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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Method" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Method"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Method</span> </div> </a> <ul id="toc-Method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Assumptions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Assumptions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Assumptions</span> </div> </a> <ul id="toc-Assumptions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Significance_testing" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Significance_testing"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Significance testing</span> </div> </a> <ul id="toc-Significance_testing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_of_two-sided_sign_test_for_matched_pairs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Example_of_two-sided_sign_test_for_matched_pairs"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Example of two-sided sign test for matched pairs</span> </div> </a> <ul id="toc-Example_of_two-sided_sign_test_for_matched_pairs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Example_of_one-sided_sign_test_for_matched_pairs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_of_one-sided_sign_test_for_matched_pairs"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Example of one-sided sign test for matched pairs</span> </div> </a> <ul id="toc-Example_of_one-sided_sign_test_for_matched_pairs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_of_sign_test_for_median_of_a_single_sample" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_of_sign_test_for_median_of_a_single_sample"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Example of sign test for median of a single sample</span> </div> </a> <ul id="toc-Example_of_sign_test_for_median_of_a_single_sample-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Software_implementations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Software_implementations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Software implementations</span> </div> </a> <button aria-controls="toc-Software_implementations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Software implementations subsection</span> </button> <ul id="toc-Software_implementations-sublist" class="vector-toc-list"> <li id="toc-Excel_software_for_the_sign_test" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Excel_software_for_the_sign_test"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Excel software for the sign test</span> </div> </a> <ul id="toc-Excel_software_for_the_sign_test-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-R_software_for_the_sign_test" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#R_software_for_the_sign_test"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>R software for the sign test</span> </div> </a> <ul id="toc-R_software_for_the_sign_test-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationship_to_other_statistical_tests" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relationship_to_other_statistical_tests"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Relationship to other statistical tests</span> </div> </a> <button aria-controls="toc-Relationship_to_other_statistical_tests-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Relationship to other statistical tests subsection</span> </button> <ul id="toc-Relationship_to_other_statistical_tests-sublist" class="vector-toc-list"> <li id="toc-Wilcoxon_signed-rank_test" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wilcoxon_signed-rank_test"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Wilcoxon signed-rank test</span> </div> </a> <ul id="toc-Wilcoxon_signed-rank_test-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Paired_t-test" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Paired_t-test"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Paired t-test</span> </div> </a> <ul id="toc-Paired_t-test-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-McNemar's_test" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#McNemar's_test"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>McNemar's test</span> </div> </a> <ul id="toc-McNemar's_test-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Friedman_test" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Friedman_test"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.4</span> <span>Friedman test</span> </div> </a> <ul id="toc-Friedman_test-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Trinomial_test" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Trinomial_test"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.5</span> <span>Trinomial test</span> </div> </a> <ul id="toc-Trinomial_test-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet 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href="https://eu.wikipedia.org/wiki/Zeinuen_proba" title="Zeinuen proba – Basque" lang="eu" hreflang="eu" data-title="Zeinuen proba" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A2%D8%B2%D9%85%D9%88%D9%86_%D8%B9%D9%84%D8%A7%D9%85%D8%AA" title="آزمون علامت – Persian" lang="fa" hreflang="fa" data-title="آزمون علامت" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Test_des_signes" title="Test des signes – French" lang="fr" hreflang="fr" data-title="Test des signes" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Test_dei_segni" title="Test dei segni – Italian" lang="it" hreflang="it" data-title="Test dei segni" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Tekentoets" title="Tekentoets – Dutch" lang="nl" hreflang="nl" data-title="Tekentoets" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Test_znak%C3%B3w" title="Test znaków – Polish" lang="pl" hreflang="pl" data-title="Test znaków" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a 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searchaux" style="display:none">Statistical test with teststatistic the number of signs of one type</div> <p>The <b>sign test</b> is a <a href="/wiki/Statistical_test" class="mw-redirect" title="Statistical test">statistical test</a> for consistent differences between pairs of observations, such as the weight of subjects before and after treatment. Given pairs of observations (such as weight pre- and post-treatment) for each subject, the sign test determines if one member of the pair (such as pre-treatment) tends to be greater than (or less than) the other member of the pair (such as post-treatment). </p><p>The paired observations may be designated <i>x</i> and <i>y</i>. For comparisons of paired observations (<i>x</i>,y), the sign test is most useful if comparisons can only be expressed as <i>x</i> > <i>y</i>, <i>x</i> = <i>y</i>, or <i>x</i> < <i>y</i>. If, instead, the observations can be expressed as numeric quantities (<i>x</i> = 7, <i>y</i> = 18), or as ranks (rank of <i>x</i> = 1st, rank of <i>y</i> = 8th), then the paired <a href="/wiki/T-test" class="mw-redirect" title="T-test">t-test</a><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> or the <a href="/wiki/Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Wilcoxon signed-rank test</a><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> will usually have greater power than the sign test to detect consistent differences. </p><p>If X and Y are quantitative variables, the <b>sign test</b> can be used to <a href="/wiki/Hypothesis_test" class="mw-redirect" title="Hypothesis test">test the hypothesis</a> that the difference between the <i>X</i> and <i>Y</i> has zero median, assuming continuous distributions of the two <a href="/wiki/Random_variable" title="Random variable">random variables</a> <i>X</i> and <i>Y</i>, in the situation when we can draw <a href="/wiki/Paired_difference_test" title="Paired difference test">paired samples</a> from <i>X</i> and <i>Y</i>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The sign test can also test if the median of a collection of numbers is significantly greater than or less than a specified value. For example, given a list of student grades in a class, the sign test can determine if the median grade is significantly different from, say, 75 out of 100. </p><p>The sign test is a <a href="/wiki/Non-parametric_test" class="mw-redirect" title="Non-parametric test">non-parametric test</a> which makes very few assumptions about the nature of the distributions under test – this means that it has very general applicability but may lack the <a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">statistical power</a> of the alternative tests. </p><p>The two conditions for the paired-sample sign test are that a sample must be randomly selected from each population, and the samples must be dependent, or paired. Independent samples cannot be meaningfully paired. Since the test is nonparametric, the samples need not come from normally distributed populations. Also, the test works for left-tailed, right-tailed, and two-tailed tests. