Testing Conditional Slopes in Interaction Models

Why Conditional Slopes Need Their Own Tests

In an interaction model, the association between a predictor and a response depends on the value or group of another variable. A coefficient test for the predictor alone therefore does not necessarily answer whether its slope is zero for a particular group or at a particular moderator value. To answer that question, first express the conditional slope as a combination of model coefficients, then test that combination.

The method is a direct application of linear contrasts. The estimate of a conditional slope may combine several estimated coefficients, and its standard error must reflect both their uncertainty and their covariance. Ignoring that covariance can produce the wrong standard error and test result.

Writing the Conditional Slope as a Contrast

Consider a model with quantitative response \(Y\), predictor \(x\), and moderator \(z\):

Here \(x_iz_i\) is the interaction term. For a fixed moderator value \(z=z_0\), the terms involving \(x\) are \((\beta_1+\beta_3z_0)x\). Thus the slope of the model mean with respect to \(x\), at \(z_0\), is \(\beta_1+z_0\beta_3\). The term \(\beta_2z_0\) changes the model mean at that moderator value, but not its slope with respect to \(x\).

A binary group indicator is a special case. Let \(G=0\) denote one group and \(G=1\) the other, and fit \(Y_i=\beta_0+\beta_1x_i+\beta_2G_i+\beta_3x_iG_i+\varepsilon_i\). The slope for \(G=0\) is \(\beta_1\); for \(G=1\), it is \(\beta_1+\beta_3\). The difference between the \(G=1\) and \(G=0\) slopes is \(\beta_3\).

More generally, a linear contrast is \(L=\mathbf{c}^{\mathsf T}\boldsymbol{\beta}\), a specified weighted sum of regression coefficients. If \(\widehat{\mathbf{V}}\) is the estimated covariance matrix of the coefficient estimates, the estimated contrast and its standard error are:

For the slope at \(z_0\), the relevant weights on \((\beta_1,\beta_3)\) are \((1,z_0)\). For a binary group slope at \(G=1\), the weights on those coefficients are \((1,1)\). A test of whether a conditional slope equals a benchmark \(L_0\) uses \(H_0:L=L_0\) and the statistic \(t=(\hat{L}-L_0)/SE(\hat{L})\), with the regression residual degrees of freedom.

Why the Covariance Is Part of the Standard Error

For a conditional slope \(L=\beta_1+z_0\beta_3\), the estimated slope is a sum of two estimates. The variance of a sum includes a covariance term:

This follows by expanding the variance of a linear combination: for random quantities \(A\) and \(B\), \(\operatorname{Var}(A+cB)=\operatorname{Var}(A)+c^2\operatorname{Var}(B)+2c\operatorname{Cov}(A,B)\). Setting \(A=\hat{\beta}_1\), \(B=\hat{\beta}_3\), and \(c=z_0\) gives the expression above. The covariance may be positive, negative, or zero; it cannot generally be omitted.

Assumptions for the Tests

The formulas for a contrast are algebraically valid for fitted coefficients, but the usual regression \(t\) test and its reference distribution depend on the model assumptions. For the interaction model, assume the conditional mean has the stated linear form, observations are independent, errors have mean zero and a common variance, and errors are normally distributed for an exact \(t\) reference distribution. The predictor and moderator values must also provide information to estimate the coefficients; in particular, the model terms must not be perfectly collinear.

In practice, assess whether the interaction model is a reasonable description of the data and whether the residual behavior is consistent with these assumptions. The worked examples below stipulate that the interaction form and residual checks are adequate, observations are independent, and the model is estimable. Without such justification, a computed \(t\) statistic does not by itself establish that the reported test is trustworthy.

Testing a Slope at a Chosen Moderator Value

When \(z\) is quantitative, a conditional slope can be tested at any specified value \(z_0\) for which the fitted model is meaningful. The null hypothesis for a zero slope is \(H_0:\beta_1+z_0\beta_3=0\). To test another benchmark, replace zero with that benchmark. The test concerns the slope of the fitted mean response with respect to \(x\) at \(z_0\), not the coefficient \(\beta_1\) alone except when \(z_0=0\).

Testing Whether Two Conditional Slopes Differ

To compare slopes at two moderator values, subtract their conditional-slope expressions. At \(z_a\) and \(z_b\), the slopes are \(\beta_1+z_a\beta_3\) and \(\beta_1+z_b\beta_3\). Their difference, defined as the slope at \(z_b\) minus the slope at \(z_a\), simplifies to \((z_b-z_a)\beta_3\). The common \(\beta_1\) term cancels. Thus testing whether these two slopes are equal is equivalent to testing whether this contrast is zero. In the binary-group model, the corresponding slope difference is simply \(\beta_3\).

A Reliable Testing Sequence

Common Mistake: Testing the Wrong Coefficient

A frequent error is to use the \(t\) test for \(\beta_1\) as if it tested the slope for every group or moderator value. In the quantitative-moderator model, \(\beta_1\) is the slope only at \(z=0\); at \(z=2\), for example, the slope is \(\beta_1+2\beta_3\). Likewise, in a binary-group model, \(\beta_1\) is the slope for \(G=0\), while the slope for \(G=1\) is \(\beta_1+\beta_3\). The correction is to write the requested slope explicitly and test that linear contrast with its own standard error.

A related mistake is to calculate the variance of a sum by adding only the coefficient variances. The correct expression includes twice the covariance, multiplied by the contrast weights. Because the covariance can materially change the result, a slope contrast cannot generally be tested by combining coefficient standard errors alone.