Q 572. What is the purpose of utilising Levene's Test and how does it differ from Bartlett's Test in assessing the equality of variances among multiple groups? Explain using example(s).

 

Note for website visitors -

• This platform hosts two questions per week, one on Tuesday and the other on Friday. 
• All previous questions can be found here: https://www.benchmarksixsigma.com/forum/lean-six-sigma-business-excellence-questions/. 
• To participate in the current question, please visit the forum homepage at https://www.benchmarksixsigma.com/forum/.
• The question will be open until the following Tuesday or Friday evening at 5 PM Indian Standard Time, depending on the launch day. 
• Responses will not be visible until they are reviewed, and only non-plagiarised answers with less than 5-10% plagiarism will be approved. 
• If you are unsure about plagiarism, please check your answer using a plagiarism checker tool such as https://smallseotools.com/plagiarism-checker/ before submitting. 
• All correct answers shall be published, and the top-rated answer will be displayed first. The author will receive an honourable mention in our Business Excellence dictionary at https://www.benchmarksixsigma.com/forum/business-excellence-dictionary-glossary/ along with the related term.

• Some people seem to be using ChatGPT to find forum answers. This is a risky approach as ChatGPT is error prone as our questions are application questions (they are never straightforward). Have a look at this funny example - https://www.benchmarksixsigma.com/forum/topic/39458-using-ai-to-respond-to-forum-questions/

 

Levene's Test:  It is a statistical test used to assess if the variances of two or more groups are statistically different. The test determines if the differences in the variances of the groups are more than what would be expected by chance. It is commonly used in analysis of variance (ANOVA) to confirm the assumption of homogeneity of variance across different groups. If significant differences in variances are found through Levene's test, it may impact the results and interpretation of ANOVA, and alternative approaches may be necessary.

 

Purpose: The purpose of using Levene's Test is to assess whether the hypothesis of equal variance between groups is valid or not. This helps to determine whether a parametric statistical test (such as a t-test or ANOVA) which assumes equal variances can be used to analyze the data or whether a non-parametric test which does not assume equal variances should be used instead. Overall, the purpose of utilizing Levene's Test is to ensure the validity and accuracy of statistical analyses.

 

Bartlett’s Test:  It is also a statistical test used to determine whether the variances of two or more populations are equal. Specifically, it tests the null hypothesis that the variances are equal against the alternative hypothesis that at least one variance is different from the others. This test is commonly used in analysis of variance (ANOVA) to ensure that the assumption of homogeneity of variances is met before performing further tests. Bartlett's test is sensitive to departures from normality and can give erroneous results if the data are not normally distributed.

 

Equality of variance:  This refers to the assumption that the variance of the population is the same across different groups or samples being compared. This assumption is important in many statistical methods, such as analysis of variance (ANOVA) and t-tests, because violating the assumption can lead to incorrect conclusions or inaccurate results.

 

Hypotheses:  When performing Levene's Test or Bartlett's Test to assess homogeneity of variance, we are testing for two hypotheses. These can be simply stated as follows:

Null Hypothesis – which assumes that the variances are equal across all samples or groups;

Alternative Hypothesis – which proposes that the variances are not equal across all samples or groups.

 

To test for equality of variance, a common method is to use either Levene's test or Bartlett's test. These tests compare the variances of the groups or samples being compared and calculate a p-value. If the p-value is less than a chosen significance level (such as 0.05), then the null hypothesis of equal variances is rejected, indicating that the variances are significantly different and that the assumption of equality of variance has been violated. In such cases, alternative statistical methods that do not rely on the assumption of equal variances can be used instead.

 

By above definitions, it is clear that both Levene's Test and Bartlett's Test, are statistical methods used to determine the equality of variances among multiple groups. However, they differ in their assumptions about the distribution of the data. Levene's Test is a robust method that is less sensitive to outliers, whereas Bartlett's Test assumes that the data is normally distributed. The purpose of utilising Levene's Test is to determine whether the variances of the groups are equal or not. If the variances are not equal, it can have an impact on the results of further statistical analyses, such as ANOVA. By using Levene's Test, researchers can ensure that they are using appropriate statistical methods to analyse their data.

 

For example, let's say a researcher wants to compare the math test scores of three different high schools. Before running an ANOVA to compare the means of the three schools, they should first use Levene's Test to determine whether the variances of the scores are equal. If the variances are not equal, they may need to use a different statistical method, such as Welch's ANOVA, which is designed to handle unequal variances. In summary, Levene's Test is a valuable tool for researchers to assess the equality of variances among multiple groups. While it differs from Bartlett's Test in its assumptions about the data, it provides a robust and reliable method for ensuring that appropriate statistical methods are used in further analyses.

