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<?xml version="1.0"?>
<!--
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<?xml-stylesheet type="text/xsl" href="./xdoc.xsl"?>
<!-- $Revision$ $Date$ -->
<document url="stat.html">
<properties>
<title>The Commons Math User Guide - Statistics</title>
</properties>
<body>
<section name="1 Statistics">
<subsection name="1.1 Overview" href="overview">
<p>
The statistics package provides frameworks and implementations for
basic Descriptive statistics, frequency distributions, bivariate regression,
and t-, chi-square and ANOVA test statistics.
</p>
<p>
<a href="#1.2 Descriptive statistics">Descriptive statistics</a><br></br>
<a href="#1.3 Frequency distributions">Frequency distributions</a><br></br>
<a href="#1.4 Simple regression">Simple Regression</a><br></br>
<a href="#1.5 Statistical tests">Statistical Tests</a><br></br>
</p>
</subsection>
<subsection name="1.2 Descriptive statistics" href="univariate">
<p>
The stat package includes a framework and default implementations for
the following Descriptive statistics:
<ul>
<li>arithmetic and geometric means</li>
<li>variance and standard deviation</li>
<li>sum, product, log sum, sum of squared values</li>
<li>minimum, maximum, median, and percentiles</li>
<li>skewness and kurtosis</li>
<li>first, second, third and fourth moments</li>
</ul>
</p>
<p>
With the exception of percentiles and the median, all of these
statistics can be computed without maintaining the full list of input
data values in memory. The stat package provides interfaces and
implementations that do not require value storage as well as
implementations that operate on arrays of stored values.
</p>
<p>
The top level interface is
<a href="../apidocs/org/apache/commons/math/stat/descriptive/UnivariateStatistic.html">
org.apache.commons.math.stat.descriptive.UnivariateStatistic.</a>
This interface, implemented by all statistics, consists of
<code>evaluate()</code> methods that take double[] arrays as arguments
and return the value of the statistic. This interface is extended by
<a href="../apidocs/org/apache/commons/math/stat/descriptive/StorelessUnivariateStatistic.html">
StorelessUnivariateStatistic</a>, which adds <code>increment(),</code>
<code>getResult()</code> and associated methods to support
"storageless" implementations that maintain counters, sums or other
state information as values are added using the <code>increment()</code>
method.
</p>
<p>
Abstract implementations of the top level interfaces are provided in
<a href="../apidocs/org/apache/commons/math/stat/descriptive/AbstractUnivariateStatistic.html">
AbstractUnivariateStatistic</a> and
<a href="../apidocs/org/apache/commons/math/stat/descriptive/AbstractStorelessUnivariateStatistic.html">
AbstractStorelessUnivariateStatistic</a> respectively.
</p>
<p>
Each statistic is implemented as a separate class, in one of the
subpackages (moment, rank, summary) and each extends one of the abstract
classes above (depending on whether or not value storage is required to
compute the statistic). There are several ways to instantiate and use statistics.
Statistics can be instantiated and used directly, but it is generally more convenient
(and efficient) to access them using the provided aggregates,
<a href="../apidocs/org/apache/commons/math/stat/descriptive/DescriptiveStatistics.html">
DescriptiveStatistics</a> and
<a href="../apidocs/org/apache/commons/math/stat/descriptive/SummaryStatistics.html">
SummaryStatistics.</a>
</p>
<p>
<code>DescriptiveStatistics</code> maintains the input data in memory
and has the capability of producing "rolling" statistics computed from a
"window" consisting of the most recently added values.
</p>
<p>
<code>SummaryStatisics</code> does not store the input data values
in memory, so the statistics included in this aggregate are limited to those
that can be computed in one pass through the data without access to
the full array of values.
