CTTWO

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

Required Arguments

TABLE — 2 by 2 matrix containing the observed counts in the contingency table. (Input)

EXPECT — 3 by 3 matrix containing the expected values of each cell in TABLE under the null hypothesis of independence, in the first 2 rows and 2 columns, and the marginal totals in the last row and column. (Output)

CHICTR — 3 by 3 matrix containing the contributions to chi‑squared for each cell in TABLE in the first 2 rows and 2 columns. (Output)
The last row and column contain the total contribution to chi‑squared for that row or column.

CHISQ — Vector of length 15 containing statistics associated with this contingency table. (Output)

I	CHISQ(I)
1	Pearson chi‑squared statistic
2	Probability of a larger Pearson chi‑squared
3	Degrees of freedom for chi‑squared
4	Likelihood ratio G2 (chi‑squared)
5	Probability of a larger G2
6	Yates corrected chi-squared
7	Probability of a larger corrected chi‑squared
8	Fisher’s exact test (one tail)
9	Fisher’s exact test (two tail)
10	Exact mean
11	Exact standard deviation

The following statistics are based upon the chi‑squared statistic CHISQ(1)

.

I	CHISQ(I)
12	Phi (Φ)
13	The maximum possible Φ
14	Contingency coefficient P
15	The maximum possible contingency coefficient

STAT — 24 by 5 matrix containing statistics associated with this table. (Output)
Each row of the matrix corresponds to a statistic.

 

Row	Statistic
1	Gamma
2	Kendall’s b
3	Stuart’s c
4	Somers’ D (row)
5	Somers’ D (column)
6	Product moment correlation
7	Spearman rank correlation
8	Goodman and Kruskal (row)
9	Goodman and Kruskal (column)
10	Uncertainty coefficient U (normed)
11	Uncertainty Ur|c (row)
12	Uncertainty Uc|r (column)
13	Optimal prediction λ (symmetric)
14	Optimal prediction λr|c (row)
15	Optimal prediction λc|r (column)
16	Optimal prediction λ*r|c (row)
17	Optimal prediction λ*c|r (column)
18	Yule’s Q
19	Yule’s Y
20	Crossproduct ratio
21	Log of crossproduct ratio
22	Test for linear trend
23	Kappa
24	McNemar test of symmetry

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

Column	Statistic
1	Estimated statistic
2	Its estimated standard error for any parameter value
3	Its estimated standard error under the null hypothesis
4	z‑score for testing the null hypothesis
5	p‑value for the test in column 4

In the McNemar test, column 1 contains the statistic, column 2 contains the chi‑squared degrees of freedom, column 4 contains the exact p‑value, and column 5 contains the chi‑squared asymptotic p‑value.

Optional Arguments

LDTABL — Leading dimension of TABLE exactly as specified in the dimension statement of the calling program. (Input)
Default: LDTABL = size (TABLE,1).

ICMPT — Computing option. (Input)
If ICMPT = 0, all of the values in CHISQ and STAT are computed. ICMPT = 1 means compute only the first 11 values of CHISQ, and no values of STAT are computed.
Default: ICMPT = 0.

IPRINT — Printing option. (Input)
IPRINT = 0 means no printing is performed. If IPRINT = 1, printing is performed.
Default: IPRINT = 0.

LDEXPE — Leading dimension of EXPECT exactly as specified in the dimension statement of the calling program. (Input)
Default: LDEXPE = size (EXPECT,1).

LDCHIC — Leading dimension of CHI exactly as specified in the dimension statement of the calling program. (Input)
Default: LDCHI = size (CHI,1).

LDSTAT — Leading dimension of STAT exactly as specified in the dimension statement of the calling program. (Input)
Default: LDSTAT = size (STAT,1).

FORTRAN 90 Interface

Generic: CALL CTTWO (TABLE, EXPECT, CHICTR, CHISQ, STAT [, …])

Specific: The specific interface names are S_CTTWO and D_CTTWO.

FORTRAN 77 Interface

Single: CALL CTTWO (TABLE, LDTABL, ICMPT, IPRINT, EXPECT, LDEXPE, CHICTR, LDCHIC, CHISQ, STAT, LDSTAT)

Double: The double precision name is DCTTWO.

Description

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).

Notation

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.

The Chi-squared Statistics

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.

Measures Related to Chi-squared (Phi and the Contingency Coefficient)

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

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.

Standard Errors and p-values for Some Measures of Association

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.

Measures of Association for Ranked Rows and Columns

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.

Other measures of associations, Gamma, Kendall’s b, Stuart’s c and Somers’ D, are also computed similarly to a correlation coefficient in that the numerator in these statistics in some sense is a “covariance.” In fact, these measures differ only in their denominators, their numerators being the “covariance” between the auv’s and the buv’s defined earlier. The numerator is computed as

 

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.

