AP Statistics Curriculum 2007 NonParam ANOVA - Revision history

Tdlee: Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

2020-03-03T17:21:21Z

Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

Dinov at 13:58, 18 June 2014

2014-06-18T13:58:29Z

Jenny at 21:06, 28 June 2010

2010-06-28T21:06:32Z

Jenny at 17:42, 28 June 2010

2010-06-28T17:42:12Z

PriscillaChui: /* Notes */

2008-03-10T23:19:03Z

‎Notes

PriscillaChui: /* Calculations */

2008-03-10T23:18:10Z

‎Calculations

PriscillaChui: /* The Kruskal-Wallis Test */

2008-03-10T23:17:06Z

‎The Kruskal-Wallis Test

PriscillaChui: /* Motivational Example */

2008-03-10T23:16:16Z

‎Motivational Example

IvoDinov: /* The Kruskal-Wallis Test using SOCR Analyses */

2008-03-03T17:52:08Z

‎The Kruskal-Wallis Test using SOCR Analyses

IvoDinov at 20:46, 2 March 2008

2008-03-02T20:46:52Z

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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

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← Older revision		Revision as of 13:58, 18 June 2014
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−	==[[AP_Statistics_Curriculum_2007 \| General Advance-Placement (AP) Statistics Curriculum]] - ~~Means~~ of Several Independent Samples==	+	==[[AP_Statistics_Curriculum_2007 \| General Advance-Placement (AP) Statistics Curriculum]] - Medians of Several Independent Samples==

	In this section we extend the [[EBook#Chapter_XI:_Analysis_of_Variance_.28ANOVA.29 \| multi-sample inference which we discussed in the ANOVA section]], to the situation where the [[AP_Statistics_Curriculum_2007_ANOVA_1Way#ANOVA_Conditions\| ANOVA assumptions]] are invalid. Hence we use a non-parametric analysis to study differences in centrality between two or more populations.		In this section we extend the [[EBook#Chapter_XI:_Analysis_of_Variance_.28ANOVA.29 \| multi-sample inference which we discussed in the ANOVA section]], to the situation where the [[AP_Statistics_Curriculum_2007_ANOVA_1Way#ANOVA_Conditions\| ANOVA assumptions]] are invalid. Hence we use a non-parametric analysis to study differences in centrality between two or more populations.

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 ===Motivational Example===
-Suppose four groups of students were randomly assigned to be taught with four different techniques, and their achievement test scores were recorded. Are the distributions of test scores the same, or do they differ in location? The data is presented in the table below.
+Suppose four groups of students are randomly assigned to be taught with four different techniques, and their achievement test scores are recorded. Are the distributions of test scores the same, or do they differ in location? The data is presented in the table below.
 <center>

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 Let ''N'' be the total number of observations, then <math>N = \sum_{i=1}^k {n_i}</math>.
-Let <math>R(X_{ij})</math> denote the rank assigned to <math>X_{ij}</math> and let <math>R_i</math> be the sum of ranks assigned to the <math>i^{th}</math> sample.
+Let <math>R(X_{ij})</math> denotes the rank assigned to <math>X_{ij}</math> and let <math>R_i</math> be the sum of ranks assigned to the <math>i^{th}</math> sample.
 : <math>R_i = \sum_{j=1}^{n_i} {R(X_{ij})}, i = 1, 2,  ... , k</math>.

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 ==Notes==
-* The [http://en.wikipedia.org/wiki/Friedman_test Friedman Fr test] is the rank equivalent of the randomized block design alternative to the [[AP_Statistics_Curriculum_2007_ANOVA_2Way |two-way analysis of variance F test]]. [[SOCR_EduMaterials_AnalysisActivities_Friedman | The SOCR Friedman Test activity ]] demonstrates how  to use [http://socr.ucla.edu/htmls/SOCR_Analyses.html SOCR Analyses] to compute the Friedman test statistics and p-value.
+* The [http://en.wikipedia.org/wiki/Friedman_test Friedman Fr Test] is the rank equivalent of the randomized block design alternative to the [[AP_Statistics_Curriculum_2007_ANOVA_2Way |Two-Way Analysis of Variance F Test]]. [[SOCR_EduMaterials_AnalysisActivities_Friedman | The SOCR Friedman Test Activity ]] demonstrates how to use [http://socr.ucla.edu/htmls/SOCR_Analyses.html SOCR Analyses] to compute the Friedman Test statistics and p-value.
 ==References==

