Solved – What to do when a Mann-Whitney U assumption is violated

nonparametricspsswilcoxon-mann-whitney-test

I'm using a M-W U test to analyse some Likert scale results I have, as my data is ordinal.

One of the assumptions that keeps on appearing on references is the following: "both distributions must be the same shape (i.e., the distribution of scores for both categories of the independent variable must have the same shape).

If I fail this assumption, and therefore am unable to use M-W U, what can I use instead?

EDIT: I cannot add images, but will try to explain what my data looks like.

I have two groups based on my IV ("Yes" and "No"), charted on a boxplot with y-axis ranging from 1 – 5. For "Yes", I have a box extending from 4 to 5, with no variance. For "No", the box ranges from 3.5 to 4.5, with variance from 2.0 to 3.5, and 4.5 to 5.0.

I assume this means my shapes are "different". Does this mean I can't use MWU? If not, can you make suggestions as to what I can use instead (SPSS)?

EDIT2: Sample data:

"Yes": 5 5 5 5 5 5 4 4 4 4 5 5 4 4
"No":  4 5 5 5 3 2 4 3 4 4 4 4

I was using MWU to detect any significant differences (apparently, "yes") – but now am not sure if that's the right way to do it.

Best Answer

Here is an analysis using the R Hmisc and rms packages.

y1 <- c(5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 5, 5, 4, 4)
y0 <- c(4, 5, 5, 5, 3, 2, 4, 3, 4, 4, 4, 4)

y <- c(y0, y1)
x <- c(rep(0, length(y0)), rep(1, length(y1)))

require(rms)
r <- rcorr.cens(x, Surv(y))
r

   C Index            Dxy           S.D.              n        missing 
 0.6773399      0.3546798      0.1378018     26.0000000      0.0000000 
uncensored Relevant Pairs     Concordant      Uncertain 
26.0000000    406.0000000    275.0000000      0.0000000 

z <- r['Dxy']/r['S.D.']
z   # 2.57


orm(y ~ x)

Logistic (Proportional Odds) Ordinal Regression Model

orm(formula = y ~ x)

Frequencies of Responses

 2  3  4  5 
 1  2 12 11 

                     Model Likelihood          Discrimination          Rank Discrim.    
                        Ratio Test                 Indexes                Indexes       
Obs            26    LR chi2      4.58    R2                  0.184    rho     0.407    
Unique Y        4    d.f.            1    g                   0.885                     
Median Y        4    Pr(> chi2) 0.0323    gr                  2.424                     
max |deriv| 2e-04    Score chi2   4.43    |Pr(Y>=median)-0.5| 0.381                     
                     Pr(> chi2) 0.0354                                                  

     Coef    S.E.   Wald Z Pr(>|Z|)
y>=3  2.6045 1.0536  2.47  0.0134  
y>=4  1.3459 0.6854  1.96  0.0496  
y>=5 -1.3459 0.6854 -1.96  0.0496  
x     1.7126 0.8426  2.03  0.0421

The likelihood ratio $\chi^{2}_{1}$ test for comparing the two groups yields $P=0.0323$

I obtained an exact $P$-value from the exactRanktests package wilcox.exact function of 0.04904.

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