Confidence Limits

If you specify the CL option in the TABLES statement, PROC FREQ computes asymptotic confidence limits for all MEASURES statistics. The confidence coefficient is determined according to the value of the ALPHA= option, which, by default, equals 0.05 and produces 95% confidence limits.

The confidence limits are computed as

where is the estimate of the measure, is the th percentile of the standard normal distribution, and is the asymptotic standard error of the estimate.

Asymptotic Tests

For each measure that you specify in the TEST statement, PROC FREQ computes an asymptotic test of the null hypothesis that the measure equals zero. Asymptotic tests are available for the following measures of association: gamma, Kendall’s tau-, Stuart’s tau-, Somers’ , Somers’ , the Pearson correlation coefficient, and the Spearman rank correlation coefficient. To compute an asymptotic test, PROC FREQ uses a standardized test statistic , which has an asymptotic standard normal distribution under the null hypothesis. The test statistic is computed as

where is the estimate of the measure and is the variance of the estimate under the null hypothesis. Formulas for for the individual measures of association are given in the following sections.

Note that the ratio of to is the same for the following measures: gamma, Kendall’s tau-, Stuart’s tau-, Somers’ , and Somers’ . Therefore, the tests for these measures are identical. For example, the -values for the test of equal the -values for the test of .

PROC FREQ computes one-sided and two-sided -values for each of these tests. When the test statistic is greater than its null hypothesis expected value of zero, PROC FREQ displays the right-sided -value, which is the probability of a larger value of the statistic occurring under the null hypothesis. A small right-sided -value supports the alternative hypothesis that the true value of the measure is greater than zero. When the test statistic is less than or equal to zero, PROC FREQ displays the left-sided -value, which is the probability of a smaller value of the statistic occurring under the null hypothesis. A small left-sided -value supports the alternative hypothesis that the true value of the measure is less than zero. The one-sided -value can be expressed as

where has a standard normal distribution. The two-sided p-value is computed as

Exact Tests

Exact tests are available for two measures of association: the Pearson correlation coefficient and the Spearman rank correlation coefficient. If you specify the PCORR option in the EXACT statement, PROC FREQ computes the exact test of the hypothesis that the Pearson correlation equals zero. If you specify the SCORR option in the EXACT statement, PROC FREQ computes the exact test of the hypothesis that the Spearman correlation equals zero. See the section Exact Statistics for more information.

Gamma

The gamma () statistic is based only on the number of concordant and discordant pairs of observations. It ignores tied pairs (that is, pairs of observations that have equal values of or equal values of ). Gamma is appropriate only when both variables lie on an ordinal scale. The range of gamma is . If the row and column variables are independent, then gamma tends to be close to zero. Gamma is estimated by

and the asymptotic variance is

For tables, gamma is equivalent to Yule’s . See Goodman and Kruskal (1979) and Agresti (2002) for more information.

The variance under the null hypothesis that gamma equals zero is computed as

See Brown and Benedetti (1977) for details.

Kendall’s Tau-b

Kendall’s tau- () is similar to gamma except that tau- uses a correction for ties. Tau- is appropriate only when both variables lie on an ordinal scale. The range of tau- is . Kendall’s tau- is estimated by

and the asymptotic variance is

where

See Kendall (1955) for more information.

The variance under the null hypothesis that tau- equals zero is computed as

See Brown and Benedetti (1977) for details.

Stuart’s Tau-c

Stuart’s tau- () makes an adjustment for table size in addition to a correction for ties. Tau- is appropriate only when both variables lie on an ordinal scale. The range of tau- is . Stuart’s tau- is estimated by

and the asymptotic variance is

where and . The variance under the null hypothesis that tau- equals zero is the same as the asymptotic variance ,

See Brown and Benedetti (1977) for details.

Somers’ D

Somers’ and Somers’ are asymmetric modifications of tau-. indicates that the row variable X is regarded as the independent variable and the column variable Y is regarded as dependent. Similarly, indicates that the column variable Y is regarded as the independent variable and the row variable X is regarded as dependent. Somers’ differs from tau- in that it uses a correction only for pairs that are tied on the independent variable. Somers’ is appropriate only when both variables lie on an ordinal scale. The range of Somers’ is . Somers’ is computed as

and its asymptotic variance is

where and

See Somers (1962), Goodman and Kruskal (1979), and Liebetrau (1983) for more information.

The variance under the null hypothesis that equals zero is computed as

See Brown and Benedetti (1977) for details.

Formulas for Somers’ are obtained by interchanging the indices.

