Types of Sums of Squares

General Notes

There are different approaches to calculating sums of squares. Various software packages offer standard and proprietary alternatives. It is important to know the differences, and it is important to know the default setting for the software package being used for an analysis.

There has been much discussion on this topic. There are those who point out that the choice between types of sums of squares is really a choice of the experimental hypothesis and design. The attached paper "Exegeses on Linear Models" by Bill Venebles offers more perspective.

General Discussion (StatSoft) (NIST)

Type I (StatSoft)

Type II (StatSoft)

Type III (StatSoft)

Type IV (StatSoft)

Type V (StatSoft)

Type VI (StatSoft)

When using R for ANOVA (from http://www.statmethods.net/stats/anova.html):

• WARNING: R provides Type I sequential SS, not the default Type III marginal SS reported by SAS and SPSS. In a nonorthogonal design with more than one term on the right hand side of the equation order will matter (i.e., A+B and B+A will produce different results)! We will need use the drop1( ) function to produce the familiar Type III results. It will compare each term with the full model. Alternatively, we can use anova(fit.model1, fit.model2) to compare nested models directly.

Assumptions

References

StatSoft Electronic Statistics Textbook

Milliken, George A., "Analysis of Messy Data Volume 1: Designed Experiments", Chapman and Hall/CRC, 1998, ISBN 0-412-99081-4, Chapter 10

A related, interesting bit of R code:

install.packages("fortunes"); library(fortunes); fortune("curious")