Difference between revisions of "AP Statistics Curriculum 2007 NonParam VarIndep"
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==Differences of Variances of Independent Samples== | ==Differences of Variances of Independent Samples== | ||
| − | It is frequently necessary to test if ''k'' samples have equal variances. Equal variances across samples | + | It is frequently necessary to test if ''k'' samples have equal variances. Equal variances across samples are called ''homogeneity of variances''. Some statistical tests, for example the [[AP_Statistics_Curriculum_2007_ANOVA_1Way | analysis of variance]], assume that variances are equal across groups or samples. |
==Approach== | ==Approach== | ||
The (modified) Fligner-Killeen test provides the means for studying the homogeneity of variances of ''k'' populations { <math>X_{i,j}</math>, for <math>1\leq i \leq n_j</math> and <math>1\leq j \leq k</math>}. The test jointly ranks the absolute values of <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 provides the means for studying the homogeneity of variances of ''k'' populations { <math>X_{i,j}</math>, for <math>1\leq i \leq n_j</math> and <math>1\leq j \leq k</math>}. The test jointly ranks the absolute values of <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 <math>\Phi(.)</math> is the [[AP_Statistics_Curriculum_2007_Normal_Std | | + | 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 | cumulative distribution function for Normal distribution]]. The Fligner-Killeen test is sometimes also called the ''median-centering Fligner-Killeen test''. |
* '''Fligner-Killeen test statistics''': | * '''Fligner-Killeen test statistics''': | ||
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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. | ||
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==Computational Resources: Internet-based SOCR Tools== | ==Computational Resources: Internet-based SOCR Tools== | ||
| − | + | Still under development. | |
==Examples== | ==Examples== | ||
| − | + | Suppose we wanted to study whether the variances in certain time period (e.g., 1981 to 2006) of the consumer-price-indices (CPI) of several items were significantly different. We can use the [[SOCR_Data_Dinov_021808_ConsumerPriceIndex | SOCR CPI Dataset]] to answer this question for the '''Fuel''', '''Oil''', '''Bananas''', '''Tomatoes''', '''Orange Juice''', '''Beef''' and '''Gasoline''' items. | |
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<hr> | <hr> | ||
| − | ==Alternative tests of Variance | + | ==Alternative tests of Variance Homogeneity== |
* [http://en.wikipedia.org/wiki/Levene_test Levene Test] | * [http://en.wikipedia.org/wiki/Levene_test Levene Test] | ||
* [http://en.wikipedia.org/wiki/Bartlett's_test Bartlett Test] | * [http://en.wikipedia.org/wiki/Bartlett's_test Bartlett Test] | ||
| − | * [http://en.wikipedia.org/wiki/Brown-Forsythe_test Brown- | + | * [http://en.wikipedia.org/wiki/Brown-Forsythe_test Brown-Forsythe test] |
==References== | ==References== | ||
Revision as of 00:31, 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 are 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 provides the means for studying the homogeneity of variances of k populations { \(X_{i,j}\), for \(1\leq i \leq n_j\) and \(1\leq j \leq k\)}. The test jointly ranks the absolute values of \(|X_{i,j}-\tilde{X_j}|\) and assigns increasing scores \(a_{N,i}=\Phi^{-1}({1 + {i\over N+1} \over 2})\), based on the ranks of all observations, see the Conover, Johnson, and Johnson (1981) reference below.
In this test, \(\tilde{X_j}\) is the sample median of the jth population, and \(\Phi(.)\) is the cumulative distribution function for Normal distribution. The Fligner-Killeen test is sometimes also called the median-centering Fligner-Killeen test.
- Fligner-Killeen test statistics:
\[x_o^2 = {\sum_{j=1}^k {n_j(\bar{A_j} -\bar{a})^2} \over V^2}\],
- where \(\bar{A_j}\) is the mean score for the jth sample, a is the overall mean score of all \(a_{N,i}\), and \(V^2\) is the sample variance of all scores.
That is: \[N=\sum_{j=1}^k{n_j}\], \[\bar{A_j} = {1\over n_j}\sum_{i=1}^{n_j}{a_{N,m_i}}\], \[\bar{a} = {1\over N}\sum_{i=1}^{N}{a_{N,i}}\], \[V^2 = {1\over N-1}\sum_{i=1}^{N}{(a_{N,i}-\bar{a})^2}\].
- 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 \[x_o^2 \sim \chi_{(k-1)}^2\].
- 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.
Computational Resources: Internet-based SOCR Tools
Still under development.
Examples
Suppose we wanted to study whether the variances in certain time period (e.g., 1981 to 2006) of the consumer-price-indices (CPI) of several items were significantly different. We can use the SOCR CPI Dataset to answer this question for the Fuel, Oil, Bananas, Tomatoes, Orange Juice, Beef and Gasoline items.
Alternative tests of Variance Homogeneity
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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