
ODDS RATIO INDEPENDENCE TESTName:
= (N_{11}N22)/ (N_{12}N_{21}) where
N_{21} = number of failures in sample 1 N_{12} = number of successes in sample 2 N_{22} = number of failures in sample 2 The first definition shows the meaning of the odds ratio clearly, although it is more commonly given in the literature with the second definition. The log odds ratio is the logarithm of the odds ratio:
= LOG{(N_{11}N22)/ (N_{12}N_{21})} Alternatively, the log odds ratio can be given in terms of the proportions
= LOG{(p_{11}p_{22})/ (p_{12}p_{21})} where
= proportion of successes in sample 1 p_{21} = N_{21}/ (N_{11} + N_{21}) = proportion of failures in sample 1 p_{12} = N_{12}/ (N_{12} + N_{22}) = proportion of successes in sample 2 p_{22} = N_{22}/ (N_{12} + N_{22}) = proportion of failures in sample 2 Success and failure can denote any binary response. Dataplot expects "success" to be coded as "1" and "failure" to be coded as "0". The bias corrected version of the statistic is:
In addition to reducing bias, this statistic also has the advantage that the odds ratio is still defined even when N_{12} or N_{21} is zero (the uncorrected statistic will be undefined for these cases). Note that N_{11}, N_{21}, N_{12}, and N_{22} defines a 2x2 contingency table. These types of contingency tables are also referred to as fourfold tables. A common question with regards to a twoway contingency table is whether we have independence. By independence, we mean that the row and column variables are unassociated (i.e., knowing the value of the row variable will not help us predict the value of column variable and likewise knowing the value of the column variable will not help us predict the value of the row variable). A more technical definition for independence is that
One such test for the special case described above (i.e., we have success/failure data) is the log odds ratio test for independence.
<SUBSET/EXCEPT/FOR qualification> where <y1> is the first response variable; <y2> is the second response variable; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax is used for the case where you have raw data (i.e., the data has not yet been cross tabulated into a twoway table).
<SUBSET/EXCEPT/FOR qualification> where <m> is a matrix containing the twoway table; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax is used for the case where we the data have already been crosstabulated into a twoway contingency table.
where <n11> is a parameter containing the value for row 1, column 1 of a 2x2 table; <n12> is a parameter containing the value for row 1, column 2 of a 2x2 table; <n21> is a parameter containing the value for row 2, column 1 of a 2x2 table; <n22> is a parameter containing the value for row 2, column 2 of a 2x2 table. This syntax is used for the special case where you have a 2x2 table. In this case, you can enter the 4 values directly, although you do need to be careful that the parameters are entered in the order expected above.
ODDS RATIO INDEPENDENCE TEST M ODDS RATIO INDEPENDENCE TEST N11 N12 N21 N22
To read this information into Dataplot, enter
READ DPST1F.DAT SIGLEV UNCLOWCL UNCUPPCL CORLOWCL CORUPPCL The following internal parameters are automatically saved after running this command:
The chisquare independence test is more general in the sense that it applies to RxC contingency tables, not just 2x2 tables.
Fleiss, Levin, and Paik (2003), "Statistical Methods for Rates and Proportions," Third Edition, Wiley, pp. 234238.
