Send to

Choose Destination
Emerg Themes Epidemiol. 2018 Jun 26;15:9. doi: 10.1186/s12982-018-0077-7. eCollection 2018.

Cannons and sparrows: an exact maximum likelihood non-parametric test for meta-analysis of k 2 × 2 tables.

Author information

Somerset, USA.



The use of meta-analysis to aggregate multiple studies has increased dramatically over the last 30 years. For meta-analysis of homogeneous data where the effect sizes for the studies contributing to the meta-analysis differ only by statistical error, the Mantel-Haenszel technique has typically been utilized. If homogeneity cannot be assumed or established, the most popular technique is the inverse-variance DerSimonian-Laird technique. However, both of these techniques are based on large sample, asymptotic assumptions and are, at best, an approximation especially when the number of cases observed in any cell of the corresponding contingency tables is small.


This paper develops an exact, non-parametric test based on a maximum likelihood test statistic as an alternative to the asymptotic techniques. Further, the test can be used across a wide range of heterogeneity. Monte Carlo simulations show that for the homogeneous case, the ML-NP-EXACT technique to be generally more powerful than the DerSimonian-Laird inverse-variance technique for realistic, smaller values of disease probability, and across a large range of odds ratios, number of contributing studies, and sample size. Possibly most important, for large values of heterogeneity, the pre-specified level of Type I Error is much better maintained by the ML-NP-EXACT technique relative to the DerSimonian-Laird technique. A fully tested implementation in the R statistical language is freely available from the author.


This research has developed an exact test for the meta-analysis of dichotomous data. The ML-NP-EXACT technique was strongly superior to the DerSimonian-Laird technique in maintaining a pre-specified level of Type I Error. As shown, the DerSimonian-Laird technique demonstrated many large violations of this level. Given the various biases towards finding statistical significance prevalent in epidemiology today, a strong focus on maintaining a pre-specified level of Type I Error would seem critical.


Categorical analysis; DerSimonian–Laird; Exact solution; Inverse variance; Mantel–Haenszel; Meta-analysis

Supplemental Content

Full text links

Icon for BioMed Central Icon for PubMed Central
Loading ...
Support Center