org.hipparchus.stat.inference

Class ChiSquareTest

java.lang.Object

org.hipparchus.stat.inference.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.

Constructor Summary

Constructors

Constructor and Description
ChiSquareTest() Construct a ChiSquareTest

Method Summary

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 MathIllegalArgumentException

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:

MathIllegalArgumentException - if observed has negative entries

MathIllegalArgumentException - if expected has entries that are not strictly positive

MathIllegalArgumentException - if the arrays length is less than 2

chiSquareTest

public double chiSquareTest(double[] expected,
                            long[] observed)
                     throws MathIllegalArgumentException,
                            MathIllegalStateException

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:

MathIllegalArgumentException - if observed has negative entries

MathIllegalArgumentException - if expected has entries that are not strictly positive

MathIllegalArgumentException - if the arrays length is less than 2

MathIllegalStateException - if an error occurs computing the p-value

chiSquareTest

public boolean chiSquareTest(double[] expected,
                             long[] observed,
                             double alpha)
                      throws MathIllegalArgumentException,
                             MathIllegalStateException

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:

MathIllegalArgumentException - if observed has negative entries

MathIllegalArgumentException - if expected has entries that are not strictly positive

MathIllegalArgumentException - if the arrays length is less than 2

MathIllegalArgumentException - if alpha is not in the range (0, 0.5]

MathIllegalStateException - if an error occurs computing the p-value

chiSquare

public double chiSquare(long[][] counts)
                 throws MathIllegalArgumentException,
                        NullArgumentException

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

MathIllegalArgumentException - if the array is not rectangular

MathIllegalArgumentException - if counts has negative entries

chiSquareTest

public double chiSquareTest(long[][] counts)
                     throws MathIllegalArgumentException,
                            NullArgumentException,
                            MathIllegalStateException

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

MathIllegalArgumentException - if the array is not rectangular

MathIllegalArgumentException - if counts has negative entries

MathIllegalStateException - if an error occurs computing the p-value

chiSquareTest

public boolean chiSquareTest(long[][] counts,
                             double alpha)
                      throws MathIllegalArgumentException,
                             NullArgumentException,
                             MathIllegalStateException

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

MathIllegalArgumentException - if the array is not rectangular

MathIllegalArgumentException - if counts has any negative entries

MathIllegalArgumentException - if alpha is not in the range (0, 0.5]

MathIllegalStateException - if an error occurs computing the p-value

chiSquareDataSetsComparison

public double chiSquareDataSetsComparison(long[] observed1,
                                          long[] observed2)
                                   throws MathIllegalArgumentException

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:

MathIllegalArgumentException - the the length of the arrays does not match

MathIllegalArgumentException - if any entries in observed1 or observed2 are negative

MathIllegalArgumentException - if either all counts of observed1 or observed2 are zero, or if the count at some index is zero for both arrays

chiSquareTestDataSetsComparison

public double chiSquareTestDataSetsComparison(long[] observed1,
                                              long[] observed2)
                                       throws MathIllegalArgumentException,
                                              MathIllegalStateException

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:

MathIllegalArgumentException - the the length of the arrays does not match

MathIllegalArgumentException - if any entries in observed1 or observed2 are negative

MathIllegalArgumentException - if either all counts of observed1 or observed2 are zero, or if the count at the same index is zero for both arrays

MathIllegalStateException - if an error occurs computing the p-value

chiSquareTestDataSetsComparison

public boolean chiSquareTestDataSetsComparison(long[] observed1,
                                               long[] observed2,
                                               double alpha)
                                        throws MathIllegalArgumentException,
                                               MathIllegalStateException

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:

MathIllegalArgumentException - the the length of the arrays does not match

MathIllegalArgumentException - if any entries in observed1 or observed2 are negative

MathIllegalArgumentException - if either all counts of observed1 or observed2 are zero, or if the count at the same index is zero for both arrays

MathIllegalArgumentException - if alpha is not in the range (0, 0.5]

MathIllegalStateException - if an error occurs performing the test