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Method">Method</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sign_test&action=edit&section=1" title="Edit section: Method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>p</i> = Pr(<i>X</i> > <i>Y</i>), and then test the <a href="/wiki/Null_hypothesis" title="Null hypothesis">null hypothesis</a> H<sub>0</sub>: <i>p</i> = 0.50. In other words, the null hypothesis states that given a <a href="/wiki/Random" class="mw-redirect" title="Random">random</a> pair of measurements (<i>x</i><sub><i>i</i></sub>, <i>y</i><sub><i>i</i></sub>), then <i>x</i><sub><i>i</i></sub> and <i>y</i><sub><i>i</i></sub> are equally likely to be larger than the other. </p><p>To test the null hypothesis, independent pairs of sample data are collected from the populations {(<i>x</i><sub>1</sub>, <i>y</i><sub>1</sub>), (<i>x</i><sub>2</sub>, <i>y</i><sub>2</sub>), . . ., (<i>x</i><sub><i>n</i></sub>, <i>y</i><sub><i>n</i></sub>)}. Pairs are omitted for which there is no difference so that there is a possibility of a reduced sample of <i>m</i> pairs.<sup id="cite_ref-mendenhall_4-0" class="reference"><a href="#cite_note-mendenhall-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Then let <i>W</i> be the number of pairs for which <i>y</i><sub><i>i</i></sub> − <i>x</i><sub><i>i</i></sub> > 0. Assuming that H<sub>0</sub> is true, then <i>W</i> follows a <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a> <i>W</i> ~ b(<i>m</i>, 0.5). </p> <div class="mw-heading mw-heading2"><h2 id="Assumptions">Assumptions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sign_test&action=edit&section=2" title="Edit section: Assumptions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>Z</i><sub>i</sub> = <i>Y</i><sub>i</sub> – <i>X</i><sub>i</sub> for <i>i</i> = 1, ... , <i>n</i>. </p> <ol><li>The differences <i>Z<sub>i</sub></i> are assumed to be independent.</li> <li>Each <i>Z<sub>i</sub></i> comes from the same continuous population.</li> <li>The values <i>X</i><sub><i>i</i></sub> and <i>Y</i><sub><i>i</i></sub> represent are ordered (at least the <a href="/wiki/Ordinal_scale" class="mw-redirect" title="Ordinal scale">ordinal scale</a>), so the comparisons "greater than", "less than", and "equal to" are meaningful.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Significance_testing">Significance testing</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sign_test&action=edit&section=3" title="Edit section: Significance testing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since the test statistic is expected to follow a <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a>, the standard <a href="/wiki/Binomial_test" title="Binomial test">binomial test</a> is used to calculate <a href="/wiki/Statistical_significance" title="Statistical significance">significance</a>. The <a href="/wiki/Binomial_distribution#Normal_approximation" title="Binomial distribution">normal approximation to the binomial distribution</a> can be used for large sample sizes, <i>m</i> > 25.<sup id="cite_ref-mendenhall_4-1" class="reference"><a href="#cite_note-mendenhall-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>The left-tail value is computed by Pr(<i>W</i> ≤ <i>w</i>), which is the <a href="/wiki/P-value" title="P-value">p-value</a> for the alternative H<sub>1</sub>: <i>p</i> < 0.50. This alternative means that the <i>X</i> measurements tend to be higher. </p><p>The right-tail value is computed by Pr(<i>W</i> ≥ <i>w</i>), which is the p-value for the alternative H<sub>1</sub>: <i>p</i> > 0.50. This alternative means that the <i>Y</i> measurements tend to be higher. </p><p>For a two-sided alternative H<sub>1</sub> the p-value is twice the smaller tail-value. </p> <div class="mw-heading mw-heading2"><h2 id="Example_of_two-sided_sign_test_for_matched_pairs">Example of two-sided sign test for matched pairs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sign_test&action=edit&section=4" title="Edit section: Example of two-sided sign test for matched pairs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Zar gives the following example of the sign test for matched pairs. Data are collected on the length of the left hind leg and left foreleg for 10 deer.<sup id="cite_ref-Zar1999_5-0" class="reference"><a href="#cite_note-Zar1999-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable"> <tbody><tr> <th>Deer</th> <th>Hind leg length (cm)</th> <th>Foreleg length (cm)</th> <th>Difference </th></tr> <tr> <td>1</td> <td>142</td> <td>138</td> <td>+ </td></tr> <tr> <td>2</td> <td>140</td> <td>136</td> <td>+ </td></tr> <tr> <td>3</td> <td>144</td> <td>147</td> <td>− </td></tr> <tr> <td>4</td> <td>144</td> <td>139</td> <td>+ </td></tr> <tr> <td>5</td> <td>142</td> <td>143</td> <td>− </td></tr> <tr> <td>6</td> <td>146</td> <td>141</td> <td>+ </td></tr> <tr> <td>7</td> <td>149</td> <td>143</td> <td>+ </td></tr> <tr> <td>8</td> <td>150</td> <td>145</td> <td>+ </td></tr> <tr> <td>9</td> <td>142</td> <td>136</td> <td>+ </td></tr> <tr> <td>10</td> <td>148</td> <td>146</td> <td>+ </td></tr></tbody></table> <p>The null hypothesis is that there is no difference between the hind leg and foreleg length in deer. The alternative hypothesis is that there is a difference between hind leg length and foreleg length. This is a two-tailed test, rather than a one-tailed test. For the two tailed test, the alternative hypothesis is that hind leg length may be either greater than or less than foreleg length. A one-sided test could be that hind leg length is greater than foreleg length, so that the difference can only be in one direction (greater than). </p><p>There are n=10 deer. There are 8 positive differences and 2 negative differences. If the null hypothesis is true, that there is no difference in hind leg and foreleg lengths, then the expected number of positive differences is 5 out of 10. What is the probability that the observed result of 8 positive differences, or a more extreme result, would occur if there is no difference in leg lengths? </p><p>Because the test is two-sided, a result as extreme or more extreme than 8 positive differences includes the results of 8, 9, or 10 positive differences, and the results of 0, 1, or 2 positive differences. The probability of 8 or more positives among 10 deer or 2 or fewer positives among 10 deer is the same as the probability of 8 or more heads or 2 or fewer heads in 10 flips of a fair coin. The probabilities can be calculated using the <a href="/wiki/Binomial_test" title="Binomial test">binomial test</a>, with the probability of heads = probability of tails = 0.5. </p> <ul><li>Probability of 0 heads in 10 flips of fair coin = 0.00098</li> <li>Probability of 1 heads in 10 flips of fair coin = 0.00977</li> <li>Probability of 2 heads in 10 flips of fair coin = 0.04395</li> <li>Probability of 8 heads in 10 flips of fair coin = 0.04395</li> <li>Probability of 9 heads in 10 flips of fair coin = 0.00977</li> <li>Probability of 10 heads in 10 flips of fair coin = 0.00098</li></ul> <p>The two-sided probability of a result as extreme as 8 of 10 positive difference is the sum of these probabilities: </p> <dl><dd>0.00098 + 0.00977 + 0.04395 + 0.04395 + 0.00977 + 0.00098 = 0.109375.