1.3.5.7. Bartlett's Test

1. Exploratory Data Analysis
1.3. EDA Techniques
1.3.5. Quantitative Techniques

1.3.5.7. Bartlett's Test

Purpose:
Test for Homogeneity of Variances

Bartlett's test ( Snedecor and Cochran, 1983) is used to test if k samples have equal variances. Equal variances across samples is called homogeneity of variances. Some statistical tests, for example the analysis of variance, assume that variances are equal across groups or samples. The Bartlett test can be used to verify that assumption.

Bartlett's test is sensitive to departures from normality. That is, if your samples come from non-normal distributions, then Bartlett's test may simply be testing for non-normality. The Levene test is an alternative to the Bartlett test that is less sensitive to departures from normality.

Definition

The Bartlett test is defined as:

H₀:
H_a:	for at least one pair (i,j).
Test Statistic:	The Bartlett test statistic is designed to test for equality of variances across groups against the alternative that variances are unequal for at least two groups. In the above, *s_i²* is the variance of the ith group, N is the total sample size, N_i is the sample size of the ith group, k is the number of groups, and *s_p²* is the pooled variance. The pooled variance is a weighted average of the group variances and is defined as:
Significance Level:
Critical Region:	The variances are judged to be unequal if, where is the upper critical value of the chi-square distribution with k - 1 degrees of freedom and a significance level of . In the above formulas for the critical regions, the Handbook follows the convention that is the upper critical value from the chi-square distribution and is the lower critical value from the chi-square distribution. Note that this is the opposite of some texts and software programs. In particular, Dataplot uses the opposite convention.

An alternate definition (Dixon and Massey, 1969) is based on an approximation to the F distribution. This definition is given in the Product and Process Comparisons chapter (chapter 7).

Sample Output

Dataplot generated the following output for Bartlett's test using the GEAR.DAT data set:

               BARTLETT TEST
           (STANDARD DEFINITION)
 NULL HYPOTHESIS UNDER TEST--ALL SIGMA(I) ARE EQUAL
  
 TEST:
    DEGREES OF FREEDOM          =    9.000000
  
    TEST STATISTIC VALUE        =    20.78580
    CUTOFF: 95% PERCENT POINT   =    16.91898
    CUTOFF: 99% PERCENT POINT   =    21.66600
  
    CHI-SQUARE CDF VALUE        =    0.986364
  
   NULL          NULL HYPOTHESIS        NULL HYPOTHESIS
   HYPOTHESIS    ACCEPTANCE INTERVAL    CONCLUSION
 ALL SIGMA EQUAL    (0.000,0.950)         REJECT

Interpretation of Sample Output	We are testing the hypothesis that the group variances are all equal. The output is divided into two sections. The first section prints the value of the Bartlett test statistic, the degrees of freedom (k-1), the upper critical value of the chi-square distribution corresponding to significance levels of 0.05 (the 95% percent point) and 0.01 (the 99% percent point). We reject the null hypothesis at that significance level if the value of the Bartlett test statistic is greater than the corresponding critical value. The second section prints the conclusion for a 95% test. Output from other statistical software may look somewhat different from the above output.
Question	Bartlett's test can be used to answer the following question: Is the assumption of equal variances valid?
Importance	Bartlett's test is useful whenever the assumption of equal variances is made. In particular, this assumption is made for the frequently used one-way analysis of variance. In this case, Bartlett's or Levene's test should be applied to verify the assumption.
Related Techniques	Standard Deviation Plot Box Plot Levene Test Chi-Square Test Analysis of Variance
Case Study	Heat flow meter data
Software	The Bartlett test is available in many general purpose statistical software programs, including Dataplot.