Woolf Statistics for 2 by 2 Tables and Series
Menu location: Analysis_Chisquare_Woolf.
In casecontrol studies observed frequencies can often be represented by a series of two by two tables. Each stratum of this series represents observations taken at different times, different places or another system of subgrouping within one large study.
A pooled odds ratio for all strata can be calculated by the method of Mantel and Haenszel or that of Woolf. The MantelHaenszel method is more robust when some of the strata contain small frequencies.
Results are given for individual tables and for the combined statistics (Haldane corrected), including chisquare for heterogeneity between the tables.
DATA INPUT:
Observed frequencies should be entered as multiple fourfold tables:
feature present  feature absent  
outcome positive:  a  b 
outcome negative:  c  d 
Example
From Armitage and Berry (1994, p. 516).
The following data compare the smoking status of lung cancer patients with controls. Ten different studies are combined in an attempt to improve the overall estimate of relative risk. The matching of controls has been ignored because there was not enough information about matching from each study to be sure that the matching was the same in each study.
Lung cancer  Controls  
smoker  nonsmoker  smoker  nonsmoker 
83  3  72  14 
90  3  227  43 
129  7  81  19 
412  32  299  131 
1350  7  1296  61 
60  3  106  27 
459  18  534  81 
499  19  462  56 
451  39  1729  636 
260  5  259  28 
To analyse these data in StatsDirect you must select the Woolf function from the chisquare section of the analysis menu. Then enter each row of the table above as a separate 2 by 2 contingency table:
i.e. The first row is entered as:

Smkr 
Non 
Lung cancer 
83 
3 
Control 
72 
14 
... this is then repeated for each of the ten rows.
For this example:
Statistics from combined values without Haldane correction:
Odds ratio = 4.519207
Approximate 95% CI = 3.752994 to 5.441851
Chi² for E(LOR) = 0 is 253.2108, P < 0.0001
Chi² for Heterogeneity = 6.634122, P = 0.6752
Statistics from combined values with Haldane correction:
Odds ratio = 4.510211
Approximate 95% CI = 3.747642 to 5.427948
Chi² for E(LOR) = 0 is 254.0865, P < 0.0001
Chi² for Heterogeneity = 6.532662, P = 0.6856
Here we can say that there was no convincing evidence of heterogeneity between the separate estimates of relative risk from each of the different studies. The pooled estimate suggested that with 95% confidence that the true population odds for being a smoker were between 3.7 and 5.4 times greater in lung cancer patients compared with controls.
The equivalent analysis using the MantelHaenszel method gave a confidence interval for the pooled odds ratio of 3.9 to 5.6; the difference is partly accounted for by the Haldane correction. You should use the more robust MantelHaenszel for most analyses of this kind. Woolf's method is included for further investigation of intertable relationships under expert statistical guidance.