Case-control studies of dichotomous outcomes (e.g. healed or not healed) can by represented by arranging the observed counts into fourfold (2 by 2) tables. The separation of data into different tables or strata represents a sub-grouping, e.g. into age bands. Stratification of this kind is sometimes used to reduce confounding.
The Mantel-Haenszel method provides a pooled odds ratio across the strata of fourfold tables. Meta-analysis is used to investigate the combination or interaction of a group of independent studies, for example a series of fourfold tables from similar studies conducted at different centres.
This StatsDirect function examines the odds ratio for each stratum (a single fourfold table) and for the group of studies as a whole. Exact methods are used here in addition to conventional approximations.
For a single stratum odds ratio is estimated as follows:
Sample estimate of the odds ratio = (ad)/(bc)
For each table, the observed odds ratio is displayed with an exact confidence interval (Martin and Austin, 1991; Sahai and Kurshid, 1996). With very large numbers these calculations can take an appreciable amount of time. If the ’try exact’ option is not selected then the logit (Woolf) interval is given instead.
The Mantel-Haenszel method is used to estimate the pooled odds ratio for all strata, assuming a fixed effects model:
- where ni = ai+bi+ci+di.
Alternative methods, such Woolf and inverse variance, can be used to estimate the pooled odds ratio with fixed effects but the Mantel-Haenszel method is generally the most robust. A confidence interval for the Mantel-Haenszel odds ratio in StatsDirect is calculated using the Robins, Breslow and Greenland variance formula (Robins et al., 1986) or by the method of Sato (1990) if the estimate of the odds ratio can not be determined. A chi-square test statistic is given with its associated probability that the pooled odds ratio is equal to one.
If any cell count in a table is zero then a continuity correction is applied to each cell in that table – if you have selected the ’delay continuity correction’ option then no continuity correction is applied to the Mantel-Haenszel calculation unless all of the ’a’ cells or all of the ’d’ cells are zero across the studies. The type of continuity correction used is set in the options.
An exact conditional likelihood method is optionally used to evaluate the pooled odds ratio (Martin and Austin, 2000). The exact method may take an appreciable time to compute with large numbers. The exact results should be used in preference to the Mantel-Haenszel approximation, especially if some categories involve few observations (less than 15 or so).
The inconsistency of results across studies is summarised in the I² statistic, which is the percentage of variation across studies that is due to heterogeneity rather than chance – see the heterogeneity section for more information.
Note that the results from StatsDirect may differ slightly from other software or from those quoted in papers; this is due to differences in the variance formulae. StatsDirect employs the most robust practical approaches to variance according to accepted statistical literature.
Observed frequencies may be entered in a workbook (see example in relative risk meta-analysis) or directly via the screen as multiple fourfold tables:
|feature present||feature absent|
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.
To analyse these data in StatsDirect you may select the Mantel-Haenszel function from the chi-square section of the analysis menu. Select the default 95% confidence interval. Enter the number of tables as 10. Then enter each row of the table above as a separate 2 by 2 contingency table:
i.e. The first row is entered as:
... this is then repeated for each of the ten rows.
For this example:
Fixed effects (Mantel-Haenszel, Robins-Breslow-Greenland)
Pooled odds ratio = 4.681639 (95% CI = 3.865935 to 5.669455)
Chi² (test odds ratio differs from 1) = 292.379352 P < 0.0001
Fixed effects (conditional maximum likelihood)
Pooled odds ratio = 4.713244
Exact Fisher 95% CI = 3.888241 to 5.747141
Exact Fisher one sided P < 0.0001, two sided P < 0.0001
Exact mid-P 95% CI = 3.904839 to 5.719768
Exact mid-P one sided P < 0.0001, two sided P < 0.0001
Non-combinability of studies
Breslow-Day = 6.766765 (df = 9) P = 0.6614
Cochran Q = 6.641235 (df = 9) P = 0.6744
Moment-based estimate of between studies variance = 0
I² (inconsistency) = 0% (95% CI = 0% to 52.7%)
Random effects (DerSimonian-Laird)
Pooled odds ratio = 4.625084 (95% CI = 3.821652 to 5.597423)
Chi² (test odds ratio differs from 1) = 247.466729 (df = 1) P < 0.0001
Begg-Mazumdar: Kendall's tau = 0.111111 P = 0.7275 (low power)
Egger: bias = 0.476675 (95% CI = -0.786168 to 1.739517) P = 0.4094
Harbord-Egger: bias = 0.805788 (92.5% CI = -0.686033 to 2.297609) P = 0.3013
Here we can say with 95% confidence that the true population odds in favour of being a smoker were between 3.9 and 5.7 times greater in patients who had lung cancer compared with controls.