</dd></dl> <p>Thus, the probability of observing a results as extreme as 8 of 10 positive differences in leg lengths, if there is no difference in leg lengths, is <i>p</i> = 0.109375. The null hypothesis is not rejected at a significance level of <i>p</i> = 0.05. With a larger sample size, the evidence might be sufficient to reject the null hypothesis. </p><p>Because the observations can be expressed as numeric quantities (actual leg length), the paired t-test or Wilcoxon signed rank test will usually have greater power than the sign test to detect consistent differences. For this example, the paired t-test for differences indicates that there is a significant difference between hind leg length and foreleg length (<i>p</i> = 0.007). </p><p>If the observed result was 9 positive differences in 10 comparisons, the sign test would be significant. Only coin flips with 0, 1, 9, or 10 heads would be as extreme as or more extreme than the observed result. </p> <ul><li>Probability of 0 heads in 10 flips of fair coin = 0.00098</li> <li>Probability of 1 heads in 10 flips of fair coin = 0.00977</li> <li>Probability of 9 heads in 10 flips of fair coin = 0.00977</li> <li>Probability of 10 heads in 10 flips of fair coin = 0.00098</li></ul> <p>The probability of a result as extreme as 9 of 10 positive difference is the sum of these probabilities: </p> <dl><dd>0.00098 + 0.00977 + 0.00977 + 0.00098 = 0.0215.</dd></dl> <p>In general, 8 of 10 positive differences is not significant (<i>p</i> = 0.11), but 9 of 10 positive differences is significant (<i>p</i> = 0.0215). </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sign_test&action=edit&section=5" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Example_of_one-sided_sign_test_for_matched_pairs">Example of one-sided sign test for matched pairs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sign_test&action=edit&section=6" title="Edit section: Example of one-sided sign test for matched pairs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Conover<sup id="cite_ref-Conover1999_6-0" class="reference"><a href="#cite_note-Conover1999-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> gives the following example using a one-sided sign test for matched pairs. A manufacturer produces two products, A and B. The manufacturer wishes to know if consumers prefer product B over product A. A sample of 10 consumers are each given product A and product B, and asked which product they prefer. </p><p>The null hypothesis is that consumers do not prefer product B over product A. The alternative hypothesis is that consumers prefer product B over product A. This is a one-sided (directional) test. </p><p>At the end of the study, 8 consumers preferred product B, 1 consumer preferred product A, and one reported no preference. </p> <ul><li>Number of +'s (preferred B) = 8</li> <li>Number of –'s (preferred A) = 1</li> <li>Number of ties (no preference) = 1</li></ul> <p>The tie is excluded from the analysis, giving n = number of +'s and –'s = 8 + 1 = 9. </p><p>What is the probability of a result as extreme as 8 positives in favor of B in 9 pairs, if the null hypothesis is true, that consumers have no preference for B over A? This is the probability of 8 or more heads in 9 flips of a fair coin, and can be calculated using the binomial distribution with p(heads) = p(tails) = 0.5. </p><p>P(8 or 9 heads in 9 flips of a fair coin) = 0.0195. The null hypothesis is rejected, and the manufacturer concludes that consumers prefer product B over product A. </p> <div class="mw-heading mw-heading3"><h3 id="Example_of_sign_test_for_median_of_a_single_sample">Example of sign test for median of a single sample</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sign_test&action=edit&section=7" title="Edit section: Example of sign test for median of a single sample"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sprent <sup id="cite_ref-Sprent1989_7-0" class="reference"><a href="#cite_note-Sprent1989-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> gives the following example of a sign test for a median. In a clinical trial, survival time (weeks) is collected for 10 subjects with non-Hodgkin's lymphoma. The exact survival time was not known for one subject who was still alive after 362 weeks, when the study ended. The subjects' survival times were </p> <dl><dd>49, 58, 75, 110, 112, 132, 151, 276, 281, 362+</dd></dl> <p>The plus sign indicates the subject still alive at the end of the study. The researcher wished to determine if the median survival time was less than or greater than 200 weeks. </p><p>The null hypothesis is that median survival is 200 weeks. The alternative hypothesis is that median survival is not 200 weeks. This is a two-sided test: the alternative median may be greater than or less than 200 weeks. </p><p>If the null hypothesis is true, that the median survival is 200 weeks, then, in a random sample approximately half the subjects should survive less than 200 weeks, and half should survive more than 200 weeks. Observations below 200 are assigned a minus (−); observations above 200 are assigned a plus (+). For the subject survival times, there are 7 observations below 200 weeks (−) and 3 observations above 200 weeks (+) for the n=10 subjects. </p><p>Because any one observation is equally likely to be above or below the population median, the number of plus scores will have a binomial distribution with mean = 0.5. What is the probability of a result as extreme as 7 in 10 subjects being below the median? This is exactly the same as the probability of a result as extreme as 7 heads in 10 tosses of a fair coin. Because this is a two-sided test, an extreme result can be either three or fewer heads or seven or more heads. </p><p>The probability of observing k heads in 10 tosses of a fair coin, with p(heads) = 0.5, is given by the binomial formula: </p> <dl><dd>Pr(Number of heads = <i>k</i>) = Choose(10, <i>k</i>) × 0.5<sup>10</sup></dd></dl> <p>The probability for each value of <i>k</i> is given in the table below. </p> <table class="wikitable"> <tbody><tr> <th><i>k</i></th> <th>0</th> <th>1</th> <th>2</th> <th>3</th> <th>4</th> <th>5</th> <th>6</th> <th>7</th> <th>8</th> <th>9</th> <th>10 </th></tr> <tr> <th>Pr </th> <td>0.0010</td> <td>0.0098</td> <td>0.0439</td> <td>0.1172</td> <td>0.2051</td> <td>0.2461</td> <td>0.2051</td> <td>0.1172</td> <td>0.0439</td> <td>0.0098</td> <td>0.0010 </td></tr></tbody></table> <p>The probability of 0, 1, 2, 3, 7, 8, 9, or 10 heads in 10 tosses is the sum of their individual probabilities: </p> <dl><dd>0.0010 + 0.0098 + 0.0439 + 0.1172 + 0.1172 + 0.0439 + 0.0098 + 0.0010 = 0.3438.</dd></dl> <p>Thus, the probability of observing 3 or fewer plus signs or 7 or more plus signs in the survival data, if the median survival is 200 weeks, is 0.3438. The expected number of plus signs is 5 if the null hypothesis is true. Observing 3 or fewer, or 7 or more pluses is not significantly different from 5. The null hypothesis is not rejected. Because of the extremely small sample size, this sample has low power to detect a difference. </p> <div class="mw-heading mw-heading2"><h2 id="Software_implementations">Software implementations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sign_test&action=edit&section=8" title="Edit section: Software implementations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The sign test is a special case of the binomial test where the probability of success under the null hypothesis is p=0.5. Thus, the sign test can be performed using the binomial test, which is provided in most statistical software programs. On-line calculators for the sign test can be founded by searching for "sign test calculator". Many websites offer the binomial test, but generally offer only a two-sided version. </p> <div class="mw-heading mw-heading3"><h3 id="Excel_software_for_the_sign_test">Excel software for the sign test</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sign_test&action=edit&section=9" title="Edit section: Excel software for the sign test"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A template for the sign test using Excel is available at <a rel="nofollow" class="external