1. Overview

The varequal R package provides statistical methods for assessing homoscedasticity, that is, whether the variances of a response variable are equal across independent groups.

Homogeneity of variance is an important assumption for many classical statistical procedures, including one-way ANOVA and related linear-model analyses. When the group variances differ substantially, the validity of procedures that rely on a common variance can be affected.

varequal provides a unified interface to several classical, robust, rank-based, and variance-outlier-oriented procedures. The package also provides high-level functions for automatically combining several tests into a single variance-homogeneity decision.

The main functionality can be divided into four levels:

Individual tests for explicit selection of a homogeneity-of-variance procedure.
check_var_equal() as a wrapper for selecting an individual test by a short method code.
is_var_equal() as a higher-level assessment combining five tests.
Supporting datasets and utilities for demonstrating and investigating the behavior of the procedures.

The currently implemented individual tests are:

Code	Function	Main characteristic
`AB`	`Ansari_Bradley_test()`	Rank-based dispersion test
`BL`	`Bartlett_test()`	Classical parametric test
`BF`	`Brown_Forsythe_test()`	Robust Levene-type procedure
`FK`	`Fligner_Killeen_test()`	Robust rank-based procedure
`LG`	`Lam_G_test()`	Variance-outlier-oriented procedure
`LV`	`Levene_test()`	ANOVA-based robust procedure
`MBF`	`Brown_Forsythe_test(method = "MBF")`	Mehrotra-modified Brown–Forsythe
`OB`	`O.Brien_test()`	O’Brien variance test
`OM`	`O.Neill_Mathews_test()`	Weighted least-squares Levene-type test

The package source currently describes varequal as a toolkit for classical and robust tests of variance homogeneity, together with supporting measures and utilities for exploratory analysis and hypothesis testing.

2. Installation

Install the released version from CRAN:

install.packages("varequal")

Or install the development version from GitHub:

if (!require("pak")) install.packages("pak")
pak::pak("P10911004-NPUST/varequal")

Then load the package:

library(varequal)

# For reproducibility
set.seed(123)

3. Quick start

For a quick overall assessment, use is_var_equal().

The package includes the roGFP dataset, which contains redox-index measurements from three experimental batches. Each batch contains a response variable ro and a grouping variable grp.

data("roGFP")

df1 <- roGFP[[1]]

is_var_equal(df1, ro ~ grp)
#> [1] FALSE

The result is a logical value:

is_var_equal(df1, ro ~ grp)
#> [1] FALSE

If you want to inspect the component tests, request a summary:

is_var_equal(df1, ro ~ grp, summary = TRUE)
#> $is_var_equal
#> [1] FALSE
#> 
#> $summary
#>                                  is_var_equal         pval statistic
#> Modified Brown-Forsythe (Fvalue)        FALSE 2.918003e-02  3.976918
#> Fligner-Killeen (ChiSquare)              TRUE 6.677704e-02  7.166465
#> 't Lam's G (Fvalue)                     FALSE 1.326224e-05  8.351380
#> Levene's (Fvalue)                       FALSE 3.250316e-02  3.183717
#> O'Neill-Mathews (Fvalue)                FALSE 4.693756e-02  2.861920
#>                                  critical_value
#> Modified Brown-Forsythe (Fvalue)       3.310109
#> Fligner-Killeen (ChiSquare)            7.814728
#> 't Lam's G (Fvalue)                    3.400135
#> Levene's (Fvalue)                      2.806845
#> O'Neill-Mathews (Fvalue)               2.806845

For explicit test selection, use check_normality():

check_var_equal(df1, ro ~ grp, method = "LV")
#> 
#> ---------------------------------------
#> Levene's homogeneity of variance test
#> 
#> Response: ro
#> 
#>           DF   SS MS Fvalue pvalue
#> Group      3 0.01  0 3.1837 0.0325
#> Residuals 46 0.03  0              
#> Total     49 0.03                 
#> 
#> #> Group variances are unequal.
#> ---------------------------------------

If you only require a quick check for data heteroscedasticity, use is_var_equal and feel free to skip the rest of this guide.

If you only require a high-level decision about variance homogeneity, start with is_var_equal() and feel free to skip the rest of this guide. If you need to justify a particular statistical procedure, use check_var_equal() or one of the individual test functions.

4. Understanding homogeneity-of-variance testing

Most procedures in this package test the null hypothesis

\[ H_0:\sigma_1^2 = \sigma_2^2 = \cdots = \sigma_k^2 \]

against an alternative in which at least one group variance differs:

\[ H_1:\text{not all }\sigma_i^2\text{ are equal}. \]

The interpretation follows the usual hypothesis-testing framework.

A small p-value provides evidence against homogeneity of variance. A large p-value means that the test did not detect sufficient evidence of unequal variances.

A large p-value does not prove that all population variances are exactly equal. The result should be interpreted together with:

the sample size in each group;
the presence or absence of outliers;
the distributional shape of the response;
the balance of the experimental design;
the scientific magnitude of the variance differences; and
the assumptions of the downstream analysis.

Different tests have different sensitivities. In particular, Bartlett’s test can be very sensitive to non-normality, whereas robust and rank-based procedures are generally less affected by departures from normality.

5. Data input

The package accepts either:

a data frame together with a formula; or
a list of numeric vectors representing groups.

5.1 Data frame and formula

The standard interface is:

response ~ group

For example:

Levene_test(df1, ro ~ grp)
#> 
#> ---------------------------------------
#> Levene's homogeneity of variance test
#> 
#> Response: ro
#> 
#>           DF   SS MS Fvalue pvalue
#> Group      3 0.01  0 3.1837 0.0325
#> Residuals 46 0.03  0              
#> Total     49 0.03                 
#> 
#> #> Group variances are unequal.
#> ---------------------------------------

The response is the dependent variable and the right-hand side identifies the independent grouping variable.

A data frame containing several experimental variables can therefore be used directly:

check_var_equal(df1, ro ~ grp, method = "BF")
#> 
#> --------------------------------------------
#> Brown-Forsythe homogeneity of variance test
#> 
#> Response: ro
#> 
#>              DF   SS   MS Fvalue  pvalue
#> Group         3 0.02 0.01 3.9769 0.01598
#> Residuals 32.86 0.05    0               
#> Total     35.86 0.07                    
#> 
#> #> Group variances are unequal.
#> --------------------------------------------
#> 
#> --------------------------------------------
#> Brown-Forsythe homogeneity of variance test
#> 
#> Response: ro
#> 
#>              DF   SS   MS Fvalue  pvalue
#> Group         3 0.02 0.01 3.9769 0.01598
#> Residuals 32.86 0.05    0               
#> Total     35.86 0.07                    
#> 
#> #> Group variances are unequal.
#> --------------------------------------------

5.2 A list of numeric vectors

A list can be used when the observations are already separated into groups.

x <- list(A = rnorm(12), B = rnorm(12), C = rnorm(12))
is_var_equal(x)
#> [1] TRUE

6. The individual tests

6.1 Ansari–Bradley test

Ansari_Bradley_test() is a rank-based procedure for assessing equality of dispersion.

ab <- Ansari_Bradley_test(df1, ro ~ grp)
#> 
#> ---------------------------------------------
#>  Ansari-Bradley homogeneity of variance test
#> 
#>  Response: ro
#> 
#>  Chi-Square: 6.0855
#>  Chi-Square critical: 7.8147
#>  p-value: 0.10752
#> 
#>  #> Group variances are equal. 
#> ---------------------------------------------

The implementation uses an approximate chi-square statistic. The package documentation notes that results may be inaccurate for very small total sample sizes or when there are many tied values. The test is useful as a nonparametric comparison of dispersion, but it should not automatically be treated as interchangeable with a robust ANOVA-based homogeneity test.

6.2 Bartlett’s test

Bartlett_test() implements the classical Bartlett test.

Bartlett’s test is the most powerful test when no outliers exist and the observations follow normal distribution, but it is sensitive to departures from normality and to outliers.

For data that satisfy the normality assumption reasonably well and are free of important outliers, Bartlett’s test is an appropriate classical procedure.

bl <- Bartlett_test(df1, ro ~ grp, summary = TRUE)
#> 
#> ----------------------------------------
#>  Bartlett's homogeneity of variance test
#> 
#>  Response: y
#> 
#>  Chi-Square: 18.4724
#>  p-value: 0.00035
#> 
#>  #> Group variances are unequal. 
#> ----------------------------------------

6.3 Brown–Forsythe test

Brown_Forsythe_test() provides both the original Brown–Forsythe procedure and the Mehrotra-modified version.

The default method is "MBF":

mbf <- Brown_Forsythe_test(df1, ro ~ grp)
#> 
#> --------------------------------------------
#> Brown-Forsythe homogeneity of variance test
#> 
#> Response: ro
#> 
#>              DF   SS   MS Fvalue  pvalue
#> Group      1.96 0.01 0.01 3.9769 0.02918
#> Residuals 32.86 0.05    0               
#> Total     34.81 0.06                    
#> 
#> #> Group variances are unequal.
#> --------------------------------------------

The original Brown–Forsythe procedure can be selected with method = "BF":

bf <- Brown_Forsythe_test(df1, ro ~ grp, method = "BF")
#> 
#> --------------------------------------------
#> Brown-Forsythe homogeneity of variance test
#> 
#> Response: ro
#> 
#>              DF   SS   MS Fvalue  pvalue
#> Group         3 0.02 0.01 3.9769 0.01598
#> Residuals 32.86 0.05    0               
#> Total     35.86 0.07                    
#> 
#> #> Group variances are unequal.
#> --------------------------------------------

The Brown–Forsythe approach is a robust modification of Levene’s procedure that uses a robust location measure, conventionally the group median.

The package also permits a custom transformation of the response. The default is based on absolute deviations from the group median:

\[ y'_{ij} = |y_{ij} - \tilde y_i|. \]

For example:

Brown_Forsythe_test(df1, ro ~ grp, transform = function(x) abs(x - mean(x)))
#> 
#> --------------------------------------------
#> Brown-Forsythe homogeneity of variance test
#> 
#> Response: ro
#> 
#>              DF   SS   MS Fvalue  pvalue
#> Group      1.89 0.01 0.01 5.0469 0.01406
#> Residuals 30.46 0.04    0               
#> Total     32.35 0.05                    
#> 
#> #> Group variances are unequal.
#> --------------------------------------------

The method argument distinguishes the two implementations:

"BF": original Brown–Forsythe procedure;
"MBF": Mehrotra modification, which adjusts the degrees of freedom.

The modified procedure is designed to reduce Type I error inflation, with a potential trade-off in power.

6.4 Fligner–Killeen test

Fligner_Killeen_test() is a rank-based procedure that is particularly useful when robustness to non-normality and outliers is important.

fk <- Fligner_Killeen_test(df1, ro ~ grp)
#> 
#> ---------------------------------------------
#>  Fligner-Killeen homogeneity of variance test
#> 
#>  Response: ro
#> 
#>  Chi-Square: 7.1665
#>  Chi-Square critical: 7.8147
#>  p-value: 0.06678
#> 
#>  #> Group variances are equal. 
#> ---------------------------------------------

The procedure works with ranks of absolute deviations from the group median and uses a normal-score transformation. Fligner--Killeen is an important general-purpose choice when the normality assumption is questionable.

6.5 ’t Lam’s G test

Lam_G_test() implements ’t Lam’s G procedure. The procedure is related to Cochran-type variance-outlier assessment rather than being a conventional homogeneity test in the same sense as Levene’s or Bartlett’s test.

lg <- Lam_G_test(df1, ro ~ grp)
#> 
#> ---------------------------------------------
#> 't Lam's G homogeneity of variance test
#> 
#> Response: ro
#> 
#> Outlying group: C
#> 
#> F-value: 8.3514
#> F-critical: 3.4001
#> p-value: 1e-05
#> 
#> #> Group variances are unequal.
#> ---------------------------------------------

It supports three alternatives:

