Performs a chi‑squared analysis of a 2 by 2 contingency table.

Routine CTTWO computes statistics associated with 2 × 2 contingency tables. Always computed are chi‑squared tests of independence, expected values based upon the independence assumption, contributions to chi‑squared in a test of independence, and row and column marginal totals. Optionally, when ICMPT = 0, CTTWO 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, “Correlation” (for computing the tetrachoric correlation coefficient) and CTCHI in this chapter (for computing statistics in other than 2 × 2 contingency tables).

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 relative marginal 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 of the four cells in the table, the contribution to chi‑squared is given as (xij ‑ eij)2/eij. The Pearson chi‑squared statistic (denoted as X2) is computed as the sum of the cell contributions to chi‑squared. It has, of course, 1 degree of freedom and tests the null hypothesis of independence, i.e., of H0 : pij = pi∙p∙j. Reject the null hypothesis 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.

Two measures related to chi‑squared but which do not depend upon sample size are phi,

and the contingency coefficient,

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 P has a range between 0.0 and 1.0 for any given table, the upper bound of P is actually somewhat less than 1.0 (see Kendall and Stuart 1979, page 577). In order to understand association within a table, consider also the maximum possible P(CHISQ(15)) and the maximum possible ɸ (CHISQ(13)). The significance of both 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 expected 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.

Fisher’s exact test is a conservative but uniformly most powerful unbiased test of equal row (or column) cell probabilities in the 2 × 2 table. In this test, the row and column marginals are assumed fixed, and the hypergeometric distribution is used to obtain the significance level of the test. A one- or a two‑sided test is possible. See Kendall and Stuart (1979, page 582) for a discussion.

In rows 1 through 7 of STAT, estimated standard errors and asymptotic p‑values are reported. Routine CTTWO computes these standard errors 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 STAT, is only correct under the null hypothesis of no association. The z‑scores in column 4 are computed using this second estimate of the standard errors, and the p‑values in column 5 are computed from these z‑scores. See Brown and Benedetti (1977) for a discussion and formulas for the standard errors in column 3.

The measures of association ɸ and P do not require any ordering of the row and column categories. Routine CTTWO also computes several measures of association for tables in which the rows and column categories correspond to ranked observations. Two of these measures, the product‑moment correlation and the Spearman correlation, are correlation coefficients that are computed using assigned scores for the row and column categories. In the product‑moment correlation, this score is the cell index, while in the Spearman rank correlation, this score is the average of the tied ranks of the row or column marginals. Other scores are possible.

Since the product auvbuv = 1 if both auv and buv are 1 or ‑1, it is easy to show that the “covariance” is twice the total number of agreements minus the number disagreements between the row and column variables 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 two times the number of agreements minus the number of 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 (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 crossproduct ratio is also sometimes thought of as a measure of association (see Bishop, Feinberg and Holland 1975, page 14). It is computed as:

The log of the crossproduct ratio is the log of this quantity.

The Yule’s Q and Yule’s Y are related to the cross product ratio. They are computed as:

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 variable for a given value of the row variable. The best classification for each row under the null hypothesis of independence is the column that has the highest marginal probability (and thus the highest probability for the row under the independence assumption). The probability of misclassification is then one minus this marginal probability. On the other hand, if independence is not assumed so that the row and columns variables are dependent, then within each row one would classify the column variables according to the category with the highest row conditional probability. The probability of misclassification for the row is then 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 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 and 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

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

Define the total variation for rows within columns as the sum of the qj’s. Consistent with the usual methods in the analysis of variance, the reduction in the total variation is 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 that the column variable is monotonically ordered. It tests the null hypothesis that the row populations are identical, using average ranks for the column variable. This amounts to a test of Ho : p1∙ = p2∙. The Kruskal‑Wallis statistic for columns is similarly defined. Conover (1980) discusses the Kruskal‑Wallis test.

The test for a linear trend in the column probabilities assumes that the row variable is monotonically ordered. In this test, the probability for column 1 is predicted by the row index using weighted simple linear regression. The slope is given by

where

is the average row index. An asymptotic test that the slope is zero may be obtained as the usual large sample regression test of zero slope.

Kappa is a measure of agreement. 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 po = p11 + p22 denote the probability that the two judges agree, and let pc = p1∙p∙1 + p2∙p∙2 denote the expected probability of agreement under the independence model. Kappa is then given by (po ‑ pc)/(1 ‑ pc).

The McNemar test is also a test of symmetry in square contingency tables. It tests the null hypothesis Ho : θ ij = θ ji. The test statistic with 1 degree of freedom is computed as

Its exact probability may be computed via the binomial distribution.

Informational errors

The following example from Kendall and Stuart (1979, pages 582‑583) compares the teeth in breast‑fed versus bottle‑fed babies.