*! version 1.1.5 03may2018
help conovertest

Title

conovertest -- Conover-Iman test of multiple comparisons using rank sums

Syntax

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

conovertest 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 the Conover-Iman 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

conovertest reports the results of the Conover-Iman test (Conover & Iman, 1979; Conover, 1999)
for stochastic dominance among multiple pairwise comparisons following
rejection of a Kruskal-Wallis test for stochastic dominance among k groups
(Kruskal and Wallis, 1952) using kwallis.
The Conover-Iman test is akin to Dunn's test (1964)
but is based on the t distribution and is derived from the Kruskal-Wallis,
test statistic, rather than the z distribution as Dunn's test statistic is,
and can provide much greater statistical power than Dunn's test. The
interpretation of stochastic dominance requires an assumption that the CDF of
one group does not cross the CDF of the other. conovertest 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, the Conover-Iman 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. conovertest accounts for tied ranks. by() is required.

conovertest outputs both t 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(T>|t|) for each. 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(|T| >=|t|)
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).

conovertest reports both t test statistics for each pairwise comparison
and the p-value = P(T>|t|) for each.

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.

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). conovertest 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 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). conovertest will report
a maximum Sidák-adjusted p-value of 1.

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). conovertest 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).
conovertest 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).
conovertest 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).
conovertest 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). conovertest 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 Conover-Iman test tables.

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

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

rmc requests that conovertest reports t 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 t 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 conovertest to express p-values in alternative
format. The default is to express p = P(T >= |t|), and reject Ho if p >= alpha\2. When
the altp option is used, p-values are instead expressed as p = P(|T| >= |t|),
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)

Conover-Iman multiple-comparison test for stochastic dominance using a
Bonferroni correction

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

Saved results

conovertest 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 Conover-Iman t test statistics

r(P) vector of adjusted p-values for Conover-Iman t test statistics, --OR--

r(altP) vector of adjusted p-values for Conover-Iman t 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. 2017. conovertest: Conover-Iman 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.

Conover, W. J. and Iman, R. L. 1979. On multiple-comparisons procedures.
Technical Report LA-7677-MS, Los Alamos Scientific Laboratory.

Conover, W. J. 1999. Practical Nonparametric Statistics. Wiley, Hoboken,
NJ, 3rd edition.

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, dunntest