# Why an overall sum of squares does not tell the whole story

When there are 3 or more groups the overall F ratio from the ANOVA can miss specific relationships. For example it can miss trends between the means. The sums of squares term comparing k groups in an ANOVA represents the sum of k-1 orthogonal contrasts between the group means. It is possible for *one* of these contrasts to be present and for the overall F ratio to be non-significant especially when k is large. Note that when k=2 the test for the difference in group means is equivalent to a test of linear trend. In general the relationships between k group means can be exactly explained by contrasts representing trends upto order k-1. So three group means can be exactly explained by the sum of linear and quadratic contrasts and four means by the sum of linear, quadratic and cubic contrasts.

For example suppose we wish to compare three group means. The linear contrast between the group means is represented by the coefficients -1,0,1 and the quadratic contrast by the coefficients 1,-2,1. If we perform a linear regression on the response with an intercept and each person's linear and quadratic contrast coefficient based on which group they are in and add together the sums of squares for the two contrasts we will obtain the sum of squares from the usual ANOVA comparing the difference between the three group means. The reason for this additivity stems from the *orthogonality* of the contrasts.

So knowing the total sum of squares for the k-1 contrasts does *not* tell us anything about its component k-1 contrast sums of squares when k>2. It follows that, for example, a linear trend could be statistically significant even if the F ratio based on k-1 contrasts is non-significant and vice-versa. A SPSS example data set of six groups is given here where there is an overall linear trend on 1 degree of freedom (df) but the overall ANOVA sum of squares on 5 df is not statistically significant. You can see this by opening the attached SPSS data file and then running the syntax below in a syntax window in SPSS.

ONEWAY score BY group /POLYNOMIAL= 1 /MISSING ANALYSIS.

The means form a perfect linear trend going up by '1' moving from group to adjacent group but adjacent pairs of group means do not differ statistically from each other hence the overall non-significance of the group comparison in the ANOVA. The data in the above example were actually generated using Normal distributions with means of 1,2,3,4,5 and 6 for each of the six groups respectively and a common group standard deviation of 3.

See the post-hoc grad talk here for further examples of fitting orthogonal contrasts including a test of a trend.