Package com.imsl.stat

Class ContingencyTable

java.lang.Object
com.imsl.stat.ContingencyTable
All Implemented Interfaces:
Serializable, Cloneable
public class ContingencyTable extends Object implements Serializable, Cloneable
Performs a chi-squared analysis of a two-way contingency table.

Class ContingencyTable computes statistics associated with an \(r \times c\) contingency table. The function computes the chi-squared test of independence, expected values, contributions to chi-squared, row and column marginal totals, some measures of association, correlation, prediction, uncertainty, the McNemar test for symmetry, a test for linear trend, the odds and the log odds ratio, and the kappa statistic (if the appropriate optional arguments are selected).

Notation

Let \(x_{ij}\) denote the observed cell frequency in the ij cell of the table and n denote the total count in the table. Let \(p_{ij} = p_{i\bullet} p_{j\bullet}\) denote the predicted cell probabilities under the null hypothesis of independence, where \(p_{i\bullet}\) and \(p_{j\bullet}\) are the row and column marginal relative frequencies. Next, compute the expected cell counts as \(e_{ij} = np_{ij}\).

Also required in the following are \(a_{uv}\) and \(b_{uv}\) for \(u, v = 1, \ldots, n\). Let \((r_s, c_s)\) denote the row and column response of observation s. Then, \(a_{uv} = 1, 0\), or -1, depending on whether \(r_u \lt r_v, r_u = r_v\), or \(r_u \gt r_v\), respectively. The \(b_{uv}\) are similarly defined in terms of the \(c_s\) variables.

Chi-squared Statistic

For each cell in the table, the contribution to \(\chi ^2\) is given as \((x_{ij} - e_{ij})^2/e_{ij}\). The Pearson chi-squared statistic (denoted \(\chi ^2\)) is computed as the sum of the cell contributions to chi-squared. It has (r - 1) (c - 1) degrees of freedom and tests the null hypothesis of independence, i.e., \(H_0:p_{ij} = p_{i\bullet}p_{j\bullet}\). The null hypothesis is rejected if the computed value of \(\chi ^2\) is too large.

The maximum likelihood equivalent of \(\chi ^2, G^2\) is computed as follows:

$$G^2 = - 2\sum\limits_{i,j} {x_{ij} } \ln \left( {x_{ij} /np_{ij} } \right)$$

\(G^2\) is asymptotically equivalent to \(\chi ^2\) and tests the same hypothesis with the same degrees of freedom.

Measures Related to Chi-squared (Phi, Contingency Coefficient, and Cramer's V)

There are three measures related to chi-squared that do not depend on sample size:

$${\rm{phi,}} \,\, \phi \, {\rm{=}} \, \sqrt {\chi ^2 {\rm{/}}n}$$

$${\rm{contingency \,\,coefficient, \,\,}} P{\rm{ = }}\sqrt {\chi ^2 /\left( {n + \chi ^2 } \right)}$$

$${\rm{Cramer's }}\,V{\rm{, }}\,\,V = \sqrt {\chi ^2 /\left( {n\min \left( {r,c} \right)} \right)}$$

Since these statistics do not depend on sample size and are large when the hypothesis of independence is rejected, they can be thought of as measures of association and can be compared across tables with different sized samples. While both P and V have a range between 0.0 and 1.0, the upper bound of P is actually somewhat less than 1.0 for any given table (see Kendall and Stuart 1979, p. 587). The significance of all three statistics is the same as that of the \(\chi ^2\) statistic, return value from the getChiSquared method.

The distribution of the \(\chi ^2\) statistic in finite samples approximates a chi-squared distribution. To compute the exact mean and standard deviation of the \(\chi ^2\) statistic, Haldane (1939) uses the multinomial distribution with fixed table marginals. The exact mean and standard deviation generally differ little from the mean and standard deviation of the associated chi-squared distribution.

