http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_NonParam_ANOVA&feed=atom&action=historyAP Statistics Curriculum 2007 NonParam ANOVA - Revision history2024-03-28T20:38:46ZRevision history for this page on the wikiMediaWiki 1.15.1http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_NonParam_ANOVA&diff=10415&oldid=prevJenny at 21:06, 28 June 20102010-06-28T21:06:32Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Motivational Example===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Motivational Example===</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Suppose four groups of students <del class="diffchange diffchange-inline">were </del>randomly assigned to be taught with four different techniques, and their achievement test scores <del class="diffchange diffchange-inline">were </del>recorded. Are the distributions of test scores the same, or do they differ in location? The data is presented in the table below.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Suppose four groups of students <ins class="diffchange diffchange-inline">are </ins>randomly assigned to be taught with four different techniques, and their achievement test scores <ins class="diffchange diffchange-inline">are </ins>recorded. Are the distributions of test scores the same, or do they differ in location? The data is presented in the table below.</div></td></tr>
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</table>Jennyhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_NonParam_ANOVA&diff=10311&oldid=prevJenny at 17:42, 28 June 20102010-06-28T17:42:12Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Let ''N'' be the total number of observations, then <math>N = \sum_{i=1}^k {n_i}</math>.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Let ''N'' be the total number of observations, then <math>N = \sum_{i=1}^k {n_i}</math>.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Let <math>R(X_{ij})</math> <del class="diffchange diffchange-inline">denote </del>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.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Let <math>R(X_{ij})</math> <ins class="diffchange diffchange-inline">denotes </ins>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.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: <math>R_i = \sum_{j=1}^{n_i} {R(X_{ij})}, i = 1, 2, ... , k</math>.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: <math>R_i = \sum_{j=1}^{n_i} {R(X_{ij})}, i = 1, 2, ... , k</math>.</div></td></tr>
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</table>Jennyhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_NonParam_ANOVA&diff=7008&oldid=prevPriscillaChui: /* Notes */2008-03-10T23:19:03Z<p><span class="autocomment">Notes</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Notes==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Notes==</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>* The [http://en.wikipedia.org/wiki/Friedman_test Friedman Fr <del class="diffchange diffchange-inline">test</del>] is the rank equivalent of the randomized block design alternative to the [[AP_Statistics_Curriculum_2007_ANOVA_2Way |<del class="diffchange diffchange-inline">two</del>-<del class="diffchange diffchange-inline">way analysis </del>of <del class="diffchange diffchange-inline">variance </del>F <del class="diffchange diffchange-inline">test</del>]]. [[SOCR_EduMaterials_AnalysisActivities_Friedman | The SOCR Friedman Test <del class="diffchange diffchange-inline">activity </del>]] demonstrates how <del class="diffchange diffchange-inline"> </del>to use [http://socr.ucla.edu/htmls/SOCR_Analyses.html SOCR Analyses] to compute the Friedman <del class="diffchange diffchange-inline">test </del>statistics and p-value.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>* The [http://en.wikipedia.org/wiki/Friedman_test Friedman Fr <ins class="diffchange diffchange-inline">Test</ins>] is the rank equivalent of the randomized block design alternative to the [[AP_Statistics_Curriculum_2007_ANOVA_2Way |<ins class="diffchange diffchange-inline">Two</ins>-<ins class="diffchange diffchange-inline">Way Analysis </ins>of <ins class="diffchange diffchange-inline">Variance </ins>F <ins class="diffchange diffchange-inline">Test</ins>]]. [[SOCR_EduMaterials_AnalysisActivities_Friedman | The SOCR Friedman Test <ins class="diffchange diffchange-inline">Activity </ins>]] demonstrates how to use [http://socr.ucla.edu/htmls/SOCR_Analyses.html SOCR Analyses] to compute the Friedman <ins class="diffchange diffchange-inline">Test </ins>statistics and p-value.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==References==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==References==</div></td></tr>
