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 do the Levenes 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

$H_{0}=\sigma _{1}^{2}=\sigma _{2}^{2}=......=\sigma _{t}^{2}$.
$H_{1}$=Not all of the variances are equal. 

• Normally there are three versions of the Levenes test.
• There are
• 1.Use of Mean.
• 2.Use of Median.
• 3.Use of 10% of Trimmed Mean.
• The Levene test statistic is:

$W={\frac {(N-k)\sum _{i=1}^{k}N_{i}(Z_{i}-Z)^{2}}{(k-1)\sum _{i=1}^{k}\sum _{i=1}^{k}\sum _{j=1}^{N_{i}}(Z_{ij}-Z_{i})^{2}}}$.

• • where $W$ is the result of the test.
  • $k$ 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 <math>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,.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
F-Test	0.38245649772919
a	0.05
F	0.32726010523405
p 1 & 2 Tail	0.1524069466470822	0.3048138932941644

RELATED VIDEOS

Levene's Test

=SEE ALSO

• SIGNTEST
• FRIEDMANTEST

REFERENCES

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