Interpret Main Effects at the Other Predictor's Zero

Why a Main Effect Has a Reference Point

When a regression model includes an interaction between two quantitative predictors, the model’s association between one predictor and the outcome can depend on the value of the other predictor. As a result, a main-effect coefficient does not automatically describe an overall change that applies at every value of the other predictor. Its interpretation has a specific reference point: the other predictor equals zero.

That interpretation is built into the product-term equation. It is important both when reading fitted coefficients and when deciding whether their numerical values answer a useful question. Zero might represent a real, relevant condition, such as no prior experience. Or it might be far outside the values observed in the study, making the corresponding interpretation an extrapolation.

The Interaction Model and Its Assumptions

For two quantitative predictors \(x_1\) and \(x_2\), a linear regression model with their product term is:

Y_i=\beta_0+\beta_1x_{1i}+\beta_2x_{2i}+\beta_3x_{1i}x_{2i}+\varepsilon_i.

Here \(Y_i\) is the outcome for individual \(i\), \(\beta_0,\beta_1,\beta_2,\beta_3\) are model coefficients, and \(\varepsilon_i\) is the error. The model says that the conditional mean outcome is represented by the equation without the error term. A fitted model uses estimates \(b_0,b_1,b_2,b_3\) in place of the unknown coefficients.

Model assumptions: For the usual linear-model interpretation, the conditional mean of \(Y_i\), given the predictors, follows the stated equation; errors have mean zero and constant variance; and errors for different individuals are independent. Normal errors are additionally assumed when exact small-sample \(t\)-based inference is required. These assumptions concern the model and data; they do not make an association causal.

The coefficients \(b_1\) and \(b_2\) are called main-effect coefficients. To interpret \(b_1\), set \(x_2=0\). The product \(x_1x_2\) then equals zero, so the fitted mean is \(b_0+b_1x_1\). Thus \(b_1\) describes the change in fitted mean outcome per one-unit increase in \(x_1\) when \(x_2\) is zero. Likewise, \(b_2\) describes the change per one-unit increase in \(x_2\) when \(x_1=0\).

Key interpretation: In a model with \(x_1\), \(x_2\), and \(x_1x_2\), the main-effect coefficient for \(x_1\) refers to \(x_2=0\), and the main-effect coefficient for \(x_2\) refers to \(x_1=0\). Each coefficient’s units are outcome units per unit of its own predictor.

Why Setting the Other Predictor to Zero Works

The reference-point interpretation follows directly from comparing fitted means. For \(b_1\), set \(x_2=0\), then compare the fitted mean at \(x_1=a\) with the fitted mean at \(x_1=a+1\). The product term is zero at both values:

\begin{aligned} \hat{y}(a,0)&=b_0+b_1a,\\ \hat{y}(a+1,0)&=b_0+b_1(a+1),\\ \hat{y}(a+1,0)-\hat{y}(a,0)&=b_1. \end{aligned}

The fitted mean changes by \(b_1\) outcome units for a one-unit increase in \(x_1\), specifically at \(x_2=0\). For \(b_2\), set \(x_1=0\) and compare \(x_2=c\) with \(x_2=c+1\):

\begin{aligned} \hat{y}(0,c)&=b_0+b_2c,\\ \hat{y}(0,c+1)&=b_0+b_2(c+1),\\ \hat{y}(0,c+1)-\hat{y}(0,c)&=b_2. \end{aligned}

These comparisons justify the main-effect interpretations without requiring calculus. The interaction term contributes nothing at the relevant zero value. Whether that comparison is substantively informative is a separate question: it depends on what zero means and whether the model is supported near that value.

Worked Example: Both Zero Values Have a Practical Meaning

Suppose, hypothetically, a fitted model relates a performance score to weekly training hours \(H\) and years of experience \(E\):

\widehat{\text{Score}}=50+2.0H+1.5E+0.3HE.

The outcome is measured in score points, \(H\) in hours per week, and \(E\) in years. Assume the data include beginners with zero years of experience and participants who report zero weekly training hours, and that the fitted model is considered reasonable around those values.

Interpret the coefficient of \(H\). Set \(E=0\). At \(H=0\), the fitted score is \(50+2.0(0)+1.5(0)+0.3(0)(0)=50\) points. At \(H=1\), with \(E=0\), it is \(50+2.0(1)+1.5(0)+0.3(1)(0)=52\) points. The difference is \(52-50=2.0\) points. Thus, among the model’s predictions for participants with zero years of experience, one additional weekly training hour corresponds to a fitted score 2.0 points higher.

Interpret the coefficient of \(E\). Set \(H=0\). At \(E=0\), the fitted score is 50 points. At \(E=1\), with \(H=0\), it is \(50+2.0(0)+1.5(1)+0.3(0)(1)=51.5\) points. The difference is \(51.5-50=1.5\) points. Thus, at zero weekly training hours, one additional year of experience corresponds to a fitted score 1.5 points higher.

The interpretations are tied to different reference conditions. Neither coefficient alone describes the corresponding predictor’s relationship with score for every participant.

When Zero Is Outside the Observed Range

A coefficient can still be defined algebraically when the corresponding zero value is not observed. But its interpretation then refers to a fitted comparison at that value, not necessarily to a comparison supported by the data. If the model was fitted only for predictor values far from zero, its predictions at zero can be unreliable.

