PMCMR (version 4.3)

posthoc.friedman.nemenyi.test: Pairwise post-hoc Test for Multiple Comparisons of Mean Rank Sums for Unreplicated Blocked Data (Nemenyi-Test)


Calculate pairwise comparisons using Nemenyi post-hoc test for unreplicated blocked data. This test is usually conducted post-hoc after significant results of the friedman.test. The statistics refer to upper quantiles of the studentized range distribution (Tukey).


posthoc.friedman.nemenyi.test(y, …)

# S3 method for default posthoc.friedman.nemenyi.test (y, groups, blocks, …)

# S3 method for formula posthoc.friedman.nemenyi.test (formula, data, subset, na.action, …)



either a numeric vector of data values, or a data matrix.


a vector giving the group for the corresponding elements of y if this is a vector; ignored if y is a matrix. If not a factor object, it is coerced to one.


a vector giving the block for the corresponding elements of y if this is a vector; ignored if y is a matrix. If not a factor object, it is coerced to one.


a formula of the form a ~ b | c, where a, b and c give the data values and corresponding groups and blocks, respectively.


an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).


an optional vector specifying a subset of observations to be used.


a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

further arguments to be passed to or from methods.


A list with class "PMCMR"


The applied method.

The name of the data.


The p-value according to the studentized range distribution.


The estimated upper quantile of the studentized range distribution.


Defaults to "none"

%% ...


A one-way ANOVA with repeated measures that is also referred to as ANOVA with unreplicated block design can also be conducted via the friedman.test. The consequent post-hoc pairwise multiple comparison test according to Nemenyi is conducted with this function.

If y is a matrix, than the columns refer to the treatment and the rows indicate the block.

See vignette("PMCMR") for details.

Let \(R_j\) and \(n_j\) denote the sum of Friedman-ranks and the sample size of the \(j\)-th group, respectively, then a difference between two groups is significant on the level of \(\alpha\), if the following inequality is met:

$$\left| \frac{R_i}{n_i} - \frac{R_j}{n_j} \right| > \frac{q_{\infty;k;\alpha}}{\sqrt{2}} \sqrt{\frac{k \left( k + 1 \right)}{6 n}}$$

with \(k\) the number of groups (or treatments) and \(n\) the total number of data.


Janez Demsar (2006), Statistical comparisons of classifiers over multiple data sets, Journal of Machine Learning Research, 7, 1-30.

P. Nemenyi (1963) Distribution-free Multiple Comparisons. Ph.D. thesis, Princeton University.

Lothar Sachs (1997), Angewandte Statistik. Berlin: Springer. Pages: 668-675.

See Also

friedman.test, kruskal.test, posthoc.kruskal.nemenyi.test, Tukey


Run this code
## Sachs, 1997, p. 675
## Six persons (block) received six different diuretics (A to F, treatment).
## The responses are the Na-concentration (mval)
## in the urine measured 2 hours after each treatment.
y <- matrix(c(
3.88, 5.64, 5.76, 4.25, 5.91, 4.33, 30.58, 30.14, 16.92,
23.19, 26.74, 10.91, 25.24, 33.52, 25.45, 18.85, 20.45, 
26.67, 4.44, 7.94, 4.04, 4.4, 4.23, 4.36, 29.41, 30.72,
32.92, 28.23, 23.35, 12, 38.87, 33.12, 39.15, 28.06, 38.23,
26.65),nrow=6, ncol=6, 
# }

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