*! version 1.3.5 03may2018
help dunntest

Title

dunntest -- Dunn's test of multiple comparisons using rank sums

Syntax

dunntest varname [if] [in] , "by(varlist:groupvar)"
[ma(method) nokwallis nolabel wrap list rmc level(#) altp]

dunntest options

Main
by(varlist:groupvar) variable defining the k groups.
Missing observations in groupvar are ignored.

ma(method) which method to adjust for multiple comparisons

nokwallis suppress Kruskal-Wallis test output

nolabel display data values, rather than data value labels

wrap do not break wide tables

list include results of Dunn's test in a list format.

rmc report row - col rather than col - row

level(#) set confidence level; default is level(95)

altp use alternative expression of p-values

Missing observations in varname are ignored.

Description

dunntest reports the results of Dunn's test (1964) for
stochastic dominance among multiple pairwise comparisons following a
Kruskal-Wallis test of stochastic dominance among k groups
(Kruskal and Wallis, 1952) using kwallis.
The interpretation of stochastic dominance requires an assumption that the CDF
of one group does not cross the CDF of the other.
dunntest performs m = k(k-1)/2 multiple pairwise comparisons.
The null hypothesis in each pairwise comparison is that the probability of
observing a random value in the first group that is larger than a random value
in the second group equals one half; this null hypothesis corresponds to that of
the Wilcoxon-Mann-Whitney rank-sum test. Like the rank-sum test,
if the data can be assumed to be continuous, and the distributions are assumed
identical except for a shift in centrality, Dunn's test may be understood as a
test for median difference. In the syntax diagram above, varname refers to
the variable recording the outcome, and groupvar refers to
the variable denoting the population. dunntest accounts for tied ranks.
by() is required.

dunntest outputs both z test statistics for each pairwise comparison
(corresponding to the column mean minus the row mean, unless the rmc
option is used) and the p-value = P(Z>=|z|) for each, where
z is the difference of mean ranks divided by the population standard
error of that difference. Reject Ho based on p <= alpha/2 (and in combination with
p-value ordering for stepwise ma options). If you prefer to work with p-values
expressed as p = P(|Z| >=|z|) use the altp option, and reject Ho
based on p <= alpha (and in combination with p-value ordering for stepwise
ma options). These are exactly equivalent rejection decisions).

Options

"by(varlist:groupvar)" is required. It specifies a variable
that identifies the groups.

ma(method) is required. It specifies the method of adjustment used
for multiple comparisons, and must take one of the following values: none,
bonferroni, sidak, hochberg, hs, bh, or by.
none is the default method assumed if the ma option is omitted.
These methods perform as follows:

none specifies no adjustment for multiple comparisons be made. Those
comparisons rejected without adjustment at the alpha level (two-sided test) are
underlined in the output table, and starred in the list when using the list
option.

bonferroni specifies a "Bonferroni adjustment" where the
family-wise error rate (FWER) is adjusted by multiplying the p-values in
each pairwise test by m (the total number of pairwise tests) as per Dunn
(1961). dunntest will report a maximum
Bonferroni-adjusted p-value of 1. Those comparisons rejected with this
method at the alpha level (two-sided test) are underlined in the output table,
and starred in the list when using the list option.

sidak specifies a "Sidák adjustment" where the FWER is adjusted
by replacing the p-value of each pairwise test with 1 - (1 - p)^m
as per Sidák (1967). dunntest will report a maximum
Sidák-adjusted p-value of 1. Those comparisons rejected with this
method at the alpha level (two-sided test) are underlined in the output table,
and starred in the list when using the list option.

holm specifies a "Holm adjustment" where the FWER is adjusted
sequentially by adjusting the p-values of each pairwise test as
ordered from smallest to largest with p(m+1-i), where i is the
position in the ordering as per Holm (1979). dunntest will
report a maximum Holm-adjusted p-value of 1. Because in sequential tests
the decision to reject the null hypothesis depends both on the p-values and
their ordering, those comparisons rejected with this method at the alpha level
(two-sided test) are underlined in the output table, and starred in the list
when using the list option.

hs specifies a "Holm-Sidák adjustment" where the FWER is
adjusted sequentially by adjusting the p-values of each pairwise test as
ordered from smallest to largest with 1 - (1 - p)^(m+1-i), where
i is the position in the ordering (see Holm, 1979).
dunntest will report a maximum Holm-Sidák-adjusted p-value of 1. Because
in sequential tests the decision to reject the null hypothesis depends both on
the p-values and their ordering, those comparisons rejected with this
method at the alpha level (two-sided test) are underlined in the output table,
and starred in the list when using the list option.

hochberg specifies a "Hochberg adjustment" where the FWER is adjusted
sequentially by adjusting the p-values of each pairwise test as
ordered from largest to smallest with p*i, where i is the
position in the ordering as per Hochberg (1988).
dunntest will report a maximum Hochberg-adjusted p-value of 1. Because in
sequential tests the decision to reject the null hypothesis depends both on the
p-values and their ordering, those comparisons rejected with this method at
the alpha level (two-sided test) are underlined in the output table, and starred
in the list when using the list option.