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 * Note: If there are no ties, then the test statistic is reduced to:
 ::<math>T = \left(12 / N(N+1) \right) \sum_{i=1}^{k} {{R_i}^{2}} / {n_i} {-}  3 \left(N+1\right)</math>.
 However, the SOCR implementation allows for the possibility of having ties; so it uses the non-simplified, exact method of computation.
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 The SOCR computation employs the exact method instead of the approximate one (Conover 1980), since computation is easy and fast to implement and the exact method is somewhat more accurate.
-===The Kruskal-Wallis Test using SOCR Analyses===
+===The Kruskal-Wallis Test Using SOCR Analyses===
-It is much quicker to use [http://socr.ucla.edu/htmls/SOCR_Analyses.html SOCR Analyses] to compute the statistical significance of this test. This [[SOCR_EduMaterials_AnalysisActivities_KruskalWallis | SOCR KruskalWallis Test activity]] may also be helpful in understanding how to use this test in SOCR.
+It is much quicker to use [http://socr.ucla.edu/htmls/SOCR_Analyses.html SOCR Analyses] to compute the statistical significance of this test. This [[SOCR_EduMaterials_AnalysisActivities_KruskalWallis | SOCR KruskalWallis Test Activity]] may also be helpful in understanding how to use this test in SOCR.
 For the teaching-methods example above, we can easily compute the statistical significance of the differences between the group medians (centers):

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 ==The Kruskal-Wallis Test==
-'''Kruskal-Wallis one-way analysis of variance''' by ranks is a non-parametric method for testing equality of two or more population medians. Intuitively, it is identical to a [[AP_Statistics_Curriculum_2007_ANOVA_1Way | one-way analysis of variance]] with the raw data (observed measurements) replaced by their ranks.
+'''Kruskal-Wallis One-Way Analysis of Variance''' by ranks is a non-parametric method for testing equality of two or more population medians. Intuitively, it is identical to a [[AP_Statistics_Curriculum_2007_ANOVA_1Way | One-Way Analysis of Variance]] with the raw data (observed measurements) replaced by their ranks.
-Since it is a non-parametric method, the Kruskal-Wallis test '''does not''' assume a normal population, unlike the analogous one-way ANOVA.  However, the test does assume identically-shaped distributions for all groups, except for any difference in their centers (e.g., medians).
+Since it is a non-parametric method, the Kruskal-Wallis Test '''does not''' assume a normal population, unlike the analogous one-way ANOVA.  However, the test does assume identically-shaped distributions for all groups, except for any difference in their centers (e.g., medians).
 ==Calculations==

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 </center>
-The small sample sizes, and the lack of information about the distribution of each of the four samples, imply that ANOVA may not be appropriate for analyzing these data.
+The small sample sizes and the lack of distribution information of each sample illustrate how ANOVA may not be appropriate for analyzing these types of data.
 ==The Kruskal-Wallis Test==

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 <center>[[Image:SOCR_EBook_Dinov_KruskalWallis_030108_Fig1.jpg|600px]]</center>
-Clearly, there are significant differences between the group medians, even after the multiple testing correction, all groups appear different from each other.
+Clearly, there is only one significant group difference between medians, after the multiple testing correction, for the group1 vs. group4 comparison (see below):
 : Group Method1 vs. Group Method2: 	1.0	<	5.2056
 : Group Method1 vs. Group Method3: 	4.0	<	5.2056
-: Group Method1 vs. Group Method4: 	6.0	>	5.2056
+: '''Group Method1 vs. Group Method4: 	6.0	>	5.2056'''
 : Group Method2 vs. Group Method3: 	5.0	<	5.2056
 : Group Method2 vs. Group Method4: 	5.0	<	5.2056

← Older revision		Revision as of 20:46, 2 March 2008
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	==References==		==References==
		+	Conover W (1980). Practical Nonparametric Statistics. John Wiley & Sons, New York, second edition.

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