Performs a chi-squared analysis of a two-way contingency table.

The type double function is imsls_d_contingency_table.

Pearson chi-squared p-value for independence of rows and columns.

Function imsls_f_contingency_table computes statistics associated with an r × c (n_rows × n_columns) 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).

Let xij denote the observed cell frequency in the ij cell of the table and n denote the total count in the table. Let pij = pi∙p∙j denote the predicted cell probabilities under the null hypothesis of independence, where pi∙ and p∙j are the row and column marginal relative frequencies. Next, compute the expected cell counts as eij = npij.

Also required in the following are auv and buv for u, v = 1, …, n. Let (rs, cs) denote the row and column response of observation s. Then, auv = 1, 0, or −1, depending on whether ru < rv, ru = rv, or ru > rv, respectively. The buv are similarly defined in terms of the cs variables.

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

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 array statistics, 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.

The measures of association, ɸ, P, and V, do not require any ordering of the row and column categories. Function imsls_f_contingency_table 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 τb, Stuart’s τc, 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 auv variables and buv variables defined above, i.e., as follows:

Recall that auv and buv can take values −1, 0, or 1. Since the product auvbuv = 1 only if auv and buv 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 auvbuv = −1.

Gamma can be motivated in a slightly different manner. Because the “covariance” of the auv variables and the buv 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 γ = (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 auv from buv. Moreover, Somer’s D for rows is the probability of agreement minus the probability of disagreement, given that the column variable, buv, 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).

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 λc|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

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

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

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

is similarly defined.

Note that this is 1/(2n) times the sums of squares of the auv variables.

The total variation for rows within columns is the sum of the qj 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.

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:

The uncertainty coefficient for columns is similarly defined. The symmetric uncertainty coefficient contains the same numerator as Ur|c and Uc|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

where

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

denote the probability that the two judges agree, and let

denote the expected probability of agreement under the independence model. Kappa is then given by (p0 − pc)/(1 − pc).

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 H0:θij = θji. The multiple degrees-of-freedom version of the McNemar test with r (r − 1)/2 degrees of freedom is computed as follows:

The single degree-of-freedom test assumes that the differences, xij − xji, 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:

The exact probability can be computed by the binomial distribution.

The following example is taken from Kendall and Stuart (1979) and involves the distance vision in the right and left eyes. Output contains only the p-value.

The following example, which illustrates the use of Kappa and McNemar tests, uses the same distance vision data as the previous example. The available statistics are output using optional arguments.