Common Misreadings of Conditional and Average Effects

Why Effects in Interaction Models Are Easy to Misread

An interaction model does not have one universal slope for every setting of its predictors. A coefficient attached to an interaction term describes how one slope changes with another predictor; it is not automatically the slope for every observation. Related mistakes arise when an average effect is treated as an effect at mean covariates, or when two separate confidence intervals are used to judge whether two effects differ.

The key is to state precisely which quantity is being interpreted: a model coefficient, a conditional effect at specified covariate values, an average over a defined set of observations, or a contrast between two effects. These quantities can be related, but they are not interchangeable.

Interaction Coefficients and Conditional Slopes

Consider a linear model for an outcome \(Y\), a continuous predictor \(X\), and a binary group indicator \(G\), coded 0 or 1. Suppose the conditional mean is modeled as

Here \(\beta_0,\beta_1,\beta_2,\beta_3\) are population model coefficients. The fitted model replaces them with estimates \(\hat{\beta}_0,\hat{\beta}_1,\hat{\beta}_2,\hat{\beta}_3\). The conditional slope of \(Y\) with respect to \(X\) is the change in the modeled mean outcome for a one-unit increase in \(X\), holding \(G\) fixed.

For \(G=0\), the slope is \(\beta_1\). For \(G=1\), it is \(\beta_1+\beta_3\). Thus, \(\beta_3\) is the difference between the group-1 slope and the group-0 slope. It is not, by itself, the slope in both groups. This interpretation follows directly from the model algebra; a causal interpretation would require additional design and assumptions beyond this equation.

Average Effects and Effects at Mean Covariates

Averages need a target: an average over which observations, and with what weights? In a model where the marginal effect is a linear function of one moderator, averaging that effect over a specified sample gives the same result as evaluating it at that moderator’s sample mean. For example, if the slope is \(\hat{\beta}_1+\hat{\beta}_3 Z_i\), its unweighted sample average is \(\hat{\beta}_1+\hat{\beta}_3\bar{Z}\). This special equality should not be generalized to every interaction model.

With multiple interacting moderators, the marginal effect can include products of covariates. The average of a product is not generally the product of averages. Consequently, an average marginal effect over observations can differ from the effect evaluated at the vector of mean covariates. They also answer different questions: the former averages over the chosen observations; the latter evaluates the model at one covariate profile, which may not represent any actual observation.

Why Overlapping Intervals Do Not Test a Difference

A confidence interval for one effect describes uncertainty about that effect. To assess whether two effects differ, the relevant quantity is their contrast and the uncertainty of that contrast. The variance of a difference depends on both individual variances and their covariance. Looking only at whether two separate intervals overlap discards that covariance and is not a valid substitute for testing the difference.

For a contrast, the standard error is the square root of its estimated variance. A confidence interval or test for the contrast is then based on that standard error, using the appropriate reference distribution for the analysis. In an interaction model, a group-slope difference is often precisely the interaction coefficient, but its uncertainty still needs to be assessed directly.

A Short Derivation: The Interaction Coefficient Is a Slope Difference

For the group-by-continuous model, substitute \(G=1\) and \(G=0\) into the conditional mean. The mean functions are \(\beta_0+\beta_1X+\beta_2+\beta_3X\) and \(\beta_0+\beta_1X\), respectively. Differentiating each with respect to \(X\) gives slopes \(\beta_1+\beta_3\) and \(\beta_1\). Subtracting the second slope from the first yields

This algebra explains the coefficient’s interpretation. If the scientific question concerns the slope within group 1, report \(\hat{\beta}_1+\hat{\beta}_3\), with its uncertainty calculated for that sum. If the question concerns how the slopes differ between groups, report \(\hat{\beta}_3\), with its own standard error or interval.

Common Misreadings and Corrections

• Wrong: “The interaction coefficient is the slope in the non-reference and reference groups.” Correction: In the binary-group model, it is the difference between the group slopes. Calculate each conditional slope from the model.
• Wrong: “The average effect must equal the effect at average covariate values.” Correction: That equality holds for some linear effect functions, but interaction products can make the average product differ from the product of averages. Specify the target average and compute it accordingly.
• Wrong: “The two confidence intervals overlap, so the effects are not significantly different.” Correction: Form the contrast and calculate its standard error using the covariance between estimates; then assess the contrast directly.
• Wrong: “The estimated effects are in points per hour, so their covariance is in squared points per hour.” Correction: A covariance has the product of the two quantities’ units. For two slopes measured in points per hour, the covariance is in (points per hour)\(^2\), equivalently points\(^2\) per hour\(^2\).

These distinctions matter because an accurate coefficient calculation can still answer the wrong question if its conditioning values, averaging population, or contrast are left unstated. Clear interpretation names the outcome and predictor units, identifies the group or covariate values, and says whether the result is a conditional effect, an average, or a difference between effects.