</p>
<p>
<table>
<tr><th>Aggregate</th><th>Statistics Included</th><th>Values stored?</th>
<th>"Rolling" capability?</th></tr><tr><td>
<a href="../apidocs/org/apache/commons/math/stat/descriptive/DescriptiveStatistics.html">
DescriptiveStatistics</a></td><td>min, max, mean, geometric mean, n,
sum, sum of squares, standard deviation, variance, percentiles, skewness,
kurtosis, median</td><td>Yes</td><td>Yes</td></tr><tr><td>
<a href="../apidocs/org/apache/commons/math/stat/descriptive/SummaryStatistics.html">
SummaryStatistics</a></td><td>min, max, mean, geometric mean, n,
sum, sum of squares, standard deviation, variance</td><td>No</td><td>No</td></tr>
</table>
</p>
<p>
There is also a utility class,
<a href="../apidocs/org/apache/commons/math/stat/StatUtils.html">
StatUtils</a>, that provides static methods for computing statistics
directly from double[] arrays.
</p>
<p>
Here are some examples showing how to compute Descriptive statistics.
<dl>
<dt>Compute summary statistics for a list of double values</dt>
<br></br>
<dd>Using the <code>DescriptiveStatistics</code> aggregate
(values are stored in memory):
<source>
// Get a DescriptiveStatistics instance using factory method
DescriptiveStatistics stats = DescriptiveStatistics.newInstance();
// Add the data from the array
for( int i = 0; i < inputArray.length; i++) {
stats.addValue(inputArray[i]);
}
// Compute some statistics
double mean = stats.getMean();
double std = stats.getStandardDeviation();
double median = stats.getMedian();
</source>
</dd>
<dd>Using the <code>SummaryStatistics</code> aggregate (values are
<strong>not</strong> stored in memory):
<source>
// Get a SummaryStatistics instance using factory method
SummaryStatistics stats = SummaryStatistics.newInstance();
// Read data from an input stream,
// adding values and updating sums, counters, etc.
while (line != null) {
line = in.readLine();
stats.addValue(Double.parseDouble(line.trim()));
}
in.close();
// Compute the statistics
double mean = stats.getMean();
double std = stats.getStandardDeviation();
//double median = stats.getMedian(); <-- NOT AVAILABLE
</source>
</dd>
<dd>Using the <code>StatUtils</code> utility class:
<source>
// Compute statistics directly from the array
// assume values is a double[] array
double mean = StatUtils.mean(values);
double std = StatUtils.variance(values);
double median = StatUtils.percentile(50);
// Compute the mean of the first three values in the array
mean = StatuUtils.mean(values, 0, 3);
</source>
</dd>
<dt>Maintain a "rolling mean" of the most recent 100 values from
an input stream</dt>
<br></br>
<dd>Use a <code>DescriptiveStatistics</code> instance with
window size set to 100
<source>
// Create a DescriptiveStats instance and set the window size to 100
DescriptiveStatistics stats = DescriptiveStatistics.newInstance();
stats.setWindowSize(100);
// Read data from an input stream,
// displaying the mean of the most recent 100 observations
// after every 100 observations
long nLines = 0;
while (line != null) {
line = in.readLine();
stats.addValue(Double.parseDouble(line.trim()));
if (nLines == 100) {
nLines = 0;
System.out.println(stats.getMean());
}
}
in.close();
</source>
</dd>
<dt>Compute statistics in a thread-safe manner</dt>
<br/>
<dd>Use a <code>SynchronizedDescriptiveStatistics</code> instance
<source>
// Create a SynchronizedDescriptiveStatistics instance and
// use as any other DescriptiveStatistics instance
DescriptiveStatistics stats = DescriptiveStatistics.newInstance(SynchronizedDescriptiveStatistics.class);
</source>
</dd>
</dl>
</p>
</subsection>
<subsection name="1.3 Frequency distributions" href="frequency">
<p>
<a href="../apidocs/org/apache/commons/math/stat/Frequency.html">
org.apache.commons.math.stat.descriptive.Frequency</a>
provides a simple interface for maintaining counts and percentages of discrete
values.
