Levene's Test for Comparing Variances Across Groups

A test for homogeneity of variances across two or more groups, more robust to departures from normality than Bartlett's test.

Usage

leveneTest(x, ...)

# S3 method for class 'formula'
leveneTest(formula, data, subset, na.action = na.pass, center = median, ...)

# Default S3 method
leveneTest(x, g, center = median, .centerName = NULL, ...)

Arguments

x: a numeric vector of data values (default method), or a formula (formula method).
...: arguments to be passed down, e.g. data for the formula method; can also be used to pass arguments to the function given by center (e.g. trim = 0.1 for a trimmed mean when center = mean).
formula: a formula of the form lhs ~ rhs where lhs gives the data values and rhs 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").
center: the name of a function to compute the center of each group; mean gives the original Levene's test, the default, median, provides the more robust Brown-Forsythe test.
g: factor defining the groups; ignored if x is a list.
.centerName: internal, not intended to be set by the user. Used to pass the deparsed name of the center function through the method dispatch chain (from leveneTest.formula to leveneTest.default), since substitute(center) would otherwise only resolve to the literal symbol "center" rather than the original expression (e.g. mean or median) supplied by the caller.

Value

An object of class "htest" representing the result of the hypothesis test.

Details

Let $X_{ij}$ be the jth observation of X for the ith group. Let $Z_{ij} = |X_{ij} - X_i|$, where $X_i$ is the center (by default the median) of X in the ith group. Levene's test statistic is $$W_0 = \frac{\sum_i n_i (\bar{Z}_i - \bar{Z})^2 / (g - 1)}{\sum_i \sum_j (Z_{ij} - \bar{Z}_i)^2 / \sum_i (n_i - 1)}$$ where $n_i$ is the number of observations in group i and g is the number of groups. Using the median instead of the mean for $X_i$ yields the more robust variant known as the Brown-Forsythe test, which is the default here.

Note

Based on car::leveneTest() by John Fox, with contributions by Derek Ogle and Brian Ripley, adapted to conform to package standards.

References

Fox, J. and Weisberg, S. (2019) An R Companion to Applied Regression, 3rd ed., Thousand Oaks, CA: Sage.

Levene, H. (1960) Robust tests for equality of variances. In: Olkin, I. et al., eds.: Contributions to Probability and Statistics: Essays in Honor of Harold Hotelling, Stanford University Press, pp. 278-292.

Examples

## example from ansari.test:
## Hollander & Wolfe (1973, p. 86f):
## Serum iron determination using Hyland control sera
serum <- data.frame(
  grp = rep(c("ramsay", "jung.parekh"), each = 20),
  x   = c(111, 107, 100, 99, 102, 106, 109, 108, 104, 99,
          101, 96, 97, 102, 107, 113, 116, 113, 110, 98,
          107, 108, 106, 98, 105, 103, 110, 105, 104,
          100, 96, 108, 103, 104, 114, 114, 113, 108, 106, 99)
)

leveneTest(x ~ grp, data = serum)
#> 
#> 	Levene's Test for Homogeneity of Variance (center = median)
#> 
#> data:  x ~ grp
#> F = 1.7865, num df = 1, denom df = 38, p-value = 0.1893
#> 
leveneTest(x ~ grp, data = serum, center = mean)
#> 
#> 	Levene's Test for Homogeneity of Variance (center = mean)
#> 
#> data:  x ~ grp
#> F = 1.7879, num df = 1, denom df = 38, p-value = 0.1891
#> 
leveneTest(x ~ grp, data = serum, center = mean, trim = 0.1)
#> 
#> 	Levene's Test for Homogeneity of Variance (center = mean(trim=0.1))
#> 
#> data:  x ~ grp
#> F = 1.7854, num df = 1, denom df = 38, p-value = 0.1894
#> 

leveneTest(c(rnorm(10), rnorm(10, 0, 2)),
           factor(rep(c("A", "B"), each = 10)))
#> 
#> 	Levene's Test for Homogeneity of Variance (center = median)
#> 
#> data:  c(rnorm(10), rnorm(10, 0, 2)) and factor(rep(c("A", "B"), each = 10))
#> F = 1.0614, num df = 1, denom df = 18, p-value = 0.3165
#> 

leveneTest(Ozone ~ factor(Month), data = airquality)
#> 
#> 	Levene's Test for Homogeneity of Variance (center = median)
#> 
#> data:  Ozone ~ factor(Month)
#> F = 3.9558, num df = 4, denom df = 111, p-value = 0.004863
#> 

leveneTest(count ~ spray, data = InsectSprays)
#> 
#> 	Levene's Test for Homogeneity of Variance (center = median)
#> 
#> data:  count ~ spray
#> F = 3.8214, num df = 5, denom df = 66, p-value = 0.004223
#> 
# Compare this to fligner.test() and bartlett.test()