free" href="http://www.real-statistics.com/non-parametric-tests/sign-test/">http://www.real-statistics.com/non-parametric-tests/sign-test/</a> </p> <div class="mw-heading mw-heading3"><h3 id="R_software_for_the_sign_test">R software for the sign test</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sign_test&action=edit&section=10" title="Edit section: R software for the sign test"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/R_(programming_language)" title="R (programming language)">R</a>, the binomial test can be performed using the function <code>binom.test()</code>. </p><p>The syntax for the function is </p> <div class="mw-highlight mw-highlight-lang-r mw-content-ltr" dir="ltr"><pre><span></span><span class="nf">binom.test</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0.5</span><span class="p">,</span><span class="w"> </span><span class="n">alternative</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">c</span><span class="p">(</span><span class="s">"two.sided"</span><span class="p">,</span><span class="w"> </span><span class="s">"less"</span><span class="p">,</span><span class="w"> </span><span class="s">"greater"</span><span class="p">),</span><span class="w"> </span><span class="n">conf.level</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0.95</span><span class="p">)</span> </pre></div> <p>where </p> <ul><li><code>x</code> = number of successes, or a vector of length 2 giving the numbers of successes and failures, respectively</li> <li><code>n</code> = number of trials; ignored if x has length 2</li> <li><code>p</code> = hypothesized probability of success</li> <li><code>alternative</code> =indicates the alternative hypothesis and must be one of "two.sided", "greater" or "less"</li> <li><code>conf.level</code> = confidence level for the returned confidence interval.</li></ul> <p>Examples of the sign test using the R function binom.test </p><p>The sign test example from Zar <sup id="cite_ref-Zar1999_5-1" class="reference"><a href="#cite_note-Zar1999-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> compared the length of hind legs and forelegs of deer. The hind leg was longer than the foreleg in 8 of 10 deer. Thus, there are x=8 successes in n=10 trials. The hypothesized probability of success (defined as hind leg longer than foreleg) is <i>p</i> = 0.5 under the null hypothesis that hind legs and forelegs do not differ in length. The alternative hypothesis is that hind leg length may be either greater than or less than foreleg length, which is a two sided test, specified as alternative="two.sided". </p><p>The R command <code class="mw-highlight mw-highlight-lang-r mw-content-ltr" dir="ltr"><span class="nf">binom.test</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="m">8</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="o">=</span><span class="m">10</span><span class="p">,</span><span class="w"> </span><span class="n">p</span><span class="o">=</span><span class="m">0.5</span><span class="p">,</span><span class="w"> </span><span class="n">alternative</span><span class="o">=</span><span class="s">"two.sided"</span><span class="p">)</span></code> gives p=0.1094, as in the example. </p><p>The sign test example in Conover <sup id="cite_ref-Conover1999_6-1" class="reference"><a href="#cite_note-Conover1999-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> examined consumer preference for product A vs. product B. The null hypothesis was that consumers do not prefer product B over product A. The alternative hypothesis was that consumers prefer product B over product A, a one-sided test. In the study, 8 of 9 consumers who expressed a preference preferred product B over product A. </p><p>The R command <code class="mw-highlight mw-highlight-lang-r mw-content-ltr" dir="ltr"><span class="nf">binom.test</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="m">8</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="o">=</span><span class="m">9</span><span class="p">,</span><span class="w"> </span><span class="n">p</span><span class="o">=</span><span class="m">0.5</span><span class="p">,</span><span class="w"> </span><span class="n">alternative</span><span class="o">=</span><span class="s">"greater"</span><span class="p">)</span></code> gives p=0.01953, as in the example. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sign_test&action=edit&section=11" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/History_of_statistics" title="History of statistics">History of statistics</a></div> <p>Conover <sup id="cite_ref-Conover1999_6-2" class="reference"><a href="#cite_note-Conover1999-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> and Sprent <sup id="cite_ref-Sprent1989_7-1" class="reference"><a href="#cite_note-Sprent1989-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> describe <a href="/wiki/John_Arbuthnot" title="John Arbuthnot">John Arbuthnot</a>'s use of the sign test in 1710. Arbuthnot examined birth records in London for each of the 82 years from 1629 to 1710. In every year, the number of males born in London exceeded the number of females. If the null hypothesis of equal number of births is true, the probability of the observed outcome is 1/2<sup>82</sup>, leading Arbuthnot to conclude that the probability of male and female births were not exactly equal. </p><p>For his publications in 1692 and 1710, Arbuthnot is credited with "… the first use of significance tests …" <sup id="cite_ref-Bellhouse2001_8-0" class="reference"><a href="#cite_note-Bellhouse2001-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> , the first example of reasoning about statistical significance and moral certainty, <sup id="cite_ref-Hald1998_9-0" class="reference"><a href="#cite_note-Hald1998-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> and "… perhaps the first published report of a nonparametric test …".<sup id="cite_ref-Conover1999_6-3" class="reference"><a href="#cite_note-Conover1999-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Hald <sup id="cite_ref-Hald1998_9-1" class="reference"><a href="#cite_note-Hald1998-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> further describes the impact of Arbuthnot's research. </p><p>"Nicholas Bernoulli (1710–1713) completes the analysis of Arbuthnot's data by showing that the larger part of the variation of the yearly number of male births can be explained as binomial with <i>p</i> = 18/35. This is the first example of fitting a binomial to data. Hence we here have a test of significance rejecting the hypothesis <i>p</i> = 0.5 followed by an estimation of p and a discussion of the goodness of fit …" </p> <div class="mw-heading mw-heading2"><h2 id="Relationship_to_other_statistical_tests">Relationship to other statistical tests</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sign_test&action=edit&section=12" title="Edit section: Relationship to other statistical tests"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Wilcoxon_signed-rank_test">Wilcoxon signed-rank test</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sign_test&action=edit&section=13" title="Edit section: Wilcoxon signed-rank test"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The sign test requires only that the observations in a pair be ordered, for example <i>x</i> > <i>y</i>. In some cases, the observations for all subjects can be assigned a rank value (1, 2, 3, ...). If the observations can be ranked, and each observation in a pair is a random sample from a symmetric distribution, then the <a href="/wiki/Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Wilcoxon signed-rank test</a> is appropriate. The Wilcoxon test will generally have greater power to detect differences than the sign test. The <a href="/wiki/Asymptotic_relative_efficiency" class="mw-redirect" title="Asymptotic relative efficiency">asymptotic relative efficiency</a> of the sign test to the Wilcoxon signed rank test, under these circumstances, is 0.67.