Menu location: Analysis_Meta-Analysis_Relative Risk.
Cohort studies of dichotomous outcomes (e.g. dead or alive) can by represented by arranging the observed counts into fourfold (2 by 2) tables. Meta-analysis may be 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 relative risk for each stratum (a single fourfold table) and for the group of studies as a whole.
For a single stratum relative risk is defined as follows:
Relative risk = [a/(a+c)] / [b/(b+d)]
For each table the observed relative risk is displayed with a confidence interval. The likelihood score-based method of Koopman (1984) recommended by Gart and Nam is used to construct the confidence interval (Gart and Nam 1988; Sahai and Kurshid, 1996). If the ’try exact’ option is not selected then a normal approximation to the confidence interval is given instead.
The Mantel-Haenszel type method of Rothman and Boice (Rothman, 1998) is used to estimate the pooled risk ratio for all strata under the assumption of a fixed effects model:
- where ni = ai+bi+ci+di.
A confidence interval for the pooled relative risk is calculated using the Greenland-Robins variance formula(Greenland and Robins, 1985). A chi-square test statistic is given with associated probability of the pooled relative risk being equal to one. Note that some packages give a z statistic; this is equal to the square root of the chi-square statistic with one degree of freedom, i.e. it gives the same P.
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 should be entered in a workbook as follows:
|Total number||Number of cases||Total number||Number of cases|
...where total number = cases + non-cases
From Fleiss and Gross (1991).
Test workbook (Meta-analysis worksheet: Exposed total, Exposed cases, Non-exposed total, Non-exposed cases, Study).
The following data combine seven placebo-controlled randomized trials of the effect of aspirin in preventing death after myocardial infarction:
To analyse these data in StatsDirect first prepare them in four workbook columns and label these columns appropriately. Alternatively, open the test workbook using the file open function of the file menu. Then select relative risk from the meta-analysis section of the analysis menu. Select the columns marked "Exposed total", "Exposed cases", "Non-exposed total" and "Non-exposed cases" when prompted for data. Note that "exposed" and "experimental" groups are the same.
For this example:
|Study||Relative risk||95% CI (Koopman)||M-H weight|
Fixed effects (Mantel-Haenszel, Rothman-Boice)
Pooled relative risk = 0.913608 (95% CI = 0.8657 to 0.964168)
Chi² (test relative risk differs from 1) = 10.809386 (df = 1) P = 0.001
Non-combinability of studies
Cochran Q = 9.928487 (df = 6) P = 0.1277
Moment-based estimate of between studies variance = 0.007437
I² (inconsistency) = 39.6% (95% CI = 0% to 73.3%)
Random effects (DerSimonian-Laird)
Pooled relative risk = 0.892922 (95% CI = 0.800632 to 0.995851)
Chi² (test relative risk differs from 1) = 4.139819 (df = 1) P = 0.0419
Begg-Mazumdar: Kendall's tau = -0.428571 P = 0.1361 (low power)
Egger: bias = -0.736651 (95% CI = -2.653392 to 1.180091) P = 0.3685
Harbord-Egger: bias = -0.734002 (92.5% CI = -2.402692 to 0.934687) P = 0.3693
Here we can say with 95% confidence, assuming a random effects model, that for those given aspirin the true population risk of dying in the specified interval after a heart attack is at most 0.99 of the risk for those not given aspirin. Assuming a fixed effects model a stronger inference could be made about a relative risk of .96 (the upper confidence limit) but the high inter-study variation makes the fixed effects model less appropriate.