help umbrella-------------------------------------------------------------------------------

Title

O'Brien's Umbrella Test

Syntax

umbrella[varlist] [if] [in] [,options]

optionsDescription -------------------------------------------------------------------------by(groupvar)is required. It specifies the name of the grouping variable.highlow(string)gives a string of space-delimited letters that are either H or L. There must be as many letters as there are dependent variables. If the k-th such letter is an H then higher values of the k-th dependent variable denote a better outcome than lower values; if is is an L then the converse is true. When this option is omitted higher values of all of the response variables are assumed to be better than lower values.ranktablegenerates a table of the ranks of each dependent variable grouped by groupvar. The sum of ranks accross the dependent variables is also given. By default, this table is omitted.id(idvar)specifies the data set's identification variable. If the data set has a unique identifer and you wish this identifer to appear in the table of ranks then specify that variable here. Ifranktableis not specified then this option has no effect.

Description

umbrellaperforms O'Brien's (1984) Umbrella Test (also known as O'Brien's Multiple Endpoints Test). It is used to test whether a continuous response vector differs between groups of subjects and may be used as a non-parametric alternative to Hotelling's T test. It is particularly useful when each dependent variable denotes an outcome in which higher values are better than lower values or vice versa. See O'Brien (1984) for more information.

RemarksThis test reduces to a Kruskal-Wallace one way analysis of variance if there is a single response variable.

This program gives results that are identical to those of the S+ program, umbrella.s, that is posted at http://mayoresearch.mayo.edu/mayo/research/biostat/splusfunctions.cfm . umbrella.s was written by Susan Kunselman.

A related program is obrien, by Richard Goldstein (sg43 from http://www.stata.com/stb/stb28). obrien provides extensions to the t and ranksum tests described in O'Brien (1988).

MethodsRank each dependent variable (ignoring the grouping variable). For dependent variables where lower values are better than higher values invert the associated ranks. For each patient, sum these ranks accross all dependent variables. Perform a one-way analysis of variance on the sums of these ranks.

In this program we do this analysis of variance using a Kruskal-Wallace test.

AuthorsWilliam D. Dupont W. Dale Plummer, Jr. Department of Biostatistics Vanderbilt University School of Medicine

e-mail: william.dupont@vanderbilt.edu dale.plummer@vanderbilt.edu

ReferenceO'Brien, P.C. Procedures for comparing samples with multiple endpoints. Biometrics 1984; 40: 1079-1087. O'Brien, P.C. Comparing two samples: extensions of the t, rank-sum, and log-rank tests. J Am Stat Assoc 1988; 83: 52-61.

Key WordsO'Brien, Peter C. umbrella multiple endpoints multivariate analysis of variance Hotelling's T test

Example. * Test whether the vector of mpg, weight and length differs between . * foreign and domestic cars in the auto data set. Low values of mpg and . * high values or weight and length are best. . *

. sysuse auto (1978 Automobile Data)

. umbrella mpg weight length, by(foreign) highlow(L H H)

Number of response variables: 3

Variable | Outcome -------------+----------------------------------------------------- mpg | lower values are better weight | higher values are better length | higher values are better

------------------------------------------------------------------------------- -> foreign = Domestic

Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- mpg | 52 19.82692 4.743297 12 34 weight | 52 3317.115 695.3637 1800 4840 length | 52 196.1346 20.04605 147 233

------------------------------------------------------------------------------- -> foreign = Foreign

Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- mpg | 22 24.77273 6.611187 14 41 weight | 22 2315.909 433.0035 1760 3420 length | 22 168.5455 13.68255 142 193

Missing observations dropped from analysis = 0

O'Brien's Umbrella test is the following Kruskal-Wallis test on the sum of the ranks across the dependent variables.

Kruskal-Wallis equality-of-populations rank test

+---------------------------+ | foreign | Obs | Rank Sum | |----------+-----+----------| | Domestic | 52 | 2336.00 | | Foreign | 22 | 439.00 | +---------------------------+

chi-squared = 20.839 with 1 d.f. probability = 0.0001

chi-squared with ties = 20.841 with 1 d.f. probability = 0.0001