homogeneity

Conducts Bartlett’s and Levene’s tests of the homogeneity of variance assumption in analysis of variance.

Synopsis

#include <imsls.h>

float *imsls_f_homogeneity (int n, int n_treatment, int treatment[], float y[], …, 0)

The type double is imsls_d_homogeneity.

Required Arguments

int n (Input)
Number of experimental observations.

int n_treatment (Input)
Number of treatments. n_treatment must be greater than one.

int treatment[] (Input)
An array of length n containing the treatment identifiers for each observation in y. Each level of the treatment must be assigned a different integer. imsls_f_homogeneity verifies that the number of unique treatment identifiers is equal to n_treatment.

float y[] (Input)
An array of length n containing the experimental observations and any missing values. Missing values can be included in this array, although they are ignored in the analysis. They are indicated by placing a NaN (not a number) in y. The NaN value can be set using either the function imsls_f_machine(6) or imsls_d_machine(6), depending upon whether single or double precision is being used, respectively.

Return Value

Address of a pointer to the memory location of an array of length 2 containing the p-values for Bartletts and Levene’s tests.

Synopsis with Optional Arguments

#include <imsl.h>

float *imsls_f_homogeneity (int n, int n_treatment, int n_treatment[], float y[],

IMSLS_RETURN_USER, float p_value[],

IMSLS_LEVENES_MEAN, or

IMSLS_LEVENES_MEDIAN,

IMSLS_N_MISSING, int *n_missing,

IMSLS_CV, float *cv,

IMSLS_GRAND_MEAN, float *grand_mean,

IMSLS_TREATMENT_MEANS, float **treatment_means,

IMSLS_TREATMENT_MEANS_USER, float treatment_means[],

IMSLS_RESIDUALS, float **residuals,

IMSLS_RESIDUALS_USER, float residuals[],

IMSLS_STUDENTIZED_RESIDUALS, float **studentized_residuals,

IMSLS_STUDENTIZED_RESIDUALS_USER, float studentized_residuals[],

IMSLS_STD_DEVS, float **std_devs,

IMSLS_STD_DEVS_USER, float std_devs[],

IMSLS_BARTLETTS, float *bartletts,

IMSLS_LEVENES, float *levenes,

Optional Arguments

IMSLS_RETURN_USER, float p_value[] (Output)
User defined array of length 2 for storage of the p-values from Bartlett’s and Levene’s tests for homogeneity of variance. The first value returned contains the p-value for Bartlett’s test and the second value contains the p-value for Levene’s test.

IMSLS_LEVENES_MEAN (Input)

IMSLS_LEVENES_MEDIAN (Input)
Calculates Levene’s test using either the treatment means or medians. IMSLS_LEVENES_MEAN indicates that Levene’s test is calculated using the mean, and IMSLS_LEVENES_MEDIAN indicates that it is calculated using the median.

Default: IMSLS_LEVENES_MEAN

IMSLS_N_MISSING, int *n_missing (Output)
Number of missing values, if any, found in y. Missing values are denoted with a NaN (Not a Number) value in y. In these analyses, any missing values are ignored.

IMSLS_CV, float *cv (Output)
The coefficient of variation computed using the grand mean and pooled within treatment standard deviation.

IMSLS_GRAND_MEAN, float grand_mean (Output)
Mean of all the data across every location.

IMSLS_TREATMENT_MEANS, float **treatment_means (Output)
Address of a pointer to an internally allocated array of size n_treatment containing the treatment means.

IMSLS_TREATMENT_MEANS_USER, float treatment_means[] (Output)
Storage for the array treatment_means, provided by the user.

IMSLS_RESIDUALS, float **residuals (Output)
Address of a pointer to an internally allocated array of length n containing the residuals for non-missing observations. The ordering of the values in this array corresponds to the ordering of values in y and identified by the values in treatments.

IMSLS_RESIDUALS_USER, float residuals[] (Output)
Storage for the array residuals, provided by the user.

IMSLS_STUDENTIZED_RESIDUALS, float **studentized_residuals (Output)
Address of a pointer to an internally allocated array of length n containing the studentized residuals for non-missing observations. The ordering of the values in this array corresponds to the ordering of values in y and identified by the values in treatments.

IMSLS_STUDENTIZED_RESIDUALS_USER, float studentized_residuals[] (Output)
Storage for the array studentized_residuals, provided by the user.

IMSLS_STD_DEVS, float **std_devs (Output)
Address of a pointer to an internally allocated array of length n_treatment containing the treatment standard deviations.

IMSLS_STD_DEVS_USER, float std_devs[] (Output)
Storage for the array std_devs, provided by the user.

IMSLS_BARTLETTS, float *bartletts (Output)
Test statistic for Bartlett’s test.

IMSLS_LEVENES, float *levenes (Output)
Test statistic for Levene’s test.

Description

Traditional analysis of variance assumes that variances within treatments are equal. This is referred to as homogeneity of variance. The function imsls_f_homogeneity conducts both the Bartlett’s and Levene’s tests for this assumption:

versus

for at least one pair (i ≠ j), where t=n_treatments.

Bartlett’s test, Bartlett (1937), uses the test statistic:

where

and is the variance of the non-missing observations in the i‑th treatment. is referred to as the pooled variance, and it is also known as the error mean squares from a 1‑way analysis of variance.

If the usual assumptions associated with the analysis of variance are valid, then Bartlett’s test statistic is a chi-squared random variable with degrees of freedom equal to t‑1.

The original Levene’s test, Levene (1960) and Snedecor & Cochran (1967), uses a different test statistic, F0, equal to:

where

is the j‑th observation from the i‑th treatment and is the mean for the i‑th treatment. Conover, Johnson, and Johnson (1981) compared over 50 similar tests for homogeneity and concluded that one of the best tests was Levene’s test when the treatment mean, is replaced with the treatment median, . This version of Levene’s test can be requested by setting IMSLS_LEVENES_MEDIAN. In either case, Levene’s test statistic is treated as a F random variable with numerator degrees of freedom equal to (t-1) and denominator degrees of freedom (N-t).

The residual for the j-th observation within the i‑th treatment, , returned from IMSLS_RESIDUALS is unstandardized, i.e. . For investigating problems of homogeneity of variance, the studentized residuals returned by IMSLS_STUDENTIZED_RESIDUALS are recommended since they are standardized by the standard deviation of the residual. The formula for calculating the studentized residual is:

where the coefficient of variation, returned from IMSLS_CV, is also calculated using the pooled variance and the grand mean:

Example

This example applies Bartlett’s and Levene’s test to verify the homogeneity assumption for a one-way analysis of variance. There are eight treatments, each with 3 replicates for a total of 24 observations. The estimated treatment standard deviations range from 5.35 to 13.17.

In this case, Bartlett's test is not statistically significant for a stated significance level of .05; whereas Levene's test is significant with p = 0.006.

#include <imsls.h>

#include <stdio.h>

int main()

{

int i, page_width = 132;

int n = 24;

int n_treatment = 8;

int treatment[]={

1, 2, 3, 4, 5, 6, 7, 8,

1, 2, 3, 4, 5, 6, 7, 8

};

float y[] ={

30.0, 40.0, 38.9, 38.2,

41.8, 52.2, 54.8, 58.2,

20.5, 26.9, 21.4, 25.1,

26.4, 36.7, 28.9, 35.9,

21.0, 25.4, 24.0, 23.3,

34.4, 41.0, 33.0, 34.9

};

float bartletts;

float levenes;

float grand_mean;

float cv;

float *treatment_means;

float *residuals;

float *studentized_residuals;

float *std_devs;

int n_missing = 0;

float *p;

p = imsls_f_homogeneity(n, n_treatment, treatment, y,

IMSLS_BARTLETTS, &bartletts,

IMSLS_LEVENES, &levenes,

IMSLS_LEVENES_MEDIAN,

IMSLS_N_MISSING, &n_missing,

IMSLS_GRAND_MEAN, &grand_mean,

IMSLS_CV, &cv,

IMSLS_TREATMENT_MEANS, &treatment_means,

IMSLS_STD_DEVS, &std_devs,

0);

printf("\n\n\n *** Bartlett\'s Test ***\n\n");

printf("Bartlett\'s p-value = %10.3f\n", p[0]);

printf("Bartlett\'s test statistic = %10.3f\n", bartletts);

printf("\n\n\n *** Levene\'s Test ***\n\n");

printf("Levene\'s p-value = %10.3f\n", p[1]);

printf("Levene\'s test statistic = %10.3f\n", levenes);

imsls_f_write_matrix("Treatment means", n_treatment, 1,

treatment_means,

0);

imsls_f_write_matrix("Treatment std devs", n_treatment, 1, std_devs,

0);

printf("\ngrand_mean = %10.3f\n", grand_mean);

printf("cv = %10.3f\n", cv);

printf("n_missing = %d\n", n_missing);

}

Output

*** Bartlett's Test ***

Bartlett's p-value = 0.944

Bartlett's test statistic = 2.257

*** Levene's Test ***

Levene's p-value = 0.994

Levene's test statistic = 0.135

Treatment means

1 23.83

2 30.77

3 28.10

4 28.87

5 34.20

6 43.30

7 38.90

8 43.00

Treatment std devs

1 5.35

2 8.03

3 9.44

4 8.13

5 7.70

6 8.00

7 13.92

8 13.17

grand_mean = 33.871

cv = 28.378

n_missing = 0