# GLM Hypothesis Testing - Type IV Sums of Squares

Type IV sums of squares were designed to test "balanced" hypotheses for lower-order effects in ANOVA designs with missing cells. Type IV sums of squares are computed by equitably distributing cell contrast coefficients for lower-order effects across the levels of higher-order containing interactions.

Type IV sums of squares are not recommended for testing hypotheses for lower-order effects in ANOVA designs with missing cells, even though this is the purpose for which they were developed. This is because Type IV sum-of-squares are invariant to some but not all g2 inverses of X'X that could be used to solve the normal equations. Specifically, Type IV sums of squares are invariant to the choice of a g2 inverse of X'X given a particular ordering of the levels of the categorical predictor variables, but are not invariant to different orderings of levels. Furthermore, as with Type III sums of squares, Type IV sums of squares test hypotheses that are complex functions of the patterns of missing cells in higher-order containing interactions and that are ordinarily not meaningful.

Statisticians who have examined the usefulness of Type IV sums of squares have concluded that Type IV sums of squares are not up to the task for which they were developed:

• Milliken & Johnson (1992, p. 204) write: "It seems likely that few, if any, of the hypotheses tested by the Type IV analysis of [some programs] will be of particular interest to the experimenter."

• Searle (1987, p. 463-464) writes: "In general, [Type IV] hypotheses determined in this nature are not necessarily of any interest."; and (p. 465) "This characteristic of Type IV sums of squares for rows depending on the sequence of rows establishes their non-uniqueness, and this in turn emphasizes that the hypotheses they are testing are by no means necessarily of any general interest."

• Hocking (1985, p. 152), in an otherwise comprehensive introduction to general linear models, writes: "For the missing cell problem, [some programs] offers a fourth analysis, Type IV, which we shall not discuss. "

So, we recommend that you use the Type IV sums of squares solution with caution, and that you understand fully the nature of the (often non-unique) hypotheses that are being testing, before attempting interpretations of the results. Furthermore, in ANOVA designs with no missing cells, Type IV sums of squares are always equal to Type III sums of squares, so the use of Type IV sums of squares is either (potentially) inappropriate, or unnecessary, depending on the presence of missing cells in the design.

Whole Model Tests

Partitioning of Sums of Squares

Six Types of Sums of Squares

Contained Effects

Error Terms for Tests

Lack-of-Fit Tests Using Pure Error

Testing Specific Hypotheses

Estimability of Hypotheses

Testing Hypotheses for Repeated Measures and Dependent Variables