Type V sums of squares were developed as an alternative to Type IV sums of squares for testing hypotheses in ANOVA designs in missing cells. Also, this approach is widely used in industrial experimentation, to analyze fractional factorial designs; these types of designs are discussed in detail in the 2(k-p) Fractional Factorial Designs section of the Introductory Overview to the Experimental Design module. In effect, for effects for which tests are performed all population marginal means (least squares means) are estimable.

Type V sums of squares involve a combination
of the methods employed in computing Type I and Type III sums of squares. Specifically, whether
or not an effect is eligible to be dropped from the model is determined
using Type I procedures, and then hypotheses are tested for effects not
dropped from the model using Type III procedures. Type V sums of squares
can be illustrated by using a simple example. Suppose that the effects
considered are A, B,
and A by B,
in that order, and that A
and B are both categorical
predictor with, say, 3 and 2 levels, respectively. The intercept
is first entered into the model. Then A
is entered into the model, and its degrees of freedom are determined (i.e.,
the number of non-redundant columns for A
in X'X, given
the intercept). If A's degrees
of freedom are less than 2 (i.e., its number of levels minus 1), it is
eligible to be dropped. Then B
is entered into the model, and its degrees of freedom are determined (i.e.,
the number of non-redundant columns for B
in X'X, given
the intercept and A). If

Note that Type V sums of squares involve determining a reduced model for which all effects remaining in the model have at least as many degrees of freedom as they would have if there were no missing cells. This is equivalent to finding a subdesign with no missing cells such that the Type III sums of squares for all effects in the subdesign reflect differences in least squares means.

Appropriate caution should be exercised when using Type V sums of squares. Dropping an effect from a model is the same as assuming that the effect is unrelated to the outcome (see, e.g., Hocking, 1996). The reasonableness of the assumption does not necessarily insure its validity, so when possible the relationships of dropped effects to the outcome should be inspected. It is also important to note that Type V sums of squares are not invariant to the order in which eligibility for dropping effects from the model is evaluated. Different orders of effects could produce different reduced models.

In spite of these limitations, Type V sums of squares for the reduced model have all the same properties of Type III sums of squares for ANOVA designs with no missing cells. Even in designs with many missing cells (such as fractional factorial designs, in which many high-order interactions effects are assumed to be zero), Type V sums of squares provide tests of meaningful hypotheses, and sometimes hypotheses that cannot be tested using any other method.

Whole Model Tests

Partitioning of Sums of Squares

Limitations of whole model tests

Type VI (Effective Hypothesis) Sums of Squares

Error Terms for Tests

Lack-of-Fit Tests Using Pure Error

Designs with Zero Degrees of Freedom for Error

Tests of Hypotheses in Mixed-Model Designs

Hypotheses about Linear Combinations of Effects

Planned Comparisons of Least Square Means

Testing Hypotheses for Repeated Measures and Dependent Variables

See also GLM - Index.