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Frequency Tables and Statistics

The FREQ procedure provides easy access to statistics for testing for association in a crosstabulation table.

In this example, high school students applied for courses in a summer enrichment program: these courses included journalism, art history, statistics, graphic arts, and computer programming. The students accepted were randomly assigned to classes with and without internships in local companies. The following table contains counts of the students who enrolled in the summer program by gender and whether they were assigned an internship slot.

The SAS data set SummerSchool is created by inputting the summer enrichment data as cell count data, or providing the frequency count for each combination of variable values. The following DATA step statements create the SAS data set SummerSchool.

   data SummerSchool; 
      input Gender $ Internship $ Enrollment $ Count @@; 
      datalines;
   boys  yes yes 35   boys  yes no 29                  
   boys   no yes 14   boys   no no 27
   girls yes yes 32   girls yes no 10  
   girls  no yes 53   girls  no no 23
   ;

Researchers are interested in whether there is an association between internship status and summer program enrollment. The Pearson chi-square statistic is an appropriate statistic to assess the association in the corresponding 2×2 table. The following PROC FREQ statements specify this analysis.

You specify the table for which you want to compute statistics with the TABLES statement. You specify the statistics you want to compute with options after a slash (/) in the TABLES statement.

 
   proc freq data=SummerSchool order=data;
      weight count;  
      tables Internship*Enrollment / chisq;
   run;

The ORDER= option controls the order in which variable values are displayed in the rows and columns of the table. By default, the values are arranged according to the alphanumeric order of their unformatted values. If you specify ORDER=DATA, the data are displayed in the same order as they occur in the input data set. Here, since `yes' appears before `no' in the data, `yes' appears first in any table. Other options for controlling order include ORDER=FORMATTED, which orders according to the formatted values, and ORDER=FREQUENCY, which orders by descending frequency count.

In the TABLES statement, Internship*Enrollment specifies a table where the rows are internship status and the columns are program enrollment. Since the input data are in cell count form, the WEIGHT statement is required. The WEIGHT statement names the variable Count, which provides the frequency of each combination of data values. Finally, the CHISQ option requests chi-square statistics for assessing association.

Figure 28.1 presents the crosstabulation of Internship and Enrollment. In each cell, the values printed under the cell count are the table percentage, row percentage, and column percentage, respectively. For example, in the first cell, 63.21 percent of those offered courses with internships accepted them and 36.79 percent did not.

The next tables display the statistics produced by the CHISQ option. The Pearson chi-square statistic is labeled `Chi-Square' and has a value of 0.8189 with 1 degree of freedom. The associated p-value is 0.3655, which means that there is no significant evidence of an association between internship status and program enrollment. The other chi-square statistics have similar values and are asymptotically equivalent. The other statistics (Phi Coefficient, Contingency Coefficient, and Cramer's V) are measures of association derived from the Pearson chi-square. For Fisher's exact test, the two-sided p-value is 0.4122, which also shows no association between internship status and program enrollment.

The analysis, so far, has ignored gender. However, it may be of interest to ask whether program enrollment is associated with internship status after adjusting for gender. You can address this question by doing an analysis of a set of tables, in this case, by analyzing the set consisting of one for boys and one for girls. The Cochran-Mantel-Haenszel statistic is appropriate for this situation: it addresses whether rows and columns are associated after controlling for the stratification variable. In this case, you would be stratifying by gender.

The FREQ statements for this analysis are very similar to those for the first analysis, except that there is a third variable, Gender, in the TABLES statement. When you cross more than two variables, the two rightmost variables construct the rows and columns of the table, respectively, and the leftmost variables determine the stratification.

 
   proc freq data=SummerSchool;
      weight count;  
      tables Gender*Internship*Enrollment / chisq cmh;
   run;

This execution of PROC FREQ first produces two individual crosstabulation tables of Internship*Enrollment, one for boys and one for girls. Chi-square statistics are produced for each individual table. Note that the chi-square statistic for boys is significant at the level of significance. Boys offered a course with an internship are more likely to enroll than boys who are not.

If you look at the individual table for girls, you see that there is no evidence of association for girls between internship offers and program enrollment.

These individual table results demonstrate the occasional problems with combining information into one table and not accounting for information in other variables such as Gender. Figure 28.4 contains the CMH results. There are three summary (CMH) statistics: which one you use depends on whether your rows and/or columns have an order in r×c tables. However, in the case of 2×2 tables, ordering doesn't matter and all three statistics take the same value. The CMH statistic follows the chi-square distribution under the hypothesis of no association, and here, it takes the value 4.0186 with 1 degree of freedom. The associated p-value is 0.0450, which indicates a significant association at the level.

Thus, when you adjust for the effect of gender in these data, there is an association between internship and program enrollment. But, if you ignore gender, no association is found. Note that the CMH option also produces other statistics, including estimates and confidence limits for relative risk and odds ratios for 2×2 tables and the Breslow-Day Test. These results are not displayed here.

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