wmwTest: Wilcoxon-Mann-Whitney test with Confidence Interval on...

In asht: Applied Statistical Hypothesis Tests

Wilcoxon-Mann-Whitney test with Confidence Interval on Mann-Whitney Parameter

Description

The wmwTest function calculates the Wilcoxon-Mann-Whitney test (normal approximation, exact complete enumeration, and exact Mante Carlo implementation) together with confidence intervals on the Mann-Whitney parameter, Pr[ X<Y] + 0.5 Pr[X=Y].

Usage

wmwTest(x, ...)

## Default S3 method:
wmwTest(x, y, alternative = c("two.sided", "less", "greater"), 
   phiNull = 0.5, exact = NULL, correct = TRUE, conf.int = TRUE, conf.level = 0.95, 
   latentContinuous = FALSE, method = NULL, methodRule = methodRuleWMW, 
   tsmethod = c("central", "abs"), control = wmwControl(),...)

## S3 method for class 'formula'
wmwTest(formula, data, subset, na.action, ...)

## S3 method for class 'matrix'
wmwTest(x,...)

Arguments

x	a (non-empty) numeric vector of data values from group 1, or a contingency table matrix with 2 rows with the top row representing group 1 and the bottom row group 2
y	an optional (non-empty) numeric vector of data values from group 2
alternative	a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less". You can specify just the initial letter.
phiNull	null hypothesis value for the Mann-Whitney parameter, Pr[X<Y]+0.5*Pr[X=Y]. Defaults to 0.5.
exact	logical, should exact test be calculated? (see method)
correct	a logical indicating whether to apply continuity correction in the normal approximation for the p-value and confidence interval (when method='asymptotic')
conf.int	logical, should confidence intervals be calculated?
conf.level	confidence level of the interval.
latentContinuous	logical, should estimates and confidence intervals be presented as latent continuous parameters? (see details)
method	character defining method, one of 'asymptotic', 'exact.ce' (exact by complete enumeration), 'exact.mc' (exact by Monte Carlo approximation). NULL value defaults to result of methodRule
methodRule	function that inputs x,y, and exact and outputs a method, see methodRuleWMW
tsmethod	two-sided method, either 'central' (double the one-sided p-values) or 'abs' (test statistic uses absolute value of difference in phi estimate and phiNull)
control	a list of arguments for control of algorithms and output, see wmwControl
formula	a formula of the form lhs ~ rhs where lhs is a numeric variable giving the data values and rhs a factor with two levels giving the corresponding groups.
data	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).
subset	an optional vector specifying a subset of observations to be used.
na.action	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.

Details

The function wmwTest evaluates the Wilcoxon-Mann-Whitney test (also called the Mann-Whitney U test or the Wilcoxon rank sum test). The WMW test is a permutation two-sample rank test, and the test may be evaluated under many different sets of assumptions (Fay and Proschan, 2010). The least restrictive set of assumptions tests the null hypothesis that the two distributions of the two samples are equal versus the alternative that they are different. Unfortunately, with only those assumptions, we cannot get confidence intervals on the Mann-Whitney parameter, phi= Pr[ X<Y] + 0.5 Pr[X=Y]. In order to get confidence intervals on phi, we need additional assumptions, and for this function we use the proportional odds assumption. This assumption can be interpreted as saying that there exists some unknown monotonic transformation of the responses that leads to a location shift in a logistic distribution. This can work for discrete data (i.e., with ties allowed) if we interpret discrete responses as a grouping of some underlying latent continuous response. The proportional odds assumption is less restrictive that the assumption used in wilcox.test, which assumes a location shift on the unknown continuous distribution of the untransformed data.

In summary, the two-sided p-value can be interpreted as testing the null that the two distributions are equal, and the confidence intervals on the Mann-Whitney parameter are intrepreted under the proportional odds assumption. In general the confidence intervals are compatible with the associated p-values, for details see Fay and Malinovsky (2018).

There is a choice of three methods. When method='asymptotic', the test is implemented using a normal approximation, with (correct=TRUE) or without (correct=FALSE) a continuity correction. The resulting p-values should match wilcox.test (when paired=FALSE and exact=FALSE). When method='exact.ce', the test is implemented using complete enumeration of all permutations, and hence is only tractible for very small sample sizes (less than 10 in each group). When method='exact.mc', the test is implemented using Monte Carlo with B=10^4 replications (change B with control=controlWMW(nMC=B)). As B gets larger the p-value approaches the exact one. (See 'note' section, sometimes the method='exact.mc' will not work.)

