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

If a statistic cannot be computed, its value is reported as NaN (not a number). The columns are as follows:

In the McNemar tests, column 1 contains the statistic, column 2 contains the chi‑squared degrees of freedom, column 4 contains the exact p‑value (one degree of freedom only), and column 5 contains the chi‑squared asymptotic p‑value. The Kruskal‑Wallis test is the same except no exact p‑value is computed.

Routine CTCHI computes statistics associated with an r × c (NROW × NCOL) contingency table. The routine CTCHI always computes the chi‑squared test of independence, expected values, contributions to chi‑squared, and row and column marginal totals. Optionally, when ICMPT = 0, CTCHI can compute 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.

Other IMSL routines that may be of interest include TETCC in Chapter 3, for computing the tetrachoric correlation coefficient, CTTWO, for computing statistics in a 2 × 2 contingency table, and CTPRB, for computing the exact probability of an r × c contingency table.

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, respectively. Next, compute the expected cell counts as eij = n pij.

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

For each cell in the table, the contribution to X2 is given as (xij ‑ eij)2/eij. The Pearson chi‑squared statistic (denoted X2) 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., that H0 : pij = pi∙p∙j. The null hypothesis is rejected if the computed value of X2 is too large.

Compute G2, the maximum likelihood equivalent of X2, as

G2 is asymptotically equivalent to X2 and tests the same hypothesis with the same degrees of freedom.

Three measures related to chi‑squared but that do not depend upon the sample size are

phi,

the contingency coefficient,

and Cramer’s V,

Since these statistics do not depend upon sample size and are large when the hypothesis of independence is rejected, they may be thought of as measures of association and may 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, page 587). The significance of all three statistics is the same as that of the X2 statistic, CHISQ(1).

The distribution of the X2 statistic in finite samples approximates a chi‑squared distribution. To compute the exact mean and standard deviation of the X2 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.

In rows 1 through 7 of STAT, estimated standard errors and asymptotic p‑values are reported. Estimates of the standard errors are computed in two ways. The first estimate, in column 2 of matrix STAT, is asymptotically valid for any value of the statistic. The second estimate, in column 3 of the matrix, is only correct under the null hypothesis of no association. The z‑scores in column 4 of matrix STAT are computed using this second estimate of the standard errors. The p‑values in column 5 are computed from this z‑score. See Brown and Benedetti (1977) for a discussion and formulas for the standard errors in column 3.

The measures of association, ɸ, P, and V, do not require any ordering of the row and column categories. Routine CTCHI 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.

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’s and the buv’s 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 just 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, Somers’ D for rows is the probability of agreement minus the probability of disagreement, given that the column variable, buv, is not zero. 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, starting on page 592).

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, Feinberg, and Holland (1975, page 385).

Consider predicting (or classifying) the column for a given row in the table. Under the null hypothesis of independence, one would choose the column with the highest column marginal probability for all rows. In this case, the probability of misclassification for any row is one minus this marginal probability. If independence is not assumed, then within each row one would choose the column with the highest row conditional probability, and the probability of misclassification for the row becomes one 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 (p∙m). 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 and by dividing. Standard errors for these coefficients are given in Bishop, Feinberg, and Holland (1975, page 388).

A problem with the optimal prediction coefficients λ is that they vary with the marginal probabilities. One way to correct for this is to use row conditional probabilities. The optimal prediction λ* coefficients are defined as the corresponding λ coefficients in which one first adjusts the row (or column) marginals 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.

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 to be

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

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

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 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).

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.

When there are two rows, it is possible to test for a linear trend in the row probabilities if one assumes 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 zero 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 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 (po ‑ pc)/(1 ‑ pc).

The McNemar test is a test of symmetry in a square contingency table, that is, it is a test of the null hypothesis Ho : θ ij = θ ji. The multiple‑degrees‑of‑freedom version of the McNemar test with r(r ‑ 1)/2 degrees of freedom is computed as

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

Its exact probability may be computed via the binomial distribution.

Informational errors

The following example is taken from Kendall and Stuart (1979). It involves the distance vision in the right and left eyes, and especially illustrates the use of Kappa and McNemar tests. Most other test statistics are also computed.