Building an Indicator Table for Three Groups

Why Build an Indicator Table?

A regression model can include a categorical predictor by representing its categories with indicator variables. The model equation gives the rule, but a row-by-row table makes that rule easier to apply: each category gets a specific pattern of zeros and ones, and that pattern determines the fitted value. This is especially useful when checking calculations or reading a model with several groups.

Here we consider a predictor with three categories, called A, B, and C. We use A as the reference category. Under reference-cell coding, A gets zeros in both indicator columns; the other categories each get a one in their own column.

In this model, \(Y_i\) is the response for observation \(i\), \(\beta_0,\beta_1,\beta_2\) are model coefficients, and \(\varepsilon_i\) is the error. The fitted value, denoted \(\hat{Y}_i\), is obtained by using the model’s estimated coefficients and leaving out the error term. The indicator table itself follows from the category membership, not from the observed response.

The Three Rows of the Indicator Table

The coding rule creates one row for each possible category. The first indicator records membership in B; the second records membership in C. Because each observation belongs to exactly one of the three categories, its two indicators have exactly one of these patterns:

The hats indicate estimated coefficients. For an observation in A, substituting zeros into the fitted model leaves only \(\hat{\beta}_0\). For an observation in B, \(D_{Bi}=1\) and \(D_{Ci}=0\), so the fitted value is \(\hat{\beta}_0+\hat{\beta}_1\). For an observation in C, \(D_{Bi}=0\) and \(D_{Ci}=1\), giving \(\hat{\beta}_0+\hat{\beta}_2\). These are the three category-specific fitted values.

The coefficients have a direct interpretation in this setup: \(\hat{\beta}_0\) is the fitted value for A, while \(\hat{\beta}_1\) and \(\hat{\beta}_2\) are the differences between the fitted values for B and A, and C and A, respectively. The units of all three coefficients are the units of the response.

From Group Data to the Fitted Values

The table describes the coding; a fitted model supplies coefficient estimates. In the special case of a model with just this categorical predictor and an intercept, the least-squares fitted value for each category is its sample mean response. The estimates of the indicator coefficients can therefore be obtained from the three group means.

To see why, consider one category with observed responses \(y_1,\ldots,y_m\), and suppose its fitted value is a constant \(a\). The sum of squared residuals in that category is minimized when \(a\) equals its sample mean \(\bar{y}\). This follows by expanding the sum of squares:

The cross-product term disappears because \(\sum_{j=1}^{m}(y_j-\bar{y})=0\). The first term on the right does not depend on \(a\), and the second is nonnegative, reaching its minimum when \(a=\bar{y}\). Applied separately to the three categories, this means the fitted values are the three sample group means. Equating these fitted values to the three rows of the indicator table gives:

Here \(\bar{y}_A,\bar{y}_B,\bar{y}_C\) denote the observed mean responses in categories A, B, and C. This group-mean result is specific to the model containing an intercept and this categorical predictor alone. When other predictors are included, the fitted values also depend on those predictors.

Checking the Model and the Coding

The arithmetic is simple, but the row assignment must match the data. Before interpreting fitted values, identify the categories, choose which one is the reference in the specified coding, and check each observation’s membership. The response is quantitative, and the predictor categories are distinct and exhaustive for the observations being modeled.

The equation also has a statistical interpretation. It represents the conditional mean response as a category-specific value plus an error:

For these statements to describe the group means, the errors must have mean zero within each category. The indicator table and the substitution calculations are algebraic and do not require normally distributed errors. If conventional regression inference is later used, its additional error assumptions must also be assessed; calculating these fitted values alone does not establish those assumptions.

Common Mistake: Giving an Observation the Wrong Row

A frequent error is to code A, B, and C as \((0,0)\), \((1,1)\), and \((0,1)\). That assigns both indicators a value of 1 for category B, even though \(D_B\) indicates membership in B and \(D_C\) indicates membership in C. The correct B row is \((1,0)\). Another error is to treat the category labels as numerical scores and substitute them directly into this model. The categories are group labels; the indicator columns specify how each label enters the equation.