Is there a way to do a Mann-Whitney/Wilcoxon rank sum test on data in this format?
Yes. Calculating the test statistic is simple. Your biggest problem will be the heavy ties.
I have been using the wilcox.test function, but it requires each observation to be listed individually, rather than totalled;
There are several ways to get the data into a form that wilcox.test will take. Here's one:
exper <- read.table(stdin(),header=TRUE)
TreatA TreatB TreatC
Poor 2 2 1
Fair 1 1 1
Moder 1 2 0
Good 0 0 2
Exc 1 0 1
# at this point exper looks like your data
# make it an actual table:
exper <- as.matrix(exper)
class(exper) <- "table"
# convert it:
expertmp <- as.data.frame(exper)
# convert the counts to repetitions:
errle <- list(lengths=expertmp$Freq,values=1:length(expertmp$Freq))
class(errle) <- "rle"
experaw <- expertmp[inverse.rle(errle),1:2]
#need to make it an ordered factor.
experaw$Var1 <- ordered(experaw$Var1)
At this point, the data is in a form you can use. You can use rep rather than inverse.rle (and it's probably a bit shorter that way) to expand counts to repetitions.
e.g., you can do this:
kruskal.test(as.numeric(Var1)~Var2,experaw)
or this:
wilcox.test(as.numeric(Var1)~Var2,experaw[experaw$Var2!="TreatC",])
but (as R warns you in the second case) the p-value with ties is not exact. And unfortunately ties are heavy. You could do a permutation test (/randomization test) and get a "simulated" p-value, which will effectively be exact at a sufficiently large number of resamples.
However, since your observations-per-sample are pretty large, you should be able to get away with a normal approximation with the variance adjusted for ties. There's sufficient information in say Conover's Practical Nonparametric Statistics to do this for oneself.
Best Answer
First of all it might be useful to remember that Mann-Whitney test is also called Wilcoxon rank-sum test. Since it is the same test there is no need to explain the difference ;) A good answer to the common question about the difference between W statistic and U statistic is given here: Is the W statistic output by wilcox.test() in R the same as the U statistic?
Mann-Whitney/Wilcoxon rank-sum test (later MWW test) is defined in R through function wilcox.test (with paired=FALSE) which uses [dprq]wilcox functions.
However, people sometimes mistake MWW with Wilcoxon signed-rank test.
The difference comes from the assumptions. In the MWW test you are interested in the difference between two independent populations (null hypothesis: the same, alternative: there is a difference) while in Wilcoxon signed-rank test you are interested in testing the same hypothesis but with paired/matched samples.
For example, the Wilcoxon signed-rank test would be used if you had replicates (repeated) measurements between different time points/plates/... since it is the same sample but measured in different time/on different plates.
Wilcoxon signed-rank test is defined in R through wilcox.test function (with paired=TRUE) which uses [dprq]signrank functions.
Another implementation of MWW/Wilcoxon signed-rank test can be found in the coin package through wilcox_test function.