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. Table 3.1 contains counts of the students who enrolled in the summer program by gender and whether they were assigned an internship slot.
Table 3.1: Summer Enrichment Data
Enrollment 


Gender 
Internship 
Yes 
No 
Total 
boys 
yes 
35 
29 
64 
boys 
no 
14 
27 
41 
girls 
yes 
32 
10 
42 
girls 
no 
53 
23 
76 
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 ;
The variable Gender
takes the values ‘boys’ or ‘girls,’ the variable Internship
takes the values ‘yes’ and ‘no,’ and the variable Enrollment
takes the values ‘yes’ and ‘no.’ The variable Count
contains the number of students that correspond to each combination of data values. The double at sign (@@) indicates that
more than one observation is included on a single data line. In this DATA step, two observations are included on each line.
Researchers are interested in whether there is an association between internship status and summer program enrollment. The Pearson chisquare statistic is an appropriate statistic to assess the association in the corresponding 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; tables Internship*Enrollment / chisq; weight Count; 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, because ‘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. The CHISQ option requests
chisquare statistics for assessing association between these two variables. Because 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.
Figure 3.1 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 the students offered courses with internships accepted them and 36.79 percent
did not.
Figure 3.1: Crosstabulation Table


Figure 3.2 displays the statistics produced by the CHISQ option. The Pearson chisquare statistic is labeled 'ChiSquare' and has a value of 0.8189 with 1 degree of freedom. The associated pvalue is 0.3655, which means that there is no significant evidence of an association between internship status and program enrollment. The other chisquare statistics have similar values and are asymptotically equivalent. The other statistics (phi coefficient, contingency coefficient, and Cramér’s V) are measures of association derived from the Pearson chisquare. For Fisher’s exact test, the twosided pvalue is 0.4122, which also shows no association between internship status and program enrollment.
Figure 3.2: Statistics Produced with the CHISQ Option
Statistic  DF  Value  Prob 

ChiSquare  1  0.8189  0.3655 
Likelihood Ratio ChiSquare  1  0.8202  0.3651 
Continuity Adj. ChiSquare  1  0.5899  0.4425 
MantelHaenszel ChiSquare  1  0.8153  0.3666 
Phi Coefficient  0.0606  
Contingency Coefficient  0.0605  
Cramer's V  0.0606 
Fisher's Exact Test  

Cell (1,1) Frequency (F)  67 
Leftsided Pr <= F  0.8513 
Rightsided Pr >= F  0.2213 
Table Probability (P)  0.0726 
Twosided Pr <= P  0.4122 
The analysis, so far, has ignored gender. However, it might 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 CochranMantelHaenszel (CMH) 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 PROC 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.
The following PROC FREQ statements also request frequency plots for the crosstabulation tables. PROC FREQ produces these plots by using ODS Graphics to create graphs as part of the procedure output. ODS Graphics must be enabled before producing plots. The PLOTS(ONLY)=FREQPLOT option requests frequency plots. The TWOWAY=CLUSTER plotoption specifies a cluster layout for the twoway frequency plots.
ods graphics on; proc freq data=SummerSchool; tables Gender*Internship*Enrollment / chisq cmh plots(only)=freqplot(twoway=cluster); weight Count; run; ods graphics off;
This execution of PROC FREQ first produces two individual crosstabulation tables of Internship
by Enrollment
: one for boys and one for girls. Frequency plots and chisquare statistics are produced for each individual table. Figure 3.3, Figure 3.4, and Figure 3.5 show the results for boys. Note that the chisquare 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.
Figure 3.4 displays the frequency plot of Internship
by Enrollment
for boys. By default, frequency plots are displayed as bar charts. You can use PLOTS= options to request dot plots instead
of bar charts, to change the orientation of the bars from vertical to horizontal, and to change the scale from frequencies
to percents. You can also use PLOTS= options to specify other twoway layouts (stacked, vertical groups, or horizontal groups)
and to change the primary grouping from column levels to row levels.
Figure 3.6, Figure 3.7, and Figure 3.8 display the crosstabulation table, frequency plot, and chisquare statistics for girls. You can see that there is no evidence of association between internship offers and program enrollment for girls.
Figure 3.3: Crosstabulation Table for Boys


Figure 3.5: ChiSquare Statistics for Boys
Statistic  DF  Value  Prob 

ChiSquare  1  4.2366  0.0396 
Likelihood Ratio ChiSquare  1  4.2903  0.0383 
Continuity Adj. ChiSquare  1  3.4515  0.0632 
MantelHaenszel ChiSquare  1  4.1963  0.0405 
Phi Coefficient  0.2009  
Contingency Coefficient  0.1969  
Cramer's V  0.2009 
Fisher's Exact Test  

Cell (1,1) Frequency (F)  27 
Leftsided Pr <= F  0.9885 
Rightsided Pr >= F  0.0311 
Table Probability (P)  0.0196 
Twosided Pr <= P  0.0467 
Figure 3.6: Crosstabulation Table for Girls


Figure 3.8: ChiSquare Statistics for Girls
Statistic  DF  Value  Prob 

ChiSquare  1  0.5593  0.4546 
Likelihood Ratio ChiSquare  1  0.5681  0.4510 
Continuity Adj. ChiSquare  1  0.2848  0.5936 
MantelHaenszel ChiSquare  1  0.5545  0.4565 
Phi Coefficient  0.0688  
Contingency Coefficient  0.0687  
Cramer's V  0.0688 
Fisher's Exact Test  

Cell (1,1) Frequency (F)  23 
Leftsided Pr <= F  0.8317 
Rightsided Pr >= F  0.2994 
Table Probability (P)  0.1311 
Twosided Pr <= P  0.5245 
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 3.9 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 tables. However, in the case of tables, ordering does not matter and all three statistics take the same value. The CMH statistic follows the chisquare distribution
under the hypothesis of no association, and here, it takes the value 4.0186 with 1 degree of freedom. The associated pvalue 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 tables and the BreslowDay Test. These results are not displayed here.
Figure 3.9: Test for the Hypothesis of No Association
CochranMantelHaenszel Statistics (Based on Table Scores)  

Statistic  Alternative Hypothesis  DF  Value  Prob 
1  Nonzero Correlation  1  4.0186  0.0450 
2  Row Mean Scores Differ  1  4.0186  0.0450 
3  General Association  1  4.0186  0.0450 