org.apache.commons.math3.stat.inference

## Class ChiSquareTest

• public class ChiSquareTest
extends Object
Implements Chi-Square test statistics.

This implementation handles both known and unknown distributions.

Two samples tests can be used when the distribution is unknown a priori but provided by one sample, or when the hypothesis under test is that the two samples come from the same underlying distribution.

Version:
$Id: ChiSquareTest.java 1416643 2012-12-03 19:37:14Z tn$
• ### Constructor Summary

Constructors
Constructor and Description
ChiSquareTest()
Construct a ChiSquareTest
• ### Method Summary

Methods
Modifier and Type Method and Description
double chiSquare(double[] expected, long[] observed)
Computes the Chi-Square statistic comparing observed and expected frequency counts.
double chiSquare(long[][] counts)
Computes the Chi-Square statistic associated with a chi-square test of independence based on the input counts array, viewed as a two-way table.
double chiSquareDataSetsComparison(long[] observed1, long[] observed2)
Computes a Chi-Square two sample test statistic comparing bin frequency counts in observed1 and observed2.
double chiSquareTest(double[] expected, long[] observed)
Returns the observed significance level, or p-value, associated with a Chi-square goodness of fit test comparing the observed frequency counts to those in the expected array.
boolean chiSquareTest(double[] expected, long[] observed, double alpha)
Performs a Chi-square goodness of fit test evaluating the null hypothesis that the observed counts conform to the frequency distribution described by the expected counts, with significance level alpha.
double chiSquareTest(long[][] counts)
Returns the observed significance level, or p-value, associated with a chi-square test of independence based on the input counts array, viewed as a two-way table.
boolean chiSquareTest(long[][] counts, double alpha)
Performs a chi-square test of independence evaluating the null hypothesis that the classifications represented by the counts in the columns of the input 2-way table are independent of the rows, with significance level alpha.
double chiSquareTestDataSetsComparison(long[] observed1, long[] observed2)
Returns the observed significance level, or p-value, associated with a Chi-Square two sample test comparing bin frequency counts in observed1 and observed2.
boolean chiSquareTestDataSetsComparison(long[] observed1, long[] observed2, double alpha)
Performs a Chi-Square two sample test comparing two binned data sets.
• ### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Constructor Detail

• #### ChiSquareTest

public ChiSquareTest()
Construct a ChiSquareTest
• ### Method Detail

• #### chiSquare

public double chiSquare(double[] expected,
long[] observed)
throws NotPositiveException,
NotStrictlyPositiveException,
DimensionMismatchException
Computes the Chi-Square statistic comparing observed and expected frequency counts.

This statistic can be used to perform a Chi-Square test evaluating the null hypothesis that the observed counts follow the expected distribution.

Preconditions:

• Expected counts must all be positive.
• Observed counts must all be ≥ 0.
• The observed and expected arrays must have the same length and their common length must be at least 2.

If any of the preconditions are not met, an IllegalArgumentException is thrown.

Note: This implementation rescales the expected array if necessary to ensure that the sum of the expected and observed counts are equal.

Parameters:
observed - array of observed frequency counts
expected - array of expected frequency counts
Returns:
chiSquare test statistic
Throws:
NotPositiveException - if observed has negative entries
NotStrictlyPositiveException - if expected has entries that are not strictly positive
DimensionMismatchException - if the arrays length is less than 2
• #### chiSquareTest

public double chiSquareTest(double[] expected,
long[] observed)
throws NotPositiveException,
NotStrictlyPositiveException,
DimensionMismatchException,
MaxCountExceededException
Returns the observed significance level, or p-value, associated with a Chi-square goodness of fit test comparing the observed frequency counts to those in the expected array.

The number returned is the smallest significance level at which one can reject the null hypothesis that the observed counts conform to the frequency distribution described by the expected counts.

Preconditions:

• Expected counts must all be positive.
• Observed counts must all be ≥ 0.
• The observed and expected arrays must have the same length and their common length must be at least 2.

If any of the preconditions are not met, an IllegalArgumentException is thrown.

Note: This implementation rescales the expected array if necessary to ensure that the sum of the expected and observed counts are equal.

