Build Confidence Intervals for Conditional Slopes

Why a Conditional Slope Needs an Interval

In a regression model with an interaction between two quantitative predictors, the slope for one predictor depends on the value of the other. A fitted slope is an estimate: a different sample could produce a different value. A confidence interval combines the estimated conditional slope with its sampling uncertainty, so we can assess which slope values are reasonably compatible with the data and model.

The interval must account for the uncertainty in both coefficients that make up the conditional slope, including their covariance. Using only the standard error of either coefficient, or treating the two coefficient estimates as independent, can give the wrong result.

Model and Conditional Slope

Let \(x_1\) and \(x_2\) be quantitative predictors and \(Y\) the response. For observation \(i\), consider the interaction model

The coefficient vector is \(\boldsymbol{\beta}=(\beta_0,\beta_1,\beta_2,\beta_3)^\mathsf{T}\), and its least-squares estimate is \(\mathbf{b}=(b_0,b_1,b_2,b_3)^\mathsf{T}\). At a specified value \(x_2=c\), the conditional slope of the mean response with respect to \(x_1\) is \(\beta_1+c\beta_3\). Its estimate is \(b_1+cb_3\). The slope’s units are response units per unit of \(x_1\); \(c\) is expressed in the units of \(x_2\).

The assumptions for the exact interval used here are those of the classical normal-error linear model: the predictors are treated as fixed, the model is linear in its coefficients and correctly specified, the errors are independent with mean zero and common variance \(\sigma^2\), and the errors are normally distributed. The design matrix must have full column rank so all four coefficients are estimable. In practice, the study design should support independence, the fitted relationship should be reasonably represented by the model, and residual checks should not show strong evidence of nonconstant variance or severe departures from normality. The exact t-based coverage relies on these assumptions.

Formula and Justification

Write \(\widehat{\operatorname{Cov}}(\mathbf{b})\) for the estimated covariance matrix of the four coefficient estimates, in the order \(b_0,b_1,b_2,b_3\). It is commonly reported by regression software. The estimated variance of the conditional slope uses the entries for \(b_1\) and \(b_3\):

The estimated standard error is the square root of this quantity. The covariance term matters because the two coefficient estimates generally vary together across samples. It can increase or decrease the estimated variance depending on its sign and on \(c\).

Here is why the t critical value is appropriate. Let \(X\) be the design matrix, whose columns correspond to the intercept, \(x_1\), \(x_2\), and \(x_1x_2\). For a fixed \(c\), set \(\mathbf{a}=(0,1,0,c)^\mathsf{T}\); then the conditional slope and its estimate are \(\mathbf{a}^\mathsf{T}\boldsymbol{\beta}\) and \(\mathbf{a}^\mathsf{T}\mathbf{b}\). Under the normal-error model, the estimated coefficient vector is normally distributed with its true covariance matrix \(\sigma^2(X^\mathsf{T}X)^{-1}\). Thus the linear combination is normal with true sampling variance \(\sigma^2\mathbf{a}^\mathsf{T}(X^\mathsf{T}X)^{-1}\mathbf{a}\). This is not the same as saying it is normal with the estimated variance.

Estimate \(\sigma^2\) by the residual mean square \(s^2\), where \(s\) is the residual standard deviation and \(df=n-4\). Substitution gives the estimated covariance matrix \(s^2(X^\mathsf{T}X)^{-1}\), and hence the estimated standard error \(SE\) of the linear combination. Under the model assumptions, the residual sum of squares divided by \(\sigma^2\) has a chi-square distribution with \(n-4\) degrees of freedom and is independent of the coefficient estimates. Therefore, dividing the estimation error by its estimated standard error produces a t statistic with \(n-4\) degrees of freedom. Inverting that t statistic gives the interval:

where \(t^*_{n-4}\) is the positive t critical value for the desired two-sided confidence level. For a 95% interval, the critical value cuts off 2.5% in each tail of the t distribution.

More Conditional-Slope Intervals

Using and Interpreting the Interval

For each specified value \(c\), the interval targets one quantity, \(\beta_1+c\beta_3\). Changing \(c\) changes both the estimated slope and, generally, its standard error. A confidence interval at \(c=0\) is an interval for \(\beta_1\), because the product contribution is then zero. At a nonzero \(c\), the covariance between \(b_1\) and \(b_3\) contributes to the uncertainty.

The confidence level describes the long-run performance of the method: if the same sampling and analysis procedure were repeated many times under the model, 95% of intervals formed this way would contain the true conditional slope. It is not correct to say that, after calculating one interval, there is a 95% probability that the fixed parameter lies in it.

This is an interval for a slope, not an interval for the mean response at a particular predictor pair and not a prediction interval for an individual outcome. Its units are response units per unit of \(x_1\), and its interpretation is tied to the stated value of \(x_2\).

Common Mistake

A frequent error is to calculate the variance as \(\widehat{\operatorname{Var}}(b_1)+c^2\widehat{\operatorname{Var}}(b_3)\) and leave out \(2c\,\widehat{\operatorname{Cov}}(b_1,b_3)\). That calculation silently assumes the coefficient estimates have zero covariance. In an interaction model, that is not generally true. Use the relevant entries of the full coefficient covariance matrix, including the covariance term, before taking the square root.

A second error is to use a standard normal critical value automatically. When the residual variance is estimated and the normal-error model applies, use the t critical value with \(n-4\) residual degrees of freedom. The estimator’s true sampling variance involves \(\sigma^2\); replacing it with an estimate supplies an estimated standard error, and the studentized statistic—not the unstudentized estimator with that estimated variance—is t-distributed.