Standard Errors and p-values for Some Measures of Association

In Columns 1 through 4 of statistics, estimated standard errors and asymptotic p-values are reported. Estimates of the standard errors are computed in two ways. The first estimate, in Column 1 of the return matrix from the getStatistics method, is asymptotically valid for any value of the statistic. The second estimate, in Column 2 of the array, is only correct under the null hypothesis of no association. The z-scores in Column 3 of statistics are computed using this second estimate of the standard errors. The p-values in Column 4 are computed from this z-score. See Brown and Benedetti (1977) for a discussion and formulas for the standard errors in Column 2.

Measures of Association for Ranked Rows and Columns

The measures of association, \(\phi\), P, and V, do not require any ordering of the row and column categories. Class ContingencyTable also computes several measures of association for tables in which the rows and column categories correspond to ranked observations. Two of these tests, the product-moment correlation and the Spearman correlation, are correlation coefficients computed using assigned scores for the row and column categories. The cell indices are used for the product-moment correlation, while the average of the tied ranks of the row and column marginals is used for the Spearman rank correlation. Other scores are possible.

Gamma, Kendall's \(\tau_b\), Stuart's \(\tau_c\), and Somers' D are measures of association that are computed like a correlation coefficient in the numerator. In all these measures, the numerator is computed as the "covariance" between the \(a_{uv}\) variables and \(b_{uv}\) variables defined above, i.e., as follows:

$$\sum\limits_u {\sum\limits_v {a_{uv} } } b_{uv}$$

Recall that \(a_{uv}\) and \(b_{uv}\) can take values -1, 0, or 1. Since the product \(a_{uv}b_{uv} = 1\) only if \(a_{uv}\) and \(b_{uv}\) are both 1 or are both -1, it is easy to show that this "covariance" is twice the total number of agreements minus the number of disagreements, where a disagreement occurs when \(a_{uv}b_{uv} = -1\).

Kendall's \(\tau_b\) is computed as the correlation between the \(a_{uv}\) variables and the \(b_{uv}\) variables (see Kendall and Stuart 1979, p. 593). In a rectangular table \((r \neq c)\), Kendall's \(\tau_b\) cannot be 1.0 (if all marginal totals are positive). For this reason, Stuart suggested a modification to the denominator of \(\tau\) in which the denominator becomes the largest possible value of the "covariance." This maximizing value is approximately \(n^2m/(m - 1)\), where m = min (r, c). Stuart's \(\tau_c\) uses this approximate value in its denominator. For large \(n, \tau_c \approx m\tau_b/(m - 1)\).

Gamma can be motivated in a slightly different manner. Because the "covariance" of the \(a_{uv}\) variables and the \(b_{uv}\) variables can be thought of as twice the number of agreements minus the disagreements, 2(A - D), where A is the number of agreements and D is the number of disagreements, Gamma is motivated as the probability of agreement minus the probability of disagreement, given that either agreement or disagreement occurred. This is shown as \(\gamma = (A - D)/(A + D)\).

Two definitions of Somers' D are possible, one for rows and a second for columns. Somers' D for rows can be thought of as the regression coefficient for predicting \(a_{uv}\) from \(b_{uv}\). Moreover, Somer's D for rows is the probability of agreement minus the probability of disagreement, given that the column variable, \(b_{uv}\), is not 0. Somers' D for columns is defined in a similar manner.

A discussion of all of the measures of association in this section can be found in Kendall and Stuart (1979, p. 592).

Measures of Prediction and Uncertainty

Optimal Prediction Coefficients: The measures in this section do not require any ordering of the row or column variables. They are based entirely upon probabilities. Most are discussed in Bishop et al. (1975, p. 385).

Consider predicting (or classifying) the column for a given row in the table. Under the null hypothesis of independence, choose the column with the highest column marginal probability for all rows. In this case, the probability of misclassification for any row is 1 minus this marginal probability. If independence is not assumed within each row, choose the column with the highest row conditional probability. The probability of misclassification for the row becomes 1 minus this conditional probability.