Kendall’s b is computed as the correlation between the auv’s and the buv’s (see Kendall and Stuart 1979, page 583). Stuart suggested a modification to the denominator of in which the denominator becomes the largest possible value of the “covariance.” This value turns out to be approximately 2n2 in 2 × 2 tables, and this is the value used in the denominator of Stuart’s c. For large n, c ≈ 2 b.

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:

 

Measures of Prediction and Uncertainty

The 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, 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.

λ*r|c

is similarly defined.

Goodman and Kruskal τ

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.

With this definition of variation, the Goodman and Kruskal τ 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, define the total variation for the rows within column j as

 

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.

Goodman and Kruskal’s τ columns is similarly defined. See Bishop, Feinberg, and Holland (1975, page 391) for the standard errors.

The 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 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 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.

Test for Linear Trend

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

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).

McNemar Test

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.

Comments

Informational errors

 

Type	Code	Description
4	8	At least one marginal total is zero. The remainder of the analysis cannot proceed.
3	9	Some expected table values are less than 1.0. Some asymptotic p‑values may not be good.
3	10	Some expected table values are less than 2.0. Some asymptotic p‑values may not be good.
3	11	20% of the table expected values are less than 5.

Example

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

 

 

USE CTTWO_INT

 

IMPLICIT NONE

INTEGER IPRINT, LDCHIC, LDEXPE, LDSTAT, LDTABL

PARAMETER (IPRINT=1, LDCHIC=3, LDEXPE=3, LDSTAT=24, LDTABL=2)

!

REAL CHICTR(LDCHIC,3), CHISQ(15), EXPECT(LDEXPE,3), &

STAT(LDSTAT,5), TABLE(LDTABL,2)

!

DATA TABLE/4, 1, 16, 21/

 

 

!

CALL CTTWO (TABLE, EXPECT, CHICTR, CHISQ, STAT, IPRINT=IPRINT)

END

Output

 

TABLE

1 2

1 4.00 16.00

2 1.00 21.00

 

Expected values

Col 1 Col 2 Marginal

Row 1 2.3810 17.6190 20.0000

Row 2 2.6190 19.3810 22.0000

Marginal 5.0000 37.0000 42.0000

 

Contributions to chi-squared

Col 1 Col 2 Total

Row 1 1.1010 0.1488 1.2497

Row 2 1.0009 0.1353 1.1361

Total 2.1018 0.2840 2.3858

 

CHISQ

1

Pearson chi-squared 2.3858

p-value 0.1224

Degrees of freedom 1.0000

Likelihood ratio 2.5099

p-value 0.1131

Yates chi-squared 1.1398

p-value 0.2857

Fisher (one tail) 0.1435

Fisher (two tail) 0.1745

Exact mean 1.0244

Exact std dev 1.3267

Phi 0.2383

Max possible phi 0.3855

Contingency coef. 0.2318

Max possible coef. 0.3597

 

STAT

Statistic Std err. Std err. 0 t-value p-value

Gamma 0.6800 0.3135 0.4395 1.5472 0.1218

Kendall’s tau B 0.2383 0.1347 0.1540 1.5472 0.1218

Stuart’s tau C 0.1542 0.0997 NaN 1.5472 0.1218

Somers’ D row 0.1545 0.0999 0.0999 1.5472 0.1218

Somers’ D col 0.3676 0.1966 0.2376 1.5472 0.1218

Correlation 0.2383 0.1347 0.1540 1.5472 0.1218

Spearman rank 0.2383 0.1347 0.1540 1.5472 0.1218

GK tau row 0.0568 0.0641 NaN NaN NaN

GK tau col 0.0568 0.0609 NaN NaN NaN

U normed 0.0565 0.0661 NaN NaN NaN

U row 0.0819 0.0935 NaN NaN NaN

U col 0.0432 0.0516 NaN NaN NaN

Lamda sym 0.1200 0.0779 NaN NaN NaN

Lamda row 0.0000 0.0000 NaN NaN NaN

Lamda col 0.1500 0.1031 NaN NaN NaN

Lamda star row 0.0000 0.0000 NaN NaN NaN

Lamda star col 0.1761 0.1978 NaN NaN NaN

Yule’s Q 0.6800 0.3135 0.4770 1.4255 0.1540

Yule’s Y 0.3923 0.2467 0.2385 1.6450 0.1000

Ratio 5.2500 NaN NaN NaN NaN

Log ratio 1.6582 1.1662 0.9540 1.7381 0.0822

Linear trend -0.1545 0.1001 NaN -1.5446 0.1224

Kappa 0.1600 0.1572 0.1600 1.0000 0.3173

McNemar 13.2353 1.0000 NaN 0.0000 0.0003

*** WARNING ERROR 11 from CTTWO. Twenty percent of the table expected

*** values are less than 5.0.