Stata FAQ:
How can I test for nonadditivity in a randomized block ANOVA in Stata?

Randomized block ANOVA models assume additive block and treatment
effects, that is, that there is no treatment by block interaction. Tukey’s test for nonadditivity
is a one degree of freedom test of the hypothesis that there is a linear treatment by linear block
interaction. The nonadd command
can be added to your Stata system by installing the nonadd.ado program written by
ATS. You can download the nonadd command by
typing search nonadd (see
How can I use the search command to search for programs and get additional
help? for more information about using search).

Now, let’s read in an example dataset,
rb4a, which is adapted from Kirk (1982, 2nd Edition).
This example has eight subjects with four repeated measures on each subject.

use https://stats.idre.ucla.edu/stat/stata/faq/rb4a, clear

Let’s look at a table of the data.

table s a, contents(mean y)

----------+-----------------------
          |           a           
        s |    1     2     3     4
----------+-----------------------
        1 |    3     4     7     7
        2 |    6     5     8     8
        3 |    3     4     7     9
        4 |    3     3     6     8
        5 |    1     2     5    10
        6 |    2     3     6    10
        7 |    2     4     5     9
        8 |    2     3     6    11
----------+-----------------------

Now let’s compute the randomized block ANOVA.

anova y a s

              Number of obs =      32     R-squared     =  0.8790
              Root MSE      = 1.16496     Adj R-squared =  0.8214

    Source |  Partial SS    df       MS           F     Prob > F
-----------+----------------------------------------------------
     Model |      207.00    10       20.70      15.25     0.0000
           |
         a |      194.50     3  64.8333333      47.77     0.0000
         s |       12.50     7  1.78571429       1.32     0.2914
           |
  Residual |       28.50    21  1.35714286   
-----------+----------------------------------------------------
     Total |      235.50    31  7.59677419

Next, we’ll do the test for nonadditivity. Note that the variables are entered the
same way as for the randomized block analysis.

nonadd y a s

Tukey's test of nonadditivity for randomized block designs
F (1,20) = 7.8345468   Pr > F: .01108091

In this case the test for nonadditivity was statistically significant, the data are
nonadditive.

Here is a second example, rb4b, this time from Kirk’s 3rd edition. Again,
it has eight subjects with four repeated measures on each subject.

use https://stats.idre.ucla.edu/stat/stata/faq/rb4b, clear

table s a, contents(mean y)

----------+-----------------------
          |           a           
        s |    1     2     3     4
----------+-----------------------
        1 |    3     4     4     3
        2 |    2     4     4     5
        3 |    2     3     3     6
        4 |    3     3     3     5
        5 |    1     2     4     7
        6 |    3     3     6     6
        7 |    4     4     5    10
        8 |    6     5     5     8
----------+-----------------------

anova y a s

              Number of obs =      32     R-squared     =  0.7318
              Root MSE      = 1.18523     Adj R-squared =  0.6041

    Source |  Partial SS    df       MS           F     Prob > F
-----------+----------------------------------------------------
     Model |       80.50    10        8.05       5.73     0.0004
           |
         a |       49.00     3  16.3333333      11.63     0.0001
         s |       31.50     7        4.50       3.20     0.0180
           |
  Residual |       29.50    21   1.4047619   
-----------+----------------------------------------------------
     Total |      110.00    31   3.5483871

nonadd y a s

Tukey's test of nonadditivity for randomized block designs
F (1,20) = 1.2795813   Pr > F: .27135918

This time the test for nonadditivity was not significant, that is,
there is no indication of a treatment by block interaction.

References: