Group Differences in Fitted Outcomes Across Predictor Values

Introduction

When a quantitative predictor is related to an outcome differently across groups, a single group comparison may not tell the whole story. For example, two groups could have different fitted outcomes at one predictor value, and that difference could be larger or smaller at another value. The useful question is therefore often conditional: what is the fitted difference between the groups at this particular predictor value?

A group-by-predictor interaction model answers this by allowing each group to have its own fitted line. This tutorial focuses on calculating the difference between those fitted lines at specified predictor values and describing how the difference changes as the predictor changes. These are model-based comparisons, not automatically causal effects.

The Group-Specific Lines and Their Difference

Let \(Y\) be a quantitative outcome, \(x\) a quantitative predictor, and \(G\) an indicator for group membership. Set \(G=0\) for the reference group and \(G=1\) for the other group. A linear model with a group-by-predictor interaction is

Here, \(\varepsilon_i\) is the error for observation \(i\). The fitted value is the model's predicted outcome, calculated by replacing the coefficients with their estimates and omitting the error term. When \(G=0\), the terms involving \(G\) vanish. When \(G=1\), both the group term and the product term contribute.

The difference formula follows directly from writing both fitted lines and subtracting. For group \(G=0\), the fitted outcome is \(\hat{\beta}_0+\hat{\beta}_1x\). For group \(G=1\), it is \(\hat{\beta}_0+\hat{\beta}_1x+\hat{\beta}_2+\hat{\beta}_3x\). Therefore,

The shared terms cancel. Thus \(\hat{\beta}_2\) is the fitted group difference specifically at \(x=0\), while \(\hat{\beta}_3\) describes how the fitted difference changes for each one-unit increase in \(x\). Equivalently, \(\hat{\beta}_3\) is the difference between the groups' fitted slopes. The units of \(\hat{\beta}_2\) are outcome units; the units of \(\hat{\beta}_3\) are outcome units per predictor unit.

More generally, comparing the fitted differences at two predictor values \(x_1\) and \(x_2\) gives

This makes the change especially clear: if the predictor increases by \(x_2-x_1\) units, the fitted group difference changes by \(\hat{\beta}_3(x_2-x_1)\) outcome units. A positive value means the difference defined as group 1 minus group 0 increases; a negative value means it decreases.

Model Conditions for These Comparisons

The algebra above applies to the specified fitted lines. To use those lines as useful summaries of a relationship, the model should be appropriate for the data and the comparisons should be made where the data provide support. In particular, the conditional mean of \(Y\) should be reasonably represented by a straight line in \(x\) within each group, with group-specific intercepts and slopes. The observations should also be independent when the study design supports that assumption, and the fitted values at the requested \(x\) values should not rely on unsupported extrapolation beyond the groups' observed predictor ranges.

The usual zero-mean error condition is that the expected error is zero conditional on the predictor and group: \(E(\varepsilon_i\mid x_i,G_i)=0\). Constant error variance is commonly used for conventional standard errors and tests, but it is not needed for the subtraction that produces a fitted difference. This tutorial calculates point fitted differences; it does not assess their uncertainty or statistical significance. In each example, the stated predictor values are within the hypothetical data's observed range, and the given equations are assumed to be reasonable fitted linear summaries.

Reading the Sign and Units

Always state which group is subtracted from which. If the comparison is group 1 minus group 0, a positive fitted difference means the group 1 fitted outcome is higher at that predictor value; a negative difference means it is lower. Reversing the subtraction reverses the sign, but not the size of the comparison. State the outcome units for the difference and the predictor units for the rate at which it changes.

An intercept difference may be mathematically defined at \(x=0\), even when zero is not a substantively useful predictor value. The conditional difference can still be calculated at a meaningful value within the observed range by using \(\hat{\beta}_2+\hat{\beta}_3x\). Interpret the comparison at the value actually chosen rather than treating the intercept difference as the group difference everywhere.

Calculate at the Requested Values

A reliable calculation can be organized in three steps: write the two fitted lines, subtract them in the stated order, and substitute the requested predictor value or values. Evaluating both lines separately is a useful check, especially when signs are involved. If comparing two predictor values, calculate the difference at each and subtract, or use the slope-difference formula \(\hat{\beta}_3(x_2-x_1)\); the answers should agree.

Common Mistake: Treating the Group Coefficient as the Difference Everywhere

A common mistake is to report \(\hat{\beta}_2\) as the group difference at every predictor value. That is only correct at \(x=0\), because the fitted difference is \(\hat{\beta}_2+\hat{\beta}_3x\). For example, in the practice-time model, the group coefficient is 5 points, but the difference at 6 hours is \(5-0.8(6)=0.2\) points, not 5 points.

Another mistake is to interpret a positive interaction coefficient as meaning that group 1 always has a higher fitted outcome. The coefficient \(\hat{\beta}_3\) gives the change in the group 1 minus group 0 difference per predictor unit; it does not by itself give the difference at a particular value. Calculate \(\hat{\beta}_2+\hat{\beta}_3x\) at the value of interest, and preserve the direction of subtraction.