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, ...)

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 expression specifying a subset of observations, evaluated in data (subset = Month != 5), as in bartlett.test().
na.action: a function which indicates what should happen when the data contain NAs. Defaults to na.pass(): incomplete cases are then removed by the default method.
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.

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()