Define the optimal prediction coefficient \(\lambda _{c|r}\) for predicting columns from rows as the proportion of the probability of misclassification that is eliminated because the random variables are not independent. It is estimated by

$$\lambda _{c\,|\,r} = \frac{{\left( {1 - p_{ \bullet m} } \right) - (1 - \sum\limits_i {p_{im} } )}}{{1 - p_{ \bullet m} }}$$

where m is the index of the maximum estimated probability in the row \((p_{im})\) or row margin \((p_{\bullet m})\). A similar coefficient is defined for predicting the rows from the columns. The symmetric version of the optimal prediction \(\lambda\) is obtained by summing the numerators and denominators of \(\lambda _{r|c}\) and \(\lambda _{c|r}\) then dividing. Standard errors for these coefficients are given in Bishop et al. (1975, p. 388).

A problem with the optimal prediction coefficients \(\lambda\) is that they vary with the marginal probabilities. One way to correct this is to use row conditional probabilities. The optimal prediction \(\lambda*\) coefficients are defined as the corresponding \(\lambda\) coefficients in which first the row (or column) marginals are adjusted to the same number of observations. This yields

$$\lambda _{c\,|\,r}^ * = \frac{{\sum\limits_i {\max _j p_{j\,|\,i} - \max _j (\sum\limits_i {p_{j\,|\,i} } )} }}{{R - \max _j (\sum\limits_i {p_{j\,|\,i} )} }}$$

where i indexes the rows, j indexes the columns, and \(p_{j|i}\) is the (estimated) probability of column j given row i.

$$\lambda _{r\,|\,c}^ *$$

is similarly defined.

Goodman and Kruskal \(\tau\): A second kind of prediction measure attempts to explain the proportion of the explained variation of the row (column) measure given the column (row) measure. Define the total variation in the rows as follows:

$$n/2 - (\sum\limits_i {x_{i \bullet }^2 } )/\left( {2n} \right)$$

Note that this is 1/(2n) times the sums of squares of the \(a_{uv}\) variables.

With this definition of variation, the Goodman and Kruskal \(\tau\) coefficient for rows is computed as the reduction of the total variation for rows accounted for by the columns, divided by the total variation for the rows. To compute the reduction in the total variation of the rows accounted for by the columns, note that the total variation for the rows within column j is defined as follows:

$$q_j = x_{ \bullet j} /2 - (\sum\limits_i {x_{ij}^2 } )/\left( {2x_{i \bullet } } \right)$$

The total variation for rows within columns is the sum of the \(q_j\) variables. Consistent with the usual methods in the analysis of variance, the reduction in the total variation is given as the difference between the total variation for rows and the total variation for rows within the columns.

Goodman and Kruskal's \(\tau\) for columns is similarly defined. See Bishop et al. (1975, p. 391) for the standard errors.

Uncertainty Coefficients: The uncertainty coefficient for rows is the increase in the log-likelihood that is achieved by the most general model over the independence model, divided by the marginal log-likelihood for the rows. This is given by the following equation:

$$U_{r|c} = \frac{{\sum\limits_{i,j} {x_{ij} \log \left( {x_{i \bullet } x_{ \bullet j} /nx_{ij} } \right)} }}{{\sum\limits_i {x_{i \bullet } \log \left( {x_{i \bullet } /n} \right)} }}$$

The uncertainty coefficient for columns is similarly defined. The symmetric uncertainty coefficient contains the same numerator as \(U_{r|c}\) and \(U_{c|r}\)but averages the denominators of these two statistics. Standard errors for U are given in Brown (1983).

Kruskal-Wallis: The Kruskal-Wallis statistic for rows is a one-way analysis-of-variance-type test that assumes the column variable is monotonically ordered. It tests the null hypothesis that no row populations are identical, using average ranks for the column variable. The Kruskal-Wallis statistic for columns is similarly defined. Conover (1980) discusses the Kruskal-Wallis test.

Test for Linear Trend: When there are two rows, it is possible to test for a linear trend in the row probabilities if it is assumed that the column variable is monotonically ordered. In this test, the probabilities for row 1 are predicted by the column index using weighted simple linear regression. This slope is given by

$$\hat \beta = \frac{{\sum\limits_j {x_{ \bullet j} \left( {x_{1j} /x_{ \bullet j} - x_{1 \bullet } /n} \right)\left( {j - \bar j} \right)} }}{{\sum\limits_j {x_{ \bullet j} \left( {j - \bar j} \right)^2 } }}$$

where

$$\bar j = \sum\limits_j {x_{ \bullet j} } j/n$$

is the average column index. An asymptotic test that the slope is 0 may then be obtained (in large samples) as the usual regression test of zero slope.

