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Added G-test. JIRA: MATH-878. 

git-svn-id: https://svn.apache.org/repos/asf/commons/proper/math/trunk@1408174 13f79535-47bb-0310-9956-ffa450edef68
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Phil Steitz 2012-11-12 04:33:11 +00:00
parent c49c8c6667
commit 368050bbe6
1 changed files with 76 additions and 9 deletions
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src/site/xdoc/userguide/stat.xml
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@ -810,6 +810,7 @@ new PearsonsCorrelation().correlation(ranking.rank(x), ranking.rank(y))
Student's t</a>,
<a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda35f.htm">
Chi-Square</a>,
<a href="http://en.wikipedia.org/wiki/G-test">G Test</a>,
<a href="http://www.itl.nist.gov/div898/handbook/prc/section4/prc43.htm">
One-Way ANOVA</a>,
<a href="http://www.itl.nist.gov/div898/handbook/prc/section3/prc35.htm">
@ -818,12 +819,14 @@ new PearsonsCorrelation().correlation(ranking.rank(x), ranking.rank(y))
Wilcoxon signed rank</a> test statistics as well as
<a href="http://www.cas.lancs.ac.uk/glossary_v1.1/hyptest.html#pvalue">
p-values</a> associated with <code>t-</code>,
<code>Chi-Square</code>, <code>One-Way ANOVA</code>, <code>Mann-Whitney U</code>
<code>Chi-Square</code>, <code>G</code>, <code>One-Way ANOVA</code>, <code>Mann-Whitney U</code>
and <code>Wilcoxon signed rank</code> tests. The respective test classes are
<a href="../apidocs/org/apache/commons/math3/stat/inference/TTest.html">
TTest</a>,
<a href="../apidocs/org/apache/commons/math3/stat/inference/ChiSquareTest.html">
ChiSquareTest</a>,
<a href="../apidocs/org/apache/commons/math3/stat/inference/GTest.html">
GTest</a>,
<a href="../apidocs/org/apache/commons/math3/stat/inference/OneWayAnova.html">
OneWayAnova</a>,
<a href="../apidocs/org/apache/commons/math3/stat/inference/MannWhitneyUTest.html">
@ -864,14 +867,19 @@ new PearsonsCorrelation().correlation(ranking.rank(x), ranking.rank(y))
<li>p-values returned by t-, chi-square and Anova tests are exact, based
on numerical approximations to the t-, chi-square and F distributions in the
<code>distributions</code> package. </li>
<li>The G test implementation provides two p-values:
<code>gTest(expected, observed)</code>, which is the tail probability beyond
<code>g(expected, observed)</code> in the ChiSquare distribution with degrees
of freedom one less than the common length of input arrays and
<code>gTestIntrinsic(expected, observed)</code> which is the same tail
probability computed using a ChiSquare distribution with one less degeree
of freedom. </li>
<li>p-values returned by t-tests are for two-sided tests and the boolean-valued
methods supporting fixed significance level tests assume that the hypotheses
are two-sided. One sided tests can be performed by dividing returned p-values
(resp. critical values) by 2.</li>
<li>Degrees of freedom for chi-square tests are integral values, based on the
number of observed or expected counts (number of observed counts - 1)
for the goodness-of-fit tests and (number of columns -1) * (number of rows - 1)
for independence tests.</li>
<li>Degrees of freedom for g- and chi-square tests are integral values, based on the
number of observed or expected counts (number of observed counts - 1).</li>
</ul>
</p>
<p>
@ -1059,11 +1067,70 @@ TestUtils.chiSquareTest(counts, alpha);
hypothesis can be rejected with confidence <code>1 - alpha</code>.
</dd>
<br></br>
<dt><strong>g tests</strong></dt>
<br></br>
<dd>g tests are an alternative to chi-square tests that are recommended
when observed counts are small and / or incidence probabillities for
some cells are small. See Ted Dunning's paper,
<a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.14.5962">
Accurate Methods for the Statistics of Surprise and Coincidence</a> for
background and an empirical analysis showing now chi-square
statistics can be misldeading in the presence of low incidence probabilities.
This paper also derives the formulas used in computing g statistics and the
root log likelihood ratio provided by the <code>GTest</code> class.</dd>
<dd>
<dd>To compute a g-test statistic measuring the agreement between a
<code>long[]</code> array of observed counts and a <code>double[]</code>
array of expected counts, use:
<source>
double[] expected = new double[]{0.54d, 0.40d, 0.05d, 0.01d};
long[] observed = new long[]{70, 79, 3, 4};
System.out.println(TestUtils.g(expected, observed));
</source>
the value displayed will be
<code>2 * sum(observed[i]) * log(observed[i]/expected[i])</code>
</dd>
<dd> To get the p-value associated with the null hypothesis that
<code>observed</code> conforms to <code>expected</code> use:
<source>
TestUtils.gTest(expected, observed);
</source>
</dd>
<dd> To test the null hypothesis that <code>observed</code> conforms to
<code>expected</code> with <code>alpha</code> siginficance level
(equiv. <code>100 * (1-alpha)%</code> confidence) where <code>
0 &lt; alpha &lt; 1 </code> use:
<source>
TestUtils.gTest(expected, observed, alpha);
</source>
The boolean value returned will be <code>true</code> iff the null hypothesis
can be rejected with confidence <code>1 - alpha</code>.
</dd>
<dd>To evaluate the hypothesis that two sets of counts come from the
same underlying distribution, use long[] arrays for the counts and
<code>gDataSetsComparison</code> for the test statistic
<source>
long[] obs1 = new long[]{268, 199, 42};
long[] obs2 = new long[]{807, 759, 184};
System.out.println(TestUtils.gDataSetsComparison(obs1, obs2)); // g statistic
System.out.println(TestUtils.gTestDataSetsComparison(obs1, obs2)); // p-value
</source>
</dd>
<dd>For 2 x 2 designs, the <code>rootLogLikelihoodRaio</code> method
computes the
<a href="http://tdunning.blogspot.com/2008/03/surprise-and-coincidence.html">
signed root log likelihood ratio.</a> For example, suppose that for two events
A and B, the observed count of AB (both occurring) is 5, not A and B (B without A)
is 1995, A not B is 0; and neither A nor B is 10000. Then
<source>
new GTest().rootLogLikelihoodRatio(5, 1995, 0, 100000);
</source>
returns the root log likelihood associated with the null hypothesis that A
and B are independent.
</dd>
<br></br>
<dt><strong>One-Way Anova tests</strong></dt>
<br></br>
<dd>To conduct a One-Way Analysis of Variance (ANOVA) to evaluate the
null hypothesis that the means of a collection of univariate datasets
are the same, start by loading the datasets into a collection, e.g.
<source>
double[] classA =
{93.0, 103.0, 95.0, 101.0, 91.0, 105.0, 96.0, 94.0, 101.0 };