Lam_G_test(df1, ro ~ grp, alternative = "two.sided")
#> 
#> ---------------------------------------------
#> 't Lam's G homogeneity of variance test
#> 
#> Response: ro
#> 
#> Outlying group: C
#> 
#> F-value: 8.3514
#> F-critical: 3.4001
#> p-value: 1e-05
#> 
#> #> Group variances are unequal.
#> ---------------------------------------------
Lam_G_test(df1, ro ~ grp, alternative = "greater")
#> 
#> ---------------------------------------------
#> 't Lam's G homogeneity of variance test
#> 
#> Response: ro
#> 
#> Outlying group: 
#> 
#> F-value: 2.5435
#> F-critical: 2.6257
#> p-value: 0.06043
#> 
#> #> Group variances are equal.
#> ---------------------------------------------
Lam_G_test(df1, ro ~ grp, alternative = "less")
#> 
#> ---------------------------------------------
#> 't Lam's G homogeneity of variance test
#> 
#> Response: ro
#> 
#> Outlying group: C
#> 
#> F-value: 8.3514
#> F-critical: 3.0136
#> p-value: 1e-05
#> 
#> #> Group variances are unequal.
#> ---------------------------------------------

The interpretation is directional:

"greater" focuses on unusually large variances;
"less" focuses on unusually small variances;
"two.sided" considers both directions.

The method is highly sensitive to outliers, so it should be used with an explicit understanding of that property.

6.6 Levene’s test

Levene_test() provides an ANOVA-based test of homogeneity.

lv <- Levene_test(df1, ro ~ grp)
#> 
#> ---------------------------------------
#> Levene's homogeneity of variance test
#> 
#> Response: ro
#> 
#>           DF   SS MS Fvalue pvalue
#> Group      3 0.01  0 3.1837 0.0325
#> Residuals 46 0.03  0              
#> Total     49 0.03                 
#> 
#> #> Group variances are unequal.
#> ---------------------------------------

The implementation uses a transformation of each observation into a measure of within-group deviation. Its default transformation is the absolute deviation from the group median:

\[ y'_{ij} = |y_{ij} - \tilde y_i|. \]

A custom transformation can be supplied:

Levene_test(df1, ro ~ grp, transform = function(x) (x - median(x)) ^ 2)
#> 
#> ---------------------------------------
#> Levene's homogeneity of variance test
#> 
#> Response: ro
#> 
#>           DF SS MS Fvalue  pvalue
#> Group      3  0  0 2.6281 0.06142
#> Residuals 46  0  0               
#> Total     49  0                  
#> 
#> #> Group variances are equal.
#> ---------------------------------------

Other useful transformations include:

# Absolute deviations from the mean
Levene_test(df1, ro ~ grp, transform = function(x) abs(x - mean(x)))
#> 
#> ---------------------------------------
#> Levene's homogeneity of variance test
#> 
#> Response: ro
#> 
#>           DF   SS MS Fvalue pvalue
#> Group      3 0.01  0 4.1116 0.0115
#> Residuals 46 0.02  0              
#> Total     49 0.03                 
#> 
#> #> Group variances are unequal.
#> ---------------------------------------

# Square-root absolute deviations
Levene_test(df1, ro ~ grp, transform = function(x) sqrt(abs(x - median(x))))
#> 
#> ---------------------------------------
#> Levene's homogeneity of variance test
#> 
#> Response: ro
#> 
#>           DF   SS   MS Fvalue  pvalue
#> Group      3 0.04 0.01 2.4066 0.07934
#> Residuals 46 0.28 0.01               
#> Total     49 0.32                    
#> 
#> #> Group variances are equal.
#> ---------------------------------------

The transformation is subsequently analyzed using an ANOVA-style decomposition.

6.7 O’Brien’s test

O.Brien_test() implements O’Brien’s variance-homogeneity procedure.

ob <- O.Brien_test(df1, ro ~ grp)
#> 
#> ---------------------------------------
#> O'Brien's homogeneity of variance test
#> 
#> Response: ro
#> 
#>           DF SS MS Fvalue  pvalue
#> Group      3  0  0  2.422 0.07793
#> Residuals 46  0  0               
#> Total     49  0                  
#> 
#> #> Group variances are equal.
#> ---------------------------------------

The default transformation is based on squared deviations from the group median:

\[ y'_{ij} = (y_{ij} - \tilde y_i)^2. \]

O’Brien’s test can be conservative and may have relatively low power for some forms of heteroscedasticity. The package documentation therefore places Levene-type and Brown–Forsythe procedures ahead of O’Brien’s test for general homogeneity assessment.