</p>
<p>
Strings, integers, longs and chars are all supported as value types,
as well as instances of any class that implements <code>Comparable.</code>
The ordering of values used in computing cumulative frequencies is by
default the <i>natural ordering,</i> but this can be overriden by supplying a
<code>Comparator</code> to the constructor. Adding values that are not
comparable to those that have already been added results in an
<code>IllegalArgumentException.</code>
</p>
<p>
Here are some examples.
<dl>
<dt>Compute a frequency distribution based on integer values</dt>
<br></br>
<dd>Mixing integers, longs, Integers and Longs:
<source>
Frequency f = new Frequency();
f.addValue(1);
f.addValue(new Integer(1));
f.addValue(new Long(1));
f.addValue(2);
f.addValue(new Integer(-1));
System.out.prinltn(f.getCount(1)); // displays 3
System.out.println(f.getCumPct(0)); // displays 0.2
System.out.println(f.getPct(new Integer(1))); // displays 0.6
System.out.println(f.getCumPct(-2)); // displays 0
System.out.println(f.getCumPct(10)); // displays 1
</source>
</dd>
<dt>Count string frequencies</dt>
<br></br>
<dd>Using case-sensitive comparison, alpha sort order (natural comparator):
<source>
Frequency f = new Frequency();
f.addValue("one");
f.addValue("One");
f.addValue("oNe");
f.addValue("Z");
System.out.println(f.getCount("one")); // displays 1
System.out.println(f.getCumPct("Z")); // displays 0.5
System.out.println(f.getCumPct("Ot")); // displays 0.25
</source>
</dd>
<dd>Using case-insensitive comparator:
<source>
Frequency f = new Frequency(String.CASE_INSENSITIVE_ORDER);
f.addValue("one");
f.addValue("One");
f.addValue("oNe");
f.addValue("Z");
System.out.println(f.getCount("one")); // displays 3
System.out.println(f.getCumPct("z")); // displays 1
</source>
</dd>
</dl>
</p>
</subsection>
<subsection name="1.4 Simple regression" href="regression">
<p>
<a href="../apidocs/org/apache/commons/math/stat/regression/SimpleRegression.html">
org.apache.commons.math.stat.regression.SimpleRegression</a>
provides ordinary least squares regression with one independent variable,
estimating the linear model:
</p>
<p>
<code> y = intercept + slope * x </code>
</p>
<p>
Standard errors for <code>intercept</code> and <code>slope</code> are
available as well as ANOVA, r-square and Pearson's r statistics.
</p>
<p>
Observations (x,y pairs) can be added to the model one at a time or they
can be provided in a 2-dimensional array. The observations are not stored
in memory, so there is no limit to the number of observations that can be
added to the model.
</p>
<p>
<strong>Usage Notes</strong>: <ul>
<li> When there are fewer than two observations in the model, or when
there is no variation in the x values (i.e. all x values are the same)
all statistics return <code>NaN</code>. At least two observations with
different x coordinates are requred to estimate a bivariate regression
model.</li>
<li> getters for the statistics always compute values based on the current
set of observations -- i.e., you can get statistics, then add more data
and get updated statistics without using a new instance. There is no
"compute" method that updates all statistics. Each of the getters performs
the necessary computations to return the requested statistic.</li>
</ul>
</p>
<p>
<strong>Implementation Notes</strong>: <ul>
<li> As observations are added to the model, the sum of x values, y values,
cross products (x times y), and squared deviations of x and y from their
respective means are updated using updating formulas defined in
"Algorithms for Computing the Sample Variance: Analysis and
Recommendations", Chan, T.F., Golub, G.H., and LeVeque, R.J.
1983, American Statistician, vol. 37, pp. 242-247, referenced in
Weisberg, S. "Applied Linear Regression". 2nd Ed. 1985. All regression
statistics are computed from these sums.</li>
<li> Inference statistics (confidence intervals, parameter significance levels)
are based on on the assumption that the observations included in the model are
drawn from a <a href="http://mathworld.wolfram.com/BivariateNormalDistribution.html">
Bivariate Normal Distribution</a></li>
</ul>
</p>
<p>
Here are some examples.
