Variance Test for Testing One Variance or Comparing Two Variances

Performs a one-sample or two-sample test for variance, analogous to t.test(), with support for classical and likelihood-based lowest-density (LD) two-sided p-values.

Usage

varTest(x, ...)

# Default S3 method
varTest(
  x,
  y = NULL,
  sigma2_0 = NULL,
  alternative = c("two.sided", "less", "greater"),
  type = c("classic", "ld"),
  ...
)

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

Arguments

x

a numeric vector of data values, or a formula.

...

further arguments passed to methods.

y

an optional second numeric vector. If provided, a two-sample variance test is performed.

sigma2_0

a numeric value specifying the null hypothesis variance for the one-sample test. Required if y is NULL.

alternative

character string specifying the alternative hypothesis. Must be one of "two.sided", "less", or "greater".

type

character string specifying the test type:

• "classic": uses the conventional two-sided p-value \(2 \cdot \min(P(T \le t), P(T \ge t))\).

• "ld": uses a likelihood-based lowest-density (LD) definition, i.e. the probability of observing values with density less than or equal to the observed density under the null distribution.

formula

a formula of the form lhs ~ rhs, where lhs gives the response values and rhs the corresponding groups or explanatory variables.

data

an optional matrix or data frame (or similar; see model.frame()) containing the variables in the formula. By default the variables are taken from environment(formula).

subset

an optional vector specifying a subset of observations to be used in the analysis.

na.action

a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Value

An object of class "htest" with components:

statistic

the test statistic.

parameter

degrees of freedom.

p.value

the p-value of the test.

estimate

sample variance(s).

null.value

the null hypothesis value (one-sample only).

alternative

the alternative hypothesis.

method

a character string indicating the test performed.

data.name

description of the data.

Details

The null hypothesis is that the ratio of the variances of the populations from which x and y were drawn, or in the data to which the linear models x and y were fitted, is equal to ratio.

For the one-sample test, the test statistic follows a chi-squared distribution: $$X^2 = (n - 1) S^2 / \sigma_0^2$$

For the two-sample test, the test statistic follows an F distribution: $$F = S_x^2 / S_y^2$$

The LD method corresponds to a likelihood-ratio interpretation of extremeness, which is particularly appropriate for asymmetric null distributions such as chi-squared and F distributions.

The formula interface is only applicable for the 2-sample tests.

See also

var.test(), bartlett.test() for testing homogeneity of variances in more than two samples from normal distributions; ansari.test() and mood.test() for two rank based (nonparametric) two-sample tests for difference in scale.

Other test.variance: leveneTest(), mosesTest(), siegelTukeyTest(), varCI()

Examples

set.seed(1)
x <- rnorm(20, sd = 3)
y <- rnorm(25, sd = 2)

# One-sample test
varTest(x, sigma2_0 = 9, type = "classic")
#> 
#> 	One-sample variance test (classic)
#> 
#> data:  x
#> X-squared = 15.847, df = 19, p-value = 0.665
#> alternative hypothesis: true variance is not equal to 9
#> sample estimates:
#> variance 
#> 7.506291 
#> 
varTest(x, sigma2_0 = 9, type = "ld")
#> 
#> 	One-sample variance test (ld)
#> 
#> data:  x
#> X-squared = 15.847, df = 19, p-value = 0.8411
#> alternative hypothesis: true variance is not equal to 9
#> sample estimates:
#> variance 
#> 7.506291 
#> 

# Two-sample test
varTest(x, y, type = "classic")
#> 
#> 	Two-sample variance test (classic)
#> 
#> data:  x and y
#> F = 2.8526, df1 = 19, df2 = 24, p-value = 0.0165
#> alternative hypothesis: two.sided
#> sample estimates:
#>   var(x)   var(y) 
#> 7.506291 2.631395 
#> 
varTest(x, y, type = "ld")
#> 
#> 	Two-sample variance test (ld)
#> 
#> data:  x and y
#> F = 2.8526, df1 = 19, df2 = 24, p-value = 0.008804
#> alternative hypothesis: two.sided
#> sample estimates:
#>   var(x)   var(y) 
#> 7.506291 2.631395 
#> 

# Formula interface
df <- data.frame(
  value = c(x, y),
  group = rep(c("A", "B"), c(length(x), length(y)))
)
varTest(value ~ group, data = df)
#> 
#> 	Two-sample variance test (classic)
#> 
#> data:  
#> F = 2.8526, df1 = 19, df2 = 24, p-value = 0.0165
#> alternative hypothesis: two.sided
#> sample estimates:
#>   var(x)   var(y) 
#> 7.506291 2.631395 
#>