AP Statistics Curriculum 2007 NonParam VarIndep
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The (modified) Fligner-Killeen test for homogeneity of variances of k populations jointly ranks the absolute values <math>|X_{i,j}-\tilde{X_j}|</math> and assigns increasing '''scores''' <math>a_{N,i}=\Phi^{-1}({1 + {i\over N+1} \over 2})</math>, based on the ranks of all observations, see the Conover, Johnson, and Johnson (1981) reference below. | The (modified) Fligner-Killeen test for homogeneity of variances of k populations jointly ranks the absolute values <math>|X_{i,j}-\tilde{X_j}|</math> and assigns increasing '''scores''' <math>a_{N,i}=\Phi^{-1}({1 + {i\over N+1} \over 2})</math>, based on the ranks of all observations, see the Conover, Johnson, and Johnson (1981) reference below. | ||
- | In this test, <math>\tilde{X_j}</math> is the sample median of the ''j<sup>th</sup>'' population, and \Phi(.) is the [[AP_Statistics_Curriculum_2007_Normal_Std | cummulative distribution function for Normal distirbution]]. The Fligner-Killeen test is sometimes also called the ''median-centering Fligner-Killeen test''. | + | In this test, <math>\tilde{X_j}</math> is the sample median of the ''j<sup>th</sup>'' population, and <math>\Phi(.)</math> is the [[AP_Statistics_Curriculum_2007_Normal_Std | cummulative distribution function for Normal distirbution]]. The Fligner-Killeen test is sometimes also called the ''median-centering Fligner-Killeen test''. |
* '''Fligner-Killeen test statistics''': | * '''Fligner-Killeen test statistics''': | ||
- | : <math>x_o^2 = {\sum_{j=1}^k {n_j(\bar{A_j} -\bar{a})^2} \over V^2</math>, | + | : <math>x_o^2 = {\sum_{j=1}^k {n_j(\bar{A_j} -\bar{a})^2} \over V^2}</math>, |
: where <math>\bar{A_j}</math> is the mean score for the ''j<sup>th</sup>'' sample, ''a'' is the overall mean score of all <math>a_{N,i}</math>, and <math>V^2</math> is the sample variance of all scores. | : where <math>\bar{A_j}</math> is the mean score for the ''j<sup>th</sup>'' sample, ''a'' is the overall mean score of all <math>a_{N,i}</math>, and <math>V^2</math> is the sample variance of all scores. | ||
That is: | That is: | ||
: <math>N=\sum_{j=1}^k{n_j}</math>, | : <math>N=\sum_{j=1}^k{n_j}</math>, | ||
- | : <math>\bar{A_j} = {1\over n_j}\sum_{i=1}^{n_j}{a_{N, | + | : <math>\bar{A_j} = {1\over n_j}\sum_{i=1}^{n_j}{a_{N,m_i}}</math>, |
: <math>\bar{a} = {1\over N}\sum_{i=1}^{N}{a_{N,i}}</math>, | : <math>\bar{a} = {1\over N}\sum_{i=1}^{N}{a_{N,i}}</math>, | ||
: <math>V^2 = {1\over N-1}\sum_{i=1}^{N}{(a_{N,i}-\bar{a})^2}</math>. | : <math>V^2 = {1\over N-1}\sum_{i=1}^{N}{(a_{N,i}-\bar{a})^2}</math>. | ||
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* '''Note''': | * '''Note''': | ||
- | : Conover, Johnson, and Johnson (1981) carried a simulation comparing different ''variance homogeneity tests'' and reported that the ''modified Fligner-Killeen test'' is most robust against departures from normality. | + | : Conover, Johnson, and Johnson (1981) carried a simulation comparing different ''variance homogeneity tests'' and reported that the ''modified Fligner-Killeen test'' is most robust against departures from normality. |
==Model Validation== | ==Model Validation== |
Revision as of 04:28, 16 March 2008
General Advance-Placement (AP) Statistics Curriculum - Variances of Two Independent Samples
Contents |
Differences of Variances of Independent Samples
It is frequently necessary to test if k samples have equal variances. Equal variances across samples is called homogeneity of variances. Some statistical tests, for example the analysis of variance, assume that variances are equal across groups or samples.
Approach
The (modified) Fligner-Killeen test for homogeneity of variances of k populations jointly ranks the absolute values and assigns increasing scores , based on the ranks of all observations, see the Conover, Johnson, and Johnson (1981) reference below.
In this test, is the sample median of the j^{th} population, and Φ(.) is the cummulative distribution function for Normal distirbution. The Fligner-Killeen test is sometimes also called the median-centering Fligner-Killeen test.
- Fligner-Killeen test statistics:
- ,
- where is the mean score for the j^{th} sample, a is the overall mean score of all a_{N,i}, and V^{2} is the sample variance of all scores.
That is:
- ,
- ,
- ,
- .
- Fligner-Killeen probabilities:
For large sample sizes, the modified Fligner-Killeen test statistic has an asymptotic chi-square distribution with (k-1) degrees of freedom
- .
- Note:
- Conover, Johnson, and Johnson (1981) carried a simulation comparing different variance homogeneity tests and reported that the modified Fligner-Killeen test is most robust against departures from normality.
Model Validation
TBD
Computational Resources: Internet-based SOCR Tools
TBD
Examples
TBD
Hands-on Activities
TBD
Alternative tests of Variance Homegeneity
References
- Conover, W. J., Johnson, M.E., and Johnson M. M. (1981), A comparative study of tests for homogeneity of variances, with applications to the outer continental shelf bidding data. Technometrics 23, 351-361.
- SOCR Home page: http://www.socr.ucla.edu
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