<sup id="cite_ref-Conover1999_6-4" class="reference"><a href="#cite_note-Conover1999-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Paired_t-test">Paired t-test</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sign_test&action=edit&section=14" title="Edit section: Paired t-test"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the paired observations are numeric quantities (such as the actual length of the hind leg and foreleg in the Zar example), and the differences between paired observations are random samples from a single normal distribution, then the <a href="/wiki/Paired_t-test" class="mw-redirect" title="Paired t-test">paired t-test</a> is appropriate. The paired t-test will generally have greater power to detect differences than the sign test. The asymptotic relative efficiency of the sign test to the paired t-test, under these circumstances, is 0.637. However, if the distribution of the differences between pairs is not normal, but instead is heavy-tailed (<a href="/wiki/Platykurtic_distribution" class="mw-redirect" title="Platykurtic distribution">platykurtic distribution</a>), the sign test can have more power than the paired t-test, with <a href="/wiki/Asymptotic_relative_efficiency" class="mw-redirect" title="Asymptotic relative efficiency">asymptotic relative efficiency</a> of 2.0 relative to the paired t-test and 1.3 relative to the Wilcoxon signed rank test.<sup id="cite_ref-Conover1999_6-5" class="reference"><a href="#cite_note-Conover1999-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="McNemar's_test"><span id="McNemar.27s_test"></span>McNemar's test</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sign_test&action=edit&section=15" title="Edit section: McNemar's test"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In some applications, the observations within each pair can only take the values 0 or 1. For example, 0 may indicate failure and 1 may indicate success. There are 4 possible pairs: {0,0}, {0,1}, {1,0}, and {1,1}. In these cases, the same procedure as the sign test is used, but is known as <a href="/wiki/McNemar%27s_test" title="McNemar's test">McNemar's test</a>.<sup id="cite_ref-Conover1999_6-6" class="reference"><a href="#cite_note-Conover1999-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Friedman_test"><a href="/wiki/Friedman_test" title="Friedman test">Friedman test</a></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sign_test&action=edit&section=16" title="Edit section: Friedman test"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Instead of paired observations such as (Product A, Product B), the data may consist of three or more levels (Product A, Product B, Product C). If the individual observations can be ordered in the same way as for the sign test, for example B > C > A, then the <a href="/wiki/Friedman_test" title="Friedman test">Friedman test</a> may be used.<sup id="cite_ref-Zar1999_5-2" class="reference"><a href="#cite_note-Zar1999-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Trinomial_test">Trinomial test</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sign_test&action=edit&section=17" title="Edit section: Trinomial test"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bian, McAleer and Wong<sup id="cite_ref-Bian2011_10-0" class="reference"><a href="#cite_note-Bian2011-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> proposed in 2011 a non-parametric test for paired data when there are many ties. They showed that their trinomial test is superior to the sign test in presence of ties. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sign_test&action=edit&section=18" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Wilcoxon signed-rank test</a> – A more powerful variant of the sign test, but one which also assumes a symmetric distribution and interval data.</li> <li><a href="/wiki/Median_test" title="Median test">Median test</a> – An unpaired alternative to the sign test.</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sign_test&action=edit&section=19" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBaguley2012" class="citation cs2">Baguley, Thomas (2012), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ObUcBQAAQBAJ&pg=PA281"><i>Serious Stats: A Guide to Advanced Statistics for the Behavioral Sciences</i></a>, Palgrave Macmillan, p. 281, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780230363557" title="Special:BookSources/9780230363557"><bdi>9780230363557</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Serious+Stats%3A+A+Guide+to+Advanced+Statistics+for+the+Behavioral+Sciences&rft.pages=281&rft.pub=Palgrave+Macmillan&rft.date=2012&rft.isbn=9780230363557&rft.aulast=Baguley&rft.aufirst=Thomas&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DObUcBQAAQBAJ%26pg%3DPA281&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASign+test" class="Z3988"></span>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCorderForeman2014" class="citation cs2">Corder, Gregory W.; Foreman, Dale I. (2014), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CIxgAwAAQBAJ&pg=SA3-PA7">"3.6 Statistical Power"</a>, <i>Nonparametric Statistics: A Step-by-Step Approach</i> (2nd ed.), John Wiley & Sons, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781118840429" title="Special:BookSources/9781118840429"><bdi>9781118840429</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=3.6+Statistical+Power&rft.btitle=Nonparametric+Statistics%3A+A+Step-by-Step+Approach&rft.edition=2nd&rft.pub=John+Wiley+%26+Sons&rft.date=2014&rft.isbn=9781118840429&rft.aulast=Corder&rft.aufirst=Gregory+W.&rft.au=Foreman%2C+Dale+I.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCIxgAwAAQBAJ%26pg%3DSA3-PA7&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASign+test" class="Z3988"></span>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://onlinecourses.science.psu.edu/stat414/node/318">The Sign Test for a Median</a> // STAT 415 Intro Mathematical Statistics. Penn State University.</span> </li> <li id="cite_note-mendenhall-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-mendenhall_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-mendenhall_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMendenhallWackerlyScheaffer1989" class="citation cs2">Mendenhall W, Wackerly DD, Scheaffer RL (1989), "15: Nonparametric statistics", <i>Mathematical statistics with applications</i> (Fourth ed.), PWS-Kent, pp. 674–679, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-92026-8" title="Special:BookSources/0-534-92026-8"><bdi>0-534-92026-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=15%3A+Nonparametric+statistics&rft.btitle=Mathematical+statistics+with+applications&rft.pages=674-679&rft.edition=Fourth&rft.pub=PWS-Kent&rft.date=1989&rft.isbn=0-534-92026-8&rft.aulast=Mendenhall&rft.aufirst=W&rft.au=Wackerly%2C+DD&rft.au=Scheaffer%2C+RL&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASign+test" class="Z3988"></span></span> </li> <li id="cite_note-Zar1999-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Zar1999_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Zar1999_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Zar1999_5-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZar1999" class="citation cs2">Zar, Jerold H. (1999), "Chapter 24: More on Dichotomous Variables", <i>Biostatistical Analysis</i> (Fourth ed.), Prentice-Hall, pp. 516–570, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-081542-X" title="Special:BookSources/0-13-081542-X"><bdi>0-13-081542-X</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+24%3A+More+on+Dichotomous+Variables&rft.btitle=Biostatistical+Analysis&rft.pages=516-570&rft.edition=Fourth&rft.pub=Prentice-Hall&rft.date=1999&rft.isbn=0-13-081542-X&rft.aulast=Zar&rft.aufirst=Jerold+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASign+test" class="Z3988"></span></span> </li> <li id="cite_note-Conover1999-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Conover1999_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Conover1999_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Conover1999_6-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Conover1999_6-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Conover1999_6-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Conover1999_6-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-Conover1999_6-6"><sup><i><b>g</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConover1999" class="citation