Chi-Square Test for One-Way Tables

For one-way frequency tables, the CHISQ option in the TABLES statement provides a chi-square goodness-of-fit test. Let denote the number of classes, or levels, in the one-way table. Let denote the frequency of class (or the number of observations in class ) for . Then PROC FREQ computes the one-way chi-square statistic as

where is the expected frequency for class under the null hypothesis.

In the test for equal proportions, which is the default for the CHISQ option, the null hypothesis specifies equal proportions of the total sample size for each class. Under this null hypothesis, the expected frequency for each class equals the total sample size divided by the number of classes,

In the test for specified frequencies, which PROC FREQ computes when you input null hypothesis frequencies by using the TESTF= option, the expected frequencies are the TESTF= values that you specify. In the test for specified proportions, which PROC FREQ computes when you input null hypothesis proportions by using the TESTP= option, the expected frequencies are determined from the specified TESTP= proportions as

Under the null hypothesis (of equal proportions, specified frequencies, or specified proportions), has an asymptotic chi-square distribution with degrees of freedom.

In addition to the asymptotic test, you can request an exact one-way chi-square test by specifying the CHISQ option in the EXACT statement. See the section Exact Statistics for more information.

Pearson Chi-Square Test for Two-Way Tables

The Pearson chi-square for two-way tables involves the differences between the observed and expected frequencies, where the expected frequencies are computed under the null hypothesis of independence. The Pearson chi-square statistic is computed as

where is the observed frequency in table cell () and is the expected frequency for table cell (). The expected frequency is computed under the null hypothesis that the row and column variables are independent,

When the row and column variables are independent, has an asymptotic chi-square distribution with degrees of freedom. For large values of , this test rejects the null hypothesis in favor of the alternative hypothesis of general association.

In addition to the asymptotic test, you can request an exact Pearson chi-square test by specifying the PCHI or CHISQ option in the EXACT statement. See the section Exact Statistics for more information.

For tables, the Pearson chi-square is also appropriate for testing the equality of two binomial proportions. For and tables, the Pearson chi-square tests the homogeneity of proportions. See Fienberg (1980) for details.

Likelihood-Ratio Chi-Square Test

The likelihood-ratio chi-square involves the ratios between the observed and expected frequencies. The likelihood-ratio chi-square statistic is computed as

where is the observed frequency in table cell () and is the expected frequency for table cell ().

When the row and column variables are independent, has an asymptotic chi-square distribution with degrees of freedom.

In addition to the asymptotic test, you can request an exact likelihood-ratio chi-square test by specifying the LRCHI or CHISQ option in the EXACT statement. See the section Exact Statistics for more information.

Continuity-Adjusted Chi-Square Test

The continuity-adjusted chi-square for tables is similar to the Pearson chi-square, but it is adjusted for the continuity of the chi-square distribution. The continuity-adjusted chi-square is most useful for small sample sizes. The use of the continuity adjustment is somewhat controversial; this chi-square test is more conservative (and more like Fisher’s exact test) when the sample size is small. As the sample size increases, the continuity-adjusted chi-square becomes more like the Pearson chi-square.

The continuity-adjusted chi-square statistic is computed as

Under the null hypothesis of independence, has an asymptotic chi-square distribution with degrees of freedom.

Mantel-Haenszel Chi-Square Test

The Mantel-Haenszel chi-square statistic tests the alternative hypothesis that there is a linear association between the row variable and the column variable. Both variables must lie on an ordinal scale. The Mantel-Haenszel chi-square statistic is computed as

where is the Pearson correlation between the row variable and the column variable. For a description of the Pearson correlation, see the Pearson Correlation Coefficient. The Pearson correlation and thus the Mantel-Haenszel chi-square statistic use the scores that you specify in the SCORES= option in the TABLES statement. See Mantel and Haenszel (1959) and Landis, Heyman, and Koch (1978) for more information.

Under the null hypothesis of no association, has an asymptotic chi-square distribution with one degree of freedom.

In addition to the asymptotic test, you can request an exact Mantel-Haenszel chi-square test by specifying the MHCHI or CHISQ option in the EXACT statement. See the section Exact Statistics for more information.

Fisher’s Exact Test

Fisher’s exact test is another test of association between the row and column variables. This test assumes that the row and column totals are fixed, and then uses the hypergeometric distribution to compute probabilities of possible tables conditional on the observed row and column totals. Fisher’s exact test does not depend on any large-sample distribution assumptions, and so it is appropriate even for small sample sizes and for sparse tables.

2 2 Tables

For tables, PROC FREQ gives the following information for Fisher’s exact test: table probability, two-sided -value, left-sided -value, and right-sided -value. The table probability equals the hypergeometric probability of the observed table, and is in fact the value of the test statistic for Fisher’s exact test.

Where is the hypergeometric probability of a specific table with the observed row and column totals, Fisher’s exact -values are computed by summing probabilities over defined sets of tables,

The two-sided -value is the sum of all possible table probabilties (conditional on the observed row and column totals) that are less than or equal to the observed table probability. For the two-sided -value, the set includes all possible tables with hypergeometric probabilities less than or equal to the probability of the observed table. A small two-sided -value supports the alternative hypothesis of association between the row and column variables.

For tables, one-sided -values for Fisher’s exact test are defined in terms of the frequency of the cell in the first row and first column of the table, the (1,1) cell. Denoting the observed (1,1) cell frequency by , the left-sided -value for Fisher’s exact test is the probability that the (1,1) cell frequency is less than or equal to . For the left-sided -value, the set includes those tables with a (1,1) cell frequency less than or equal to . A small left-sided -value supports the alternative hypothesis that the probability of an observation being in the first cell is actually less than expected under the null hypothesis of independent row and column variables.

Similarly, for a right-sided alternative hypothesis, is the set of tables where the frequency of the (1,1) cell is greater than or equal to that in the observed table. A small right-sided -value supports the alternative that the probability of the first cell is actually greater than that expected under the null hypothesis.

Because the (1,1) cell frequency completely determines the table when the marginal row and column sums are fixed, these one-sided alternatives can be stated equivalently in terms of other cell probabilities or ratios of cell probabilities. The left-sided alternative is equivalent to an odds ratio less than 1, where the odds ratio equals (). Additionally, the left-sided alternative is equivalent to the column 1 risk for row 1 being less than the column 1 risk for row 2, . Similarly, the right-sided alternative is equivalent to the column 1 risk for row 1 being greater than the column 1 risk for row 2, . See Agresti (2007) for details.

Phi Coefficient

The phi coefficient is a measure of association derived from the Pearson chi-square. The range of the phi coefficient is for tables. For tables larger than , the range is (Liebetrau 1983). The phi coefficient is computed as

See Fleiss, Levin, and Paik (2003, pp. 98–99) for more information.

Contingency Coefficient

The contingency coefficient is a measure of association derived from the Pearson chi-square. The range of the contingency coefficient is , where (Liebetrau 1983). The contingency coefficient is computed as

See Kendall and Stuart (1979, pp. 587–588) for more information.

Cramer’s V

Cramer’s is a measure of association derived from the Pearson chi-square. It is designed so that the attainable upper bound is always 1. The range of Cramer’s is for tables; for tables larger than , the range is . Cramer’s is computed as

See Kendall and Stuart (1979, p. 588) for more information.