Can the type 3 SS be obtained using the ANOVA function or an adaptation that is readily available in Mathematica?

Question

The default output for the ANOVA function is that of a type 1 Sum of Squares (SS) table. This is sufficient for a one-way ANOVA, however, when running an n-way ANOVA (with n greater than 1) and allowing for interactions to be tested the "accepted" method seems to be either type 2 or 3 SS. I have been unable to locate a method allowing the output of the type 3 SS from the ANOVA function and have confirmed using XLSTAT that the output from the ANOVA function in Mathematica is a type 1 SS. I can of course make use of XLSTAT for such analyses but would like to use Mathematica for all data analysis if possible. Below is an example to showcase the effect of factor order on the type 1 SS for unbalanced data. The dataset shown below is copied from the ANOVA documentation page with the last entry removed to make the dataset unbalanced.

threewaydata = {{1, 1, 1, 50}, {1, 1, 1, 50}, {1, 1, 1, 54}, {1, 1, 2,
     40}, {1, 1, 2, 36}, {1, 1, 2, 40}, {1, 2, 1, 48}, {1, 2, 1, 
    48}, {1, 2, 1, 44}, {1, 2, 2, 14}, {1, 2, 2, 18}, {1, 2, 2, 
    14}, {2, 1, 1, 40}, {2, 1, 1, 36}, {2, 1, 1, 36}, {2, 1, 2, 
    18}, {2, 1, 2, 14}, {2, 1, 2, 18}, {2, 2, 1, 6}, {2, 2, 1, 2}, {2,
     2, 1, 2}, {2, 2, 2, 20}, {2, 2, 2, 16}};

The following code reorders the factors provided to the ANOVA function.

ANOVA[threewaydata[[All, {1, 2, 3, 4}]], {α, β, γ, All}, {α, β, γ}][[1]]
ANOVA[threewaydata[[All, {2, 1, 3, 4}]], {β, α, γ, All}, {β, α, γ}][[1]]
ANOVA[threewaydata[[All, {3, 1, 2, 4}]], {γ, α, β, All}, {γ, α, β}][[1]]

Below are the respective outputs. The p values for the factors and their interactions differ due to the order in which they are included (with the exception of the 3 way interaction as it is included last). This is not an issue when using a type 3 SS. For this reason I would like to know if the type 3 SS can be obtained either using the ANOVA function itself, or if there is another function capable of doing so.

The type 3 SS obtained from XLSTAT is shown below (this is the desired output). Below is the same analysis performed in Graphpad Prism. It appears to be an equivalent result to that of XLSTAT (with the factors in a different order and with less precision shown for the p-values)

Answer 1

Here I'll just tackle the problem when the number of levels is just 2 for each of the categorical variables.

The PartialSumOfSquares property of LinearModelFit will provide the Type III sum of squares that R, SAS, XLSTAT, and GraphPad Prism but (so far) only when the categorical variables are coded in a specific way. And that way is Deviation Coding (see https://stats.oarc.ucla.edu/r/library/r-library-contrast-coding-systems-for-categorical-variables/#DEVIATION).

For this case that means recoding the 1's and 2's to 1's and -1's.

threewaydata = {{1, 1, 1, 50}, {1, 1, 1, 50}, {1, 1, 1, 54}, {1, 1, 2, 40}, 
  {1, 1, 2, 36}, {1, 1, 2, 40}, {1, 2, 1, 48}, {1, 2, 1, 48}, {1, 2, 1, 44},
  {1, 2, 2, 14}, {1, 2, 2, 18}, {1, 2, 2, 14}, {2, 1, 1, 40}, {2, 1, 1, 36},
  {2, 1, 1, 36}, {2, 1, 2, 18}, {2, 1, 2, 14}, {2, 1, 2, 18}, {2, 2, 1, 6},
  {2, 2, 1, 2}, {2, 2, 1, 2}, {2, 2, 2, 20}, {2, 2, 2, 16}};

(* Convert 2's to -1'2 *)
data = threewaydata;
data[[All, 1]] = -2 data[[All, 1]] + 3;
data[[All, 2]] = -2 data[[All, 2]] + 3;
data[[All, 3]] = -2 data[[All, 3]] + 3;
lmf = LinearModelFit[data, {α, β, γ, α β, α γ, β γ, α β γ}, {α, β, γ}]
(* Type I sums of squares - default *)
lmf["ANOVATable"]

(* Type III sums of squares *)
lmf["PartialSumOfSquares"]
(* {2074.51, 1298.98, 882.353, 7.68627, 509.647, 98.0392, 1029.18} *)

These match with R, SAS, XLSTAT, and GraphPad Prism.

Answer 2

The documentation page ANOVA/tutorial/ANOVA gives the following example:

<<ANOVA`

threewaydata = {{1, 1, 1, 50}, {1, 1, 1, 50}, {1, 1, 1, 54}, {1, 1, 2,
     40}, {1, 1, 2, 36}, {1, 1, 2, 40}, {1, 2, 1, 48}, {1, 2, 1, 
    48}, {1, 2, 1, 44}, {1, 2, 2, 14}, {1, 2, 2, 18}, {1, 2, 2, 
    14}, {2, 1, 1, 40}, {2, 1, 1, 36}, {2, 1, 1, 36}, {2, 1, 2, 
    18}, {2, 1, 2, 14}, {2, 1, 2, 18}, {2, 2, 1, 6}, {2, 2, 1, 2}, {2,
     2, 1, 2}, {2, 2, 2, 20}, {2, 2, 2, 16}, {2, 2, 2, 20}};

ANOVA[threewaydata, {α, β, γ, α β, α γ, β γ}, {α, β, γ}]

and so it seems that all interactions could be determined with addition of a three-way product:

ANOVA[threewaydata, {α, β, γ, α β, α γ, β γ, α β γ}, {α, β, γ}]

This tutorial demonstrates a lot of options which can be employed to get you to the specific formula that you need.