Pool 2x2 tables when the events are rare enough to break the others

With a zero cell the ordinary log odds ratio is undefined and the usual repair – adding 0.5 everywhere – biases the pooled estimate. Peto's estimator never forms a ratio per study: it accumulates observed-minus-expected counts against their hypergeometric variance, so a zero cell contributes without special handling. The price is a bias when the odds ratio is far from one or the groups are badly unbalanced.

Usage

Mapeto(a, b, c, d, level = 0.95)

Arguments

a, b, c, d

Per-study cells: events and non-events in the treated arm, then in the control arm.

level

Confidence level.

Value

List with OR, log_OR, se_log, ci, O_E, V, k.

Details

Formula: ln OR = sum(O_i - E_i)/sum(V_i) with O = a, E = (a+b)(a+c)/N and V = (a+b)(c+d)(a+c)(b+d)/(N^2 (N-1)); se = 1/sqrt(sum V) – Peto et al. (1977), Appendix.

References

Peto, R. et al. (1977). British Journal of Cancer 35(1):1-39. doi:10.1038/bjc.1977.1 .

Examples

Mapeto(a = c(1, 2, 3, 4, 5, 6, 7, 8), b = c(1, 2, 3, 4, 5, 6, 7, 8),
  c = c(1, 2, 3, 4, 5, 6, 7, 8), d = c(1, 2, 3, 4, 5, 6, 7, 8))
#> $OR
#> [1] 1
#> 
#> $log_OR
#> [1] 0
#> 
#> $se_log
#> [1] 0.3235933
#> 
#> $ci
#> [1] 0.5303431 1.8855719
#> 
#> $O_E
#> [1] 0
#> 
#> $V
#> [1] 9.549948
#> 
#> $k
#> [1] 8
#> 
#> $method
#> [1] "Peto one-step pooled odds ratio"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"