visStatistics (version 0.3.0)

levene.test: Mean-centred Levene Test for Homogeneity of Variance

Description

Performs Levene's original mean-centred test. It tests the null hypothesis that all groups have equal variances by testing whether the absolute deviations from group means are equal across groups. The function reproduces the behaviour of leveneTest(y, g, center = mean, ...) of the car package.

Usage

levene.test(y, g, data = NULL)

Value

An object of class "htest" with components:

statistic

the value of the F-statistic.

parameter

degrees of freedom: df1=k-1, df3=N-k, where k is the number of groups and N the total sample size

p.value

the p-value of the test.

method

a character string indicating the test performed.

data.name

a character string giving the name(s) of the data.

Arguments

y

A numeric response vector.

g

A grouping factor.

data

Optional data frame containing `y` and `g`.

Details

For each observation \(y_{ij}\) in group \(i\), compute the absolute deviation from the group mean:

$$z_{ij} = |y_{ij} - \bar{y}_i|$$

where \(\bar{y}_i\) is the mean of group \(i\).

The test statistic is the F-statistic from a one-way ANOVA on the \(z_{ij}\) values:

$$F = \frac{(N-k) \sum_{i=1}^{k} n_i (\bar{z}_i - \bar{z})^2}{ (k-1) \sum_{i=1}^{k} \sum_{j=1}^{n_i} (z_{ij} - \bar{z}_i)^2}$$

where:

• \(k\) = number of groups

• \(N\) = total sample size

• \(n_i\) = sample size of group \(i\)

• \(\bar{z}_i\) = mean of absolute deviations in group \(i\)

• \(\bar{z}\) = overall mean of all absolute deviations

Under the null hypothesis of equal variances, the test statistic follows an F-distribution: \(F \sim F(k-1, N-k)\).

References

Levene, H. (1960). Robust tests for equality of variances. In I. Olkin (Ed.), Contributions to Probability and Statistics (pp. 278-292). Stanford University Press.

Examples

set.seed(123)
y <- c(rnorm(10), rnorm(10, sd = 2), rnorm(10, sd = 0.5))
g <- factor(rep(1:3, each = 10))
levene.test(y, g)

# Usage with data frame
df <- data.frame(response = y, group = g)
levene.test(response, group, data = df)

# Example with unequal variances (should reject null hypothesis)
set.seed(456)
y_unequal <- c(rnorm(15, sd = 1), rnorm(15, sd = 5), rnorm(15, sd = 0.2))
g_unequal <- factor(rep(c("A", "B", "C"), each = 15))
levene.test(y_unequal, g_unequal)