Package com.imsl.stat

Class ChiSquaredTest

java.lang.Object
com.imsl.stat.ChiSquaredTest
public class ChiSquaredTest extends Object
Chi-squared goodness-of-fit test.

ChiSquaredTest performs a chi-squared goodness-of-fit test that a random sample of observations is distributed according to a specified theoretical cumulative distribution. The theoretical distribution, which may be continuous, discrete, or a mixture of discrete and continuous distributions, is specified via a user-defined function F where F implements CdfFuntion. Because the user is allowed to specify a range for the observations in the setRange method, a test that is conditional upon the specified range is performed.

ChiSquaredTest can be constructed in two different ways. The intervals can be specified via the array cutpoints. Otherwise, the number of cutpoints can be given and equiprobable intervals computed by the constructor. The observations are divided into these intervals. Regardless of the method used to obtain them, the intervals are such that the lower endpoint is not included in the interval while the upper endpoint is always included. The user should determine the cutpoints when the cumulative distribution function has discrete elements since ChiSquaredTest cannot determine them in this case.

By default, the lower and upper endpoints of the first and last intervals are \(-\infty\) and \(+\infty\), respectively. The method setRange can be used to change the range.

A tally of counts is maintained for the observations in x as follows:

If the cutpoints are specified by the user, the tally is made in the interval to which \(x_i\) belongs, using the user-specified endpoints.

If the cutpoints are determined by the class then the cumulative probability at \(x_i\), \(F(x_i)\), is computed using cdf.

The tally for \(x_i\) is made in interval number \(\lfloor mF(x) + 1 \rfloor\), where m is the number of categories and \(\lfloor.\rfloor\) is the function that takes the greatest integer that is no larger than the argument of the function. If the cutpoints are specified by the user, the tally is made in the interval to which \(x_i\) belongs using the endpoints specified by the user. Thus, if the computer time required to calculate the cumulative distribution function is large, user-specified cutpoints may be preferred in order to reduce the total computing time.

If the expected count in any cell is less than 1, then a rule of thumb is that the chi-squared approximation may be suspect. A warning message to this effect is issued in this case, as well as when an expected value is less than 5.

See Also:

  • Constructor Details

    • ChiSquaredTest

      public ChiSquaredTest(CdfFunction cdf, double[] cutpoints, int nParameters) throws ChiSquaredTest.NotCDFException
      Constructor for the Chi-squared goodness-of-fit test.
      Parameters:
      cdf - a CdfFunction object that implements the CdfFunction interface
      cutpoints - a double array containing the cutpoints
      nParameters - an int which specifies the number of parameters estimated in computing the Cdf. For example, with a binomial distribution nParameters=1 if p is estimated from the data and nParameters=0 if p is given in advance. The degrees of freedom in \(\chi ^2\) is: $$df = n - p - 1$$ where n = number or non-empty cells and p = nParameters.
      Throws:
      ChiSquaredTest.NotCDFException

    • ChiSquaredTest

      public ChiSquaredTest(CdfFunction cdf, int nCutpoints, int nParameters) throws ChiSquaredTest.NotCDFException, InverseCdf.DidNotConvergeException
      Constructor for the Chi-squared goodness-of-fit test
      Parameters:
      cdf - a CdfFunction object that implements the CdfFunction interface
      nCutpoints - an int, the number of cutpoints
      nParameters - an int which specifies the number of parameters estimated in computing the Cdf. For example, with a binomial distribution nParameters=1 if p is estimated from the data and nParameters=0 if p is given in advance. The degrees of freedom in \(\chi ^2\) is: $$df = n - p - 1$$ where n = number or non-empty cells and p = nParameters.
      Throws:
      ChiSquaredTest.NotCDFException
      InverseCdf.DidNotConvergeException

  • Method Details

    • setRange

      public void setRange(double lower, double upper) throws ChiSquaredTest.NotCDFException
      Sets endpoints of the range of the distribution. Points outside of the range are ignored so that distributions conditional on the range can be used. In this case, the point lower is excluded from the first interval, but the point upper is included in the last interval. By default, a range on the whole real line is used.
      Parameters:
      lower - a double, the lower range limit
      upper - a double, the upper range limit
      Throws:
      ChiSquaredTest.NotCDFException

    • update

      public void update(double[] x) throws ChiSquaredTest.NotCDFException
      Adds new observations to the test.
      Parameters:
      x - a double array which contains the new observations to be added to the test. The frequencies of these observations are assumed to be 1.0.
      Throws:
      ChiSquaredTest.NotCDFException

    • update

      public void update(double x) throws ChiSquaredTest.NotCDFException
      Adds a new observation to the test.
      Parameters:
      x - a double, the new observation to be added to the test. The frequency of this observation is assumed to be 1.0.
      Throws:
      ChiSquaredTest.NotCDFException

    • update

      public void update(double[] x, double[] freq) throws ChiSquaredTest.NotCDFException
      Adds new observations to the test.
      Parameters:
      x - a double array which contains the new observations to be added to the test
      freq - a double array which contains the frequencies of the corresponding new observations in x
      Throws:
      ChiSquaredTest.NotCDFException

    • update

      public void update(double x, double freq) throws ChiSquaredTest.NotCDFException
      Adds a new observation to the test.
      Parameters:
      x - a double, the new observation to be added to the test
      freq - a double, the frequency of the new observation, x
      Throws:
      ChiSquaredTest.NotCDFException

    • getChiSquared

      public double getChiSquared() throws ChiSquaredTest.NotCDFException
      Returns the chi-squared statistic.
      Returns:
      a double, the chi-squared statistic
      Throws:
      ChiSquaredTest.NotCDFException

    • getP

      public double getP() throws ChiSquaredTest.NotCDFException
      Returns the p-value for the chi-squared statistic.
      Returns:
      a double, the p-value for the chi-squared statistic
      Throws:
      ChiSquaredTest.NotCDFException

    • getDegreesOfFreedom

      public double getDegreesOfFreedom() throws ChiSquaredTest.NotCDFException
      Returns the degrees of freedom in chi-squared. The degrees of freedom (df) in chi-squared is $$df = n - p - 1$$ where n = number or non-empty cells and p = nParameters, the number of estimated parameters.
      Returns:
      a double, the degrees of freedom in the chi-squared statistic
      Throws:
      ChiSquaredTest.NotCDFException

    • setCutpoints

      public void setCutpoints(double[] cutpoints)
      Sets the cutpoints. The intervals defined by the cutpoints are such that the lower endpoint is not included while the upper endpoint is included in the interval.
      Parameters:
      cutpoints - a double array which contains the cutpoints

    • getCutpoints

      public double[] getCutpoints()
      Returns the cutpoints.
      Returns:
      a double array which contains the cutpoints

    • getCellCounts

      public double[] getCellCounts()
      Returns the cell counts.
      Returns:
      a double array which contains the number of actual observations in each cell.

    • getExpectedCounts

      public double[] getExpectedCounts()
      Returns the expected counts.
      Returns:
      a double array which contains the number of expected observations in each cell.