不同的方差类型；一般默认III 型方差。

Type of squares

Type I Sums of Squares. Type I sums of squares involve a sequential partitioning of the whole model sums of squares. A hierarchical series of regression equations are estimated, at each step adding an additional effect into the model. In Type I sums of squares, the sums of squares for each effect are determined by subtracting the predicted sums of squares with the effect in the model from the predicted sums of squares for the preceding model not including the effect. Tests of significance for each effect are then performed on the increment in the predicted sums of squares accounted for by the effect. Type I sums of squares are therefore sometimes called sequential or hierarchical sums of squares.

Type I sums of squares are appropriate to use in balanced (equal n) ANOVA designs in which effects are entered into the model in their natural order (i.e., any main effects are entered before any two-way interaction effects, any two-way interaction effects are entered before any three-way interaction effects, and so on). Type I sums of squares are also useful in polynomial regression designs in which any lower-order effects are entered before any higher-order effects. A third use of Type I sums of squares is to test hypotheses for hierarchically nested designs, in which the first effect in the design is nested within the second effect, the second effect is nested within the third, and so on.

One important property of Type I sums of squares is that the sums of squares attributable to each effect add up to the whole model sums of squares. Thus, Type I sums of squares provide a complete decomposition of the predicted sums of squares for the whole model. This is not generally true for any other type of sums of squares. An important limitation of Type I sums of squares, however, is that the sums of squares attributable to a specific effect will generally depend on the order in which the effects are entered into the model. This lack of invariance to order of entry into the model limits the usefulness of Type I sums of squares for testing hypotheses for certain designs (e.g., fractional factorial designs).

Type II Sums of Squares. Type II sums of squares are sometimes called partially sequential sums of squares. Like Type I sums of squares, Type II sums of squares for an effect controls for the influence of other effects. Which other effects to control for, however, is determined by a different criterion. In Type II sums of squares, the sums of squares for an effect is computed by controlling for the influence of all other effects of equal or lower degree. Thus, sums of squares for main effects control for all other main effects, sums of squares for two-way interactions control for all main effects and all other two-way interactions, and so on.

Unlike Type I sums of squares, Type II sums of squares are invariant to the order in which effects are entered into the model. This makes Type II sums of squares useful for testing hypotheses for multiple regression designs, for main effect ANOVA designs, for full-factorial ANOVA designs with equal cell ns, and for hierarchically nested designs.

There is a drawback to the use of Type II sums of squares for factorial designs with unequal cell ns. In these situations, Type II sums of squares test hypotheses that are complex functions of the cell ns that ordinarily are not meaningful. Thus, a different method for testing hypotheses is usually preferred.

Type III Sums of Squares. Type I and Type II sums of squares usually are not appropriate for testing hypotheses for factorial ANOVA designs with unequal ns. For ANOVA designs with unequal ns, however, Type III sums of squares test the same hypothesis that would be tested if the cell ns were equal, provided that there is at least one observation in every cell. Specifically, in no-missing-cell designs, Type III sums of squares test hypotheses about differences in subpopulation (or marginal) means. When there are no missing cells in the design, these subpopulation means are least squares means, which are the best linear-unbiased estimates of the marginal means for the design (see, Milliken and Johnson, 1986).

Tests of differences in least squares means have the important property that they are invariant to the choice of the coding of effects for categorical predictor variables (e.g., the use of the sigma-restricted or overparameterized model) and to the choice of the particular g2 inverse of X'X used to solve the normal equations. Thus, tests of linear combinations of least squares means in general, including Type III tests of differences in least squares means, are said to not depend on the parameterization of the design. This makes Type III sums of squares useful for testing hypotheses for any design for which Type I or Type II sums of squares are appropriate, as well as for any unbalanced ANOVA design with no missing cells.

The Type III sums of squares attributable to an effect is computed as the sums of squares for the effect controlling for any effects of equal or lower degree and orthogonal to any higher-order interaction effects (if any) that contain it. The orthogonality to higher-order containing interactions is what gives Type III sums of squares the desirable properties associated with linear combinations of least squares means in ANOVA designs with no missing cells. But for ANOVA designs with missing cells, Type III sums of squares generally do not test hypotheses about least squares means, but instead test hypotheses that are complex functions of the patterns of missing cells in higher-order containing interactions and that are ordinarily not meaningful. In this situation Type V sums of squares or tests of the effective hypothesis (Type VI sums of squares) are preferred.

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.

Type V Sums of Squares. 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 Experimental Design chapter. 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 predictors 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 B's degrees of freedom are less than 1 (i.e., its number of levels minus 1), it is eligible to be dropped. Finally, A by B is entered into the model, and its degrees of freedom are determined (i.e., the number of non-redundant columns for A by B in X'X, given the intercept, A, and B). If B's degrees of freedom are less than 2 (i.e., the product of the degrees of freedom for its factors if there were no missing cells), it is eligible to be dropped. Type III sums of squares are then computed for the effects that were not found to be eligible to be dropped, using the reduced model in which any eligible effects are dropped. Tests of significance, however, use the error term for the whole model prior to dropping any eligible effects.

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 interaction 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.