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</table>PriscillaChuihttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_NonParam_ANOVA&diff=7007&oldid=prevPriscillaChui: /* Calculations */2008-03-10T23:18:10Z<p><span class="autocomment">Calculations</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* Note: If there are no ties, then the test statistic is reduced to:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* Note: If there are no ties, then the test statistic is reduced to:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>::<math>T = \left(12 / N(N+1) \right) \sum_{i=1}^{k} {{R_i}^{2}} / {n_i} {-} 3 \left(N+1\right)</math>.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>::<math>T = \left(12 / N(N+1) \right) \sum_{i=1}^{k} {{R_i}^{2}} / {n_i} {-} 3 \left(N+1\right)</math>.</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">\end{center}</del></div></td><td colspan="2"> </td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>However, the SOCR implementation allows for the possibility of having ties; so it uses the non-simplified, exact method of computation.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>However, the SOCR implementation allows for the possibility of having ties; so it uses the non-simplified, exact method of computation.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>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. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>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. </div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>===The Kruskal-Wallis Test <del class="diffchange diffchange-inline">using </del>SOCR Analyses===</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>===The Kruskal-Wallis Test <ins class="diffchange diffchange-inline">Using </ins>SOCR Analyses===</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>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 <del class="diffchange diffchange-inline">activity</del>]] may also be helpful in understanding how to use this test in SOCR.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>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 <ins class="diffchange diffchange-inline">Activity</ins>]] may also be helpful in understanding how to use this test in SOCR.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>For the teaching-methods example above, we can easily compute the statistical significance of the differences between the group medians (centers):</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>For the teaching-methods example above, we can easily compute the statistical significance of the differences between the group medians (centers):</div></td></tr>
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</table>PriscillaChuihttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_NonParam_ANOVA&diff=7006&oldid=prevPriscillaChui: /* The Kruskal-Wallis Test */2008-03-10T23:17:06Z<p><span class="autocomment">The Kruskal-Wallis Test</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==The Kruskal-Wallis Test==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==The Kruskal-Wallis Test==</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Kruskal-Wallis <del class="diffchange diffchange-inline">one</del>-<del class="diffchange diffchange-inline">way analysis </del>of <del class="diffchange diffchange-inline">variance</del>''' 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 | <del class="diffchange diffchange-inline">one</del>-<del class="diffchange diffchange-inline">way analysis </del>of <del class="diffchange diffchange-inline">variance</del>]] with the raw data (observed measurements) replaced by their ranks.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Kruskal-Wallis <ins class="diffchange diffchange-inline">One</ins>-<ins class="diffchange diffchange-inline">Way Analysis </ins>of <ins class="diffchange diffchange-inline">Variance</ins>''' 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 | <ins class="diffchange diffchange-inline">One</ins>-<ins class="diffchange diffchange-inline">Way Analysis </ins>of <ins class="diffchange diffchange-inline">Variance</ins>]] with the raw data (observed measurements) replaced by their ranks.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Since it is a non-parametric method, the Kruskal-Wallis <del class="diffchange diffchange-inline">test </del>'''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).</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Since it is a non-parametric method, the Kruskal-Wallis <ins class="diffchange diffchange-inline">Test </ins>'''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).</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Calculations==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Calculations==</div></td></tr>
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</table>PriscillaChuihttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_NonParam_ANOVA&diff=7005&oldid=prevPriscillaChui: /* Motivational Example */2008-03-10T23:16:16Z<p><span class="autocomment">Motivational Example</span></p>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The small sample sizes<del class="diffchange diffchange-inline">, </del>and the lack of information <del class="diffchange diffchange-inline">about the distribution </del>of each <del class="diffchange diffchange-inline">of the four samples, imply that </del>ANOVA may not be appropriate for analyzing these data.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The small sample sizes and the lack of <ins class="diffchange diffchange-inline">distribution </ins>information of each <ins class="diffchange diffchange-inline">sample illustrate how </ins>ANOVA may not be appropriate for analyzing these <ins class="diffchange diffchange-inline">types of </ins>data.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==The Kruskal-Wallis Test==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==The Kruskal-Wallis Test==</div></td></tr>
