Compare Additive and Interaction Predictions

Why Compare Predictions?

An additive regression model says that the fitted change associated with one predictor is the same at every value of the other predictor. A product-term model allows that change to vary. Comparing fitted predictions at selected combinations of the predictors makes this difference concrete: it shows what each model implies for the outcome in the units that matter.

The comparison is useful, but it needs careful interpretation. If two models were fitted separately, their coefficient estimates may differ not only because one includes a product term, but because fitting the models recalculates all of their coefficients. Thus, the difference between their predictions is not automatically equal to the product-term contribution. We will separate these two comparisons.

The Models and Their Assumptions

Let \(Y\) be a quantitative outcome and let \(x\) and \(z\) be two quantitative predictors. An additive model has a conditional mean that is a sum of a constant and separate contributions from the predictors. A product-term model includes \(xz\), allowing the fitted slope for one predictor to depend on the value of the other.

\begin{aligned} \text{Additive:}\quad & E(Y\mid x,z)=\beta_0+\beta_1x+\beta_2z,\\ \text{Product term:}\quad & E(Y\mid x,z)=\beta_0+\beta_1x+\beta_2z+\beta_3xz. \end{aligned}

In practice, the coefficients are estimated from data. Replacing each \(\beta\) with its fitted estimate gives a fitted prediction, written \(\hat{y}\), at specified values of \(x\) and \(z\). The additive model’s fitted change per one-unit increase in \(x\) is constant at \(\hat{\beta}_1\). In the product-term model it is \(\hat{\beta}_1+\hat{\beta}_3z\), so it may differ across values of \(z\).

Definition: A fitted prediction at \(x=x_0\) and \(z=z_0\) is the value obtained by substituting those predictor values and the estimated coefficients into a fitted regression equation. It is an estimated conditional mean outcome, not a guarantee about an individual outcome.

The comparison assumes that the equations are being used as models for the conditional mean of \(Y\). For ordinary regression inference, the usual conditions include a correctly specified mean relationship, errors with mean zero conditional on the predictors, independent errors, and constant error variance; normal errors are additionally used for exact small-sample \(t\)-based inference. The arithmetic of substituting values into fitted equations does not require normal errors, but a misspecified mean relationship can make the predictions misleading. Predictor combinations should also be within, or reasonably close to, combinations represented in the observed data. Otherwise, the comparison relies on extrapolation.

What the Product Term Changes

Suppose, first, that an additive equation and a product-term equation have exactly the same intercept and lower-order coefficients, and that the second equation adds only \(\beta_3xz\). Subtracting their fitted mean expressions shows the specific contribution of the product term.

\begin{aligned} &(\beta_0+\beta_1x+\beta_2z+\beta_3xz) -(\beta_0+\beta_1x+\beta_2z)\\ &\qquad=\beta_3xz. \end{aligned}

Thus, when the lower-order coefficients are held fixed, the difference at a specified combination is \(\beta_3xz\). This difference is zero whenever \(x=0\) or \(z=0\), and its sign and size depend on the product \(xz\) as well as on \(\beta_3\). This is a direct algebraic comparison, not a claim that two separately fitted models will share the same lower-order coefficient estimates.

When the models are fitted separately, denote the additive estimates by \(\hat{\beta}_{0,A},\hat{\beta}_{1,A},\hat{\beta}_{2,A}\), and the product-term estimates by \(\hat{\beta}_{0,I},\hat{\beta}_{1,I},\hat{\beta}_{2,I},\hat{\beta}_{3,I}\). Their prediction difference is

\hat{y}_I-\hat{y}_A =(\hat{\beta}_{0,I}-\hat{\beta}_{0,A}) +(\hat{\beta}_{1,I}-\hat{\beta}_{1,A})x +(\hat{\beta}_{2,I}-\hat{\beta}_{2,A})z +\hat{\beta}_{3,I}xz.

The difference includes changes to the intercept and both main-effect estimates, as well as the product term. This is why a useful comparison has two parts: calculate the predictions from the fitted models as they stand, then identify whether a claim concerns that overall difference or the product term’s isolated contribution.

Worked Example: Comparing Two Separately Fitted Models

Suppose a hypothetical analysis predicts an outcome measured in points from study hours \(x\) and a preparation score \(z\). The additive model is \(\hat{y}_A=40+5x+2z\); the product-term model fitted separately is \(\hat{y}_I=42+3x+1.5z+1.2xz\). Calculate and compare predictions at \((x,z)=(2,0)\) and \((2,4)\). These values are illustrative fitted equations, not results from a real study.

At \(x=2,z=0\), the additive prediction is \(40+5(2)+2(0)=50\) points. The product-term prediction is \(42+3(2)+1.5(0)+1.2(2)(0)=48\) points. The product-term model predicts 2 fewer points at this combination.

At \(x=2,z=4\), the additive prediction is \(40+5(2)+2(4)=58\) points. The product-term prediction is \(42+3(2)+1.5(4)+1.2(2)(4)=42+6+6+9.6=63.6\) points. Here, the product-term model predicts 5.6 more points.

The comparison changes across the selected combinations: the product-term model predicts less at \(z=0\) and more at \(z=4\). It would be incorrect to attribute the entire difference at either point to \(1.2xz\). The models were fitted separately, and their intercepts and main-effect coefficients differ as well. For instance, at \((2,0)\) the product-term contribution itself is zero, yet the predictions are 2 points apart because the other fitted coefficients differ.

Isolating the Product-Term Contribution

A second comparison holds the lower-order coefficients fixed so that only the product term changes. This is a clean way to see the term’s algebraic contribution at particular values, but it is not the same as comparing two separately fitted models.

