Anovan gives contradicting results

I was playing around with the toy example on this page: http://www.mathworks.co.uk/help/stats/anovan.html And I tried the following:

y = [52.7 57.5 45.9 44.5 53.0 57.0 45.9 44.0]';
g1 = [1 2 1 2 1 2 1 2]; 
g2 = {'hi';'hi';'lo';'lo';'hi';'hi';'lo';'lo'}; 
g3 = {'may';'may';'may';'may';'june';'june';'june';'june'};

p_0 = anovan(y, {g1})
p_1 = anovan(y, {g1 g2})
p_2 = anovan(y, {g1 g2 g3})

which gives:

p_0 =

    0.7577

p_1 =

    0.3596
    0.0007

p_3 =

    0.4174
    0.0028
    0.9140

But I would expect p_0[0], p_1[0], p_2[0] to be the same. But they are not? Can anybody explain me why?

Answer 1

[Tom Lane]

This is a balanced design, so the sum of squares explained by g1 is indeed the same in each case. But its F statistic and p-value are not. That's because the F statistic is the ratio of the terms mean square to the error mean square. As you add more terms, the error mean square changes.