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</table>PriscillaChuihttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_NonParam_ANOVA&diff=6855&oldid=prevIvoDinov: /* The Kruskal-Wallis Test using SOCR Analyses */2008-03-03T17:52:08Z<p><span class="autocomment">The Kruskal-Wallis Test using SOCR Analyses</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center>[[Image:SOCR_EBook_Dinov_KruskalWallis_030108_Fig1.jpg|600px]]</center></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center>[[Image:SOCR_EBook_Dinov_KruskalWallis_030108_Fig1.jpg|600px]]</center></div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Clearly, there <del class="diffchange diffchange-inline">are </del>significant <del class="diffchange diffchange-inline">differences between the </del>group medians, <del class="diffchange diffchange-inline">even </del>after the multiple testing correction, <del class="diffchange diffchange-inline">all groups appear different from each other</del>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Clearly, there <ins class="diffchange diffchange-inline">is only one </ins>significant group <ins class="diffchange diffchange-inline">difference between </ins>medians, after the multiple testing correction, <ins class="diffchange diffchange-inline">for the group1 vs</ins>. <ins class="diffchange diffchange-inline">group4 comparison (see below):</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: Group Method1 vs. Group Method2: 1.0 < 5.2056</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: Group Method1 vs. Group Method2: 1.0 < 5.2056</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: Group Method1 vs. Group Method3: 4.0 < 5.2056</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: Group Method1 vs. Group Method3: 4.0 < 5.2056</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>: Group Method1 vs. Group Method4: 6.0 > 5.2056</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>: <ins class="diffchange diffchange-inline">'''</ins>Group Method1 vs. Group Method4: 6.0 > 5.2056<ins class="diffchange diffchange-inline">'''</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: Group Method2 vs. Group Method3: 5.0 < 5.2056</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: Group Method2 vs. Group Method3: 5.0 < 5.2056</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: Group Method2 vs. Group Method4: 5.0 < 5.2056</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: Group Method2 vs. Group Method4: 5.0 < 5.2056</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_NonParam_ANOVA&diff=6830&oldid=prevIvoDinov at 20:46, 2 March 20082008-03-02T20:46:52Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==References==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==References==</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Conover W (1980). Practical Nonparametric Statistics. John Wiley & Sons, New York, second edition.</ins></div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_NonParam_ANOVA&diff=6829&oldid=prevIvoDinov: /* Calculations */2008-03-02T20:44:10Z<p><span class="autocomment">Calculations</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Calculations==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Calculations==</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"># Rank all data from all groups together; i.e., rank the data from 1 to N ignoring group membership. Assign any tied values the average of the ranks they would have received had they not been tied.</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Let </ins>''<ins class="diffchange diffchange-inline">N</ins>'' <ins class="diffchange diffchange-inline">be </ins>the total number of observations, <ins class="diffchange diffchange-inline">then </ins><math>N = \sum_{i=1}^<ins class="diffchange diffchange-inline">k {</ins>n_i}</math>.</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"># The test statistic is given by: </del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">: <math>K = (N-1)\frac{\sum_{i=1}^g n_i(\bar{r}_{i\cdot} - \bar{r})^2}{\sum_{i=1}^g\sum_{j=1}^{n_i}(r_{ij} - \bar{r})^2}</math>, where:</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">#*<math>n_g</math> is the number of observations in group <math>g</math></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">#*<math>r_{ij}</math> is the rank (among all observations) of observation </del>''<del class="diffchange diffchange-inline">j</del>'' <del class="diffchange diffchange-inline">from group ''i''</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">#*<math>N</math> is </del>the total number of observations <del class="diffchange diffchange-inline">across all groups</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">#*<math>\bar{r}_{i\cdot} = \frac{\sum_{j=1}^{n_i}{r_{ij}}}{n_i}</math></del>,</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">#*</del><math><del class="diffchange diffchange-inline">\bar{r} =(</del>N<del class="diffchange diffchange-inline">+1)/2</math> is the average of all the <math>r_{ij}</math>.</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">#*Notice that the denominator of the expression for <math>K</math> is exactly <math>(N-1)N(N+1)/12</math>. Thus <math>K </del>= <del class="diffchange diffchange-inline">\frac{12}{N(N+1)}</del>\sum_{i=1}^<del class="diffchange diffchange-inline">g </del>n_i<del class="diffchange diffchange-inline">(\bar{r</del>}<del class="diffchange diffchange-inline">_{i\cdot} - \bar{r})^2</del></math><del class="diffchange diffchange-inline">.