Levene's Test vs Bartlett's Test

 

Levene's test and Bartlett's test are both statistical tests used to assess the homogeneity of variances in a dataset.

Both are commonly employed in the field of statistics, especially in analysis of variance (ANOVA) or regression analysis.

Levene's Test: Levene's test is called parametric test that evaluates whether the variances of different groups in a dataset are significantly different. Levene’s test is also based on the absolute deviations from the group means. Levene's test is less sensitive to departures from normality.	Bartlett's test, making it more robust when dealing with non-normal data.
Also, the null hypothesis for Levene's test states that the variances of all groups should equal. If the p-value associated with Levene's test is below a predetermined significance level (e.g., 0.05), we reject the null hypothesis and conclude that there is evidence of unequal variances among the groups.	Bartlett's Test: Bartlett's test is also a parametric test used to assess the homogeneity of variances, but it assumes that the data in each group are normally distributed. It is based on the logarithms of the variances, making it sensitive to departures from normality. Thus, Bartlett's test is more appropriate when dealing with normally distributed data.
	The null hypothesis for Bartlett's test states that the variances of all groups are equal. If the p-value associated with Bartlett's test is below a predetermined significance level (e.g., 0.05), we reject the null hypothesis and conclude that there is evidence of unequal variances among the groups.

 

·         In summary, Levene's test is more robust to non-normality, while Bartlett's test assumes normality. The choice between the two tests depends on the distributional properties of the data and the assumptions of the analysis being performed.

·         If the data are normally distributed, Bartlett's test is suitable, but if the data are non-normal, Levene's test is often preferred.

Levene’s test and Bartellet’s test is used to test if samples have equal variances. Certain tests like ANOVA , assume that variances are same across the sample. These tests can help in verifying the assumption.

Similarities between Levene’s and Bartlett’s Test:

1.    Both have the same Null and Alternate Hypothesis. i.e.

Null = Variance are equal

Alternate = Variances are not equal

Differences

Levene’s Test	Bartlett’s Test
Alternative to Bartlett’s test	Alternative to Levene’s test
Works if sample data is not normal.	Assumes the sample data is normal
This test is robust in case the data set has ordinal data.
It uses the median of the sample data sets.	It uses the mean of the sample data sets.
Non parametric test	Parametric test

 

Example:

Suppose a researcher is interested in comparing the effectiveness of three different trainings (A, B, and C) in reducing defect density. The defect density is measured as defects / hr, Higher defect density indicates less effective training. The researcher collects defect density data from three groups of participants: Group A (n = 30), Group B (n = 25), and Group C (n = 28).

To assess the equality of variances, the researcher can perform both Levene's and Bartlett's tests based on data normality. The analysis can be as below:

Ho = The variances between the 3 groups are equal

Ha = The variances between the 3 groups are not equal

If p value > 0.05, the variances between the 3 groups are equal and if p value < 0.05 , the variances between the 3 groups are not equal.

If the data set is normal then the researcher should use Bartlett’s test and if the data set is not normal, the researcher should use Levene’s test.

Levene's test and Bartlett's test are both statistical tests used to assess the equality of variances among multiple groups. They are commonly employed in the field of statistics, including in the analysis of data. Let's examine the purpose of utilizing Levene's test and how it differs from Bartlett's test, along with relevant examples from the service industry.

 

The purpose of Levene's test is to determine if the variances of different groups or samples are equal or not. It is a robust test, meaning it is less sensitive to departures from normality compared to Bartlett's test. Levene's test is typically used when the assumption of normality may not hold or when there is concern about outliers influencing the test results. It is based on absolute deviations from the group means, rather than squared deviations as in Bartlett's test.

 

On the other hand, Bartlett's test is also used to evaluate the equality of variances among groups, but it assumes that the data are normally distributed. It is more sensitive to departures from normality compared to Levene's test. Bartlett's test calculates the sum of squared deviations from the group means, which can be affected by outliers or non-normality.

 

Let's consider an example from the service industry to illustrate the difference between the two tests. Suppose we want to compare the customer satisfaction scores of three different airlines: Airline A, Airline B, and Airline C. We collect data from a random sample of customers from each airline, and we want to assess if there is a difference in the variances of customer satisfaction scores among the three airlines.

 

Using Levene's test, we calculate the absolute deviations of individual scores from their respective group means for each airline. The test will then evaluate whether these deviations are significantly different among the three groups, indicating unequal variances.

