Derive the Conditional Slope of One Predictor

Why a Predictor’s Slope Can Depend on Another Predictor

In a regression model with two quantitative predictors and their product, the fitted relationship between the outcome and one predictor is not necessarily the same at every value of the other predictor. A useful way to express that relationship is to find the slope of the fitted mean with respect to the predictor of interest, while holding the other predictor fixed.

This is a derivative of the model’s fitted mean, not a claim that changing a predictor causes an outcome to change. The derivative describes the model’s fitted association at a specified value of the other predictor. To interpret it well, identify which predictor is being varied, which is being held fixed, and the units of all three variables.

The Model and the Conditional Slope

Let \(Y\) be a quantitative outcome, and let \(x\) and \(z\) be two quantitative predictors. The product-term model for the conditional mean is

Here \(\beta_0,\beta_1,\beta_2,\beta_3\) are model parameters. In a fitted model, the estimated conditional mean is

The conditional mean is the model’s mean outcome at specified predictor values. The fitted conditional mean is its estimate, calculated using estimated coefficients. For the rest of this tutorial, “fitted mean” refers to this estimated quantity unless stated otherwise.

The model is a statement about the conditional mean. A standard regression formulation writes \(Y_i\) as this mean plus an error, with errors having conditional mean zero: \(E(\varepsilon_i\mid x_i,z_i)=0\). This condition connects the regression equation to the conditional mean. Independent observations and suitable variance assumptions are relevant to standard regression inference, but normal errors are not needed for the algebraic derivative derived below.

Deriving the Conditional Slope

Differentiate the fitted mean with respect to \(x\), treating \(z\) as a constant. The intercept has derivative zero. The term \(\hat{\beta}_1x\) contributes \(\hat{\beta}_1\), and \(\hat{\beta}_2z\) contributes zero because it does not vary with \(x\). In the product term \(\hat{\beta}_3xz\), the factor \(\hat{\beta}_3z\) is constant with respect to \(x\), so the derivative is \(\hat{\beta}_3z\).

At a particular value \(z_0\), substitute that value into the derivative:

The result does not depend on \(x\). Thus, for a fixed \(z_0\), the model’s fitted mean has a constant slope as \(x\) varies. The slope can depend on \(z_0\), because the product term contributes a quantity involving \(z_0\).

The units provide a useful check. If \(Y\) is measured in points, \(x\) in hours, and \(z\) in tutoring sessions, then \(\hat{\beta}_1\) has units of points per hour. The product coefficient \(\hat{\beta}_3\) has units of points per hour per session; multiplying it by \(z\), measured in sessions, gives points per hour. Both terms in the conditional-slope formula therefore have matching units.

Worked Examples

A Reliable Calculation Process

For any fitted equation of this form, keep the differentiation and the contextual interpretation as separate steps. This reduces errors such as differentiating with respect to the wrong predictor or forgetting to hold the other predictor fixed.

Common Mistake: Using Only the Main-Effect Coefficient

A frequent error is to report \(\hat{\beta}_1\) as the slope of \(x\) for every value of \(z\). That omits the contribution of the product term. In the fitted model, the derivative includes both the \(x\) term and the product term:

For example, in the quiz-score equation, reporting 3.2 points per study hour at every tutoring level would ignore \(0.4z\). At \(z=2\), the fitted slope is \(3.2+0.4(2)=4.0\), not 3.2. The coefficient \(\hat{\beta}_1\) is the formula’s value when \(z=0\); a slope at another specified \(z\) requires substitution into the full derivative.

Another mistake is to describe the derivative as the change in a particular person’s observed outcome. The derivative applies to the fitted conditional mean. Individual observations can lie above or below that mean because the regression model includes unexplained variation.