6.8 O’Neill–Mathews test

O.Neill_Mathews_test() implements the weighted least-squares approach to Levene’s test.

om <- O.Neill_Mathews_test(df1, ro ~ grp)
#> 
#> ---------------------------------------------
#> O'Neill-Mathews homogeneity of variance test
#> 
#> Response: ro
#> 
#>           DF   SS   MS Fvalue  pvalue
#> Group      3 0.02 0.01 2.8619 0.04694
#> Residuals 46 0.09    0               
#> Total     49  0.1                    
#> 
#> #> Group variances are unequal.
#> ---------------------------------------------

Its default transformation is again based on absolute deviations from the group median:

\[ y'_{ij} = |y_{ij} - \tilde y_i|. \]

This procedure is useful when a weighted least-squares formulation is desired, particularly when group sizes are unequal.

7. Selecting a test with `check_var_equal()`

check_var_equal() provides a common wrapper around the individual tests. The available method codes are:

Code	Procedure
`AB`	Ansari–Bradley
`BL`	Bartlett
`FK`	Fligner–Killeen
`LG`	’t Lam’s G
`LV`	Levene
`MBF`	Mehrotra–Brown–Forsythe
`BF`	Brown–Forsythe
`OB`	O’Brien
`OM`	O’Neill–Mathews

This is particularly useful when the method is selected programmatically. For example:

method <- "FK"
result <- check_var_equal(df1, ro ~ grp, method = method, silent = TRUE)
result
#> $method
#> [1] "Fligner-Killeen homogeneity of variance test"
#> 
#> $is_var_equal
#> [1] TRUE
#> 
#> $alpha
#> [1] 0.05
#> 
#> $summary
#> [1] NA
#> 
#> $statistic
#> ChiSquare 
#>  7.166465 
#> 
#> $pvalue
#> [1] 0.06677704

8. Automatic assessment with `is_var_equal()`

is_var_equal() combines five procedures:

Brown–Forsythe;
Fligner–Killeen;
’t Lam’s G;
Levene;
O’Neill–Mathews.

The sensitivity argument ranges from 1 to 5. A larger value requires more component tests to agree that the variances are equal:

for (s in 1:5) {
  cat("sensitivity =", s, "\n")
  print(is_var_equal(df1, ro ~ grp, sensitivity = s))
}
#> sensitivity = 1 
#> [1] TRUE
#> sensitivity = 2 
#> [1] FALSE
#> sensitivity = 3 
#> [1] FALSE
#> sensitivity = 4 
#> [1] FALSE
#> sensitivity = 5 
#> [1] FALSE

Conceptually:

sensitivity = 1 is permissive;
sensitivity = 3 is the default;
sensitivity = 5 is conservative.

Set summary = TRUE to obtain the component-test results:

result <- is_var_equal(df1, ro ~ grp, summary = TRUE)
result$is_var_equal
#> [1] FALSE
result$summary
#>                                  is_var_equal         pval statistic
#> Modified Brown-Forsythe (Fvalue)        FALSE 2.918003e-02  3.976918
#> Fligner-Killeen (ChiSquare)              TRUE 6.677704e-02  7.166465
#> 't Lam's G (Fvalue)                     FALSE 1.326224e-05  8.351380
#> Levene's (Fvalue)                       FALSE 3.250316e-02  3.183717
#> O'Neill-Mathews (Fvalue)                FALSE 4.693756e-02  2.861920
#>                                  critical_value
#> Modified Brown-Forsythe (Fvalue)       3.310109
#> Fligner-Killeen (ChiSquare)            7.814728
#> 't Lam's G (Fvalue)                    3.400135
#> Levene's (Fvalue)                      2.806845
#> O'Neill-Mathews (Fvalue)               2.806845

The summary contains the individual test decisions, p-values, test statistics, and critical values.

9. Comparing the tests

No single homogeneity-of-variance test is uniformly optimal.