cs2">Conover, W.J. (1999), "Chapter 3.4: The Sign Test", <i>Practical Nonparametric Statistics</i> (Third ed.), Wiley, pp. 157–176, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-16068-7" title="Special:BookSources/0-471-16068-7"><bdi>0-471-16068-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+3.4%3A+The+Sign+Test&rft.btitle=Practical+Nonparametric+Statistics&rft.pages=157-176&rft.edition=Third&rft.pub=Wiley&rft.date=1999&rft.isbn=0-471-16068-7&rft.aulast=Conover&rft.aufirst=W.J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASign+test" class="Z3988"></span> </span> </li> <li id="cite_note-Sprent1989-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Sprent1989_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Sprent1989_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSprent1989" class="citation cs2">Sprent, P. (1989), <i>Applied Nonparametric Statistical Methods</i> (Second ed.), Chapman & Hall, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-412-44980-3" title="Special:BookSources/0-412-44980-3"><bdi>0-412-44980-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Nonparametric+Statistical+Methods&rft.edition=Second&rft.pub=Chapman+%26+Hall&rft.date=1989&rft.isbn=0-412-44980-3&rft.aulast=Sprent&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASign+test" class="Z3988"></span></span> </li> <li id="cite_note-Bellhouse2001-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bellhouse2001_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBellhouse2001" class="citation cs2">Bellhouse, P. (2001), "John Arbuthnot", in <a href="/wiki/Chris_Heyde" title="Chris Heyde">C.C. Heyde</a>; <a href="/wiki/Eugene_Seneta" title="Eugene Seneta">E. Seneta</a> (eds.), <i>in Statisticians of the Centuries</i>, Springer, pp. 39–42, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-95329-9" title="Special:BookSources/0-387-95329-9"><bdi>0-387-95329-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=John+Arbuthnot&rft.btitle=in+Statisticians+of+the+Centuries&rft.pages=39-42&rft.pub=Springer&rft.date=2001&rft.isbn=0-387-95329-9&rft.aulast=Bellhouse&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASign+test" class="Z3988"></span></span> </li> <li id="cite_note-Hald1998-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hald1998_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hald1998_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHald1998" class="citation cs2">Hald, Anders (1998), "Chapter 4. Chance or Design: Tests of Significance", <i>A History of Mathematical Statistics from 1750 to 1930</i>, Wiley, p. 65</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+4.+Chance+or+Design%3A+Tests+of+Significance&rft.btitle=A+History+of+Mathematical+Statistics+from+1750+to+1930&rft.pages=65&rft.pub=Wiley&rft.date=1998&rft.aulast=Hald&rft.aufirst=Anders&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASign+test" class="Z3988"></span></span> </li> <li id="cite_note-Bian2011-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bian2011_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBianMcAleerWong2011" class="citation cs2">Bian G, McAleer M, Wong WK (2011), <i>A trinomial test for paired data when there are many ties.</i>, Mathematics and Computers in Simulation, 81(6), pp. 1153–1160</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+trinomial+test+for+paired+data+when+there+are+many+ties.&rft.pages=1153-1160&rft.pub=Mathematics+and+Computers+in+Simulation%2C+81%286%29&rft.date=2011&rft.aulast=Bian&rft.aufirst=G&rft.au=McAleer%2C+M&rft.au=Wong%2C+WK&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASign+test" class="Z3988"></span></span> </li> </ol></div></div> <ul><li><a href="/wiki/Jean_D._Gibbons" title="Jean D. Gibbons">Gibbons, J.D.</a> and Chakraborti, S. (1992). Nonparametric Statistical Inference. Marcel Dekker Inc., New York.</li> <li>Kitchens, L.J.(2003). <i>Basic Statistics and Data Analysis.</i> Duxbury.</li> <li>Conover, W. J. (1980). <i>Practical Nonparametric Statistics</i>, 2nd ed. Wiley, New York.</li> <li>Lehmann, E. L. (1975). <i>Nonparametrics: Statistical Methods Based on Ranks.</i> Holden and Day, San Francisco.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist 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href="/wiki/Outline_of_statistics" title="Outline of statistics">Outline</a></li> <li><a href="/wiki/List_of_statistics_articles" title="List of statistics articles">Index</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Descriptive_statistics" style="font-size:114%;margin:0 4em"><a href="/wiki/Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">Continuous data</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Central_tendency" title="Central tendency">Center</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mean" title="Mean">Mean</a> <ul><li><a href="/wiki/Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li> <li><a href="/wiki/Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li> <li><a href="/wiki/Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li> <li><a href="/wiki/Cubic_mean" title="Cubic mean">Cubic</a></li> <li><a href="/wiki/Generalized_mean" title="Generalized mean">Generalized/power</a></li> <li><a href="/wiki/Geometric_mean" title="Geometric mean">Geometric</a></li> <li><a href="/wiki/Harmonic_mean" title="Harmonic mean">Harmonic</a></li> <li><a href="/wiki/Heronian_mean" title="Heronian mean">Heronian</a></li> <li><a href="/wiki/Heinz_mean" title="Heinz mean">Heinz</a></li> <li><a href="/wiki/Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li> <li><a href="/wiki/Median" title="Median">Median</a></li> <li><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_dispersion" title="Statistical dispersion">Dispersion</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li> <li><a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li> <li><a href="/wiki/Interquartile_range" title="Interquartile range">Interquartile range</a></li> <li><a href="/wiki/Percentile" title="Percentile">Percentile</a></li> <li><a href="/wiki/Range_(statistics)" title="Range (statistics)">Range</a></li> <li><a href="/wiki/Standard_deviation" title="Standard deviation">Standard deviation</a></li> <li><a href="/wiki/Variance#Sample_variance" title="Variance">Variance</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Shape_of_the_distribution" class="mw-redirect" title="Shape of the distribution">Shape</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">Moments</a> <ul><li><a href="/wiki/Kurtosis" title="Kurtosis">Kurtosis</a></li> <li><a href="/wiki/L-moment" title="L-moment">L-moments</a></li> <li><a href="/wiki/Skewness" title="Skewness">Skewness</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Count_data" title="Count data">Count data</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li> <li><a href="/wiki/Grouped_data" title="Grouped data">Grouped data</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Dependence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li> <li><a href="/wiki/Rank_correlation" title="Rank correlation">Rank correlation</a> <ul><li><a href="/wiki/Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li> <li><a href="/wiki/Spearman%27s_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_graphics" title="Statistical graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bar_chart" title="Bar chart">Bar chart</a></li> <li><a href="/wiki/Biplot" title="Biplot">Biplot</a></li> <li><a href="/wiki/Box_plot" title="Box plot">Box plot</a></li> <li><a href="/wiki/Control_chart" title="Control chart">Control chart</a></li> <li><a href="/wiki/Correlogram" title="Correlogram">Correlogram</a></li> <li><a href="/wiki/Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li> <li><a href="/wiki/Forest_plot" title="Forest plot">Forest plot</a></li> <li><a href="/wiki/Histogram" title="Histogram">Histogram</a></li> <li><a href="/wiki/Pie_chart" title="Pie chart">Pie chart</a></li> <li><a href="/wiki/Q%E2%80%93Q_plot" title="Q–Q plot">Q–Q plot</a></li> <li><a href="/wiki/Radar_chart" title="Radar chart">Radar chart</a></li> <li><a href="/wiki/Run_chart" title="Run chart">Run