Author: psteitz
Date: Mon Nov 12 04:33:11 2012
New Revision: 1408174
URL: http://svn.apache.org/viewvc?rev=1408174&view=rev
Log:
Added G-test. JIRA: MATH-878.
Modified:
commons/proper/math/trunk/src/site/xdoc/userguide/stat.xml
Modified: commons/proper/math/trunk/src/site/xdoc/userguide/stat.xml
URL:
http://svn.apache.org/viewvc/commons/proper/math/trunk/src/site/xdoc/userguide/stat.xml?rev=1408174&r1=1408173&r2=1408174&view=diff
==============================================================================
--- commons/proper/math/trunk/src/site/xdoc/userguide/stat.xml (original)
+++ commons/proper/math/trunk/src/site/xdoc/userguide/stat.xml Mon Nov 12
04:33:11 2012
@@ -810,6 +810,7 @@ new PearsonsCorrelation().correlation(ra
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(ra
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(ra
<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>p-values returned by t-tests are for two-sided tests and the
boolean-valued
+ <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 < alpha < 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 };
