Levene.Test {LaplacesDemon}

Levene's Test

Description

The Levene.Test function is a Bayesian form of Levene's test (Levene, 1960) of equality of variances.

Usage

Levene.Test(x, Method="U", G=NULL, Data=NULL)

Arguments

x

This required argument must be an object of class demonoid.ppc, iterquad.ppc, laplace.ppc, pmc.ppc, or vb.ppc.

Method

The method defaults to U for a univariate dependent variable (DV), y. When the DV is multivariate, Method="C" applies Levene's test to each column associated in Y. When Method="R", Levene's test is applied to each row associated in Y.

G

This argument defaults to NULL, or is required to have the same dimensions as the DV. For example, if the DV is univariate, then G must have a length equal to y, which is usually represented with a length of N for the number of records or T for the number of time-periods. If the DV is multivariate, then G must be a matrix, like Y, and have the same number of rows and columns. The purpose of the G argument is to allow the user to specify each element of y or Y to be in a particular group, so the variance of the groups can be tested. As such, each element of G must consist of an integer from one to the number of groups desired to be tested. The reason this test allows this degree of specificity is so the user can specify groups, such as according to covariate levels. By default, 4 groups are specified, where the first quarter of the records are group 1 and the last quarter of the records are group 4.

Data

This argument is required when the DV is multivariate, hence when Method="C" or Method="R". The DV is required to be named Y.

Details

This function is a Bayesian form of Levene's test. Levene's test is used to assess the probability of the equality of residual variances in different groups. When residual variance does not differ by group, it is often called homoscedastic (or homoskedastic) residual variance. Homoskedastic residual variance is a common assumption. An advantage of Levene's test to other tests of homoskedastic residual variance is that Levene's test does not require normality of the residuals.

The Levene.Test function estimates the test statistic, W, as per Levene's test. This Bayesian form, however, estimates W from the observed residuals as W^{obs}, and W from residuals that are replicated from a homoskedastic process as W^{rep}. Further, W^{obs} and W^{rep} are estimated for each posterior sample. Finally, the probability that the distribution of W^{obs} is greater than the distribution of W^{rep} is reported (see below).

Value

The Levene.Test function returns a plot (or for multivariate Y, a series of plots), and a vector with a length equal to the number of Levene's tests conducted.

One plot is produced per univariate application of Levene's test. Each plot shows the test statistic W, both from the observed process (W.obs as a black density) and the replicated process (W.rep as a red line). The mean of W.obs is reported, along with its 95% quantile-based probability interval (see p.interval), the probability p(W^{obs} > W^{rep}), and the indicated results, either homoskedastic or heteroskedastic.

Each element of the returned vector is the probability p(W^{obs} > W^{rep}). When the probability is p < 0.025 or p > 0.975, heteroskedastic variance is indicated. Otherwise, the variances of the groups are assumed not to differ effectively.

Author(s)

Statisticat, LLC. [e-mail]

References

Levene, H. (1960). "Robust Tests for Equality of Variances". In I. Olkins, S. G. Ghurye, W. Hoeffding, W. G. Madow, & H. B. Mann (Eds.), Contributions to Probability and Statistics, p. 278–292. Stanford University Press: Stanford, CA.

See Also

IterativeQuadrature, LaplaceApproximation, LaplacesDemon, PMC, p.interval, and VariationalBayes.

Examples

#First, update the model with IterativeQuadrature, LaplaceApproximation,
#  LaplacesDemon, PMC, or VariationalBayes.
#Then, use the predict function, creating, say, object Pred.
#Finally:
#Levene.Test(Pred)