[R] ANOVA and permutation tests : beware of traps

[Stéphane Adamowicz]

Recently, I came across a strange and potentially troublesome behaviour of the lm and aov functions that ask questions about calculation accuracy. Let us consider the 2 following datasets dat1 & dat2 :

> (dat1 <- data.frame(Y=c(1:3, 10+1:3), F=c(rep("A",3), rep("B",3))))
   Y F
1  1 A
2  2 A
3  3 A
4 11 B
5 12 B
6 13 B
> (dat2 <- data.frame(Y=c(10+1:3, 1:3), F=c(rep("A",3), rep("B",3))))
   Y F
1 11 A
2 12 A
3 13 A
4  1 B
5  2 B
6  3 B

They only differ in the order of values that were exchanged between samples A and B. Thus the sd is 1 for each sample in either data sets, and the absolute mean difference |A-B| is 10 in both datasets.
Now, let us perform an anova to compare samples A and B in both datasets (of course, in such simple case, a bilateral T test would do the job, but an anova is nevertheless allowed and should give the same probability than Student's test):

> (anova1 <- anova(lm(Y~F, dat1)))
Analysis of Variance Table

Response: Y
          Df Sum Sq Mean Sq F value    Pr(>F)    
F          1    150     150     150 0.0002552 ***
Residuals  4      4       1                      

> (anova2 <- anova(lm(Y~F, dat2)))
Analysis of Variance Table

Response: Y
          Df Sum Sq Mean Sq F value    Pr(>F)    
F          1    150     150     150 0.0002552 ***
Residuals  4      4       1

As expected, both datasets give a same anova table, but this is only apparent. Indeed :

> anova1$F[1] == anova2$F[1]
[1] FALSE
> anova1$F[1] - anova2$F[1]
[1] 5.684342e-14

In fact the F values differ slightly, and this holds also for the aov function. I checked also (not shown) that both the residual and factorial sums of squares differ between dat1 and dat2. Thus, for some undocumented reason (at least for the end user), the F values depend on the order of data!
While such tiny differences (e-14 in this example) are devoid of consequences on the risk evaluation by Fisher's distribution, they may have huge consequences on the risk evaluation by the permutation method. Indeed, the shift from continuous to discrete distributions is far from being insignificant.

For instance, the following code in R is at the basis of many permutation algorithms found in the internet and in teaching because it seems quite straightforward (see for example http://www.uvm.edu/~dhowell/StatPages/More_Stuff/Permutation%20Anova/PermTestsAnova.html
http://www.usna.edu/Users/math/jager/courses/sm439/labs/Lab7.pdf
http://biostatistics1014.blogspot.fr/2013/04/one-way-anova-permutation-test-and.html
http://adn.biol.umontreal.ca/~numericalecology/Rcode/):

> nperm <- 1000	# number of permutations
> Fperm <- replicate(n=nperm, anova(lm(sample(Y) ~ F, dat1))$F[1]) # calculates nperm F values
> (prob <- (sum(anova1$F[1] <= Fperm) + 1)/(nperm +1))	# risk calculation	
[1] 0.04695305

Of course, because of the sample function, repeating this code gives different prob values, but they remain always close to 5% instead of the exact probability of 10%. Indeed, there are only choose(6,3) = 20 possible permutations, but because they are symmetric, they give only 10 absolute mean differences. Thus, the only exact probabilities are 10%, 20% ... 100%. In our example where samples do not overlap, 10% is obviously the right answer.

Thus, the use of lm and aov functions in permutation methods does not seem a good idea as it results in biases that underestimate the exact risk. In the simple case of one-way anova, it is quite simple to remedy this problem. As the total Sums of squares and the degrees of freedom do not change with permutations, it is easier and much faster to compare the residual sums of squares. For instance, the exact probabilities can be calculated that way :

> combi <- combn(6, 3, FUN=function(i) {append(dat1$Y[i], dat1$Y[-i])}) # all permutations
> SCEResid <- apply(combi, 2, FUN=function(x) sum(tapply(x, dat1$F, function(x) sum((x - mean(x))^2)))) # all resi SS
> (prob <- mean(SCEResid <= SCEResid[1])) # risk calculation
[1] 0.1

10% is indeed the exact risk.

Finally, my problem is : How can we know if R libraries that use randomization procedures are not biased ? In the basic case of one way anova, it seems easy to submit the above example (by the way, the defunct lmPerm library does not succeed ...), but how can we check more complex anova models ?




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