Custom Contrasts for a Planned Group Comparison

Introduction

A categorical predictor can describe several group means, but a research question often asks for one particular comparison among them. For example, the question might be whether the average fitted outcome in two treatment groups is higher or lower than the fitted outcome in a control group. A custom contrast expresses that planned comparison as weights on the group means.

The weights make the comparison explicit: which groups contribute, in what proportions, and in which direction. Once the group means are fitted, multiplying each mean by its weight and adding the results gives the fitted difference requested by the question.

Model and Definition

Suppose a categorical predictor has \(K\) groups. Let \(\mu_j\) denote the mean outcome for group \(j\), and let \(\hat{\mu}_j\) denote its fitted mean. A group-means model represents the outcome for observation \(i\) in group \(j\) as

Here, \(\varepsilon_{ij}\) is the error for that observation. For the standard linear-model interpretation, assume errors have mean zero, are independent across observational units, and have a common variance. Normality is additionally needed for exact small-sample inference using the usual t procedures, but not to calculate the fitted group means or their contrast. The contrast calculations below concern fitted means, not a test or confidence interval.

The zero-sum condition makes the expression a comparison rather than a weighted overall level: adding the same constant to every group mean does not change the contrast. The weights are chosen to match the question, not automatically to match the number of observations in each group.

Translating a Planned Comparison into Weights

First write the comparison in words or as a difference of averages. Then assign a positive weight to the groups on the first side and a negative weight to those on the second side. Groups not involved receive weight zero. For example, with a control group \(C\) and two treatment groups \(A\) and \(B\), the average treatment mean minus the control mean is

In the order \(C,A,B\), the weights are \((-1,\frac{1}{2},\frac{1}{2})\). They sum to zero, as a contrast’s weights must. This ordering matters: each weight must be attached to the intended category.

Why the Fitted Group Means Are Used

In a group-means model, the fitted mean for a group is its sample mean. This follows directly from least squares. For group \(j\), the sum of squared residuals as a function of a proposed mean \(m\) is \(\sum_i(Y_{ij}-m)^2\). Differentiating with respect to \(m\) and setting the derivative to zero gives

Thus, the fitted contrast is calculated by applying the planned weights to the fitted group means. A different full-rank coding of the categorical predictor can use different coefficient values, but if it represents the same group means, this contrast of fitted means is unchanged. The contrast weights are weights on the means; they should not be confused with coefficients from an arbitrary coding.

What the Weights Say About the Comparison

For an average of \(r\) groups compared with an average of \(s\) other groups, give each group in the first average weight \(1/r\), and each group in the second average weight \(-1/s\). These weights sum to \(r(1/r)+s(-1/s)=1-1=0\). This rule describes an equal average of group means. If the question instead specifies unequal weights, use those weights as stated and ensure that the positive and negative parts each have the intended total weight.

The scale of a contrast also matters. Multiplying every weight by the same nonzero number produces a rescaled contrast with the same zero-versus-nonzero comparison, but its numerical value changes by that factor. For example, \((-1,\frac{1}{2},\frac{1}{2})\) gives the treatment average minus control; \((-2,1,1)\) gives twice that difference. Use weights whose scale matches the comparison you want to report.

Common Mistakes

• Forgetting to divide within an average. The weights \((-1,1,1)\) compare the sum of two group means with the control mean. The weights \((-1,\frac{1}{2},\frac{1}{2})\) compare their average with control.
• Using group sizes when the plan calls for equal group means. Sample-size weighting answers a pooled-observation question. It is not automatically the same as averaging the group means equally.
• Failing to check the sum. If weights do not sum to zero, the expression is not a contrast of the group means as defined here. Revisit the comparison and its scale.
• Misplacing a weight. List the category order beside the weights before calculating. A correct set of numbers attached to the wrong groups describes a different comparison.
• Reading the contrast as a causal effect. A fitted difference summarizes modeled group means. By itself it does not show that group membership caused the difference.