Woolf Statistics for 2 by 2 Tables and Series


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In case-control 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 sub-grouping 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 Mantel-Haenszel 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 chi-square for heterogeneity between the tables.



Observed frequencies should be entered as multiple fourfold tables:


  feature present feature absent
outcome positive: a b
outcome negative: c d



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 non-smoker smoker non-smoker
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 chi-square 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:





Lung cancer







... 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 Mantel-Haenszel 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 Mantel-Haenszel for most analyses of this kind. Woolf's method is included for further investigation of inter-table relationships under expert statistical guidance.


P values

confidence intervals



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