A practical comparison is:

Situation	Useful starting point
Approximately normal data, no important outliers	Bartlett
Normality is uncertain	Fligner–Killeen
Outliers may be present	Brown–Forsythe or Fligner–Killeen
A conventional Levene framework is desired	Levene
Unequal group sizes with weighted least squares	O’Neill–Mathews
Variance-outlier behavior is of interest	’t Lam’s G
Explicit comparison with O’Brien’s transformation	O’Brien
Rank-based dispersion comparison	Ansari–Bradley

These are guidelines rather than universal rules. The final choice should depend on the data and the assumptions of the downstream analysis.

10. Why multiple tests can disagree

Different procedures weight distributional features differently.

For example, a dataset can contain:

approximately equal variances but strong skewness;
a single extreme observation in one group;
unequal variances that differ primarily in one group;
several small groups with low statistical power; or
unequal sample sizes.

Under such conditions, two valid tests can produce different decisions.

Therefore, the question should not simply be:

Which test gives a significant p-value?

A better question is:

Which variance-homogeneity procedure is appropriate for the distribution, sample size, outlier structure, and experimental design?

is_var_equal() is intended to provide a practical aggregate assessment, while the individual functions allow the analyst to inspect the behavior of specific procedures.

11. Inspecting the data before testing

Formal tests should not replace graphical inspection.

For example, boxplots can reveal differences in spread and possible outliers:

text = sprintf("Variance equal: %s", is_var_equal(df1, ro ~ grp))
boxplot(ro ~ grp, 
        data = df1, 
        main = text, 
        horizontal = TRUE, 
        xlab = "Redox index", 
        ylab = "Group")
points(x = df1$ro, 
       y = jitter(as.numeric(df1$grp), amount = 0.15))

The same graphical strategy can be used with the CYCB1 dataset.

data("CYCB1")
df2 <- CYCB1[[2]]
text = sprintf("Variance equal: %s", is_var_equal(df2, cells ~ grp))
boxplot(cells ~ grp,
        data = df2,
        main = text,
        horizontal = TRUE,
        xlab = "Cell number",
        ylab = "Group"
)
points(x = df2$cells, 
       y = jitter(as.numeric(df2$grp), amount = 0.15))

12. Built-in datasets

12.1 `roGFP`

roGFP is a list containing three experimental batches. Each data frame contains:

TEMP: air temperature;
RGF1: RGF1 peptide concentration;
treatment: combined treatment;
grp: group label;
ro: reduced–oxidized redox index.

The redox index ranges from -1 (reduced) to 1 (oxidized).

data("roGFP")
str(roGFP[[1]])
#> 'data.frame':    50 obs. of  5 variables:
#>  $ TEMP     : Factor w/ 2 levels "22C","31C": 1 1 1 1 1 1 1 1 1 1 ...
#>  $ RGF1     : Factor w/ 2 levels "0nM","5nM": 1 1 1 1 1 1 1 1 1 1 ...
#>  $ treatment: Factor w/ 4 levels "22C_0nM","22C_5nM",..: 1 1 1 1 1 1 1 1 1 1 ...
#>  $ grp      : Factor w/ 4 levels "A","B","C","D": 1 1 1 1 1 1 1 1 1 1 ...
#>  $ ro       : num  -0.0877 -0.0795 -0.062 -0.1316 -0.0455 ...

The dataset is based on measurements from Arabidopsis thaliana roots.

12.2 `CYCB1`

CYCB1 is a list containing three experimental batches of meristematic root cell counts. Each data frame contains:

TEMP: air temperature;
RGF1: RGF1 peptide concentration;
treatment: combined treatment;
grp: group label;
cells: number of meristematic root cells.

data("CYCB1")
str(CYCB1[[1]])
#> 'data.frame':    61 obs. of  5 variables:
#>  $ TEMP     : Factor w/ 2 levels "22C","31C": 1 1 1 1 1 1 1 1 1 1 ...
#>  $ RGF1     : Factor w/ 2 levels "0","5": 1 1 1 1 1 1 1 1 1 1 ...
#>  $ treatment: Factor w/ 4 levels "22C_0","22C_5",..: 1 1 1 1 1 1 1 1 1 1 ...
#>  $ grp      : Factor w/ 4 levels "A","B","C","D": 1 1 1 1 1 1 1 1 1 1 ...
#>  $ cells    : int  57 61 63 64 60 67 65 60 60 58 ...