let n11 = 53 let n21 = 7 let n12 = 48 let n22 = 12 . set write decimals 4 odds ratio independence test n11 n21 n12 n22The following output is generated. Log(Odds Ratio) Test for Independence 2x2 Table (Log(Odds Ratio) = 0) H0: The Two Variables Are Independent Ha: The Two Variables Are Not Independent Sample 1: Number of Observations: 60 Number of Successes: 53 Number of Failures: 7 Probability of Success: 0.8833 Probability of Failure: 0.1167 Sample 2: Number of Observations: 60 Number of Successes: 48 Number of Failures: 12 Probability of Success: 0.8000 Probability of Failure: 0.2000 Log(Odds Ratio) = Log(n11*n22/(n12*n21)): Log(Odds Ratio): 0.6381 Standard Error of Log(Odds Ratio): 0.5156 Log(Odds Ratio) (Bias Corrected): 0.6089 Standard Error (Bias Corrected): 0.5026 Large Sample Confidence Interval for Log(Odds Ratio)  Uncorrected Ratio Bias Corrected Ratio ( 0.6380874 ) ( 0.6089435 ) Confidence Lower Upper Lower Upper Value (%) Limit Limit Limit Limit  50.00 0.6381 0.6381 0.6089 0.6089 80.00 0.2041 1.0721 0.1859 1.0320 90.00 0.0227 1.2989 0.0352 1.2531 95.00 0.2101 1.4863 0.2178 1.4357 97.50 0.3726 1.6487 0.3762 1.5941 99.00 0.5615 1.8377 0.5604 1.7783 Test for Independence: ChiSquare Test Statistic: 1.5633 CDF of Test Statistic: 0.9410 Test Statistic with Yates Correction: 1.0005 CDF of Test Statistic with Yates Correction: 0.8415 Without Yates Correction:  Null Hypothesis Null Null Confidence Critical Acceptance Hypothesis Hypothesis Level Value Interval Conclusion  Independent 50.0% 0.00 (0,0.500) REJECT Independent 80.0% 0.84 (0,0.800) REJECT Independent 90.0% 1.28 (0,0.900) REJECT Independent 95.0% 1.64 (0,0.950) ACCEPT Independent 97.5% 1.96 (0,0.975) ACCEPT Independent 99.0% 2.33 (0,0.990) ACCEPT With Yates Bias Correction:  Null Hypothesis Null Null Confidence Critical Acceptance Hypothesis Hypothesis Level Value Interval Conclusion  Independent 50.0% 0.00 (0,0.500) REJECT Independent 80.0% 0.84 (0,0.800) REJECT Independent 90.0% 1.28 (0,0.900) ACCEPT Independent 95.0% 1.64 (0,0.950) ACCEPT Independent 97.5% 1.96 (0,0.975) ACCEPT Independent 99.0% 2.33 (0,0.990) ACCEPTProgram 2: let n = 1 let p = 0.9 let y1 = binomial rand numb for i = 1 1 200 let p = 0.68 let y2 = binomial rand numb for i = 1 1 130 . set write decimals 4 odds ratio independence test y1 y2The following output is generated. Log(Odds Ratio) Test for Independence 2x2 Table (Log(Odds Ratio) = 0) H0: The Two Variables Are Independent Ha: The Two Variables Are Not Independent Sample 1: Number of Observations: 200 Number of Successes: 175 Number of Failures: 25 Probability of Success: 0.8750 Probability of Failure: 0.1250 Sample 2: Number of Observations: 130 Number of Successes: 88 Number of Failures: 42 Probability of Success: 0.6769 Probability of Failure: 0.3231 Log(Odds Ratio) = Log(n11*n22/(n12*n21)): Log(Odds Ratio): 1.2062 Standard Error of Log(Odds Ratio): 0.2844 Log(Odds Ratio) (Bias Corrected): 1.1955 Standard Error (Bias Corrected): 0.2824 Large Sample Confidence Interval for Log(Odds Ratio)  Uncorrected Ratio Bias Corrected Ratio ( 1.206243 ) ( 1.195462 ) Confidence Lower Upper Lower Upper Value (%) Limit Limit Limit Limit  50.00 1.2062 1.2062 1.1955 1.1955 80.00 0.9669 1.4456 0.9578 1.4331 90.00 0.8418 1.5707 0.8336 1.5574 95.00 0.7384 1.6741 0.7310 1.6599 97.50 0.6488 1.7637 0.6420 1.7489 99.00 0.5446 1.8679 0.5385 1.8524 Test for Independence: ChiSquare Test Statistic: 19.1040 CDF of Test Statistic: 1.0000 Test Statistic with Yates Correction: 17.8995 CDF of Test Statistic with Yates Correction: 1.0000 Without Yates Correction:  Null Hypothesis Null Null Confidence Critical Acceptance Hypothesis Hypothesis Level Value Interval Conclusion  Independent 50.0% 0.00 (0,0.500) REJECT Independent 80.0% 0.84 (0,0.800) REJECT Independent 90.0% 1.28 (0,0.900) REJECT Independent 95.0% 1.64 (0,0.950) REJECT Independent 97.5% 1.96 (0,0.975) REJECT Independent 99.0% 2.33 (0,0.990) REJECT With Yates Bias Correction:  Null Hypothesis Null Null Confidence Critical Acceptance Hypothesis Hypothesis Level Value Interval Conclusion  Independent 50.0% 0.00 (0,0.500) REJECT Independent 80.0% 0.84 (0,0.800) REJECT Independent 90.0% 1.28 (0,0.900) REJECT Independent 95.0% 1.64 (0,0.950) REJECT Independent 97.5% 1.96 (0,0.975) REJECT Independent 99.0% 2.33 (0,0.990) REJECT  
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Date created: 01/07/2008 