The tsmethod='central' gives two-sided p-value that is equal to min(1,min(2*pless,2*pgreater)). Alternatively, tsmethod='abs' gives the two-sided method, which is based on the test statistic |phi - phiNull|. Under the proportional odds assumption, tsmethod='central' allows us to interpret p.value/2 as one-sided p-values (this is not allowed using tsmethod='abs'). With continuous data, the p-values will be the same, but with ties they can be different.

From the two groups x (or top row of contringency table, or first factor in rhs of formula) and y (or bottom row of contingency table, or second factor in rhs of formula) the Mann-Whitney parameter represents Pr[X<Y]+0.5Pr[X=Y]. It is also the area under the curve of an ROC curve (see Hanley and McNeil, 1982). The confidence interval when method='asymptotic' generalizes the Method 5 of Newcombe (2006), which was a score-type modification of the interval of Hanley and McNeil (1982). The generalization is that the confidence interval adjusts for ties and allows a continuity correction (see examples below).

The methodRule function allows automatic choice of the method of calculation based on the data and the exact argument.

When the data are discrete, we can treat the data as if they are a grouping of some underlying continuous responses. Using the proportional odds assumption, we can then translate the Mann-Whitney parameter on the observed discrete data into the Mann-Whitney parameter on the latent continuous data (when latentContinuous=TRUE and using the default control=controlWMW(latentOutput='mw')). You can also translate the results into the proportional odds parameter on the latent continuous responses (when latentContinuous=TRUE and using control=controlWMW(latentOutput='po')). Translation is done with latentTransform.

Value

A list with class "htest" containing the following components:

statistic	U statistic estimate of the Mann-Whitney parameter.
parameter	tie factor
p.value	the p-value for the test.
conf.int	a confidence interval for the Mann-Whitney parameter appropriate to the specified alternative hypothesis.
estimate	the estimated difference in means
null.value	the specified hypothesized value of the mean difference
alternative	a character string describing the alternative hypothesis.
method	a character string describing the test.
data.name	a character string giving the name(s) of the data.

Warning

The algorithm for calculating the confidence interval when tsmethod='abs' is not guaranteed to give the correct value. It is possible to skip over a value. For more accurate results increase control=wmwControl(rcheckgrid) and control=wmwControl(ncheckgrid)

Note

The method='exact.mc' can sometimes fail. The issue is that for some Monte Carlo simulations the one-sided p-value function is not monotonic, even in for data sets where the one-sided p-value would be monotonic if we could do complete enumeration. In this case, the confidence limit will be set to NA and a warning will suggest trying method='asymptotic' or method='exact.ce' if feasible. Here is an example where that occurs: set.seed(1); g<- c(rep(0,6),1,rep(0,4),1,rep(0,3),1,1,0,1,1,0,rep(1,5)); y<-1:26; wmwTest(y~g,exact=TRUE).

References

Fay, MP and Malinovsky, Y (2018). Confidence Intervals of the Mann-Whitney Parameter that are Compatible with the Wilcoxon-Mann-Whitney Test. Statistics in Medicine: DOI: 10.1002/sim.7890.

Fay, MP and Proschan MA (2010). Wilcoxon-Mann-Whitney of t-test? On assumptions for hypothesis tests and multiple interpretations of decision rules. Statistics Surveys 4:1-39.

Hanley, JA, and McNeil, BJ (1982). The Meaning and Use of the Area under a Receiver Operating Characteristic (ROC) Curve. Radiology 143: 29-36.

Newcombe, Robert G. (2006). Confidence intervals for an effect size measure based on the Mann-Whitney statistic. Part 2: asymptotic methods and evaluation. Statistics in medicine 25(4): 559-573.

See Also

See wilcox.test for either exact p-value or the same asymptotic p-value and confidence interval on location shift under the shift assumption.

See wilcox_test for exact p-value and exact confidence interval on location shift.

Examples

# data from Table 1 of Hanley and McNeil (also given in Table 1 of Newcombe, 2006)
HMdata<-matrix(c(33,3,6,2,6,2,11,11,2,33),nrow=2,dimnames=
   list(c("Normal","Abnormal"),
   c("Definitely Normal",
   "Probably Normal",
   "Questionable",
   "Probably Abnormal",
   "Definitely Abnormal")))
HMdata
# to match Newcombe (2006, Table 1, Method 5) exactly 
# use correct=FALSE and RemoveTeAdjustment=TRUE
wmwTest(HMdata, correct=FALSE, RemoveTieAdjustment=TRUE)
# generally smaller intervals with closer to nominal coverage with 
# tie adjustment and continuity correction
wmwTest(HMdata)

asht documentation built on Aug. 24, 2023, 5:08 p.m.