Odds ratios approximate risk ratios when the outcome under consideration is rare but can diverge substantially from risk ratios when the outcome is common. In this paper, we derive optimal analytic conversions of odds ratios and hazard ratios to risk ratios that are minimax for the bias ratio when outcome probabilities are specified to fall in any fixed interval. The results for hazard ratios are derived under a proportional hazard assumption for the exposure. For outcome probabilities specified to lie in symmetric intervals centered around 0.5, it is shown that the square‐root transformation of the odds ratio is the optimal minimax conversion for the risk ratio. General results for any nonsymmetric interval are given both for odds ratio and for hazard ratio conversions. The results are principally useful when odds ratios or hazard ratios are reported in papers, and the reader does not have access to the data or to information about the overall outcome prevalence.
Optimal Allocation Ratios: A Square Root Relationship between the Ratios of Symbiotic Costs and Benefits
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Abstract -
Shaffer, Bradley (Ed.)Abstract Rising global temperatures threaten to disrupt population sex ratios, which can in turn cause mate shortages, reduce population growth and adaptive potential, and increase extinction risk, particularly when ratios are male biased. Sex ratio distortion can then have cascading effects across other species and even ecosystems. Our understanding of the problem is limited by how often studies measure temperature effects in both sexes. To address this, the current review surveyed 194 published studies of heat tolerance, finding that the majority did not even mention the sex of the individuals used, with <10% reporting results for males and females separately. Although the data are incomplete, this review assessed phylogenetic patterns of thermally induced sex ratio bias for 3 different mechanisms: sex-biased heat tolerance, temperature-dependent sex determination (TSD), and temperature-induced sex reversal. For sex-biased heat tolerance, documented examples span a large taxonomic range including arthropods, chordates, protists, and plants. Here, superior heat tolerance is more common in females than males, but the direction of tolerance appears to be phylogenetically fluid, perhaps due to the large number of contributing factors. For TSD, well-documented examples are limited to reptiles, where high temperature usually favors females, and fishes, where high temperature consistently favors males. For temperature-induced sex reversal, unambiguous cases are again limited to vertebrates, and high temperature usually favors males in fishes and amphibians, with mixed effects in reptiles. There is urgent need for further work on the full taxonomic extent of temperature-induced sex ratio distortion, including joint effects of the multiple contributing mechanisms.more » « less
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1086/716182