Worked Example: Holding the Lower-Order Terms Fixed

Consider the illustrative equations \(\hat{y}_A=30+4x+2z\) and \(\hat{y}_I=30+4x+2z+0.8xz\), where the outcome is measured in units. Their intercept and main-effect coefficients match. Compare predictions at \((x,z)=(3,2)\) and \((3,-2)\).

At \((3,2)\), the additive prediction is \(30+4(3)+2(2)=46\). The product-term prediction is \(30+4(3)+2(2)+0.8(3)(2)=46+4.8=50.8\). The difference is \(50.8-46=4.8\) units, exactly \(0.8(3)(2)=4.8\).

At \((3,-2)\), the additive prediction is \(30+4(3)+2(-2)=38\). The product-term prediction is \(30+4(3)+2(-2)+0.8(3)(-2)=38-4.8=33.2\). The difference is \(33.2-38=-4.8\) units, exactly \(0.8(3)(-2)=-4.8\).

The product term raises the prediction in the first combination and lowers it in the second because the products \(xz\) have opposite signs. These are predictions from constructed equations chosen to isolate the term’s contribution. If the equations had been separately estimated from a data set, the lower-order estimates would generally not be guaranteed to match.

Compare Changes Across Predictor Values

Another clear comparison asks how much each model’s prediction changes as \(x\) moves between two specified values, while \(z\) is held at a chosen value. This makes the models’ differing implications for the \(x\)-slope visible. It compares fitted mean changes, not observed changes for particular individuals.

Worked Example: Comparing Predicted Changes at Two Values of the Second Predictor

Suppose a hypothetical outcome is a recovery rating in points, and \(x\) is an activity measure in hours while \(z\) is a centered baseline score. The fitted equations are \(\hat{y}_A=12+0.6x-0.4z\) and \(\hat{y}_I=12+0.8x-0.4z-0.05xz\). Compare the predicted change from \(x=0\) to \(x=4\) at \(z=-2\) and \(z=2\).

In the additive model, the prediction at \(x=0\) is \(12-0.4z\), and at \(x=4\) it is \(12+0.6(4)-0.4z\). The difference is \(2.4\) points at either \(z\), since the \(z\) term cancels: \(0.6(4)=2.4\).

In the product-term model at \(z=-2\), the prediction at \(x=0\) is \(12-0.4(-2)=12.8\). At \(x=4\), it is \(12+0.8(4)-0.4(-2)-0.05(4)(-2)=12+3.2+0.8+0.4=16.4\). The predicted change is \(16.4-12.8=3.6\) points.

At \(z=2\), the product-term model predicts \(12-0.4(2)=11.2\) when \(x=0\). At \(x=4\), it predicts \(12+3.2-0.8-0.4=14.0\). The predicted change is \(14.0-11.2=2.8\) points. Thus, for this fitted equation, the predicted four-hour change is 3.6 points at \(z=-2\) and 2.8 points at \(z=2\), compared with the additive model’s 2.4 points at both values. These are model-based associations; they do not by themselves establish that changing activity causes a change in recovery rating.

How to Make a Fair, Useful Comparison

A comparison is clearest when the predictor combinations are chosen for a reason and stated in context. Use values within the observed predictor ranges, and do not compare predictions at different \(x\) and \(z\) values while describing the result as though only one predictor changed. If the purpose is to show how the fitted \(x\)-relationship differs by \(z\), hold \(x\) values fixed across the chosen \(z\) values, or calculate changes in \(x\) separately at each \(z\).

Report predictions in the outcome’s units and name the predictor combination for each one. A prediction of 63.6 points is more informative than “the interaction model is higher.” If a model comparison concerns separately fitted equations, state that it compares their overall predictions. If it concerns only the product term, make clear that the lower-order coefficients are being held fixed.

Prediction differences do not, by themselves, establish that the interaction is statistically supported. The estimated product coefficient may be uncertain, and the predictions also have uncertainty. A visual or numerical difference between fitted values is a description of the models’ implications at chosen values, not a substitute for the interaction test or interval analysis.

Common Mistakes

Assigning the whole difference to the product term. With separately fitted models, the estimates of the intercept and main effects can also change. Use the full prediction-difference expression, or explicitly hold lower-order terms fixed when isolating \(\hat{\beta}_3xz\).
Calling a fitted prediction an individual outcome. A fitted value estimates a conditional mean. Actual outcomes can differ because of residual variation.
Comparing combinations that change both predictors without saying so. State both predictor values, or hold one fixed when describing a change associated with the other.
Inferring statistical support from a prediction gap. A gap is not itself a test or confidence interval for the interaction.
Ignoring where the data lie. A precise calculation can still be a poor guide when the selected predictor combination is far outside the observed data.

Key takeaway: Substitute specified predictor values into each fitted equation and compare the predictions in outcome units. For separately fitted models, the difference reflects changes in all fitted coefficients; only when the lower-order terms are held fixed is the added contribution exactly the product coefficient times \(xz\).

Check Your Understanding

Answer each question by distinguishing a model’s fitted prediction from the observed outcome.

For an additive model \(\hat{y}=10+2x+z\) and a product-term model \(\hat{y}=10+2x+z+0.5xz\) with the same lower-order terms, calculate both predictions at \(x=2,z=3\). What is the product term’s contribution?
Why might the difference between predictions from two separately fitted models not equal the product coefficient times \(xz\)?
An additive model predicts a 5-unit increase when \(x\) rises by one unit. What feature of its form makes that predicted increase the same at every \(z\)?
A product-term model predicts different changes in \(x\) at two values of \(z\). Does that comparison alone show that the interaction is statistically significant? Explain.
Why should predictions at a combination of \(x\) and \(z\) far outside the observed data be treated cautiously?