bh specifies a "Benjamini-Hochberg adjustment" where the false discovery rate
(FDR) is adjusted sequentially by adjusting the p-values of
each pairwise test as ordered from largest to smallest with p[m/(m+1-i)],
where i is the position in the ordering (see Benjamini & Hochberg, 1995).
dunntest will report a maximum Benjamini-Hochberg-adjusted p-value of 1. Such
FDR-adjusted p-values are sometimes refered to as q-values in the
literature. Because in sequential tests the decision to reject the null
hypothesis depends both on the p-values and their ordering, those
comparisons rejected with this method at the alpha level (two-sided test) are
underlined in the output table, and starred in the list when using the list
option.

by specifies a "Benjamini-Yekutieli adjustment" where the false
discovery rate (FDR) is adjusted sequentially by adjusting the p-values of
each pairwise test as ordered from largest to smallest with p[m/(m+1-i)]C,
where i is the position in the ordering, and C = 1 + 1/2 + ... + 1/m
(see Benjamini & Yekutieli, 2001). dunntest will
report a maximum Benjamini-Yekutieli-adjusted p-value of 1. Such
FDR-adjusted p-values are sometimes refered to as q-values in the
literature. Because in sequential tests the decision to reject the null
hypothesis depends both on the p-values and their ordering, those
comparisons rejected with this method at the alpha level (two-sided test) are
underlined in the output table, and starred in the list when using the list
option.

nokwallis suppresses the display of the Kruskal-Wallis test table.

nolabel causes the actual data codes to be displayed rather than the
value labels in the Dunn's test tables.

wrap requests that dunntest not break up wide tables to make
them readable.

list requests that dunntest also provide a output in list form,
one pairwise test per line.

rmc requests that dunntest reports z statistic based on the mean rank
of the row variable minus the mean rank of the column variable. The default is
to report the mean rank of the column variable minus the mean rank of the row
variable. The difference between these two is simply the sign of the z
statistic.

level(#) specifies the compliment of alpha*100. The default,
level(95) (or as set by set level) corresponds to alpha = 0.05.

altp directs dunntest to express p-values in alternative
format. The default is to express p = P(Z >= |z|), and reject Ho if p >= alpha\2. When
the altp option is used, p-values are instead expressed as p = P(|Z| >= |z|),
and reject Ho if p <= alpha. These two expressions give identical
test results. Use of altp is therefore merely a semantic choice.

Example

Setup

. webuse census

Test for equal median age by region type simultaneously

. kwallis medage, by(region)

Dunn's multiple-comparison test for stochastic dominance using a
Bonferroni correction

. dunntest medage, by(region) ma(bonferroni) nokwallis

Saved results

dunntest saves the following in r():

Scalars

r(df) degrees of freedom for the Kruskal-Wallis test

r(chi2_adj) chi-squared adjusted for ties for the Kruskal-Wallis test

Matrices

r(Z) vector of Dunn's z test statistics

r(P) vector of adjusted p-values for Dunn's z test statistics --OR--

r(altP) vector of adjusted p-values for Dunn's z test statistics when using the altp option

Author

Alexis Dinno

Portland State University

alexis.dinno@pdx.edu

Please contact me with any questions, bug reports or suggestions for
improvement. Fixing bugs will be facilitated by sending along:

(1) a copy of the data (de-labeled or anonymized is fine),

(2) a copy of the command used, and

(3) a copy of the exact output of the command.

Suggested citation

Dinno A. 2014. dunntest: Dunn's test of multiple comparisons using rank
sums. Stata software package. URL:

References

Benjamini, Y. and Hochberg, Y. 1995. Controlling the false discovery rate: A practical and powerful approach to multiple testing.
Journal of the Royal Statistical Society. Series B (Methodological). 57: 289-300.

Benjamini, Y. and Yekutieli, D. 2001. The control of the false discovery rate in multiple testing under dependency.
Annals of Statistics. 29: 1165-1188.

Dunn, O. J. 1961. Multiple comparisons among means.
Journal of the American Statistical Association. 56: 52-64.

Dunn, O. J. 1964. Multiple comparisons using rank sums. Technometrics.
6: 241-252.

Hochberg, Y. 1988. A sharper Bonferroni procedure for multiple tests of significance. Biometrika. 75: 800-802.

Holm, S. 1979. A simple sequentially rejective multiple test procedure.
Scandinavian Journal of Statistics. 6: 65-70.

Kruskal, W. H. and Wallis, A. 1952. Use of ranks in one-criterion variance analysis.
Journal of the American Statistical Association. 47: 583-621.

Sidák, Z. 1967. Rectangular confidence regions for the means of multivariate normal distributions.
Journal of the American Statistical Association. 62: 626-633.

Also See

Help: kwallis, ranksum, conovertest