The `Levene.Test`

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

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

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.

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.

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).

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.

`IterativeQuadrature`

,
`LaplaceApproximation`

,
`LaplacesDemon`

,
`PMC`

,
`p.interval`

, and
`VariationalBayes`

.

```
# NOT RUN {
#First, update the model with IterativeQuadrature, LaplaceApproximation,
# LaplacesDemon, PMC, or VariationalBayes.
#Then, use the predict function, creating, say, object Pred.
#Finally:
#Levene.Test(Pred)
# }
```

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