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I have two independent samples that are non-normally distributed and have unequal variances.

I want to do a very simple two-tailed test of equality like this:

H0: The two populations are equal versus

H1: The two populations are not equal

Is a WMW appropriate to use in this circumstance? I have read this message board and the general consensus is that it is, so long as it is the stochastic equality between the two that is being compared and not any measure of central tendency. But this is where I get confused, because I am only a high schooler, and I do not really know how to write a statement of results for stochastic equality.

Wikipedia gives a sample statement of results for a WMW test as:

"Median latencies in groups E and C were 153 and 247 ms; the distributions in the two groups differed significantly (Mann–Whitney U = 10.5, n1 = n2 = 8, P < 0.05 two-tailed)."

But is this an appropriate way for me to state my results for my teacher since I can only look at my results from a stochastic equality perspective? If it is not, can someone in very simple and plain language provide me with an example of how I should state my results? Thank you, I am only just learning statistics and any help would be appreciated.

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2 Answers 2

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Let me show results from the Mann-Whitney-Wilcoxon Signed rank test for some simulated data in R. First, the data meet your specifications because they are sampled from gamma distributions, which are right skewed and so not normal. The respective population means are $\mu_1 = 10,\, \mu_2 = 20.$ Also, the population variance from which the first sample was drawn is less than the population variance for the second sample.

Simulated Data and 'Location'. In a real life situation we would not have all of this information about the populations. The point here is to see how well the samples (both of size 20) reflect the populations from which they were drawn. In particular we want to see if a 2-sample Mann-Whitney-Wilcoxon test can detect that the location of the second population is located above the first.

set.seed(2020)
x1 = rgamma(20, 3, .3)
x2 = rgamma(20, 4, .2)
summary(x1); sd(x1)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  2.401   6.846  10.218  10.335  13.861  24.977 
[1] 5.370183  # sample SD of x1
summary(x2); sd(x2)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  8.202  10.945  16.474  17.362  22.043  34.085 
[1] 7.15061   # sample SD of x2

Location shift. Notice that the 1st quartile, median, mean, and 3rd quartiles for the first sample are all below the corresponding values for the second sample. There are many ways to express the 'location' of a sample (or distribution). The median is among them, but certainly not the only one. (Also, the maximum value in the first sample lies below the maximum value in the second sample, but in these right-skewed distributions maximum values are rather unpredictable.)

Thus, if one does not want to use the median as the specific measure of location, there are other ways to discuss location. A general term for expressing that two samples (or populations) do not have the same location, is that there is a "location shift" of one from the other. You should discuss this with your instructor, but I think location shift might be the best terminology for you to use.

Boxplots (each based on quartiles, median, min and max) show that the sample x1 tends to be 'located' below the sample x2.

boxplot(x1, x2) 

enter image description here

Mann-Whitney-Wilcoxon Test. The M-W-W test does show a significant location shift (P-value about 0.002). Notice the terminology "location shift" in the R output for this procedure.

wilcox.test(x1, x2)

        Wilcoxon rank sum test

data:  x1 and x2
W = 87, p-value = 0.001767
alternative hypothesis: 
   true location shift is not equal to 0

ECDFs and 'Stochastic Dominance'. Another technical term for saying that one sample is located above another is "stochastic dominance/" If you google around the internet you will see that there are several senses (sometimes called 'orders') of stochastic dominance. A lot of this may be above your level, but on some pages you'll find some simple gambling examples and see illustrative graphs. You might use the terminology "stochastic dominance," if you're prepared to explain it.

ECDFs. One of the simpler kinds of stochastic dominance has to do with ECDF plots. Especially for large samples, the empirical CDF (ECDF) of a sample imitates the CDF of the population from which the sample was chosen. To make an ECDF of a sample begin by sorting its values from smallest to largest. The ECDF is a stairstep function. With a sample of size $n$ it jumps up by distance $1/n$ at each sorted data value. (If $k$ observations are tiec at a value, then the jump there is $k/n.)$ So the ECDF starts at $0$ on the left and rises to $1$ at the right. One sample "dominates" another if its ECDF is to the right of the ECDF of the other.

The left panel below shows the CDFs of the two populations from which our samples were randomly chosen: blue for the first sample and maroon for the second. At right (same colors) are the ECDF for the samples. [R code below the graph.]

enter image description here

par(mfrow=c(1,2))
 curve(pgamma(x, 3, .3), 0, 40, col="blue", lwd=2, ylab="CDF", 
       main="Population CDFs")
   curve(pgamma(x, 4, .2), add=T, col="maroon", lwd=2)
 plot(ecdf(x1), col="blue", main="Sample ECDFs")
   lines(ecdf(x2), col="maroon")
par(mfrow=c(1,1))
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    $\begingroup$ Thank you Bruce, this is a very detailed answer that has really helped me to conceptualize what a location shift is and what it means in terms that I can understand. $\endgroup$
    – birdwhen05
    Commented Apr 29, 2020 at 15:15
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The wikipedia statement reports all you need here. To answer your question: Just go with that. It's fairly standard language to report your results like that.

But for I'd recommend reading BruceET's answer for a more in depth explanation of what the Mann-Whitney test actually does and how it works. It's important to understand why you're using the test that you're using, and if this is your main result, the box plot is useful to visualise the difference.

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