chart</a></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li> <li><a href="/wiki/Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li> <li><a href="/wiki/Violin_plot" title="Violin plot">Violin plot</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_collection" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Design_of_experiments" title="Design of experiments">Study design</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Effect_size" title="Effect size">Effect size</a></li> <li><a href="/wiki/Missing_data" title="Missing data">Missing data</a></li> <li><a href="/wiki/Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li> <li><a href="/wiki/Statistical_population" title="Statistical population">Population</a></li> <li><a href="/wiki/Replication_(statistics)" title="Replication (statistics)">Replication</a></li> <li><a href="/wiki/Sample_size_determination" title="Sample size determination">Sample size determination</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">Sampling</a> <ul><li><a href="/wiki/Cluster_sampling" title="Cluster sampling">Cluster</a></li> <li><a href="/wiki/Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li> <li><a href="/wiki/Opinion_poll" title="Opinion poll">Opinion poll</a></li> <li><a href="/wiki/Questionnaire" title="Questionnaire">Questionnaire</a></li> <li><a href="/wiki/Standard_error" title="Standard error">Standard error</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li> <li><a href="/wiki/Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li> <li><a href="/wiki/Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li> <li><a href="/wiki/Random_assignment" title="Random assignment">Random assignment</a></li> <li><a href="/wiki/Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li> <li><a href="/wiki/Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li> <li><a href="/wiki/Scientific_control" title="Scientific control">Scientific control</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li> <li><a href="/wiki/Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li> <li><a href="/wiki/Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Observational_study" title="Observational study">Observational studies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohort_study" title="Cohort study">Cohort study</a></li> <li><a href="/wiki/Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li> <li><a href="/wiki/Natural_experiment" title="Natural experiment">Natural experiment</a></li> <li><a href="/wiki/Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a></li> <li><a href="/wiki/Sampling_distribution" title="Sampling distribution">Sampling distribution</a> <ul><li><a href="/wiki/Order_statistic" title="Order statistic">Order statistic</a></li></ul></li> <li><a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a> <ul><li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li></ul></li> <li><a href="/wiki/Statistical_model" title="Statistical model">Statistical model</a> <ul><li><a href="/wiki/Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li> <li><a href="/wiki/Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li> <li><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameter</a> <ul><li><a href="/wiki/Location_parameter" title="Location parameter">location</a></li> <li><a href="/wiki/Scale_parameter" title="Scale parameter">scale</a></li> <li><a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></li></ul></li> <li><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric family</a> <ul><li><a href="/wiki/Likelihood_function" title="Likelihood function">Likelihood</a> <a href="/wiki/Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size:85%;">(monotone)</span></a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential family</a></li></ul></li> <li><a href="/wiki/Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li> <li><a href="/wiki/Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/U-statistic" title="U-statistic">U</a></li> <li><a href="/wiki/V-statistic" title="V-statistic">V</a></li></ul></li> <li><a href="/wiki/Optimal_decision" title="Optimal decision">Optimal decision</a> <ul><li><a href="/wiki/Loss_function" title="Loss function">loss function</a></li></ul></li> <li><a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li> <li><a href="/wiki/Statistical_distance" title="Statistical distance">Statistical distance</a> <ul><li><a href="/wiki/Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li> <li><a href="/wiki/Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li> <li><a href="/wiki/Robust_statistics" title="Robust statistics">Robustness</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Estimating_equations" title="Estimating equations">Estimating equations</a> <ul><li><a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li> <li><a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li> <li><a href="/wiki/M-estimator" title="M-estimator">M-estimator</a></li> <li><a href="/wiki/Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li> <li><a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a> <ul><li><a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a> <ul><li><a href="/wiki/Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li> <li><a href="/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li> <li><a href="/wiki/Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Plug-in</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Confidence_interval" title="Confidence interval">Confidence interval</a></li> <li><a href="/wiki/Pivotal_quantity" title="Pivotal quantity">Pivot</a></li> <li><a href="/wiki/Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li> <li><a href="/wiki/Prediction_interval" title="Prediction interval">Prediction interval</a></li> <li><a href="/wiki/Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li> <li><a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">Resampling</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li> <li><a href="/wiki/Power_(statistics)" title="Power (statistics)">Power</a> <ul><li><a href="/wiki/Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li> <li><a href="/wiki/Permutation_test" title="Permutation test">Permutation test</a> <ul><li><a href="/wiki/Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li> <li><a href="/wiki/Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li> <li><a href="/wiki/Score_test" title="Score test">Score/Lagrange multiplier</a></li> <li><a href="/wiki/Wald_test" title="Wald test">Wald</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Z-test" title="Z-test"><i>Z</i>-test <span style="font-size:85%;">(normal)</span></a></li> <li><a href="/wiki/Student%27s_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li> <li><a href="/wiki/F-test" title="F-test"><i>F</i>-test</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chi-squared_test" title="Chi-squared test">Chi-squared</a></li> <li><a href="/wiki/G-test" title="G-test"><i>G</i>-test</a></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li> <li><a href="/wiki/Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li> <li><a href="/wiki/Lilliefors_test" title="Lilliefors test">Lilliefors</a></li> <li><a href="/wiki/Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li> <li><a href="/wiki/Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size:85%;">(Shapiro–Wilk)</span></a></li> <li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li> <li><a href="/wiki/Model_selection" title="Model selection">Model selection</a> <ul><li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li> <li><a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">AIC</a></li> <li><a href="/wiki/Bayesian_information_criterion" title="Bayesian information criterion">BIC</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Sign</a> <ul><li><a href="/wiki/Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li> <li><a href="/wiki/Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size:85%;">(Wilcoxon)</span></a> <ul><li><a href="/wiki/Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li> <li><a href="/wiki/Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size:85%;">(Mann–Whitney)</span></a></li> <li><a