Parameters:
observed - array of observed frequency counts
expected - array of expected frequency counts
Returns:
p-value
Throws:
NotPositiveException - if observed has negative entries
NotStrictlyPositiveException - if expected has entries that are not strictly positive
DimensionMismatchException - if the arrays length is less than 2
MaxCountExceededException - if an error occurs computing the p-value
• #### chiSquareTest

public boolean chiSquareTest(double[] expected,
long[] observed,
double alpha)
throws NotPositiveException,
NotStrictlyPositiveException,
DimensionMismatchException,
OutOfRangeException,
MaxCountExceededException
Performs a Chi-square goodness of fit test evaluating the null hypothesis that the observed counts conform to the frequency distribution described by the expected counts, with significance level alpha. Returns true iff the null hypothesis can be rejected with 100 * (1 - alpha) percent confidence.

Example:
To test the hypothesis that observed follows expected at the 99% level, use

chiSquareTest(expected, observed, 0.01)

Preconditions:

• Expected counts must all be positive.
• Observed counts must all be ≥ 0.
• The observed and expected arrays must have the same length and their common length must be at least 2.
•  0 < alpha < 0.5

If any of the preconditions are not met, an IllegalArgumentException is thrown.

Note: This implementation rescales the expected array if necessary to ensure that the sum of the expected and observed counts are equal.

Parameters:
observed - array of observed frequency counts
expected - array of expected frequency counts
alpha - significance level of the test
Returns:
true iff null hypothesis can be rejected with confidence 1 - alpha
Throws:
NotPositiveException - if observed has negative entries
NotStrictlyPositiveException - if expected has entries that are not strictly positive
DimensionMismatchException - if the arrays length is less than 2
OutOfRangeException - if alpha is not in the range (0, 0.5]
MaxCountExceededException - if an error occurs computing the p-value
• #### chiSquare

public double chiSquare(long[][] counts)
throws NullArgumentException,
NotPositiveException,
DimensionMismatchException
Computes the Chi-Square statistic associated with a chi-square test of independence based on the input counts array, viewed as a two-way table.

The rows of the 2-way table are count[0], ... , count[count.length - 1]

Preconditions:

• All counts must be ≥ 0.
• The count array must be rectangular (i.e. all count[i] subarrays must have the same length).
• The 2-way table represented by counts must have at least 2 columns and at least 2 rows.

If any of the preconditions are not met, an IllegalArgumentException is thrown.

Parameters:
counts - array representation of 2-way table
Returns:
chiSquare test statistic
Throws:
NullArgumentException - if the array is null
DimensionMismatchException - if the array is not rectangular
NotPositiveException - if counts has negative entries
• #### chiSquareTest

public double chiSquareTest(long[][] counts)
throws NullArgumentException,
DimensionMismatchException,
NotPositiveException,
MaxCountExceededException
Returns the observed significance level, or p-value, associated with a chi-square test of independence based on the input counts array, viewed as a two-way table.

The rows of the 2-way table are count[0], ... , count[count.length - 1]

Preconditions:

• All counts must be ≥ 0.
• The count array must be rectangular (i.e. all count[i] subarrays must have the same length).
• The 2-way table represented by counts must have at least 2 columns and at least 2 rows.

If any of the preconditions are not met, an IllegalArgumentException is thrown.

Parameters:
counts - array representation of 2-way table
Returns:
p-value
Throws:
NullArgumentException - if the array is null
DimensionMismatchException - if the array is not rectangular
NotPositiveException - if counts has negative entries
MaxCountExceededException - if an error occurs computing the p-value
• #### chiSquareTest

public boolean chiSquareTest(long[][] counts,
double alpha)
throws NullArgumentException,
DimensionMismatchException,
NotPositiveException,
OutOfRangeException,
MaxCountExceededException
Performs a chi-square test of independence evaluating the null hypothesis that the classifications represented by the counts in the columns of the input 2-way table are independent of the rows, with significance level alpha. Returns true iff the null hypothesis can be rejected with 100 * (1 - alpha) percent confidence.

The rows of the 2-way table are count[0], ... , count[count.length - 1]

Example:
To test the null hypothesis that the counts in count[0], ... , count[count.length - 1]  all correspond to the same underlying probability distribution at the 99% level, use

chiSquareTest(counts, 0.01)

Preconditions:

• All counts must be ≥ 0.
• The count array must be rectangular (i.e. all count[i] subarrays must have the same length).
• The 2-way table represented by counts must have at least 2 columns and at least 2 rows.

If any of the preconditions are not met, an IllegalArgumentException is thrown.