These datasets are included as practical examples for applying variance homogeneity tests to experimental biological data.

13. Benchmarking and method performance

For comparing variance-homogeneity procedures, the benchmark uses four possible outcomes:

O_O: variance is homogeneous and the method predicts homogeneous;
O_X: variance is homogeneous but the method predicts heterogeneous;
X_X: variance is heterogeneous and the method predicts heterogeneous;
X_O: variance is heterogeneous but the method predicts homogeneous.

From these quantities:

\[ \text{Type I error} = \frac{O\_X}{O\_O + O\_X}\times100\%, \]

\[ \text{Type II error} = \frac{X\_O}{X\_X + X\_O}\times100\%, \]

\[ \text{Accuracy} = \frac{O\_O + X\_X} {O\_O + O\_X + X\_X + X\_O}\times100\%. \]

Scenario 1: normal data, moderate sample size

For normally distributed, outlier-free data with group sizes from 8 to 20:

Test	Type I error	Type II error	Accuracy
Ansari–Bradley	2.6%	25.1%	86.15%
Bartlett	6.1%	1.5%	96.20%
Modified Brown–Forsythe	2.3%	23.3%	87.20%
Brown–Forsythe	2.7%	17.7%	89.80%
Fligner–Killeen	2.4%	17.6%	90.00%
’t Lam’s G	8.1%	1.8%	95.05%
Levene	2.7%	13.6%	91.85%
O’Brien	1.7%	37.5%	80.40%
O’Neill–Mathews	1.7%	18.0%	90.15%

This scenario illustrates the advantage of Bartlett’s test when its normality assumptions are satisfied. Its low Type II error produces high accuracy in this benchmark.

Scenario 2: normal data, small sample size

For group sizes from 3 to 7, the reported Type II errors become much larger for many procedures. For example:

Test	Type I error	Type II error	Accuracy
Ansari–Bradley	3.3%	89.6%	53.55%
Bartlett	4.3%	56.0%	69.85%
Modified Brown–Forsythe	0.9%	93.9%	52.60%
Brown–Forsythe	1.4%	92.5%	53.05%
Fligner–Killeen	2.2%	90.3%	53.75%
’t Lam’s G	16.0%	39.3%	72.35%
Levene	2.7%	87.9%	54.70%
O’Brien	0.4%	97.0%	51.30%
O’Neill–Mathews	0.6%	94.2%	52.60%

The central lesson is not that one test is universally best. Rather, small samples can make variance testing intrinsically difficult. A non-significant result should therefore be interpreted cautiously when group sizes are very small.

Scenario 3: normal data with outliers

The benchmark also considers moderate sample sizes with one or two outliers. In that setting, the Bartlett and ’t Lam’s G can become extremely sensitive to the introduced outliers, while robust procedures can maintain substantially better Type I error control.

For example, the reported results include:

Test	Type I error	Type II error	Accuracy
Bartlett	100.0%	0.0%	50.00%
Modified Brown–Forsythe	0.0%	41.0%	79.50%
Brown–Forsythe	0.0%	25.5%	87.25%
Fligner–Killeen	2.6%	9.0%	94.20%
Levene	0.0%	21.0%	89.50%

Practical interpretation

The benchmark supports three general principles:

Normal, clean data: Bartlett can be highly effective.
Non-normal or contaminated data: robust procedures such as Brown–Forsythe and Fligner–Killeen are often preferable.
Very small samples: all tests can have poor ability to detect genuine heteroscedasticity, so graphical and substantive assessment becomes especially important.

The benchmark is a simulation study rather than a universal ranking of methods. Results depend on the exact simulation design, and performance under a user’s data-generating process can differ.

Summary

The varequal package provides a unified R interface for assessing homogeneity of variance across independent groups.