Worked Example: Age Zero Is Not a Useful Reference

Suppose, hypothetically, a study of adults aged 45 to 75 fits a model for systolic blood pressure using age \(A\), measured in years, and weekly exercise \(H\), measured in hours:

\widehat{\text{Pressure}}=122+0.65A-1.1H+0.02AH.

Blood pressure is measured in mmHg. Assume the study includes adults with no weekly exercise, so \(H=0\) is represented, but no participants near age zero.

Age coefficient. The coefficient \(0.65\) describes the fitted change per year of age when \(H=0\). For example, at \(H=0\), the fitted value at age 45 is \(122+0.65(45)=151.25\) mmHg. At age 46 it is \(122+0.65(46)=151.90\) mmHg. The difference is \(151.90-151.25=0.65\) mmHg per year. This reference condition is relevant to the study if adults with zero weekly exercise were observed.

Exercise coefficient. The coefficient \(-1.1\) describes the fitted change per additional weekly exercise hour when \(A=0\). At age zero, the fitted value for \(H=0\) is \(122\) mmHg, while for \(H=1\) it is \(122-1.1=120.9\) mmHg. The difference is \(120.9-122=-1.1\) mmHg per hour. However, age zero is far outside the adult study’s observed ages. This coefficient’s reference-point interpretation is algebraically correct, but it does not provide a well-supported description of the exercise association among the adults studied.

The age coefficient’s reference value and the exercise coefficient’s reference value must be checked separately. One can be meaningful and supported even when the other is not.

Worked Example: Rainfall Zero and Fertilizer Zero

Consider a hypothetical crop-yield model with rainfall \(R\), in centimeters during a growing season, and fertilizer \(F\), in kilograms per hectare:

\widehat{\text{Yield}}=4.2+0.08R+0.12F+0.003RF.

Yield is measured in tonnes per hectare. Suppose the observations cover rainfall from 30 to 90 cm and include fields receiving no fertilizer.

Rainfall coefficient. The coefficient \(0.08\) refers to the fitted change per centimeter of rainfall when \(F=0\). At zero fertilizer, raising rainfall from 40 to 41 cm changes the fitted yield from \(4.2+0.08(40)=7.4\) tonnes per hectare to \(4.2+0.08(41)=7.48\) tonnes per hectare. The difference is \(7.48-7.4=0.08\) tonnes per hectare per centimeter, a comparison at a fertilizer level represented in the data.

Fertilizer coefficient. The coefficient \(0.12\) refers to the fitted change per kilogram per hectare of fertilizer when \(R=0\). At zero rainfall, the fitted yield for \(F=0\) is \(4.2\) tonnes per hectare; for \(F=1\), it is \(4.2+0.12=4.32\) tonnes per hectare. The difference is \(0.12\) tonnes per hectare per kilogram per hectare. But zero rainfall is outside the observed range of 30 to 90 cm. The numerical coefficient therefore describes the model at an unsupported reference value, not a reliable fertilizer comparison for the observed growing conditions.

Reading Main Effects Carefully

Before giving a verbal interpretation, identify the coefficient, the other predictor’s zero value, and the units. Then ask whether zero is a realistic condition and whether the data include observations near it. A mathematically correct interpretation can be of limited practical value when it depends on extrapolation.

Coefficient	Reference value	Interpretation
\(b_1\), for \(x_1\)	\(x_2=0\)	Fitted mean change per one-unit increase in \(x_1\), at \(x_2=0\)
\(b_2\), for \(x_2\)	\(x_1=0\)	Fitted mean change per one-unit increase in \(x_2\), at \(x_1=0\)

In the examples, a coefficient measured in points per training hour is different from one measured in mmHg per exercise hour. State the outcome units and predictor units explicitly; coefficient numbers without units are easy to misread. Also describe the result as a fitted association, not as evidence that changing a predictor will cause the outcome to change.

Common Mistakes

Treating a main effect as an overall effect. Saying “one more unit of \(x_1\) changes the outcome by \(b_1\), regardless of \(x_2\)” ignores the interaction model’s reference condition. The correction is to state that \(b_1\) refers to \(x_2=0\).
Assuming zero is automatically meaningful. A coefficient always has a model-based reference interpretation, but zero might be outside the observed range. Check the variable’s meaning and the study’s data before presenting the interpretation as practically informative.
Confusing the two reference values. The coefficient of \(x_1\) is interpreted at \(x_2=0\), not at \(x_1=0\). The coefficient of \(x_2\) is interpreted at \(x_1=0\).
Claiming causation from a fitted association. A regression coefficient alone does not establish that changing a predictor causes the outcome to change.

Key takeaway: In an uncentered interaction model, each main-effect coefficient describes the fitted change for its own predictor when the other predictor equals zero. State that reference value, use the correct units, and check whether the interpretation is supported by the observed data.

Check Your Understanding

For each question, state the reference value involved and consider whether it supports a useful interpretation.

In a model with predictors \(x_1\), \(x_2\), and \(x_1x_2\), what does the coefficient of \(x_1\) describe?
A fitted model predicts exam score from study hours and years of experience. The coefficient of study hours is 1.4. Which predictor must be zero for this coefficient’s main-effect interpretation?
A study includes adults aged 50 to 80 and uses age as one of two interacting predictors. Why might the other predictor’s main-effect coefficient be difficult to interpret?
In the rainfall example, why is the fertilizer coefficient’s reference-point interpretation less useful than the rainfall coefficient’s?
Why is it incorrect to claim that a main-effect coefficient describes the predictor’s association at every value of the other predictor?