Puzzling rank effect with Friedman anova and multcompare

Hello !

I want to compare the influence of different factors on a measure for which I have different points in time.

I use a matrix in which each row is a time-point, and each column is a factor, as an input of the friedman function. (The friedman test is chosen because the different factors =columns are not independent). The number of repetitions is set to 1. Here is the precise example :

M=[-0.0001   -0.0097   -0.0097   -0.0098   -0.0098   -0.0098   -0.0098   -0.0099   -0.0098   -0.0094   -0.0095 ; 0.0261    0.0069    0.0060    0.0044    0.0029    0.0015    0.0007   -0.0005   -0.0010   -0.0006   -0.0001 ; 0.0441    0.0284    0.0256    0.0226    0.0200    0.0179    0.0169    0.0157    0.0147    0.0130    0.0130 ; 0.0261    0.0511    0.0510    0.0513    0.0503    0.0485    0.0464    0.0440    0.0412    0.0338    0.0295 ; 0.0535    0.1037    0.1068    0.1055    0.1019    0.0974    0.0936    0.0899    0.0865    0.0764    0.0687 ; 0.0404    0.0887    0.0998    0.1051    0.1060    0.1043    0.1007    0.0970    0.0949    0.0842    0.0776 ; 0.0304    0.0999    0.1124    0.1193    0.1223    0.1223    0.1208    0.1190    0.1173    0.1083    0.1012 ; 0.0726    0.1253    0.1335    0.1379    0.1410    0.1429    0.1428    0.1422    0.1400    0.1315    0.1234 ; 0.0577    0.1575    0.1659    0.1689    0.1709    0.1699    0.1682    0.1664    0.1643    0.1555    0.1472 ; 0.0429    0.1793    0.1833    0.1851    0.1857    0.1854    0.1840    0.1822    0.1809    0.1727    0.1654 ; 0.1036    0.1898    0.1931    0.1960    0.1974    0.1974    0.1964    0.1945    0.1925    0.1850    0.1770 ; 0.1081    0.2167    0.2189    0.2210    0.2222    0.2214    0.2195    0.2178    0.2158    0.2061    0.1970 ; 0.1078    0.2244    0.2368    0.2413    0.2441    0.2445    0.2441    0.2428    0.2411    0.2321    0.2243 ; 0.0894    0.2477    0.2524    0.2567    0.2588    0.2599    0.2597    0.2583    0.2578    0.2510    0.2451 ; 0.1363    0.2638    0.2659    0.2696    0.2717    0.2715    0.2717    0.2710    0.2705    0.2638    0.2575 ; 0.1397    0.2605    0.2637    0.2664    0.2684    0.2690    0.2689    0.2679    0.2668    0.2613    0.2565];

[p,table,stats]=friedman(M,1);

multcompare(stats)

Thus, there are 16 time-points and 11 different factors.

If I plot the evolution in time for the different factors, I can see that the first one (first column) is consistently well below the others. However, when I look at the ranks of the different factors in the stats structure given by the friedman function, or in the output of the mutlcompare function, I can see that the rank of the first column is slightly (though not significantly) superior to the rank of the last one.

Can anyone explain me why it is the case ?

Maybe it is because the 'row effect' (beta j in the help for this function) is dominated by the columns 2 to 11, leading to a significant but negative column effect alpha i for the first column ?

Or is it due to the effect of the 3 first time-points ?

Thanks in advance for helping...

Laureline LOGIACO

[Laureline Logiaco]

Question on the question by the author :

Is this question too specific / irrelevant for this forum ?

[Walter Roberson]

Not necessarily -- but the people who understand stats and answer questions do not seem to check the forum every day.

Answer 1

[Laureline Logiaco]

Hello,

I think I found the right answer to this problem.

Friedman anova is sensitive to the ranks between the columns, not to their absolute values.

If on a small subset of rows, one column is slightly above the others, but well below the other columns on other rows, one could think that this column is on average lower than the others.

However, because it was the highest on some rows, its overall rank will possibly be rather high.

Thus, the ranks given by the Friedman table can appear at odds with intuition.

I hope this may help others better interpret Friedman anova results...

Laureline