The principal functions are:

is_var_equal() for an aggregate assessment based on multiple tests;
check_var_equal() for selecting a specific procedure by method code;
Brown_Forsythe_test() for a robust Levene-type approach;
Fligner_Killeen_test() for a robust rank-based procedure;
Levene_test() for an ANOVA-based approach;
Bartlett_test() for classical normal-theory testing;
Ansari_Bradley_test() for rank-based dispersion assessment;
Lam_G_test() for variance-outlier-oriented assessment;
O.Brien_test() for O’Brien’s variance test; and
O.Neill_Mathews_test() for a weighted least-squares Levene-type approach.

No single test should be treated as universally best. The appropriate method depends on distributional assumptions, outliers, sample size, group balance, and the statistical model that follows.

References

Ansari, A. R., & Bradley, R. A. (1960). Rank-sum tests for dispersions. The Annals of Mathematical Statistics, 31, 1174–1189.

Bartlett, M. S. (1937). Properties of sufficiency and statistical tests. Proceedings of the Royal Society of London. Series A, 160(901), 268–282. https://doi.org/10.1098/rspa.1937.0109

Brown, M. B., & Forsythe, A. B. (1974). Robust tests for the equality of variances. Journal of the American Statistical Association, 69(346), 364–367. https://doi.org/10.1080/01621459.1974.10482955

Conover, W. J., Johnson, M. E., & Johnson, M. M. (1981). A comparative study of tests for homogeneity of variances, with applications to the outer continental shelf bidding data. Technometrics, 23, 351–361. https://doi.org/10.1080/00401706.1981.10487680

Fligner, M. A., & Killeen, T. J. (1976). Distribution-free two-sample tests for scale. Journal of the American Statistical Association, 71(353), 210–213. https://doi.org/10.1080/01621459.1976.10481517

Mehrotra, D. V. (1997). Improving the Brown–Forsythe solution to the generalized Behrens–Fisher problem. Communications in Statistics–Simulation and Computation, 26, 1139–1145. https://doi.org/10.1080/03610919708813431

O’Brien, R. G. (1981). A simple test for variance effects in experimental designs. Psychological Bulletin, 89(3), 570–574. https://doi.org/10.1037/0033-2909.89.3.570

O’Neill, M. E., & Mathews, K. (2000). Theory & Methods: A Weighted Least Squares Approach to Levene’s Test of Homogeneity of Variance. Australian & New Zealand Journal of Statistics, 42(1), 81–100. https://doi.org/10.1111/1467-842X.00109

Sharma, D., & Kibria, B. M. G. (2013). On some test statistics for testing homogeneity of variances: A comparative study. Journal of Statistical Computation and Simulation, 83, 1944–1963. https://doi.org/10.1080/00949655.2012.675336

’t Lam, R. U. E. (2010). Scrutiny of variance results for outliers: Cochran’s test optimized. Analytica Chimica Acta, 659(1–2), 68–84. https://doi.org/10.1016/j.aca.2009.11.032

Zhou, Y., Zhu, Y., & Wong, W. K. (2023). Statistical tests for homogeneity of variance for clinical trials and recommendations. Contemporary Clinical Trials Communications, 33, 101119. https://doi.org/10.1016/j.conctc.2023.101119

varequal

1. Overview

2. Installation

3. Quick start

4. Understanding homogeneity-of-variance testing

5. Data input

5.1 Data frame and formula

5.2 A list of numeric vectors

6. The individual tests

6.1 Ansari–Bradley test

6.2 Bartlett’s test

6.3 Brown–Forsythe test

6.4 Fligner–Killeen test

6.5 ’t Lam’s G test

6.6 Levene’s test

6.7 O’Brien’s test

6.8 O’Neill–Mathews test

7. Selecting a test with check_var_equal()

8. Automatic assessment with is_var_equal()

9. Comparing the tests

10. Why multiple tests can disagree

11. Inspecting the data before testing

12. Built-in datasets

12.1 roGFP

12.2 CYCB1

13. Benchmarking and method performance

Scenario 1: normal data, moderate sample size

Scenario 2: normal data, small sample size

Scenario 3: normal data with outliers

Practical interpretation

Summary

References

7. Selecting a test with `check_var_equal()`

8. Automatic assessment with `is_var_equal()`

12.1 `roGFP`

12.2 `CYCB1`