In two-column data, a similar test for a linear trend in the column probabilities is computed. This test assumes that the rows are monotonically ordered.

Kappa: Kappa is a measure of agreement computed on square tables only. In the kappa statistic, the rows and columns correspond to the responses of two judges. The judges agree along the diagonal and disagree off the diagonal. Let

$$p_0 = \sum\limits_i {x_{ii} } /n$$

denote the probability that the two judges agree, and let

$$p_c = \sum\limits_i {e_{ii} } /n$$

denote the expected probability of agreement under the independence model. Kappa is then given by \((p_0 - p_c)/(1 - p_c)\).

McNemar Tests: The McNemar test is a test of symmetry in a square contingency table. In other words, it is a test of the null hypothesis \(H_0:\theta_{ij} = \theta_{ji}\). The multiple degrees-of-freedom version of the McNemar test with r (r - 1)/2 degrees of freedom is computed as follows:

$$ \sum\limits_{i \lt j} \frac{\left( x_{ij}-x_{ji} \right)^2}{\left( x_{ij}+x_{ji} \right)} $$

The single degree-of-freedom test assumes that the differences, \(x_{ij} - x_{ji}\), are all in one direction. The single degree-of-freedom test will be more powerful than the multiple degrees-of-freedom test when this is the case. The test statistic is given as follows:

$$\frac{{\left( {\sum\limits_{i \lt j} {\left( {x_{ij} - x_{ji} } \right)} } \right)^2 }} {{\sum\limits_{i \lt j} {\left( {x_{ij} + x_{ji} } \right)} }}$$

The exact probability can be computed by the binomial distribution.

See Also:

  • Constructor Summary

    Constructors
    Constructor
    Description
    ContingencyTable(double[][] table)
    Constructs and performs a chi-squared analysis of a two-way contingency table.

  • Method Summary

    Modifier and Type
    Method
    Description
    double
    getChiSquared()
    Returns the Pearson chi-squared test statistic.
    double
    getContingencyCoef()
    Returns contingency coefficient.
    double[][]
    getContributions()
    Returns the contributions to chi-squared for each cell in the table.
    double
    getCramersV()
    Returns Cramer's V.
    int
    getDegreesOfFreedom()
    Returns the degrees of freedom for the chi-squared tests associated with the table.
    double
    getExactMean()
    Returns exact mean.
    double
    getExactStdev()
    Returns exact standard deviation.
    double[][]
    getExpectedValues()
    Returns the expected values of each cell in the table.
    double
    getGSquared()
    Returns the likelihood ratio G2 (chi-squared).
    double
    getGSquaredP()
    Returns the probability of a larger G2 (chi-squared).
    double
    getP()
    Returns the Pearson chi-squared p-value for independence of rows and columns.
    double
    getPhi()
    Returns phi.
    double[][]
    getStatistics()
    Returns the statistics associated with this table.

    Methods inherited from class java.lang.Object

    clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

  • Constructor Details

    • ContingencyTable

      public ContingencyTable(double[][] table)
      Constructs and performs a chi-squared analysis of a two-way contingency table.
      Parameters:
      table - A double matrix containing the observed counts in the contingency table.

  • Method Details

    • getExactMean

      public double getExactMean()
      Returns exact mean.
      Returns:
      A double scalar containing the exact mean based on Pearson's chi-square statistic.

    • getExactStdev

      public double getExactStdev()
      Returns exact standard deviation.
      Returns:
      A double scalar containing the exact standard deviation based on Pearson's chi-square statistic.

    • getPhi

      public double getPhi()
      Returns phi.
      Returns:
      A double scalar containing the phi based on Pearson chi-squared statistic.