 

Using Bartlett's test, we calculate the squared deviations of individual scores from their respective group means for each airline. The test will assess whether these squared deviations significantly differ among the three groups, indicating unequal variances. However, Bartlett's test assumes that the data are normally distributed, so if the data deviate from normality, the test results may not be reliable.

 

In summary, Levene's test is more suitable when the assumption of normality may be violated or when there are concerns about outliers. It is based on absolute deviations and is considered a robust test. On the other hand, Bartlett's test assumes normality and is based on squared deviations. It may be more appropriate when the data are known or assumed to be normally distributed.

Levene’s test helps to assess to test the variance across the groups under study. This test helps to establish if the datasets that we are studying conforms to the assumption of homogenous variance, before conducting Analysis Of Variance tests. Levene’s test can be an alternate for Barlett’s test, if being sensitive to normality of the data is not a focus. If we have confidence in data coming from a normal distribution then Barlett’s test can give much better performance of the test. For example lets say we want to assess the across various students from different disciplines of engineering on their reading duration in a day. We are looking at population data sets that may not follow a normal distribution. In this case Levene’s test will help to validate the hypothesis. Null hypothesis will that variance across these groups of student will be same. Alternate hypothesis will be that the variance across these groups on reading hours per day will not be same.

Levene's test is widely used in many statistical analyses, including analysis of variance (ANOVA) and regression analyses, making it a valuable tool in data analysis. Its ability to account for non-normality and outliers in the data make it a preferred option over other tests that assume normal distribution.

 

When Comparing to Bartlett's Test, Levene's Test is more robust and does not rely on the assumption of normality. This makes it a better choice when the underlying distribution is not known or is suspected to be non-normal.
Levene's test uses the absolute deviations from the group means as the metric for assessing the equality of variances. The test compares the average absolute deviation of each observation from its group mean across the different groups. If the average absolute deviations are similar across the groups, it suggests that the variances are equal. Levene's test is more robust to moderate deviations from normality and is less sensitive to outliers 

 

Levene's test is generally more robust than Bartlett's test when the assumption of normality is violated or when the sample sizes are unequal. Bartlett's test is more sensitive to departures from normality and may provide misleading results in such cases Also, Bartlett's test is known to be sensitive to unequal sample sizes. Ie If the sample sizes are significantly different across the groups, Bartlett's test may lead to inaccurate conclusions. hence Levene's test is less affected by unequal sample sizes.

Example:

 

If we are conducting a study to compare the heights of three groups of people or category, 1)Men, 2)Women 3) Children we wanted to make sure that the variances of the heights are equal before you conduct a one-way ANOVA.

We could use Levene's test to assess the equality of variances. The results of the Levene's test might show that the variances are equal, which would mean that we could proceed with the one-way ANOVA.
Alternatively, we can use Bartlett's test also to assess the equality of variances. The results of the Bartlett's test might show that the variances are not equal, which would mean that we would need to use a more robust statistical test, such as a non-parametric test.

Generally speaking, Levene's test is a good choice for assessing the equality of variances when the data is not normally distributed. Bartlett's test is a good choice for assessing the equality of variances when the data is normally distributed.
 

 

 

 

Levene's test and Bartlett's test are used to test the equality/homogeneity of variances for two or more samples before performing ANOVA.  

 

If the sample data is normal, it is preferred to choose the Bartlett's test, as Bartlett's test is more sensitive to the violations of normality when compared to Levene's test. Irrespective of data being normal or not normal, Levene test can be conducted to check the homogeneity of the variances. 

 

Example – Attritions recorded for a process X in an organization across all locations for 2022.

 

Month	Location A	Location B	Location C	Location D	Location E
January	42	8	32	56	51
February	45	14	33	60	58
March	51	25	41	58	57
April	61	43	52	62	67
May	69	54	62	63	81
June	76	64	72	68	88
July	78	71	77	69	94
August	78	69	75	71	93
September	72	58	68	69	85
October	62	47	58	67	74
November	51	29	47	61	61
December	44	16	35	58	55

 

Assumptions: Considering data is normal and the values are independent.

H0: Variances are equal across all locations.

Ha: Variances are not equal across all locations.

 

SUMMARY
Locations	Count	Sum	Average	Variance
A	12	729	60.75	190.3864
B	12	498	41.5	504.6364
C	12	652	54.33333333	279.697
D	12	762	63.5	26.09091
E	12	864	72	248.3636

 

 

Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	6223.667	4	1555.916667	6.227781	0.000331	2.539689
Within Groups	13740.92	55	249.8348485
Total	19964.58	59

 

P value using Levene’s test – 0.000331

P value using Bartlett's test – 0.00104

 

Inference -

As p value < 0.05 (significance level), we cannot accept the null hypothesis of equal variances across locations.  

