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The FREQ Procedure

Example 28.4: Analyzing a 2x2 Contingency Table

This example computes chi-square tests and Fisher's exact test to compare the probability of coronary heart disease for two types of diet. It also estimates the relative risks and computes exact confidence limits for the odds ratio.

The data set FatComp contains hypothetical data for a case-control study of high fat diet and the risk of coronary heart disease. The data are recorded as cell counts, where the variable Count contains the frequencies for each exposure and response combination. The data is sorted in descending order by the variables Exposure and Response, so that the first cell of the 2×2 table contains the frequency of positive exposure and positive response. The FORMAT procedure creates formats to identify the type of exposure and response with character values.

   proc format;
      value ExpFmt 1='High Cholesterol Diet'
                   0='Low Cholesterol Diet';
      value RspFmt 1='Yes'
                   0='No';
   run;

   data FatComp;
      input Exposure Response Count;
      label Response='Heart Disease';
      datalines;
   0 0  6
   0 1  2
   1 0  4
   1 1 11
   ;
   proc sort data=FatComp;
      by descending Exposure descending Response;
   run;

In the following statements, the TABLES statement creates a two-way table, and the option ORDER=DATA orders the contingency table values by their order in the data set. The CHISQ option produces several chi-square tests, while the RELRISK option produces relative risk measures. The EXACT statement creates the exact Pearson chi-square test and exact confidence limits for the odds ratio. These statements produce Output 28.4.1 through Output 28.4.3.

   proc freq data=FatComp order=data;
      weight Count;
      tables Exposure*Response / chisq relrisk;
      exact pchi or;
      format Exposure ExpFmt. Response RspFmt.;
      title 'Case-Control Study of High Fat/Cholesterol Diet';
   run;

Output 28.4.1: Contingency Table
 

Case-Control Study of High Fat/Cholesterol Diet
The FREQ Procedure
Frequency
Percent
Row Pct
Col Pct
Table of Exposure by Response
Exposure Response(Heart Disease) Total
Yes No
High Cholesterol Diet 11
47.83
73.33
84.62
4
17.39
26.67
40.00
15
65.22
 
 
Low Cholesterol Diet 2
8.70
25.00
15.38
6
26.09
75.00
60.00
8
34.78
 
 
Total 13
56.52
10
43.48
23
100.00


The contingency table in Output 28.4.1 displays the variable values so that the first table cell contains the frequency for the first cell in the data set, the frequency of positive exposure and positive response.

Output 28.4.2: Chi-Square Statistics
 

Case-Control Study of High Fat/Cholesterol Diet
The FREQ Procedure
Statistics for Table of Exposure by Response
 
Statistic DF Value Prob
Chi-Square 1 4.9597 0.0259
Likelihood Ratio Chi-Square 1 5.0975 0.0240
Continuity Adj. Chi-Square 1 3.1879 0.0742
Mantel-Haenszel Chi-Square 1 4.7441 0.0294
Phi Coefficient   0.4644  
Contingency Coefficient   0.4212  
Cramer's V   0.4644  
WARNING: 50% of the cells have expected counts less than 5.
(Asymptotic) Chi-Square may not be a valid test.
 
Pearson Chi-Square Test
Chi-Square 4.9597
DF 1
Asymptotic Pr > ChiSq 0.0259
Exact Pr >= ChiSq 0.0393
 
Fisher's Exact Test
Cell (1,1) Frequency (F) 11
Left-sided Pr <= F 0.9967
Right-sided Pr >= F 0.0367
   
Table Probability (P) 0.0334
Two-sided Pr <= P 0.0393
Sample Size = 23


Since the expected counts in some of the cells are small, PROC FREQ displays a warning that the asymptotic chi-square tests may not be appropriate. In this case, the exact tests in Output 28.4.2 are appropriate. The alternative hypothesis for this analysis states that coronary heart disease is more likely to be associated with a high fat diet, so a one-sided test is desired. Fisher's exact right-sided test analyzes whether the probability of heart disease in the high fat group exceeds the probability of heart disease in the low fat group; since this p-value is small, the alternative hypothesis is supported.

Output 28.4.3: Relative Risk
 

Case-Control Study of High Fat/Cholesterol Diet
The FREQ Procedure
Statistics for Table of Exposure by Response
 
Estimates of the Relative Risk (Row1/Row2)
Type of Study Value 95% Confidence Limits
Case-Control (Odds Ratio) 8.2500 1.1535 59.0029
Cohort (Col1 Risk) 2.9333 0.8502 10.1204
Cohort (Col2 Risk) 0.3556 0.1403 0.9009
 
Odds Ratio (Case-Control Study)
Odds Ratio 8.2500
   
Asymptotic Conf Limits  
95% Lower Conf Limit 1.1535
95% Upper Conf Limit 59.0029
   
Exact Conf Limits  
95% Lower Conf Limit 0.8677
95% Upper Conf Limit 105.5488
Sample Size = 23


The odds ratio, displayed in Output 28.4.3, provides an estimate of the relative risk when an event is rare. This estimate indicates that the odds of heart disease is 8.25 times higher in the high fat diet group; however, the wide confidence limits indicate that this estimate has low precision.

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