    • getContingencyCoef

      public double getContingencyCoef()
      Returns contingency coefficient.
      Returns:
      A double scalar containing the contingency coefficient based on Pearson chi-squared statistic.

    • getCramersV

      public double getCramersV()
      Returns Cramer's V.
      Returns:
      A double scalar containing the Cramer's V based on Pearson chi-squared statistic.

    • getP

      public double getP()
      Returns the Pearson chi-squared p-value for independence of rows and columns.
      Returns:
      A double scalar containing the Pearson chi-squared p-value for independence of rows and columns.

    • getGSquaredP

      public double getGSquaredP()
      Returns the probability of a larger G2 (chi-squared).
      Returns:
      A double scalar containing the probability of a larger \(G^2\) (chi-squared).

    • getGSquared

      public double getGSquared()
      Returns the likelihood ratio G2 (chi-squared).
      Returns:
      A double scalar containing the likelihood ratio \(G^2\) (chi-squared).

    • getChiSquared

      public double getChiSquared()
      Returns the Pearson chi-squared test statistic.
      Returns:
      A double scalar containing the Pearson chi-squared test statistic.

    • getDegreesOfFreedom

      public int getDegreesOfFreedom()
      Returns the degrees of freedom for the chi-squared tests associated with the table.
      Returns:
      An int scalar containing the degrees of freedom for the chi-squared tests associated with the table.

    • getExpectedValues

      public double[][] getExpectedValues()
      Returns the expected values of each cell in the table.
      Returns:
      A double matrix of size (table.length+1) * (table[0].length+1) containing the expected values of each cell in the table, under the null hypothesis. The marginal totals are in the last row and column.

    • getStatistics

      public double[][] getStatistics()
      Returns the statistics associated with this table.
      Returns:
      A double matrix of size 23 * 5 containing statistics associated with this table. Each row corresponds to a statistic.
      Row Statistics
      0 gamma
      1 Kendall's \(\tau_b\)
      2 Stuart's \(\tau_c\)
      3 Somers' D for rows (given columns)
      4 Somers' D for columns (given rows)
      5 product moment correlation
      6 Spearman rank correlation
      7 Goodman and Kruskal \(\tau\) for rows (given columns)
      8 Goodman and Kruskal \(\tau\) for columns (given rows)
      9 uncertainty coefficient U (symmetric)
      10 uncertainty \(U_{r|c}\) (rows)
      11 uncertainty \(U_{c|r}\) (columns)
      12 optimal prediction \(\lambda\) (symmetric)
      13 optimal prediction \(\lambda_{r|c}\) (rows)
      14 optimal prediction \(\lambda_{c|r}\) (columns)
      15 optimal prediction \(\lambda^*_{r|c}\) (rows)
      16 optimal prediction \(\lambda^*_{c|r}\) (columns)
      17 test for linear trend in row probabilities if table.length = 2.  If table.length is not 2, a test for linear trend in column probabilities if table[0].length = 2.
      18 Kruskal-Wallis test for no row effect
      19 Kruskal-Wallis test for no column effect
      20 kappa (square tables only)
      21 McNemar test of symmetry (square tables only)
      22 McNemar one degree of freedom test of symmetry (square tables only)

      If a statistic cannot be computed, or if some value is not relevant for the computed statistic, the entry is NaN (Not a Number).

      The columns are as follows:

      Column Value
      0 estimated statistic
      1 standard error for any parameter value
      2 standard error under the null hypothesis
      3 t value for testing the null hypothesis
      4 p-value of the test in column 3
      In the McNemar tests, column 0 contains the statistic, column 1 contains the chi-squared degrees of freedom, column 3 contains the exact p-value (1 degree of freedom only), and column 4 contains the chi-squared asymptotic p-value. The Kruskal-Wallis test is the same except no exact p-value is computed.

    • getContributions

      public double[][] getContributions()
      Returns the contributions to chi-squared for each cell in the table.
      Returns:
      A double matrix of size (table.length+1) * (table[0].length+1) containing the contributions to chi-squared for each cell in the table. The last row and column contain the total contribution to chi-squared for that row or column.