Pearson Correlation Coefficient

The Pearson correlation coefficient () is computed by using the scores specified in the SCORES= option. This measure is appropriate only when both variables lie on an ordinal scale. The range of the Pearson correlation is . The Pearson correlation coefficient is estimated by

and its asymptotic variance is

where and are the row and column scores and

See Snedecor and Cochran (1989) for more information.

The SCORES= option in the TABLES statement determines the type of row and column scores used to compute the Pearson correlation (and other score-based statistics). The default is SCORES=TABLE. See the section Scores for details about the available score types and how they are computed.

The variance under the null hypothesis that the correlation equals zero is computed as

Note that this expression for the variance is derived for multinomial sampling in a contingency table framework, and it differs from the form obtained under the assumption that both variables are continuous and normally distributed. See Brown and Benedetti (1977) for details.

PROC FREQ also provides an exact test for the Pearson correlation coefficient. You can request this test by specifying the PCORR option in the EXACT statement. See the section Exact Statistics for more information.

Spearman Rank Correlation Coefficient

The Spearman correlation coefficient () is computed by using rank scores, which are defined in the section Scores. This measure is appropriate only when both variables lie on an ordinal scale. The range of the Spearman correlation is . The Spearman correlation coefficient is estimated by

and its asymptotic variance is

where and are the row and column rank scores and

See Snedecor and Cochran (1989) for more information.

The variance under the null hypothesis that the correlation equals zero is computed as

where

Note that the asymptotic variance is derived for multinomial sampling in a contingency table framework, and it differs from the form obtained under the assumption that both variables are continuous and normally distributed. See Brown and Benedetti (1977) for details.

PROC FREQ also provides an exact test for the Spearman correlation coefficient. You can request this test by specifying the SCORR option in the EXACT statement. See the section Exact Statistics for more information.

Polychoric Correlation

When you specify the PLCORR option in the TABLES statement, PROC FREQ computes the polychoric correlation. This measure of association is based on the assumption that the ordered, categorical variables of the frequency table have an underlying bivariate normal distribution. For tables, the polychoric correlation is also known as the tetrachoric correlation. See Drasgow (1986) for an overview of polychoric correlation. The polychoric correlation coefficient is the maximum likelihood estimate of the product-moment correlation between the normal variables, estimating thresholds from the observed table frequencies. The range of the polychoric correlation is from –1 to 1. Olsson (1979) gives the likelihood equations and an asymptotic covariance matrix for the estimates.

To estimate the polychoric correlation, PROC FREQ iteratively solves the likelihood equations by a Newton-Raphson algorithm that uses the Pearson correlation coefficient as the initial approximation. Iteration stops when the convergence measure falls below the convergence criterion or when the maximum number of iterations is reached, whichever occurs first. The CONVERGE= option sets the convergence criterion, and the default value is 0.0001. The MAXITER= option sets the maximum number of iterations, and the default value is 20.

Lambda (Asymmetric)

Asymmetric lambda, , is interpreted as the probable improvement in predicting the column variable Y given knowledge of the row variable X. The range of asymmetric lambda is . Asymmetric lambda () is computed as

and its asymptotic variance is

where

The values of and are determined as follows. Denote by the unique value of such that , and let be the unique value of such that . Because of the uniqueness assumptions, ties in the frequencies or in the marginal totals must be broken in an arbitrary but consistent manner. In case of ties, is defined as the smallest value of such that .

For those columns containing a cell for which , records the row in which is assumed to occur. Initially is set equal to –1 for all . Beginning with , if there is at least one value such that , and if , then is defined to be the smallest such value of , and is set equal to . Otherwise, if , then is defined to be equal to . If neither condition is true, then is taken to be the smallest value of such that .

The formulas for lambda asymmetric can be obtained by interchanging the indices.

See Goodman and Kruskal (1979) for more information.

Lambda (Symmetric)

The nondirectional lambda is the average of the two asymmetric lambdas, and . Its range is . Lambda symmetric is computed as

and its asymptotic variance is computed as

where

The definitions of and are given in the previous section. The values and are defined in a similar way for lambda asymmetric ().

See Goodman and Kruskal (1979) for more information.

Uncertainty Coefficients (Asymmetric)

The uncertainty coefficient measures the proportion of uncertainty (entropy) in the column variable Y that is explained by the row variable X. Its range is . The uncertainty coefficient is computed as

and its asymptotic variance is

where

The formulas for the uncertainty coefficient can be obtained by interchanging the indices.

See Theil (1972, pp. 115–120) and Goodman and Kruskal (1979) for more information.

Uncertainty Coefficient (Symmetric)

The uncertainty coefficient is the symmetric version of the two asymmetric uncertainty coefficients. Its range is . The uncertainty coefficient is computed as

and its asymptotic variance is

where , , and are defined in the previous section. See Goodman and Kruskal (1979) for more information.