Derive a Difference-in-Slopes Interaction Contrast

Why Compare Conditional Slopes?

In a regression model with an interaction between two quantitative predictors, the slope relating one predictor to the mean response can depend on the value of the other predictor. Comparing those slopes at two specified values gives a direct description of how much the relationship changes across those values. The product coefficient makes this comparison especially simple: the difference in slopes is the product coefficient multiplied by the difference between the two specified values.

The direction of the comparison matters. In this tutorial, the contrast is defined as the slope at the second value minus the slope at the first value. Reversing the order reverses the sign. The contrast describes a feature of the fitted regression mean; by itself, it does not establish that changing either predictor causes a change in the response.

Model and Assumptions

Let \(X\) and \(Z\) be two quantitative predictors and let \(Y\) be the response. Use the product-term model

Here \(\beta_0,\beta_1,\beta_2,\beta_3\) are the model coefficients, and \(\varepsilon_i\) is the error for observation \(i\). For the conditional-mean interpretation, assume the model has the form

Equivalently, the error has conditional mean zero: \(E(\varepsilon_i\mid X_i,Z_i)=0\). The result below follows from this specified mean function. It does not require an assumption about the errors being normally distributed. The predictors are treated as quantitative, and the linear product-term form is assumed to be an appropriate description over the values being compared. The coefficients may be estimated from data; the algebra applies to either the model coefficients or their fitted estimates.

Deriving the Contrast

Write the conditional mean as a function of \(x\) and \(z\). At a fixed \(z\), the terms involving \(x\) are \(\beta_1x+\beta_3xz\). Their combined coefficient on \(x\) is \(\beta_1+\beta_3z\), so this is the conditional slope of \(X\) at \(Z=z\).

The derivative is constant in \(x\) for any fixed \(z\), as expected for a linear relationship in \(X\) at that fixed value of \(Z\). Substituting \(b\) and \(a\) into this slope expression and subtracting in the stated order gives the result.

The lower-order coefficient \(\beta_1\) cancels because it contributes equally to the slope at both values of \(Z\). The contrast depends on the product coefficient and on how far apart \(a\) and \(b\) are. In particular, it is zero if \(a=b\), and it is also zero if \(\beta_3=0\). A positive product coefficient means the conditional slope of \(X\) increases as \(Z\) increases; it does not necessarily mean the slope itself is positive.

The same result can be checked without calculus. For a nonzero change \(h\) in \(X\), at fixed \(z\), the change in the conditional mean divided by \(h\) is

This is the same conditional slope for any \(h\ne 0\). Subtracting this quantity at \(z=a\) from the quantity at \(z=b\) again gives \(\beta_3(b-a)\). Thus the result is exact for the specified linear product-term mean function, not an approximation for small changes in \(X\).

Interpreting Sign, Order, and Units

A difference-in-slopes contrast has units that combine the slope units with the units of the predictor whose values are being compared. If \(Y\) is measured in response units, \(X\) in \(X\)-units, and \(Z\) in \(Z\)-units, then the slope of \(X\) has units of response units per \(X\)-unit. The product coefficient \(\beta_3\) has units of response units per \(X\)-unit per \(Z\)-unit. Multiplying it by \(b-a\), measured in \(Z\)-units, leaves response units per \(X\)-unit: the units of a difference between slopes for \(X\).

The sign must be interpreted with the contrast order in view. If \(b>a\), a positive \(\beta_3\) makes the slope at \(b\) greater than the slope at \(a\). If \(b<a\), the same positive coefficient produces a negative contrast because the first named value is lower than the second. Always write down which slope is subtracted from which before interpreting the sign.

Use the Fitted Coefficient and Preserve the Comparison

When a regression output reports an estimated product coefficient, use that estimate in the contrast. If the estimated coefficient is \(\hat{\beta}_3\), the estimated difference between the conditional slopes at \(b\) and \(a\) is \(\hat{\beta}_3(b-a)\). The other coefficients are needed if the separate slopes are requested, but they cancel from their difference. This is a calculation about the fitted model; assessing uncertainty in the contrast requires additional calculations and is a separate question.

Common Mistake

A frequent error is to report \(\beta_3\) itself as the difference between the slopes at any two chosen values. That is only correct when those values differ by exactly one unit in \(Z\), with the higher value second. In general, the difference is \(\beta_3(b-a)\). For example, a product coefficient of 0.12 and a comparison from \(Z=2\) to \(Z=5\) give \(0.12(5-2)=0.36\), not 0.12.

Another error is to interpret a positive contrast as proof that the slope at the second value is positive. The contrast only says that this slope is greater than the slope at the first value. As the energy-use example showed, both slopes can be negative. Calculate or otherwise establish the individual slopes before making claims about their signs.