Manuals/calci/LEVENESTEST

LEVENESTEST(xRange,ConfidenceLevel,LogicalValue)

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.

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

- 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 $i^{th}$ group.
- $Y_{ij}isthevalueofthemeasuredvariableforthe<math>j^{th}$ case from the $i^{th}$ 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 $i^{th}$ subgroup.
2. $z_{ij}=|y_{ij}-{\bar {y_{i}}}|$ ,Where ${\bar {y_{i}}}$ is the Median of the $i^{th}$ subgroup
3. $z_{ij}=|y_{ij}-{\bar {y_{i}}}|$ ,Where ${\bar {y_{i}}}$ is the 10%Trimmed Mean of the $i^{th}$ 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 $df_{1}=t-1$ , $df_{2}=N-t$ .
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.