<dl>
<dt>Estimate a model based on observations added one at a time</dt>
<br></br>
<dd>Instantiate a regression instance and add data points
<source>
regression = new SimpleRegression();
regression.addData(1d, 2d);
// At this point, with only one observation,
// all regression statistics will return NaN
regression.addData(3d, 3d);
// With only two observations,
// slope and intercept can be computed
// but inference statistics will return NaN
regression.addData(3d, 3d);
// Now all statistics are defined.
</source>
</dd>
<dd>Compute some statistics based on observations added so far
<source>
System.out.println(regression.getIntercept());
// displays intercept of regression line
System.out.println(regression.getSlope());
// displays slope of regression line
System.out.println(regression.getSlopeStdErr());
// displays slope standard error
</source>
</dd>
<dd>Use the regression model to predict the y value for a new x value
<source>
System.out.println(regression.predict(1.5d)
// displays predicted y value for x = 1.5
</source>
More data points can be added and subsequent getXxx calls will incorporate
additional data in statistics.
</dd>
<dt>Estimate a model from a double[][] array of data points</dt>
<br></br>
<dd>Instantiate a regression object and load dataset
<source>
double[][] data = { { 1, 3 }, {2, 5 }, {3, 7 }, {4, 14 }, {5, 11 }};
SimpleRegression regression = new SimpleRegression();
regression.addData(data);
</source>
</dd>
<dd>Estimate regression model based on data
<source>
System.out.println(regression.getIntercept());
// displays intercept of regression line
System.out.println(regression.getSlope());
// displays slope of regression line
System.out.println(regression.getSlopeStdErr());
// displays slope standard error
</source>
More data points -- even another double[][] array -- can be added and subsequent
getXxx calls will incorporate additional data in statistics.
</dd>
</dl>
</p>
</subsection>
<subsection name="1.5 Statistical tests" href="tests">
<p>
The interfaces and implementations in the
<a href="../apidocs/org/apache/commons/math/stat/inference/">
org.apache.commons.math.stat.inference</a> package provide
<a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc22.htm">
Student's t</a>,
<a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda35f.htm">
Chi-Square</a> and
<a href="http://www.itl.nist.gov/div898/handbook/prc/section4/prc43.htm">
One-Way ANOVA</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> and <code>One-Way ANOVA</code> tests. The
interfaces are
<a href="../apidocs/org/apache/commons/math/stat/inference/TTest.html">
TTest</a>,
<a href="../apidocs/org/apache/commons/math/stat/inference/ChiSquareTest.html">
ChiSquareTest</a>, and
<a href="../apidocs/org/apache/commons/math/stat/inference/OneWayAnova.html">
OneWayAnova</a> with provided implementations
<a href="../apidocs/org/apache/commons/math/stat/inference/TTestImpl.html">
TTestImpl</a>,
<a href="../apidocs/org/apache/commons/math/stat/inference/ChiSquareTestImpl.html">
ChiSquareTestImpl</a> and
<a href="../apidocs/org/apache/commons/math/stat/inference/OneWayAnovaImpl.html">
OneWayAnovaImpl</a>, respectively.
The
<a href="../apidocs/org/apache/commons/math/stat/inference/TestUtils.html">
TestUtils</a> class provides static methods to get test instances or
to compute test statistics directly. The examples below all use the
static methods in <code>TestUtils</code> to execute tests. To get
test object instances, either use e.g.,
<code>TestUtils.getTTest()</code> or use the implementation constructors
directly, e.g.,
<code>new TTestImpl()</code>.
</p>
<p>
<strong>Implementation Notes</strong>
<ul>
<li>Both one- and two-sample t-tests are supported. Two sample tests
can be either paired or unpaired and the unpaired two-sample tests can
be conducted under the assumption of equal subpopulation variances or
without this assumption. When equal variances is assumed, a pooled
variance estimate is used to compute the t-statistic and the degrees
of freedom used in the t-test equals the sum of the sample sizes minus 2.
