Worked Calculations for a Continuous Interaction

From an Interaction Model to Numerical Results

A continuous interaction means that the slope for one quantitative predictor depends on the value of another. A coefficient table gives the model estimates, but it does not directly list every conditional slope or predicted mean that might be useful. Those quantities are calculated from the coefficients at specified predictor values. Their uncertainty must also be calculated from the uncertainty in the relevant combination of coefficients.

This tutorial works through those calculations for a model with two quantitative predictors. The examples use the same hypothetical coefficient table so that the distinction between a slope, a fitted mean, and a contrast is explicit. The numbers are illustrative, not results from a real study.

The model assumes that the conditional mean has the stated form over the predictor values being considered. For the usual \(t\)-based confidence intervals in these examples, also assume independent errors with mean zero, constant variance, and a normal error distribution conditional on the predictors. The coefficient estimates must be identifiable, and the chosen predictor profiles should be supported by the data rather than requiring extrapolation.

The Hypothetical Coefficient Table

Suppose a hypothetical analysis predicts an end-of-course score \(Y\), measured in points, from weekly study time \(X\), measured in hours, and a workload index \(Z\), measured in index points. The model includes their product \(XZ\). Assume \(n=45\), giving \(df=45-4=41\) residual degrees of freedom for the four fitted coefficients.

For these worked calculations, suppose the estimated covariances between distinct coefficient estimates are all zero. This is a simplifying feature of the hypothetical example, not a general property of interaction models. In the coefficient order \((\hat{\beta}_0,\hat{\beta}_1,\hat{\beta}_2,\hat{\beta}_3)\), the estimated covariance matrix is therefore:

Each diagonal entry is the square of the corresponding standard error. In an actual coefficient table, standard errors alone do not generally determine the uncertainty of a combination of coefficients: the covariances may also be needed. Here, the stated zero covariances make each variance calculation transparent.

Conditional Slopes and Their Confidence Intervals

At a fixed workload index \(z\), the model's slope for study time is the rate at which the conditional mean score changes with one additional study hour. It is \(\beta_1+\beta_3z\). The interaction estimate is negative, so the fitted study-time slope gets smaller as the workload index increases.

Fitted Means at a Predictor Profile

A fitted mean is the model's estimated conditional mean outcome at specified predictor values. Substituting a profile \((x,z)\) into the fitted equation gives that estimate. To form a confidence interval, treat the prediction as a linear combination of coefficient estimates. With zero covariances in this example, its variance is the sum of each coefficient variance multiplied by the square of that coefficient's multiplier.

Contrasts Between Two Predicted Profiles

A profile contrast is the difference between two fitted means. If profile A is \((x_A,z_A)\) and profile B is \((x_B,z_B)\), define the contrast as fitted mean A minus fitted mean B. The intercept cancels, but the remaining coefficient multipliers depend on both profiles. Calculate the difference in this specified order so the sign has a clear interpretation.

Why the Calculation Details Matter

Conditional slopes, fitted means, and profile contrasts are all functions of the same coefficient estimates, but they answer different questions. A conditional slope describes a one-unit change in one predictor while holding the other fixed. A fitted mean describes the model's estimated average outcome at one predictor profile. A profile contrast compares two such estimated averages. Their uncertainty depends on the coefficient multipliers for the specific quantity being calculated.

In this example, every variance used the diagonal covariance matrix. In a general analysis, include the covariance terms: for a combination \(\sum_j c_j\hat{\beta}_j\), its estimated variance is \(\sum_j c_j^2\widehat{\operatorname{Var}}(\hat{\beta}_j)+2\sum_{j<k}c_jc_k\widehat{\operatorname{Cov}}(\hat{\beta}_j,\hat{\beta}_k)\). Omitting nonzero covariance terms can produce an incorrect standard error and interval.

• Common mistake: treating \(\hat{\beta}_1\) as the study-time slope at every workload level. The correction is to calculate \(\hat{\beta}_1+\hat{\beta}_3z\) at the particular \(z\) of interest.
• Common mistake: using only the standard error of one coefficient for a conditional slope or contrast. The correction is to identify all coefficient multipliers first, then calculate the uncertainty of that full linear combination.
• Common mistake: calling a fitted mean an individual prediction. The correction is to state that its confidence interval concerns the conditional mean at the specified profile, not the spread of individual outcomes.
• Common mistake: interpreting a contrast without naming its order. The correction is to report, for example, “profile A minus profile B”; reversing the order reverses the sign and interval endpoints.

Always attach units and predictor values to an interpretation. Here, the conditional slope is measured in score points per additional study hour, while fitted means and profile contrasts are measured in score points. Also check that the selected predictor values and combinations are adequately supported before interpreting the calculations.