Ho rejected and Ha accepted.

Levene's test is used to test if a specified number of samples, which have a normal distribution, have the same variance. This test is used to identify the homogeneity of the samples. During analyse phase, sometimes, we assume that the variances are equal for different sets of samples. Levene's test is used to validate this assumption. It is done before ANOVA. If Levene's test is failed, that is, if we are not able to validate that our assumption that variances are equal is not true, then ANOVA should NOT be conducted. 

 

Bartlett's test is also used to check if variances among multiple sample sets are equal. Levene's assumes that the data is normal, whereas for Bartlett's normality assumption is not a requirement. 

 

Levene's test is less sensitive to violations of normality whereas Bartlett's isnt.

 

On basis of the data & the requirement, either of the test will be used. 

 

 

 

 

 

 

The Levene’s test is generally used to test for equality of variance in a dataset. It is used to determine if two or more samples have equal variances. If the results of the test indicate that the samples do not have equal variances, then it means that one sample has different variance than other samples. An advantage of Levene’s test is, it is highly stable for the data set which is not normally distributed.

Null Hypothesis: - Data Groups have equal variances.

Alternate Hypothesis: - Data Groups have different variances.

If the p-value for the Levene’s test is greater than .05, then the variances are not significantly different from each other and assumption of equal variance is met however If the p-value for the Levene's test is less than .05, then variances for one or more sample data set is not equal.

Difference Between the Levene’s test and Bartlett's Test: -

Both tests are used to test the assumptions of variance equality. However, the main difference is Bartlett test requires data of each group to be normally distributed and Levene’s test to be used when data is not normally distributed. For Normality check Anderson Darling can be performed.

Example: -

Data of cost of tickets sold in thousands in as how for a month are tabulated for five different competent Circus groups.

The P value for Levene’s test and bartlett test are highly different as Data is not normally distributed and Levene’s test is more stable for non-normal distribution.

Gem	Joyride	Starlite	Fantasy	Fun
39.3	23.3	7.3	10	36
42	60	11.3	180	40.7
40.7	150	18.7	36	40
43.3	36.7	30.7	120	46.7
44	70	38	48	56
47.3	110	44.7	52.7	60.7
48	53.3	49.3	54	64.7
49.3	52	48	54	64
48	20	40.7	50	58.7
46.7	40.7	33.3	43.3	51.3
42.7	5	21.3	36	42.7
40.7	80	12.7	150	38.7

 

 Levene’s Test Steps and Result in Minitab: -

 

Levene's Test.docx

Levene’ s Test is statistical test to compare the variances of two or more samples and test the homogeneity of variance . Specifically To get to know if there are significant differences between sample variances and can be used as a precursor while conducting inferential tests such as  ANOVA.

 

Levene’ s Test tells us if samples taken for testing have equal variances. It tests the equality of variance whether they are close enough to be called as “Homogeneity of variances”

As generally ,ANOVA assumption , that variances across samples are equal – Levene’s test is used to verify this assumption. Ideally one would want a non- significant result for this test – which means assumption of equal variances shall hold good before proceeding to ANOVA.

 Let us say 2 groups , 1.Control group & 2. Experimental group.

In case of t – test we want that the variances shall be different in the 2 groups.( as of course, because one group has treatment).while independent sample t -tests takes for granted that variances are equal , Levene's test really tests the equality.

 

In case of Levene’s Test, we want variances to be the same. we would like it to be non-significant because we don’t want difference in variances.

 

If Levene’s test is significant ( p < 0.05) – it means there is no equality in variance or sufficient variance available between 2 or more samples. Levene’s test is highly useful to make sure that our assumptions equality of variances are scrutinized /judged.

In Hypothesis testing ,while comparing 2 or more samples with each other–Here ‘s an example of comparing 2 or more groups.

 

Example :

Let’s say a Human resource specialist doing his doctorate wants to know Job satisfaction index % of 2 groups of genders Male & female. ( another group Transgenders may also be included – if the company of research focusses on Diversity, inclusivity).

§Here running one-way ANOVA, based on mean(research studies running Levene’s test can have p<0.01 if sample size is huge)

The null hypothesis is that there is homogeneity in variance. Alternate hypothesis is there variances are unequal

However for this case we will assume  alpha value of 0.05, a significant result here < 0.05 indicates that we are not inline with homogeneity of variance.

In the other case a non-significant result would mean that we are inline with our assumption of homogeneity of variance between 2 or more groups..

Interesting answers from all respondents. Best answer has been provided by Vijay Tomar.