When equal variances is not assumed, the t-statistic uses both sample
variances and the
<a href="http://www.itl.nist.gov/div898/handbook/prc/section3/gifs/nu3.gif">
Welch-Satterwaite approximation</a> is used to compute the degrees
of freedom. Methods to return t-statistics and p-values are provided in each
case, as well as boolean-valued methods to perform fixed significance
level tests. The names of methods or methods that assume equal
subpopulation variances always start with "homoscedastic." Test or
test-statistic methods that just start with "t" do not assume equal
variances. See the examples below and the API documentation for
more details.</li>
<li>The validity of the p-values returned by the t-test depends on the
assumptions of the parametric t-test procedure, as discussed
<a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html">
here</a></li>
<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
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>
</ul>
</p>
<p>
<strong>Examples:</strong>
<dl>
<dt><strong>One-sample <code>t</code> tests</strong></dt>
<br></br>
<dd>To compare the mean of a double[] array to a fixed value:
<source>
double[] observed = {1d, 2d, 3d};
double mu = 2.5d;
System.out.println(TestUtils.t(mu, observed);
</source>
The code above will display the t-statisitic associated with a one-sample
t-test comparing the mean of the <code>observed</code> values against
<code>mu.</code>
</dd>
<dd>To compare the mean of a dataset described by a
<a href="../apidocs/org/apache/commons/math/stat/descriptive/StatisticalSummary.html">
org.apache.commons.math.stat.descriptive.StatisticalSummary</a> to a fixed value:
<source>
double[] observed ={1d, 2d, 3d};
double mu = 2.5d;
SummaryStatistics sampleStats = null;
sampleStats = SummaryStatistics.newInstance();
for (int i = 0; i < observed.length; i++) {
sampleStats.addValue(observed[i]);
}
System.out.println(TestUtils.t(mu, observed);
</source>
</dd>
<dd>To compute the p-value associated with the null hypothesis that the mean
of a set of values equals a point estimate, against the two-sided alternative that
the mean is different from the target value:
<source>
double[] observed = {1d, 2d, 3d};
double mu = 2.5d;
System.out.println(TestUtils.tTest(mu, observed);
</source>
The snippet above will display the p-value associated with the null
hypothesis that the mean of the population from which the
<code>observed</code> values are drawn equals <code>mu.</code>
</dd>
<dd>To perform the test using a fixed significance level, use:
<source>
TestUtils.tTest(mu, observed, alpha);
</source>
where <code>0 < alpha < 0.5</code> is the significance level of
the test. The boolean value returned will be <code>true</code> iff the
null hypothesis can be rejected with confidence <code>1 - alpha</code>.
To test, for example at the 95% level of confidence, use
<code>alpha = 0.05</code>
</dd>
<br></br>
<dt><strong>Two-Sample t-tests</strong></dt>
<br></br>
<dd><strong>Example 1:</strong> Paired test evaluating
the null hypothesis that the mean difference between corresponding
(paired) elements of the <code>double[]</code> arrays
<code>sample1</code> and <code>sample2</code> is zero.
<p>
To compute the t-statistic:
<source>
TestUtils.pairedT(sample1, sample2);
</source>
</p>
<p>
To compute the p-value:
<source>
TestUtils.pairedTTest(sample1, sample2);
</source>
</p>
<p>
To perform a fixed significance level test with alpha = .05:
<source>
TestUtils.pairedTTest(sample1, sample2, .05);
</source>
</p>
The last example will return <code>true</code> iff the p-value
returned by <code>TestUtils.pairedTTest(sample1, sample2)</code>
is less than <code>.05</code>
</dd>
<dd><strong>Example 2: </strong> unpaired, two-sided, two-sample t-test using
<code>StatisticalSummary</code> instances, without assuming that
subpopulation variances are equal.
<p>
First create the <code>StatisticalSummary</code> instances. Both
<code>DescriptiveStatistics</code> and <code>SummaryStatistics</code>
implement this interface. Assume that <code>summary1</code> and
<code>summary2</code> are <code>SummaryStatistics</code> instances,
each of which has had at least 2 values added to the (virtual) dataset that
it describes. The sample sizes do not have to be the same -- all that is required
is that both samples have at least 2 elements.
