LEVENESTEST

LEVENESTEST(xRange, ConfidenceLevel, NewTableFlag)

• $xRange$ is the set of values for the test.
• $ConfidenceLevel$ is the value from 0 to 1.
• $LogicalValue$ is either TRUE or FALSE. TRUE for getting results in a new cube. FALSE will display results in the same cube.
  
  

DESCRIPTION

• This function used to test the Homogeneity of variances.
• Levene's test is used to test the Samples have equal variances.
• Equal variances across samples is called homogeneity of variance or homoscedasticity.

To perform the Levene's test we need the following assumptions:

1. The Samples from the populations are independent of one another.
2. The population under consideration are Normally Distributed.

For three or more variables the following statistical tests for homogeneity of variances are commonly used:

  1.Levene's Test.
  2.Bartlett Test.

• Levene's test is an alternative to the Bartlett test.
• If the data surely is of normally distributed or nearly to normally distributed then we can use the Bartlett test.
• The Levene's test is defined as

:Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_0 = \sigma_1^2 = \sigma_2^2=...... = \sigma_t^2} . :Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1} =Not all of the variances are equal.

Normally there are three versions of the Levenes test. These are:

1. Use of Mean.
2. Use of Median.
3. Use of 10% of Trimmed Mean.
   
   

The Levene test statistic is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W=\frac{(N-k)\sum_{i=1}^k N_i(Z_i-Z)^2}{(k-1)\sum_{i=1}^k \sum_{i=1}^k \sum_{j=1}^{N_i} (Z_{ij}-Z_i)^2}} .

• where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} is the result of the test.
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} is the number of different groups to which the sampled cases belong.
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} is the total number of cases in all groups.
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_i} is the number of cases in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{th}} group.
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_{ij}} is the value of the measured variable for the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^{th}} case from the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{th}} group.
  
  

Zij is satisfying the one of the following conditions:

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_{ij}=|y_{ij}-\bar{y_i}|} ,Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{y_i}} is the Mean of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{th}} subgroup.
2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_{ij}=|y_{ij}-\bar{y_i}|} ,Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{y_i}} is the Median of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{th}} subgroup
3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_{ij}=|y_{ij}-\bar{y_i}|} ,Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{y_i}} is the 10%Trimmed Mean of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{th}} subgroup.
   
   

Levene's Testing Procedure:

1. checking the assumptions.
2. State the Null(H0) and alternative(H1) hypothesis.
3. Decide on the Significance level (α).
4. Finding the Critical value and Rejection Region.Here Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df_1=t-1} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df_2=N-t} .
5. Compute the Levenes statistic using the formula.
6. Then decision of the value of the test statistic,W is falls in the rejection region or if p-value ≤ α, then reject Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_0} .Otherwise, fail to reject Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_0} . For the computation p-value we have to use the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df_2}
7. Finally we have to conclude that the rejection of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_0} or fail to rejection Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_0} according to the test statistic at the significance level.
   
   

EXAMPLE

X1	X2
3067	3200
2730	2777
2840	2623
2913	3044
2789	2834

=LEVENESTEST(B1:C5, 0.05, 0)

LEVENES TEST

	DATA-0	DATA-1
Median	2840	2834
Mean	2867.8	2895.6
Variance	16923.7	51713.3
Count	5	5
df	4	4

SUMMARY OUTPUT

LEVENESTEST	STATISTICS
W	1.0439235110342522
P-Value	0.3368108674971864
α	0.05
Result	Accept Null Hypothesis that variances are equal for all groups

RELATED VIDEOS

Levene's Test

SEE ALSO

• SIGNTEST
• FRIEDMANTEST

REFERENCES

• Levene's test documentation on Wikipedia
• Levene's test for variance in Excel