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I have a series of data (4 numerical variables) and 4 categorical variables (with multiple factors each). I need to test for differences between various subsets based on the categorical variables. All 4 numerical variables are nonparametric. In my example, I tested for differences of one numerical variable and one group of 8 factors using the Kruskal-Wallis test. This is the result (in R):

Kruskal-Wallis rank sum test

data:  limestone$Surfaceand limestone$Massif
Kruskal-Wallis chi-squared = 43.3924, df = 9, p-value = 1.826e-06

I followed with rejection of the test and performed the Duun's test (I got different sample sizes) with no adjustments for multiple comparisons. I got this result:

Col Mean-|
Row Mean |     Bucegi      Buila     Cernei    Fagaras     Hasmas     Parang   Piatra_C   Piatra_C      Piule
---------+---------------------------------------------------------------------------------------------------
   Buila |  -2.779234
         |     0.0027
         |
  Cernei |  -0.737853   2.695350
         |     0.2303     0.0035
         |
 Fagaras |  -3.264245  -1.507075  -3.085395
         |     0.0005     0.0659     0.0010
         |
  Hasmas |  -0.718388   1.763448  -0.143859   2.550436
         |     0.2363     0.0389     0.4428     0.0054
         |
  Parang |  -1.015773   0.131363  -0.733286   0.858709  -0.635328
         |     0.1549     0.4477     0.2317     0.1953     0.2626
         |
Piatra_C |   0.761830   1.985903   1.113509   2.558423   1.124042   1.343742
         |     0.2231     0.0235     0.1327     0.0053     0.1305     0.0895
         |
Piatra_C |   0.288611   3.805670   1.266603   3.814943   1.083942   1.176788  -0.654423
         |     0.3864     0.0001     0.1026     0.0001     0.1392     0.1196     0.2564
         |
   Piule |  -1.226132   1.672145  -0.675704   2.488801  -0.380582   0.472088  -1.347346  -1.777345
         |     0.1101     0.0472     0.2496     0.0064     0.3518     0.3184     0.0889     0.0378
         |
   Tarcu |  -3.840893  -1.895752  -3.833724  -0.011279  -2.971822  -0.904673  -2.682276  -4.667904  -3.020135
         |     0.0001     0.0290     0.0001     0.4955     0.0015     0.1828     0.0037     0.0000     0.0013

I also used the Bonferroni adjustment, but the results are very different (many comparisons are now non-significant, and some have p-value of 1), and I don't know what to believe. Do I always need to adjust with Bonferroni? I surfed some websites that said that this is not always a good option. I know I can use pairwise Wilcoxon tests, but I got too much data to compare, and it will take me a lot of time (and nerves), so I need to know if I can trust the Dunn's test with or without Bonferroni. Thanks for any help.

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  • $\begingroup$ "All 4 numerical variables are nonparametric." -- sorry, but there's no such thing as "nonparametric variables". I see many questions here that try to use this terminology and I doubt quite so many people could all conjure up essentially identical terminology from scratch -- if you can recall, where have you seen the term? Does it exist in a book somewhere? $\endgroup$ – Glen_b Jul 10 '15 at 1:21
  • $\begingroup$ They follow a non-Gaussian distribution. Sorry for my terminology, I have recently started studying statistics. $\endgroup$ – Litwos Jul 10 '15 at 7:44
  • $\begingroup$ (There's no need to apologize for the terminology - many people say exactly the same thing; I'm just very keen to find where it originates, if I can.) There's nothing that means you need to use parametric methods with Gaussian data, and there are many parametric methods suitable for non-Gaussian data. The issues you note above can happen with post hoc multiple comparisons using ANOVA as well. $\endgroup$ – Glen_b Jul 10 '15 at 9:42
  • $\begingroup$ And what would be the workaround here in my situation? How can I discern between the two situations I exemplified? $\endgroup$ – Litwos Jul 13 '15 at 10:48

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