</p>
<p><strong>Note:</strong> The <code>SummaryStatistics</code> class does
not store the dataset that it describes in memory, but it does compute all
statistics necessary to perform t-tests, so this method can be used to
conduct t-tests with very large samples. One-sample tests can also be
performed this way.
(See <a href="#1.2 Descriptive statistics">Descriptive statistics</a> for details
on the <code>SummaryStatistics</code> class.)
</p>
<p>
To compute the t-statistic:
<source>
TestUtils.t(summary1, summary2);
</source>
</p>
<p>
To compute the p-value:
<source>
TestUtils.tTest(sample1, sample2);
</source>
</p>
<p>
To perform a fixed significance level test with alpha = .05:
<source>
TestUtils.tTest(sample1, sample2, .05);
</source>
</p>
<p>
In each case above, the test does not assume that the subpopulation
variances are equal. To perform the tests under this assumption,
replace "t" at the beginning of the method name with "homoscedasticT"
</p>
</dd>
<br></br>
<dt><strong>Chi-square tests</strong></dt>
<br></br>
<dd>To compute a chi-square statistic measuring the agreement between a
<code>long[]</code> array of observed counts and a <code>double[]</code>
array of expected counts, use:
<source>
long[] observed = {10, 9, 11};
double[] expected = {10.1, 9.8, 10.3};
System.out.println(TestUtils.chiSquare(expected, observed));
</source>
the value displayed will be
<code>sum((expected[i] - observed[i])^2 / 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.chiSquareTest(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.chiSquareTest(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 compute a chi-square statistic statistic associated with a
<a href="http://www.itl.nist.gov/div898/handbook/prc/section4/prc45.htm">
chi-square test of independence</a> based on a two-dimensional (long[][])
<code>counts</code> array viewed as a two-way table, use:
<source>
TestUtils.chiSquareTest(counts);
</source>
The rows of the 2-way table are
<code>count[0], ... , count[count.length - 1]. </code><br></br>
The chi-square statistic returned is
<code>sum((counts[i][j] - expected[i][j])^2/expected[i][j])</code>
where the sum is taken over all table entries and
<code>expected[i][j]</code> is the product of the row and column sums at
row <code>i</code>, column <code>j</code> divided by the total count.
</dd>
<dd>To compute the p-value associated with the null hypothesis that
the classifications represented by the counts in the columns of the input 2-way
table are independent of the rows, use:
<source>
TestUtils.chiSquareTest(counts);
</source>
</dd>
<dd>To perform a chi-square test of independence with <code>alpha</code>
siginficance level (equiv. <code>100 * (1-alpha)%</code> confidence)
where <code>0 < alpha < 1 </code> use:
<source>
TestUtils.chiSquareTest(counts, 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>
<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 };
double[] classB =
{99.0, 92.0, 102.0, 100.0, 102.0, 89.0 };
double[] classC =
{110.0, 115.0, 111.0, 117.0, 128.0, 117.0 };
List classes = new ArrayList();
classes.add(classA);
classes.add(classB);
classes.add(classC);
</source>
Then you can compute ANOVA F- or p-values associated with the
null hypothesis that the class means are all the same
using a <code>OneWayAnova</code> instance or <code>TestUtils</code>
methods:
<source>
double fStatistic = TestUtils.oneWayAnovaFValue(classes); // F-value
double pValue = TestUtils.oneWayAnovaPValue(classes); // P-value
</source>
To test perform a One-Way Anova test with signficance level set at 0.01
(so the test will, assuming assumptions are met, reject the null
hypothesis incorrectly only about one in 100 times), use
<source>
TestUtils.oneWayAnovaTest(classes, 0.01); // returns a boolean
// true means reject null hypothesis
</source>
</dd>
</dl>
</p>
</subsection>
</section>
</body>
</document>