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

$H_{0}=\sigma _{1}^{2}=\sigma _{2}^{2}=......=\sigma _{t}^{2}$.
$H_{1}$=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.
  • $N$ is the total number of cases in all groups.
  • $N_{i}$ is the number of cases in the $i^{t}h$ group.
  • $Y_{ij}isthevalueofthemeasuredvariableforthe<math>j_{t}h$ case from the $i^{t}h$ group.
• Zij is satisfying the one of the following conditions:
• 1.$z_{ij}=|y_{ij}-{\bar {y_{i}}}|$,Where ${\bar {y_{i}}}$ is the Mean of the $i^{t}h$ subgroup.
• 2.$z_{ij}=|y_{ij}-{\bar {y_{i}}}|$,Where ${\bar {y_{i}}}$ is the Median of the $i^{t}h$ subgroup
• 3.$z_{ij}=|y_{ij}-{\bar {y_{i}}}|$,Where ${\bar {y_{i}}}$ is the 10%Trimmed Mean of the $i^{t}h$ 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 $df_{1}=t-1$,$df_{2}=N-t$.
• 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 $H_{0}$.Otherwise, fail to reject $H_{0}$. For the computation p-value we have to use the value of $df_{1}$ and $df_{2}$.
• 7. Finally we have to conclude that the rejection of $H_{0}$ or fail to rejection $H_{0}$ according to the test statistic at the significance level.