For a square matrix, is it appropriate to use a chi-squared distribution when each level of the variables are assumed to have the same overall frequency?
Specifically, I'm analyzing a dataset of the number of genes that have increased expression in an experimental treatment in two related species. My data look like this, with species 1 on the columns and species 2 on the rows:
Low Intermediate High Low 2594 163 405 Intermediate 1350 558 155 High 467 65 322
I a priori expect each class to have been the same in the common ancestor of the species (off-diagonals = 0). That is:
Low Intermediate High Low 3786 0 0 Intermediate 0 1425 0 High 0 0 868
My question is which of the off-diagonal cells have diverged more than expected by chance.
As a simple first pass, I've modified the standard
chisq.test (in R) to use the overall total for each class (Low, Intermediate, High) rather than the marginal total for each class (assuming species independent...which they aren't).
# data d <- matrix(c(2594L, 1350L, 467L, 163L, 558L, 65L, 405L, 155L, 322L), nrow=3, ncol=3) # row and column sums rs <- rowSums(d) cs <- colSums(d) # grand mean for each class gm <- (rs + cs) / sum(d * 2) Ec <- outer(gm, gm, "*") * sum(d)
Ec is the Expected value for each cell using the grand mean for each class.
Is it reasonable to use a chisq distribution to determine if the observed values deviate from the expected values by more than chance?
Ec.chistat <- sum((data-Ec)^2 / Ec) pchisq(Ec.chistat, df=(nrow(data)-1) * (ncol(data)-1), lower.tail = FALSE)
I realize I could probably use a GLM for this, but it's convenient to keep in table format to directly address which of the off-diagonals have increased more than expected by chance.
Note: for comparison, the standard chisq assuming independence of variable would be:
rs <- rowSums(d) cs <- colSums(d) n <- sum(d) (E <- outer(rs, cs, "*")/n) chistat <- sum((d - E) ^ 2 / E) pchisq(chistat, df=(nrow(d)-1) * (ncol(d)-1), lower.tail = FALSE)