
ODDS RATIO CHISQUARE 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. The odds ratio chisquare test is applied in the situation where we have a series of fourfold tables. That is, the two variables for the fourfold tables are the same, but data is collected from different populations or groups with regards to these variables. Fleiss, Levin, and Paik (p. 234) list the following questions that are typically asked about these type of data: Suppose we have g fourfold tables. Then
This test is based on decomposing the total chisquare in the following way:
The \( \chi_{\mbox{homogeneity}}^{2} \) assesses the degree of homogeneity (i.e., equality) among the g measures of association. The \( \chi_{\mbox{association}}^{2} \) assesses the significance of the average degree of association. The overall measure of association (across all groups) is the weighted average of the g individual measures:
Under the hypothesis of zero overall association, \( \bar{Y} \) has an average value of zero and a standard error of
From this
follows an approximately a standard normal distribution under the null hypothesis and
follows an approximately chisquare distribution with one degree of freedom. In addition,
follows an approximately chisquare distribution with g  1 degrees of freedom. Note that \( \chi_{\mbox{association}}^{2} \) and \( \chi_{\mbox{homogeneity}}^{2} \) are uncorrelated. Based on the above formulas, we can answer the above questions as follows.
The above discussion is based on a generic statistic for the measure of association. For the odds ratio chisquare test, the specific measure of association is the bias corrected log odds ratio (given above). Note that the standard error of the bias corrected log odds ratio is:
The ODDS RATIO CHISQUARE TEST generates the following output:
<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 <y1> and <y2> denote a series of 2x2 tables (i.e., rows 1 and 2 are group 1, rows 3 and 4 are group 2, and so on).
<SUBSET/EXCEPT/FOR qualification> where <y1> is the first response variable; <y2> is the second response variable; <groupid> is a group id 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). In this case, the two response variables have an equal number of cases for each group.
<SUBSET/EXCEPT/FOR qualification> where <y1> is the first response variable; <groupid1> is a group id variable corresponding to <y1>; <y2> is the second response variable; <groupid2> is a group id variable corresponding to <y2>; 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). In this case, the two response variables may have an unequal number of cases for each group, so <y1> and <y2> require different group id variables.
ODDS RATIO CHISQUARE TEST Y1 Y2 X ODDS RATIO CHISQUARE TEST Y1 X1 Y2 X2
To read this information into Dataplot, enter
READ DPST1F.DAT SIGLEV LOGLOWCL LOGUPPCL ODDLOWCL ODDUPPCL Dataplot saves the following internal parameters:
let n1 = 105 let n2 = 192 let n3 = 145 let n = n1 + n2 + n3 let x = 3 for i = 1 1 n let istop = n1 + n2 let x = 2 for i = 1 1 istop let x = 1 for i = 1 1 n1 . set statistic missing value 99 . . Group 1 values . let y1 = 0 for i = 1 1 n let y2 = 0 for i = 1 1 n let y1 = 1 for i = 1 1 81 let y2 = 1 for i = 1 1 34 . . Group 2 values (have unequal samples here, so fill . with missing values . let istrt = n1 + 1 let istop1 = istrt + 118  1 let istop2 = istrt + 69  1 let y1 = 1 for i = istrt 1 istop1 let y2 = 1 for i = istrt 1 istop2 let istrt2 = n1 + 174 + 1 let istop2 = n1 + n2 let y2 = 99 for i = istrt2 1 istop2 . . Group 3 values . let istrt = n1 + n2 + 1 let istop1 = istrt + 82  1 let istop2 = istrt + 52  1 let y1 = 1 for i = istrt 1 istop1 let y2 = 1 for i = istrt 1 istop2 . odds ratio chisquare test y1 y2 xThe following output is generated. SUMMARY OF LOG(ODDS RATIO)  LOG OF STANDARD  ODDS RATIO ODDS RATIO ERROR 1/SE(L(i))**2 w(i)* GROUP  O(i) L(i) SE(L(i)) w(i) L(i)**2 =============================================================================== 1.  6.894114 1.930668 0.3099319 10.41040 38.80455 2.  2.414514 0.8814980 0.2138429 21.86806 16.99233 3.  2.313836 0.8389067 0.2400251 17.35748 12.21558 =============================================================================== TOTAL  49.63593 68.01245 CHISQUARE ANALYSIS OF LOG(ODDS RATIO) NUMBER OF GROUPS = 3 ESTIMATE OF COMBINED LOG(ODDS RATIO) = 1.086652 STANDARD ERROR OF COMBINED LOG(ODDS RATIO) = 0.1419390 CHISQUARE TEST STATISTIC (TOTAL) = 68.01245 DEGRESS OF FREEDOM = 3 CDF OF TEST STATISTIC = 1.000000 CHISQUARE TEST STATISTIC (ASSOCIATION) = 58.61073 DEGRESS OF FREEDOM = 1 CDF OF TEST STATISTIC = 1.000000 CHISQUARE TEST STATISTIC (HOMOGENEITY) = 9.401718 DEGRESS OF FREEDOM = 2 CDF OF TEST STATISTIC = 0.9978321 CHISQUARE TEST FOR CONSISTENCY OF ASSOCIATION (HOMOGENEITY) NULL HYPOTHESIS NULL NULL CONFIDENCE CRITICAL ACCEPTANCE HYPOTHESIS HYPOTHESIS LEVEL VALUE INTERVAL CONCLUSION =================================================================== CONSISTENT 50.0% 1.39 (0,0.500) REJECT CONSISTENT 80.0% 3.22 (0,0.800) REJECT CONSISTENT 90.0% 4.61 (0,0.900) REJECT CONSISTENT 95.0% 5.99 (0,0.950) REJECT CONSISTENT 97.5% 7.38 (0,0.975) REJECT CONSISTENT 99.0% 9.21 (0,0.990) REJECT CHISQUARE TEST FOR OVERALL DEGREE OF ASSOCIATION NULL HYPOTHESIS NULL NULL CONFIDENCE CRITICAL ACCEPTANCE HYPOTHESIS HYPOTHESIS LEVEL VALUE INTERVAL CONCLUSION =================================================================== NO ASSOCIATION 50.0% 0.45 (0,0.500) REJECT NO ASSOCIATION 80.0% 1.64 (0,0.800) REJECT NO ASSOCIATION 90.0% 2.71 (0,0.900) REJECT NO ASSOCIATION 95.0% 3.84 (0,0.950) REJECT NO ASSOCIATION 97.5% 5.02 (0,0.975) REJECT NO ASSOCIATION 99.0% 6.63 (0,0.990) REJECT LARGE SAMPLE CONFIDENCE INTERVAL FOR LOG(ODDS RATIO) LOG(ODDS RATIO) ODDS RATIO ( 1.086652 ) ( 2.964333 ) CONFIDENCE LOWER UPPER LOWER UPPER VALUE (%) LIMIT LIMIT LIMIT LIMIT  50.000 0.990915 1.18239 2.69370 3.26216 80.000 0.904750 1.26855 2.47131 3.55571 90.000 0.853183 1.32012 2.34711 3.74387 95.000 0.808457 1.36485 2.24444 3.91513 97.500 0.768509 1.40479 2.15655 4.07469 99.000 0.721041 1.45226 2.05657 4.27277  
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Date created: 10/10/2008 