Why does Dunn's post hoc test & CDL output shows different CLD for two treatments with the same mean?

Question

I've been working with a Kruskal wallis non-parametric test in R and following it up with Dunn's multiple comparison's test. Looking at the results of glyphosate H&S(abc) comparisons with others it is showing as not significantly different from any other treatment. However, triclopyr H&S(bc) shows significantly different from Imazapyr H&S(a), Hexazinone Soil(a), and the untreated control. Triclopyr and Glyphosate have the same mean with triclopyr having a greater SE. Is this difference in multiple comparisons results and subsequent CLD from the differences in median and SE, or is there something wrong?

Kt<-kruskal.test(mortality~treatment, data=callerydata)
> Kt

    Kruskal-Wallis rank sum test

data:  mortality by treatment
Kruskal-Wallis chi-squared = 40.897, df = 7, p-value = 8.473e-07


> library(dunn.test)
> #compact letter display
> library(rcompanion)
> ##https://rcompanion.org/handbook/F_08.html
> CLD1<-DT1$res
> CLD1
                           Comparison           Z      P.unadj        P.adj
1             Control - Glyphosate HS -2.05136182 4.023172e-02 0.0804634485
2          Control - Glyphosate Stump -4.21069005 2.545918e-05 0.0002376191
3    Glyphosate HS - Glyphosate Stump -2.15932823 3.082471e-02 0.0663916861
4                Control - Hexazinone -0.87452793 3.818308e-01 0.4648375135
5          Glyphosate HS - Hexazinone  1.17683389 2.392618e-01 0.3349665403
6       Glyphosate Stump - Hexazinone  3.33616212 8.494361e-04 0.0039640353
7               Control - Imazapyr HS -0.88532458 3.759816e-01 0.4785220354
8         Glyphosate HS - Imazapyr HS  1.16603725 2.435994e-01 0.3247991829
9      Glyphosate Stump - Imazapyr HS  3.32536548 8.830270e-04 0.0027471951
10           Hexazinone - Imazapyr HS -0.01079664 9.913857e-01 1.0000000000
11           Control - Imazapyr Stump -4.21069005 2.545918e-05 0.0003564286
12     Glyphosate HS - Imazapyr Stump -2.15932823 3.082471e-02 0.0719243266
13  Glyphosate Stump - Imazapyr Stump  0.00000000 1.000000e+00 1.0000000000
14        Hexazinone - Imazapyr Stump -3.33616212 8.494361e-04 0.0047568424
15       Imazapyr HS - Imazapyr Stump -3.32536548 8.830270e-04 0.0030905945
16             Control - Triclopyr HS -2.73155021 6.303713e-03 0.0176503976
17       Glyphosate HS - Triclopyr HS -0.68018839 4.963852e-01 0.5791160442
18    Glyphosate Stump - Triclopyr HS  1.47913984 1.391029e-01 0.2049938130
19          Hexazinone - Triclopyr HS -1.85702228 6.330798e-02 0.1181749043
20         Imazapyr HS - Triclopyr HS -1.84622564 6.485945e-02 0.1135040371
21      Imazapyr Stump - Triclopyr HS  1.47913984 1.391029e-01 0.2163823582
22          Control - Triclopyr Stump -4.21069005 2.545918e-05 0.0007128572
23    Glyphosate HS - Triclopyr Stump -2.15932823 3.082471e-02 0.0784629017
24 Glyphosate Stump - Triclopyr Stump  0.00000000 1.000000e+00 1.0000000000
25       Hexazinone - Triclopyr Stump -3.33616212 8.494361e-04 0.0059460530
26      Imazapyr HS - Triclopyr Stump -3.32536548 8.830270e-04 0.0035321080
27   Imazapyr Stump - Triclopyr Stump  0.00000000 1.000000e+00 1.0000000000
28     Triclopyr HS - Triclopyr Stump -1.47913984 1.391029e-01 0.2291107322
> cldList(P.adj~Comparison, data=CLD1, threshold = 0.05)
            Group Letter MonoLetter
1         Control      a        a  
2    GlyphosateHS    abc        abc
3 GlyphosateStump      b         b 
4      Hexazinone     ac        a c
5      ImazapyrHS     ac        a c
6   ImazapyrStump      b         b 
7     TriclopyrHS     bc         bc
8  TriclopyrStump      b         b 

I've checked my data set to be certain the data has been input correctly. Graph attached is mean+/-SE % mortality one year post treatment with compact letter display from the output. Thanks for your thoughts in advance. I'm no means a statistician and always assume there's user error here.

I need to work on the box plot for visualizing. The significance takes up so much space that the y axis extends too much- we can only have 100% of trees die. I'm not sure if this visualization with box plot is intuitive. Perhaps better without the significance?

Answer 1

The K-W test is a nonparametric test that doesn't test means. The H0 hypothesis that it is testing is whether two samples come from the same distribution, not that the means of the samples are the same. If you did a parametric test for means, then you'd be right to be concerned. However, it's entirely possible for two samples to have the same mean but different distributions as detected by nonparametric tests. Your barplots aren't really helping to distinguish the difference between hypotheses either. Maybe try a box plot instead.