Manuals/calci/LEVENESTEST

LEVENESTEST(xRange,ConfidenceLevel,LogicalValue)

• $xRange$ is the set of values for the test.
• $ConfidenceLevel$ is the value from 0 to 1.
• $LogicalValue$ is either TRUE or FALSE.

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 independently 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

$H_{0}=\sigma _{1}^{2}=\sigma _{2}^{2}=......=\sigma _{t}^{2}$.

$H_{1}=Notallofthevariancesareequal$.

• 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 <math>W< is the result of the test, k is the number of different groups to which the sampled cases belong,N is the total number of cases in all groups,Ni is the number of cases in the th group, Yij is the value of the measured variable for the jth case from the ith group.Zij is satifying the one of the following conditions:1.(Formula)Where yi(bar)is the Mean of the ith subgroup.2.(Formula)Where yi(bar)is the Median of the ith subgroup3.(Formula)Where yi(bar)is the 10%Trimmed Mean of the ith 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 df1=t-1 ,df2=N-t.5.Compute the Levenes statistic using the formula.6.Then decision of the value of the test statistic, FLevene, falls in the rejection region or if p-value ≤ α, then reject H0 ;

otherwise, fail to reject H0 .For the computation p-value we have to use the value of df1 and df2.7. Finally we have to conclude that the rejection of H0 or fail to rejection H0 according to the test statistic at the significance level.