Parameters:
counts - array representation of 2-way table
alpha - significance level of the test
Returns:
true iff null hypothesis can be rejected with confidence 1 - alpha
Throws:
NullArgumentException - if the array is null
DimensionMismatchException - if the array is not rectangular
NotPositiveException - if counts has any negative entries
OutOfRangeException - if alpha is not in the range (0, 0.5]
MaxCountExceededException - if an error occurs computing the p-value
• #### chiSquareDataSetsComparison

public double chiSquareDataSetsComparison(long[] observed1,
long[] observed2)
throws DimensionMismatchException,
NotPositiveException,
ZeroException

Computes a Chi-Square two sample test statistic comparing bin frequency counts in observed1 and observed2. The sums of frequency counts in the two samples are not required to be the same. The formula used to compute the test statistic is

 ∑[(K * observed1[i] - observed2[i]/K)2 / (observed1[i] + observed2[i])]  where
K = &sqrt;[&sum(observed2 / ∑(observed1)]

This statistic can be used to perform a Chi-Square test evaluating the null hypothesis that both observed counts follow the same distribution.

Preconditions:

• Observed counts must be non-negative.
• Observed counts for a specific bin must not both be zero.
• Observed counts for a specific sample must not all be 0.
• The arrays observed1 and observed2 must have the same length and their common length must be at least 2.

If any of the preconditions are not met, an IllegalArgumentException is thrown.

Parameters:
observed1 - array of observed frequency counts of the first data set
observed2 - array of observed frequency counts of the second data set
Returns:
chiSquare test statistic
Throws:
DimensionMismatchException - the the length of the arrays does not match
NotPositiveException - if any entries in observed1 or observed2 are negative
ZeroException - if either all counts of observed1 or observed2 are zero, or if the count at some index is zero for both arrays
Since:
1.2
• #### chiSquareTestDataSetsComparison

public double chiSquareTestDataSetsComparison(long[] observed1,
long[] observed2)
throws DimensionMismatchException,
NotPositiveException,
ZeroException,
MaxCountExceededException

Returns the observed significance level, or p-value, associated with a Chi-Square two sample test comparing bin frequency counts in observed1 and observed2.

The number returned is the smallest significance level at which one can reject the null hypothesis that the observed counts conform to the same distribution.

See chiSquareDataSetsComparison(long[], long[]) for details on the formula used to compute the test statistic. The degrees of of freedom used to perform the test is one less than the common length of the input observed count arrays.

Preconditions:
• Observed counts must be non-negative.
• Observed counts for a specific bin must not both be zero.
• Observed counts for a specific sample must not all be 0.
• The arrays observed1 and observed2 must have the same length and their common length must be at least 2.

If any of the preconditions are not met, an IllegalArgumentException is thrown.

Parameters:
observed1 - array of observed frequency counts of the first data set
observed2 - array of observed frequency counts of the second data set
Returns:
p-value
Throws:
DimensionMismatchException - the the length of the arrays does not match
NotPositiveException - if any entries in observed1 or observed2 are negative
ZeroException - if either all counts of observed1 or observed2 are zero, or if the count at the same index is zero for both arrays
MaxCountExceededException - if an error occurs computing the p-value
Since:
1.2
• #### chiSquareTestDataSetsComparison

public boolean chiSquareTestDataSetsComparison(long[] observed1,
long[] observed2,
double alpha)
throws DimensionMismatchException,
NotPositiveException,
ZeroException,
OutOfRangeException,
MaxCountExceededException

Performs a Chi-Square two sample test comparing two binned data sets. The test evaluates the null hypothesis that the two lists of observed counts conform to the same frequency distribution, with significance level alpha. Returns true iff the null hypothesis can be rejected with 100 * (1 - alpha) percent confidence.

See chiSquareDataSetsComparison(long[], long[]) for details on the formula used to compute the Chisquare statistic used in the test. The degrees of of freedom used to perform the test is one less than the common length of the input observed count arrays.

Preconditions:
• Observed counts must be non-negative.
• Observed counts for a specific bin must not both be zero.
• Observed counts for a specific sample must not all be 0.
• The arrays observed1 and observed2 must have the same length and their common length must be at least 2.
•  0 < alpha < 0.5

If any of the preconditions are not met, an IllegalArgumentException is thrown.

Parameters:
observed1 - array of observed frequency counts of the first data set
observed2 - array of observed frequency counts of the second data set
alpha - significance level of the test
Returns:
true iff null hypothesis can be rejected with confidence 1 - alpha
Throws:
DimensionMismatchException - the the length of the arrays does not match
NotPositiveException - if any entries in observed1 or observed2 are negative
ZeroException - if either all counts of observed1 or observed2 are zero, or if the count at the same index is zero for both arrays
OutOfRangeException - if alpha is not in the range (0, 0.5]
MaxCountExceededException - if an error occurs performing the test
Since:
1.2