# What type of statistics test do I use for the following

I have three sets of means. For instance

mean.a = [3,2,7,3,8,2,5,7,3,2,5]
mean.b = [9,7,9,10,7,13,30,17,16,9,9]
mean.c = [23,24,25,23,22,22,21,9,25,34,12]


And for each mean, I have standard deviations.

sd.a = [.1,.18,.12,.24,.12,.16,.25,.1,.1,.23,.11]
sd.b = [.3,.13,.1,.24,.2,.1,.24,.11,.1,.23,.1]
sd.c = [.3,.1,.33,.21,.32,.35,.31,.3,.33,.42,.42]


I want to statistically show that a is less than b and c. How can I do this, and find a p value for this? Note that the values are paired. So, for instance, the 5th value of a should be less than the 5th value of b and 5th value of c.

It was suggested to me to use a Wilcoxon signed rank test to show, firstly, that c > b then that b > a. I found the R function wilcox.test(a, b, paired=T). However, I have means and SDs for both a and b, so how would I do this?

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Does the order of these points have meaning? Really I am asking if the data are naturally paired. If so the paired t test or more generally the Wilcoxon signed rank test can be applied to looking at differences of b with a or c with b. –  Michael Chernick Aug 25 '12 at 3:02
The order does have meaning yes. So ideally, b and c would be less at all positions than a. –  CodeGuy Aug 25 '12 at 4:05
Although it makes little difference here, because the differences between the sets are blatantly obvious, it is strange to have means reported to less precision than indicated by the standard deviations. This renders the SDs almost irrelevant. –  whuber Aug 25 '12 at 4:25
@codeguy: I think what Michael was asking is whether there is pairing. Is the (say) fifth value of group A matched to the fifth value of group B and C? Your answer seemed to be about the logical order of the three treatment groups, not about pairing or matching of the values in the groups. –  Harvey Motulsky Aug 25 '12 at 17:04
@Peter because the SD of the (rounded or truncated) data will be orders of magnitude greater than the reported SD of the data themselves. That implies replacing the reported SDs by zero will create only inconsequential changes. –  whuber Aug 25 '12 at 21:54

Given that the order is meaningful then a paired test such as the Wilcoxon signed rank test could be applied. It does not compare means. It is the original pair measurements that are differenced. Because of random variation it is not necessary for all paired differences for b-a to be positive for the test to be meaningful.

Since you want to show c>b and b>a you want to do two tests and should make a p-value adjustment for multiple testing.

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A minor point - I think he wants to show $c>a$ and $b>a$, not the full ordering, but your point still applies. –  jbowman Aug 25 '12 at 16:00
@jbowman Thanks. All I was suggesting was the two comparisons so the full ordering would require a third test and so I made the change you suggest. –  Michael Chernick Aug 25 '12 at 16:16
how do I do this given that I have means and standard deviations? –  CodeGuy Aug 25 '12 at 21:03
I found the R function wilcox.test(array1, array2, paired=TRUE), but I have means and SDs for each of array1 and array2. –  CodeGuy Aug 25 '12 at 21:15
@CodeGuy You can't. You need the raw data. If you want to do component-wise tests and are willing to assume normality (not sure it is a good assumption) you can do unpaired t tests. That would give 11 tests for each of the comparisons (c with b and b with a). That would be a total of 22 hypothesis tests. The multiplicity correction becomes very important now. Of course given the magnitude of the difference in means relative to their standard deviations the p-values should be very small. –  Michael Chernick Aug 25 '12 at 21:15