Dunn.test - interpretation p-value and Ho rejection I am using the Dunn test (the package Dunn.test in R) as a post-hoc test to my Kruskal Wallis analysis. My alpha value is 0.05, so I would reject the Ho for p - values <= 0.05. 
When printing my results, R prints the note: 

alpha = 0.05
  Reject Ho if p <= alpha/2

Does anybody happen to know why R advises me that I should reject Ho if the p-value is 0.025 even at a set alpha value of 0.05? (For clarity: why is R using this particular decision rule of alpha/2 instead of just alpha?)
My guess would be that it has something to do with testing one- or two sided. I couldn't find if the default of Dunn.test and therewith the given results is for a one or two sided approach. 
 A: With dunn.test you have an argument altp which sets how the p-value will be expressed. If in function call you set altp=TRUE, then the p values will be expressed in alternative format. 
Test default is to express p-value = P(Z ≥ |z|), and reject Ho if p ≤ α/2. So what you describe sounds like normal behaviour. You still can change it - if the altp option is used, p-values are instead expressed as p-value = P(|Z| ≥ |z|), and Ho is rejected if p ≤ α. 
As it´s said in documentation, both expressions should anyway give identical test results so the use of altp is therefore merely a semantic choice".
A: If you read the original O. J. Dunn (1964) paper, decisions in the test are made using a threshold based on α / 2.  My understanding is that the author of the dunn.test package wrote the dunn.test function to be faithful to the test as it was designed in the original paper.  
I find reporting the p values as relative to an α / 2 threshold to be confusing for most users, but as the OP points out, the function output tells you what decision rule to use.  And as @Oka points out, the function includes an option to report results relative to an α threshold.
Therefore, if the altp = TRUE option is used, the results will match those from the dunnTest function in the FSA package.
if(!require(dunn.test)){install.packages("dunn.test")}
if(!require(FSA)){install.packages("FSA")}

set.seed(sum(utf8ToInt("Endofconfusion")))

A = rnorm(20, 10, 3)
Y = c(A, A+2, A+4)
G = factor(rep(c("A", "B", "C"), 1, each = length(Y)/3))

plot(Y ~ G)

library(FSA)
dunnTest(Y ~ G, method="holm")

   ### Dunn (1964) Kruskal-Wallis multiple comparison
   ### p-values adjusted with the Holm method.
   ###
   ###   Comparison         Z     P.unadj       P.adj
   ### 1      A - B -1.602483 0.109048909 0.218097818
   ### 2      A - C -3.204965 0.001350787 0.004052361
   ### 3      B - C -1.602483 0.109048909 0.109048909

library(dunn.test)
dunn.test(Y, G, method="holm", altp=TRUE)

   ### Comparison of Y by G                              
   ### (Holm)
   ###
   ### Col Mean-|
   ### Row Mean |          A          B
   ### ---------+----------------------
   ###        B |  -1.602482
   ###          |     0.2181
   ###          |
   ###        C |  -3.204965  -1.602482
   ###          |    0.0041*     0.1090
   ### 
   ### alpha = 0.05
   ### Reject Ho if p <= alpha

A: The proper decision rule is to reject Ho when the pvalue does not exceed alpha, so when alpha and p-value are equal, you still reject Ho. 
This is consistent with what R is telling you. 
