Sum of Squares
The ANOVA can be calculated using one of three types of sums of squares (SS).
Type III or Partial SS is the default when there are no multilevel, categoric factors. It considers all other terms in the model before calculating the SS for an individual term.
Type II or Classical SS is used when there is at least one multilevel, categoric factor. With this method a factor’s main effect SS calculation is done assuming that the factor does not take part in interactions.
Type I or Sequential SS is not used as a default. SS calculations are done in the order the factors were added to the design. The process begins with the main effects then proceeds up through the interactions.
All SS methods will provide the same value when the design is balanced and orthogonal.
More Details
For the following SS Types,
SS(A) is the Sum of Squares associated with A SS(A|B) is the sum of squares for A given that B is already in the model R(A,B,C) is the residual sum of squares for a model when A, B, and the C terms are in the model
Yhat = Intercept + A + B + C + AB + AC + BC + ABC, is the model.
Type I Sum of Squares (aka Sequential)
Hierarchical decomposition
Type I SS is the SS corresponding to each effect adjusted for every other effect preceding it in the model.
The Type I SS for the B term considers A to come before it,
SS(B) = SS(B|A) = R(B) - R(A, B)
The Type I SS for the AC term considers A, B, C, and AB to come before it,
SS(AC) = SS(AC|A,B,C,AB) = R(AC) - R(A,B,C,AB,AC)
Type II Sum of Squares (aka Classical)
Type II SS is the reduction in the SSerror due to adding an effect after all other terms have been added to the model except effects that contain the effect being tested.
The Type II SS for the B term compares the extra sum of squares from B with A, C, and AC (note: no interactions involving B)
SS(B) = SS(B|A, C, AC) = R(B) - R(A, B, C, AC)
The Type II SS for the AC term compares the extra sum of square from AC with A, B, C, AB, and BC (note: ABC contains AC therefore not part of the comparison)
SS(AC) = SS(AC|A, B, C, AB, BC) = R(AC) - R(A, B, C, AB, AC, BC)
Type III Sum of Squares (aka Partial)
Type III SS is the SS corresponding to each effect adjusted for every other effect in the model.
The Type III SS for the B term compares the extra sum of squares from B with A, C, AB, AC, BC, and ABC
SS(B) = SS(B|A, C, AB, AC, BC, ABC) = R(B) - R(A, B, C, AB, AC, BC, ABC)
The Type III SS for the AC term compares the extra sum of squares from AC with A, B, C, AB, BC, ABC
SS(AC) = SS(AC|A, B, C, AB, BC, ABC) = R(AC) - R(A, B, C, AB, AC, BC, ABC)