Difference between revisions of "Manuals/calci/LEVENESTEST"
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*The Levene test statistic is: | *The Levene test statistic is: | ||
<math>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}</math>. | <math>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}</math>. | ||
| − | *where <math>W< is the result of the test | + | **where <math>W</math> is the result of the test. |
| − | + | **<math>k</math> is the number of different groups to which the sampled cases belong. | |
| + | **<math>N</math> is the total number of cases in all groups. | ||
| + | **<math>N_i</math> is the number of cases in the <math>i^th</math> group. | ||
| + | **<math>Y_{ij} is the value of the measured variable for the <math>j_th</math> case from the <math>i^th</math> group. | ||
| + | *Zij is satisfying the one of the following conditions: | ||
| + | *1.<math>z_{ij}=|y_{ij}-\bar{y_i}|</math>,Where <math>\bar{y_i}</math> is the Mean of the <math>i^th</math> subgroup. | ||
| + | *2.<math>z_{ij}=|y_{ij}-\bar{y_i}|</math>,Where <math>\bar{y_i}</math> is the Median of the <math>i^th</math> subgroup | ||
| + | *3.<math>z_{ij}=|y_{ij}-\bar{y_i}|</math>,Where <math>\bar{y_i}</math> is the 10%Trimmed Mean of the <math>i^th</math> 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 <math>df_1=t-1</math>,<math>df_2=N-t</math>. | ||
| + | *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 <math>H_0</math>.Otherwise, fail to reject <math>H_0</math>. For the computation p-value we have to use the value of <math>df_1</math> and <math>df_2</math>. | ||
| + | *7. Finally we have to conclude that the rejection of <math>H_0</math> or fail to rejection <math>H_0</math> according to the test statistic at the significance level. | ||
| + | |||
| + | ==Example== | ||
Revision as of 04:24, 30 April 2014
LEVENESTEST(xRange,ConfidenceLevel,LogicalValue)
- 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 xRange} is the set of values for 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 ConfidenceLevel} is the value from 0 to 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 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
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}
.
=Not all of the variances are equal.
=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:
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 <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.