a finite target set of \(N\) individuals, one such standardized model-based contrast is
This standardization makes the target covariate distribution explicit. A coefficient that holds \(C_i\) fixed and a standardized average need not answer the same question, especially when the model's contrast varies across covariate values.
Worked Example: Contrasting Two Levels of Peer Exposure
In a hypothetical study, \(G_i\) is the proportion of an individual's neighbors assigned treatment, on a 0-to-1 scale. Suppose the fitted mean model is \(\widehat{E}(Y\mid Z,G,C)=20-3Z-8G+1.5C\), where \(Y\) is a symptom score in points and \(C\) is a pre-treatment severity score. For an individual with \(C=4\) and own treatment \(Z=1\), compare peer exposure \(G=0.25\) with \(G=0.75\).
At \(G=0.25\), the fitted mean is \(20-3(1)-8(0.25)+1.5(4)=21\) points. At \(G=0.75\), it is \(20-3(1)-8(0.75)+1.5(4)=17\) points. The model-based contrast, higher peer exposure minus lower peer exposure, is \(17-21=-4\) points. Equivalently, \(\hat{\beta}_G(0.75-0.25)=-8(0.5)=-4\) points.
This is a fitted contrast for the specified own treatment and covariate value. If the target is an average over individuals with different \(C_i\), predictions should be calculated for each target individual and averaged. In this particular additive model, the contrast is constant across \(C_i\); a richer model could imply otherwise.
Functional Form and Sensitivity
The numerical scale assigned to peer exposure influences the regression specification. Treating \(G_i\) as a continuous proportion and using one linear term assumes that equal increases in the proportion correspond to equal changes in the fitted mean. Treating exposure as categories instead allows the fitted differences between categories to vary. Either form can be useful, but the choice should reflect the exposure conditions and the scientific contrast of interest.
A practical sensitivity analysis fits a small number of scientifically interpretable alternatives, then compares predictions for the same target conditions and population. For instance, one model might use the peer-treatment proportion as a single linear term, while another uses indicators for low, medium, and high exposure. Compare the implied contrasts—not just whether individual coefficients are statistically significant. Large changes in a target contrast signal that the conclusion depends on functional-form choices. This exercise does not establish which form is correct, and it cannot repair an exposure mapping that fails to represent relevant peer assignments.
Worked Example: Linear Versus Categorical Peer Exposure
Suppose a hypothetical analysis estimates the peer-exposure contrast in symptom-score points while holding own treatment fixed. A linear model for the peer-treatment proportion predicts a difference of \(-4.0\) points between proportions 0.25 and 0.75. A second model, using low, medium, and high exposure categories, predicts a difference of \(-5.2\) points between the categories containing those proportions.
The difference between the two model-based estimates is \(-5.2-(-4.0)=-1.2\) points. Both estimates describe contrasts generated by their respective fitted models; neither calculation alone identifies the causal effect. A useful report would state both fitted contrasts, explain the exposure scale and target conditions, and note that the estimated difference is sensitive to the chosen representation of peer exposure. The choice between specifications should not be made solely by selecting whichever produces the more favorable estimate.
Regression Contrasts and Design-Based Estimates
A design-based estimator uses the assignment probabilities and observed outcomes to estimate an exposure-specific outcome mean or contrast for the randomized population. For example, the Horvitz–Thompson mean for exposure condition \((z,g)\), when each individual's probability \(\pi_i(z,g)\) of receiving that condition is known and positive, is
The indicator is 1 when individual \(i\) is observed in that condition and 0 otherwise. This estimator uses the assignment mechanism directly. A regression instead uses a chosen outcome model to produce fitted contrasts. Under a randomized design, both may target the same causal contrast, but they can give different estimates in the realized sample. They also rely on different ingredients: the design-based calculation uses known assignment probabilities, while a model-based contrast relies on the fitted outcome regression and its form.
Worked Example: A Regression Contrast and a Design-Based Contrast
Consider a hypothetical randomized assignment with \(N=4\) individuals and a binary exposure condition \(A\), where \(A=1\) denotes a specified combination of own treatment and peer exposure and \(A=0\) denotes the comparison condition. The observed outcomes, in points, are 12 and 16 for the two individuals assigned \(A=1\), and 8 and 10 for the two assigned \(A=0\). Suppose every individual has assignment probability \(\pi_i(1)=0.4\) and \(\pi_i(0)=0.6\), both positive.
A regression of \(Y\) on an intercept and \(A\), with no other predictors, gives the difference in observed group means: \((12+16)/2-(8+10)/2=14-9=5\) points. This is the model-based contrast from the fitted two-group mean model.
The Horvitz–Thompson mean for \(A=1\) is
For \(A=0\), it is
The design-based contrast is \(17.5-7.5=10\) points, not 5 points. These estimates differ because the weighting in the design-based estimator differs from the unweighted regression, and the observed assignment pattern is only one realization of the randomization. This difference does not show that either calculation is erroneous. The estimators can target the same finite-population contrast while having different realized values and precision.
Assumptions and Common Pitfalls
For an outcome-regression contrast to support a causal interpretation under randomized exposure assignment, state the conditions that connect the data to the target. The exposure mapping must define the relevant exposure conditions; observed outcomes must agree with the corresponding potential outcomes (consistency); each target exposure condition must have positive assignment probability; and the regression must represent the conditional means well enough for the desired contrast. If the design induces dependence among individuals' assignments or outcomes, uncertainty calculations must also respect that design and dependence. A regression's conventional standard errors should not be assumed valid without checking how they relate to the assignment structure.
Randomization can justify design-based comparisons without requiring a correct outcome regression. It does not make every fitted regression contrast automatically unbiased or make an inadequate functional form harmless. Conversely, a regression can improve precision or summarize patterns, but its causal meaning should be tied to the design and estimand rather than inferred from coefficient labels.
- Mistake: “The treatment coefficient is the direct causal effect.” In an additive model it is the fitted own-treatment contrast, holding the other included predictors fixed. Call it causal only with appropriate identification conditions and a model adequate for that contrast.
- Mistake: “Including peer exposure accounts for all interference.” It accounts only for the peer-treatment information represented by the chosen mapping. The regression cannot recover distinctions the mapping omits.
- Mistake: “Regression and design-based estimates must match.” They can target the same estimand and still differ in a realized sample because their weighting and estimation procedures differ.
- Mistake: “A good-looking fit proves the functional form.” Fit diagnostics and sensitivity comparisons can reveal problems, but they do not establish that a specification is causally correct.
Check Your Understanding
Use the definitions and distinctions in this tutorial to answer the following questions.
- In an additive model with binary \(Z\) and \(G\), what fitted contrast does the coefficient of \(G\) represent, and what assumptions are needed before calling it causal?
- A linear model estimates a peer-exposure coefficient of \(-6\) outcome points per unit of a 0-to-1 exposure proportion. What model-based contrast does it imply for an increase from 0.2 to 0.7?
- Why can an additive regression fail to reproduce every observed mean across the four combinations of binary own treatment and peer exposure?
- Give one reason a design-based estimate and an outcome-regression contrast may differ in the same randomized study.
- What should be compared when evaluating a linear peer-exposure term against a categorical exposure specification?