Calculating Fitted Means for Two Categories

From a Two-Category Model to Two Fitted Values

A fitted regression equation lets us calculate predicted outcomes for specified predictor values. With a two-category predictor, the calculation is especially direct: the reference category is coded zero, and the comparison category is coded one. Substituting those two values into the same equation gives the fitted outcome for each category. Comparing the results also provides a useful check on the meaning of the indicator coefficient.

Suppose the response \(Y\) is a quantitative outcome and the two-level factor is coded with an indicator \(x\): \(x=0\) for the reference category and \(x=1\) for the comparison category. The fitted model is

Here \(b_0\) is the estimated intercept and \(b_1\) is the estimated indicator coefficient. The fitted value \(\hat{y}\) is the outcome predicted by the model for the specified category. It is not a guarantee of the outcome for every individual in that category: individuals can have outcomes above or below the fitted value.

The coefficient is a difference in the outcome’s units. A positive \(b_1\) means the comparison category has the larger fitted outcome; a negative \(b_1\) means it has the smaller fitted outcome. The sign describes the comparison relative to the chosen reference category, so category labels and coding must be stated when interpreting it.

Calculate Each Fitted Outcome

The calculation follows the coding. For the reference category, substitute \(x=0\). For the comparison category, substitute \(x=1\). Then subtract the reference fitted value from the comparison fitted value to check the coefficient.

This procedure assumes the model being used is the stated two-category model, with no additional predictors or interaction terms. It also assumes the indicator is coded as described. The arithmetic does not require a significance test or a claim that the category itself causes a change in the outcome. The calculation describes fitted values from this model.

Why the Difference Must Equal the Coefficient

The equality is a direct consequence of substitution and subtraction, not a coincidence in a particular data set. Write the comparison fitted value minus the reference fitted value, using the fitted model:

The intercept appears in both fitted values and therefore cancels when they are subtracted. The indicator coefficient is what remains. This is why the coefficient represents the fitted difference for the comparison category relative to the reference category under this coding.

The subtraction order matters. Comparison minus reference gives \(b_1\); reference minus comparison gives \(-b_1\). Neither calculation is inherently wrong, but its sign must match the stated order. A coefficient of \(-8.2\), for example, means the comparison fitted outcome is 8.2 outcome units lower than the reference fitted outcome, not 8.2 units higher.

The interpretation also depends on the model. In the two-category model shown here, both category fitted values share the same intercept and differ only by the indicator term. If a different model includes additional predictors or interactions, its fitted values may depend on those terms as well. The calculation on this page is for the stated model, not a rule to ignore other terms in a larger equation.

Fitted Values and Group Averages

When an ordinary least-squares regression includes only an intercept and a two-category indicator, the fitted value for each category equals that category’s sample mean outcome. This fact can help check a calculation, but it applies to this simple model, where the only predictor is the indicator. In models containing other predictors, the fitted category values need not equal the raw group averages.

The reason follows from how least squares chooses the coefficients. For the reference group, the fitted value is \(b_0\); its residual sum of squares is minimized when \(b_0\) is the reference group’s sample mean. For the comparison group, the fitted value is \(b_0+b_1\); its residual sum of squares is minimized when that fitted value is the comparison group’s sample mean. Thus the two fitted values match the two group averages, and their difference is the estimated indicator coefficient.

Common Mistakes and a Reliable Check

A frequent mistake is to report the intercept as the fitted value for both categories. The intercept is the reference-category fitted value because its indicator is zero; the comparison category’s fitted value includes the indicator coefficient. Another mistake is to call \(b_1\) the comparison fitted value. It is the difference between the two fitted values, not generally the fitted value itself.

• Wrong: “The comparison group’s fitted outcome is \(b_1\).” Correction: It is \(b_0+b_1\); \(b_1\) is the comparison-minus-reference difference.
• Wrong: “The comparison group is higher because the coefficient is negative.” Correction: With comparison-minus-reference ordering, a negative coefficient means the comparison fitted outcome is lower.
• Wrong: “The fitted value is the outcome every member of the group will have.” Correction: It is the model’s predicted value for that category; individual outcomes can vary.
• Wrong: “A positive fitted difference proves the category caused the outcome to increase.” Correction: A fitted association is not by itself a causal conclusion.

A compact check is to write both substitutions before interpreting anything: \(x=0\) gives \(b_0\), while \(x=1\) gives \(b_0+b_1\). Then subtract in the same order used to describe the coefficient. If the difference is not \(b_1\), check the arithmetic, the coding, and the subtraction order.