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Sum Of Squares

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A measure of the variability in a set of data, it is actually the sum of the squared deviations of the observed data from the mean. It forms the numerator of the formula for calculating variance. It is used in the Analysis of Variance (ANOVA) procedure to calculate the mean sum of squares (MSS) for the F-statistic.


To get the Total sum of squares (SST) we take the deviations of each individual observed value (yij) from the overall mean of the combined sample (ybar), whereas for the Treatment sum of squares the deviations are between the individual treatment means (ybari) and the overall mean (ybar). If the analysis involves only one factor, then:

Total SS - Treatment SS = Error SS.


Different approaches are used to calculate sums of squares depending on the type of design and the analysis objective. The resulting sums of squares are named Types I to VI. The first four types are most commonly used:

Type I: The SS for an effect is computed after adjusting for all the effects previously entered into the model. Also called Sequential SS, it depends on the order in which the effects are entered in the model. If you change the order, the resulting SS values will also change.
Type II: This is a called a Partially Sequential SS, computed as the reduction in the error SS obtained by adding each effect to a model consisting of all other effects that do not contain it. E.g.: In a two-factor design with factors A and B, the SS for main effect A is not adjusted for effect AB. Similarly, in a 3-factor design the SS for effect AB is adjusted for effects A, B, C, AC and BC, but not for effect ABC.
Type III: Also called Orthogonal SS, this SS for each effect is adjusted for all other effects in the model, regardless of order.
Type IV: Balanced SS, this SS is computed like Type III SS but are more appropriate if the design suffers from missing data.

External Links

Types of Sums of Squares from NIMH (National Institute for Mental Health): -