href="/wiki/Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="/wiki/Analysis_of_variance" title="Analysis of variance">anova</a> <ul><li><a href="/wiki/Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size:85%;">(Kruskal–Wallis)</span></a></li> <li><a href="/wiki/Friedman_test" title="Friedman test">2-way <span style="font-size:85%;">(Friedman)</span></a></li> <li><a href="/wiki/Jonckheere%27s_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size:85%;">(Jonckheere–Terpstra)</span></a></li></ul></li> <li><a href="/wiki/Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian probability</a> <ul><li><a href="/wiki/Prior_probability" title="Prior probability">prior</a></li> <li><a href="/wiki/Posterior_probability" title="Posterior probability">posterior</a></li></ul></li> <li><a href="/wiki/Credible_interval" title="Credible interval">Credible interval</a></li> <li><a href="/wiki/Bayes_factor" title="Bayes factor">Bayes factor</a></li> <li><a href="/wiki/Bayes_estimator" title="Bayes estimator">Bayesian estimator</a> <ul><li><a href="/wiki/Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li> <li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Confounding" title="Confounding">Confounding variable</a></li> <li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li> <li><a href="/wiki/Regression_validation" title="Regression validation">Regression validation</a></li> <li><a href="/wiki/Mixed_model" title="Mixed model">Mixed effects models</a></li> <li><a href="/wiki/Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li> <li><a href="/wiki/Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li> <li><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li> <li><a href="/wiki/General_linear_model" title="General linear model">General linear model</a></li> <li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li> <li><a href="/wiki/Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li> <li><a href="/wiki/Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li> <li><a href="/wiki/Isotonic_regression" title="Isotonic regression">Isotonic</a></li> <li><a href="/wiki/Robust_regression" title="Robust regression">Robust</a></li> <li><a href="/wiki/Heteroscedasticity" class="mw-redirect" title="Heteroscedasticity">Heteroscedasticity</a></li> <li><a href="/wiki/Homoscedasticity" class="mw-redirect" title="Homoscedasticity">Homoscedasticity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Exponential_family" title="Exponential family">Exponential families</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic <span style="font-size:85%;">(Bernoulli)</span></a> / <a href="/wiki/Binomial_regression" title="Binomial regression">Binomial</a> / <a href="/wiki/Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li> <li><a href="/wiki/Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li> <li><a href="/wiki/Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_Multivariate_/_Time-series_/_Survival_analysis" style="font-size:114%;margin:0 4em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a> / <a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a> / <a href="/wiki/Time_series" title="Time series">Time-series</a> / <a href="/wiki/Survival_analysis" title="Survival analysis">Survival analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohen%27s_kappa" title="Cohen's kappa">Cohen's kappa</a></li> <li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Graphical_model" title="Graphical model">Graphical model</a></li> <li><a href="/wiki/Poisson_regression" title="Poisson regression">Log-linear model</a></li> <li><a href="/wiki/McNemar%27s_test" title="McNemar's test">McNemar's test</a></li> <li><a href="/wiki/Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Cochran–Mantel–Haenszel statistics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_linear_model" title="General linear model">Regression</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li> <li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">Principal components</a></li> <li><a href="/wiki/Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li> <li><a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li> <li><a href="/wiki/Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li> <li><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></li> <li><a href="/wiki/Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a> <ul><li><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li> <li><a href="/wiki/Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a> <ul><li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a> <ul><li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Time_series" title="Time series">Time-series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li> <li><a href="/wiki/Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationarity</a></li> <li><a href="/wiki/Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li> <li><a href="/wiki/Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li> <li><a href="/wiki/Cointegration" title="Cointegration">Cointegration</a></li> <li><a href="/wiki/Structural_break" title="Structural break">Structural break</a></li> <li><a href="/wiki/Granger_causality" title="Granger causality">Granger causality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li> <li><a href="/wiki/Johansen_test" title="Johansen test">Johansen</a></li> <li><a href="/wiki/Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size:85%;">(Ljung–Box)</span></a></li> <li><a href="/wiki/Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li> <li><a href="/wiki/Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Time_domain" title="Time domain">Time domain</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a> <ul><li><a href="/wiki/Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li> <li><a href="/wiki/Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li> <li><a href="/wiki/Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size:85%;">(Box–Jenkins)</span></a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li> <li><a href="/wiki/Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Frequency_domain" title="Frequency domain">Frequency domain</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li> <li><a href="/wiki/Wavelet" title="Wavelet">Wavelet</a></li> <li><a href="/wiki/Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Survival_function" title="Survival function">Survival function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator (product limit)</a></li> <li><a href="/wiki/Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li> <li><a href="/wiki/Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li> <li><a href="/wiki/First-hitting-time_model" title="First-hitting-time model">First hitting time</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Failure_rate" title="Failure rate">Hazard function</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nelson%E2%80%93Aalen_estimator" title="Nelson–Aalen estimator">Nelson–Aalen estimator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Test</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Log-rank_test" class="mw-redirect" title="Log-rank test">Log-rank test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Applications" style="font-size:114%;margin:0 4em"><a href="/wiki/List_of_fields_of_application_of_statistics" title="List of fields of application of statistics">Applications</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Biostatistics" title="Biostatistics">Biostatistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bioinformatics" title="Bioinformatics">Bioinformatics</a></li> <li><a href="/wiki/Clinical_trial" title="Clinical trial">Clinical trials</a> / <a href="/wiki/Clinical_study_design" title="Clinical study design">studies</a></li> <li><a href="/wiki/Epidemiology" title="Epidemiology">Epidemiology</a></li> <li><a href="/wiki/Medical_statistics" title="Medical statistics">Medical statistics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Engineering_statistics" title="Engineering statistics">Engineering statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div 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