</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"># A correction for ties can be made by dividing <math>K</math> by <math>1 - \frac{\sum_{i=1}^G (t_{i}^3 - t_{i})}{N^3-N}</math>, where G is the number of groupings of different tied ranks, and t<sub>i</sub> is the number of tied values within group i that are tied at a particular value. This correction usually makes little difference in the value of K unless there are a large number of ties.</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"># Finally, the p-value is approximated by <math>\Pr(\chi^2_{g-1} \ge K)</math>. If some n<sub>i</sub>'s are small (i.e., less than 5) the probability distribution of K can be quite different from this [http://en.wikipedia.org/wiki/Chi-square Chi-square distribution]</del>. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The <del class="diffchange diffchange-inline">null hypothesis </del>of <del class="diffchange diffchange-inline">equal </del>population <del class="diffchange diffchange-inline">medians would </del>then <del class="diffchange diffchange-inline">be rejected if </del><math><del class="diffchange diffchange-inline">K </del>\<del class="diffchange diffchange-inline">ge </del>\<del class="diffchange diffchange-inline">chi</del>^<del class="diffchange diffchange-inline">2_</del>{\alpha<del class="diffchange diffchange-inline">: g</del>-1}</math>. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">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.</ins></div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">: <math>R_i = \sum_{j=1}^{n_i} {R(X_{ij})}, i = 1, 2, ... , k</math>.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The <ins class="diffchange diffchange-inline">SOCR program computes <math>R_i</math> for each sample. The test statistic is defined for the following formulation of hypotheses:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">: <math>H_o</math>: All </ins>of <ins class="diffchange diffchange-inline">the k </ins>population <ins class="diffchange diffchange-inline">distribution functions are identical.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">: <math>H_1</math>: At least one of the populations tends to yield larger observations than at least one of the other populations.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Suppose {<math>X_{i,1}, X_{i,2}, \cdots, X_{i,n_i}</math>} represents the values of the <math>i^{th}</math> sample, where <math>1\leq i\leq k</math>.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"> </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">: Test statistics:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:: T = <math>(1/{{S}^{2}}) (\sum_{i=1}^{k} {{R_i}^{2}} / {n_i} {-}{N {(N + 1)}^{2} }) / 4</math>,</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">where</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">::<math>{{S}^{2}} = \left( \left({1/ {N - 1}}\right) \right) \sum{{R(X_{ij})}^{2}} {-} {N {\left(N + 1)\right)}^{2} } ) / 4</math>.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">* Note: If there are no ties, </ins>then <ins class="diffchange diffchange-inline">the test statistic is reduced to:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">::</ins><math><ins class="diffchange diffchange-inline">T = </ins>\<ins class="diffchange diffchange-inline">left(12 / N(N+1) </ins>\<ins class="diffchange diffchange-inline">right) \sum_{i=1}</ins>^{<ins class="diffchange diffchange-inline">k} {{R_i}^{2}} / {n_i} {-} 3 \left(N+1\right)</math>.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">\end{center}</ins></div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">However, the SOCR implementation allows for the possibility of having ties; so it uses the non-simplified, exact method of computation.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Multiple comparisons have to be done here. For each pair of groups, the following is computed and printed at the '''Result''' Panel.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"><math>|R_{i} /n_{i} -R_{j} /n_{j} | > t_{1-</ins>\alpha <ins class="diffchange diffchange-inline">/2} (S^{2^{} } (N</ins>-1<ins class="diffchange diffchange-inline">-T)/(N-k))^{1/2_{} } /(1/n_{i} +1/n_{j} )^{1/2_{}</ins>}</math><ins class="diffchange diffchange-inline">.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">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</ins>. </div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===The Kruskal-Wallis Test using SOCR Analyses===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===The Kruskal-Wallis Test using SOCR Analyses===</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_NonParam_ANOVA&diff=6828&oldid=prevIvoDinov: /* Notes */2008-03-02T20:29:00Z<p><span class="autocomment">Notes</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Notes==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Notes==</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>* 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]].</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>* 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]]. <ins class="diffchange diffchange-inline">[[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.</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==References==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==References==</div></td></tr>
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