ODDS RATIO INDEPENDENCE TEST

ODDS RATIO INDEPENDENCE TEST (LET)

Analysis Command

Perform a log odds ratio test of independence for a 2x2 contingency table.

Given two variables where each variable has exactly two possible outcomes (typically defined as success and failure), we define the odds ratio as:
o = (N₁₁/N₁₂)/ (N₂₁/N₂₂)
    = (N₁₁N22)/ (N₁₂N₂₁)

where

N₁₁ = number of successes in sample 1
N₂₁ = number of failures in sample 1
N₁₂ = number of successes in sample 2
N₂₂ = 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:

l(o) = LOG{(N₁₁/N₁₂)/ (N₂₁/N₂₂)}
       = LOG{(N₁₁N22)/ (N₁₂N₂₁)}

Alternatively, the log odds ratio can be given in terms of the proportions

l(o) = LOG{(p₁₁/p₁₂)/ (p₂₁/p₂₂)}
       = LOG{(p₁₁p₂₂)/ (p₁₂p₂₁)}


where

p₁₁ = N₁₁/ (N₁₁ + N₂₁)
      = proportion of successes in sample 1
p₂₁ = N₂₁/ (N₁₁ + N₂₁)
      = proportion of failures in sample 1
p₁₂ = N₁₂/ (N₁₂ + N₂₂)
      = proportion of successes in sample 2
p₂₂ = N₂₂/ (N₁₂ + N₂₂)
      = 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:

l'(o) = LOG[{(N₁₁+0.5) (N₂₂+0.5)}/ {(N₁₂+0.5) (N₂₁+0.5)}]

In addition to reducing bias, this statistic also has the advantage that the odds ratio is still defined even when N₁₂ or N₂₁ is zero (the uncorrected statistic will be undefined for these cases).

Note that N₁₁, N₂₁, N₁₂, and N₂₂ defines a 2x2 contingency table. These types of contingency tables are also referred to as fourfold tables.

A common question with regards to a two-way 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

P(row i, column j) = P(row i)*P(column j)       for all i,j

One such test for the special case described above (i.e., we have success/failure data) is the log odds ratio test for independence.

H0:	The two-way table is independent
Ha:	The two-way table is not independent
Test Statistic:	The log odds ratio independence test statistic is: \( T = \frac{(N_{1} + N_{2})(N_{11}N_{22} - N_{12}N_{21})^2} {N_{1}N_{2}(N_{11} + N_{12})(N_{21} + N_{22})} \) Some analysts prefer to use the Yates corrected version of the test statistic: \( T = \frac{(N_{1} + N_{2})(|N_{11}N_{22} - N_{12}N_{21}| - 0.5(N_{1} + N_{2}))^2} {N_{1}N_{2}(N_{11} + N_{12})(N_{21} + N_{22})} \)
Significance Level:	\( \alpha \)
Critical Region:	T > \( \phi \) (\( \alpha \)) where \( \phi \) is the percent point function of the normal distribution
Conclusion:	Reject the independence hypothesis if the value of the test statistic is greater than the normal value.

ODDS RATIO INDEPENDENCE TEST <y1> <y2>
                        <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 two-way table).

ODDS RATIO INDEPENDENCE TEST <m>
                        <SUBSET/EXCEPT/FOR qualification>
where <m> is a matrix containing the two-way table;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax is used for the case where we the data have already been cross-tabulated into a two-way contingency table.

ODDS RATIO INDEPENDENCE TEST <n11> <n12> <n21> <n22>
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 Y1 Y2
ODDS RATIO INDEPENDENCE TEST M
ODDS RATIO INDEPENDENCE TEST N11 N12 N21 N22

Dataplot performs this test for both the bias corrected log(odds ratio) case and the no bias correction case.

The following information is written to the file dpst1f.dat (in the current directory):

Column 1	-	significance level
Column 2	-	lower confidence limit (uncorrected case)
Column 3	-	upper confidence limit (uncorrected case)
Column 4	-	lower confidence limit (corrected case)
Column 5	-	upper confidence limit (corrected case)

To read this information into Dataplot, enter

SET READ FORMAT F10.5,1X,4E15.7
READ DPST1F.DAT SIGLEV UNCLOWCL UNCUPPCL CORLOWCL CORUPPCL


The following internal parameters are automatically saved after running this command:

STATVAL	=	test statistic (uncorrected)
STATVALY	=	test statistic (with Yates correction)
STATCDF	=	cdf value for test statistic (uncorrected)
STATCDFY	=	cdf value for test statistic (with Yates correction)
ODDSRATI	=	value of the log(odds ratio)
ODDSRASE	=	value of the standard error of the log(odds ratio)
ODDSRABC	=	value of the bias corrected log(odds ratio)
ODDSBCSE	=	value of the bias corrected standard error of the log(odds ratio)

The CHI-SQUARE INDEPENDENCE TEST performs an alternative test for independence.

The chi-square independence test is more general in the sense that it applies to RxC contingency tables, not just 2x2 tables.

None

None

CHI-SQUARE INDEPENDENCE TEST	= Perform a chi-square test for independence.
FISHER EXACT TEST	= Perform Fisher's exact test.
ASSOCIATION PLOT	= Generate an association plot.
SIEVE PLOT	= Generate a sieve plot.
ROSE PLOT	= Generate a Rose plot.
BINARY TABULATION PLOT	= Generate a binary tabulation plot.
ROC CURVE	= Generate a ROC curve.
ODDS RATIO	= Compute the bias corrected odds ratio.
LOG ODDS RATIO	= Compute the bias corrected log(odds ratio).

Andrew Ruhkin, private communication.

Fleiss, Levin, and Paik (2003), "Statistical Methods for Rates and Proportions," Third Edition, Wiley, pp. 234-238.

Categorical Data Analysis

2007/2

 
let n11 = 53
let n21 = 7
let n12 = 48
let n22 = 12
.
set write decimals 4
odds ratio independence test n11 n21 n12 n22
    

The 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:
Chi-Square 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)         ACCEPT
    

 
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 y2
    

The 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:
Chi-Square 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