The function performs a test for a monotonic trend in variances. The test statistic
is based on a combination of the finite intersection approach and the classical Levene
procedure (using the group means), the modified Brown-Forsythe Levene-type procedure
(using the group medians) or the modified Levene-type procedure (using the group trimmed means).
More robust versions of the test using the correction factor or structural zero removal method
are also available. Two options for calculating critical values, namely, approximated
and bootstrapped, are available. By default, NAs from the data are omitted.
the default option is "median" corresponding to the robust Brown-Forsythe
Levene-type procedure; "mean" corresponds to the classical Levene's procedure,
and "trim.mean" corresponds to the robust Levene-type procedure using the group
trimmed means.
tail
the default option is "right", corresponding to an increasing trend in variances
as the one-sided alternatives; "left" corresponds to a decreasing trend in variances,
and "both" corresponds to any (increasing or decreasing) monotonic trend in variances
as the two-sided alternatves.
trim.alpha
the fraction (0 to 0.5) of observations to be trimmed from
each end of 'x' before the mean is computed.
bootstrap
the default option is FALSE, i.e., no bootstrap; if the option is set to
TRUE, the function performs the bootstrap method described in Lim and Loh (1996)
for Levene's test.
num.bootstrap
number of bootstrap samples to be drawn when bootstrap is set to TRUE;
the default value is 1000.
correction.method
procedures to make the ltrend test more robust; the default option is
"none"; "correction.factor" applies the correction factor described by O'Brien (1978) and
Keyes and Levy (1997); "zero.removal" performs the structural zero removal method by
Hines and Hines (2000); "zero.correction" performs a combination of O'Brien's correction factor
and the Hines-Hines structural zero removal method (Noguchi and Gel, 2009); note that the options
"zero.removal" and "zero.correction" are only applicable when the location is set to "median";
otherwise, "none" is applied.
correlation.method
measures of correlation; the default option is "pearson", the usual correlation
coefficient which is equivalent to the t-test; nonparametric measures of correlation such as
"kendall" (Kendall's tau) or "spearman" (Spearman's rho) may also be chosen, in which case, two
libraries, Hmisc and Kendall, are required.
Value
A list with the following vector components.
T
the statistic and p-value of the test based on the Tippett p-value combination.
F
the statistic and p-value of the test based on the Fisher p-value combination.
N
the statistic and p-value of the test based on the Liptak p-value combination.
L
the statistic and p-value of the test based on the Mudholkar-George p-value combination.
Each of the vector components contains the following numeric components.
statistic
the value of the test statistic expressed in terms of correlation (Pearson, Kendall, or Spearman).
p.value
the p-value of the test.
method
type of test performed.
data.name
a character string giving the name of the data.
non.bootstrap.statistic
the statistic of the test without bootstrap method.
non.bootstrap.p.value
the p-value of the test without bootstrap method.
Author(s)
Kimihiro Noguchi, W. Wallace Hui, Yulia R. Gel, Joseph L. Gastwirth, Weiwen Miao
References
Boos, D. D. and Brownie, C. (1989). Bootstrap methods for testing homogeneity
of variances. Technometrics 31, 69-82.
Brown, M. B. and Forsythe, A. B. (1974). Robust tests for equality of
variances. Journal of the American Statistical Association 69, 364-367.
Gastwirth, J. L., Gel, Y. R., and Miao, W. (2008). The Impact of Levene's Test of Equality
of Variances on Statistical Theory and Practice. Working paper,
Department of Statistics, George Washington University.
Hines, W. G. S. and Hines, R. J. O. (2000). Increased power with modified forms of the
Levene (med) test for heterogeneity of variance. Biometrics 56, 451-454.
Hui, W., Gel, Y. R., and Gastwirth, J. L. (2008). lawstat: an R package for law, public
policy and biostatistics. Journal of Statistical Software 28, Issue 3.
Keyes, T. K. and Levy, M. S. (1997). Analysis of Levenes test under design imbalance.
Journal of Educational and Behavioral Statistics 22, 845-858.
Levene, H. (1960). Robust Tests for Equality of Variances, in Contributions
to Probability and Statistics, ed. I. Olkin, Palo Alto, CA: Stanford Univ. Press.
Lim,T.-S., Loh, W.-Y. (1996) A comparison of tests of equality of variances
Computational Statistical & Data Analysis 22, 287-301.
Mudholkar, G. S., McDermott, M. P., and Mudholkar, A. (1995). Robust finite-intersection
tests for homogeneity of ordered variances.
Journal of Statistical Planning and Inference 43, 185-195.
Noguchi, K. and Gel, Y. R. (2009) Combination of Levene-type tests and a finite-intersection
method for testing equality of variances against ordered alternatives. Working paper,
Department of Statistics and Actuarial Science, University of Waterloo.
O'Brien, R. G. (1978). Robust techniques for testing heterogeneity of variance effects
in factorial designs. Psychometrika 43, 327-344.
library(Hmisc)
library(Kendall)
data(pot)
lnested.test(pot[,"obs"], pot[,"type"], location="median", tail="left",
correction.method="zero.correction")$N
## lnested test based on the modified Brown-Forsythe Levene-type procedure using the
## group medians with modified structural zero removal method and correction factor
## (left-tailed with Pearson correlation coefficient)
##
## data: pot[, "obs"]
## Test Statistic (N) = 4.905, p-value = 0.0002618
lnested.test(pot[,"obs"], pot[,"type"], location="median", tail="left",
correction.method="zero.correction",bootstrap=TRUE,num.bootstrap=500)$N
## bootstrap lnested test based on the modified Brown-Forsythe Levene-type procedure
## using the group medians with modified structural zero removal method and correction
## factor (left-tailed with Pearson correlation coefficient)
##
## data: pot[, "obs"]
## Test Statistic (N) = 4.9936, p-value = 0.000207