Optimal approximate conversions of odds ratios and hazard ratios to risk ratios

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.     

Sex ratios

Sex ratio theory attempts to explain variation at all levels (species, population, individual, brood) in the proportion of offspring that are male (the sex ratio). In many cases this work has been extremely successful, providing qualitative and even quantitative explanations of sex ratio variation. However, this is not always the situation, and one of the greatest remaining problems is explaining broad taxonomic patterns. Specifically, why do different organisms show so much variation in the amount and precision with which they adjust their offspring sex ratios?     

Endophage-ectophage ratios and plant defense

Summary Endophagous folivores, which are concealed inside leaf tissue for much of their life cycle, or which live externally but feed internally, should be more successful on heavily defended plants than ectophagous species. This is because endophages are more facile at feeding selectively and can manipulate tissue development to avoid physical and chemical defenses and to enhance nutrition. As a result, endophage-ectophage ratios should increase on more heavily defended hosts. This pattern will likely be strengthened by negative asymmetrical interactions with ectophages and pathogens, which may displace endophages from lightly defended hosts. The hypothesis predicts that endophages should be particularly abundant in resource-poor habitats which seem to support a preponderance of heavily defended hosts. Although data do not yet exist for a rigorous test, several observed cases where endophage distributions seem biased toward heavily defended hosts are at least consistent with the hypothesis. Plant defense levels may have little influence on the total number of herbivores associated with a host, but I suggest that guild structure can be profoundly altered.     

Inbreeding and parasite sex ratios

The breeding system of parasitic protozoa affects the evolution of drug resistance and virulence, and is relevant to disease diagnosis and the development of chemo- and immunotherapy. A major group of protozoan parasites, the phylum Apicomplexa, that includes the aetiological agents of malaria, toxoplasmosis and coccidiosis, all have dimorphic sexual stages. The sex ratio (proportion of males produced by parasites) is predicted to depend upon the inbreeding rate, and it has been suggested that sex-ratio data offer a relatively cheap and easy method for indirectly estimating inbreeding rates. Here, we exploit a new theoretical machinery to show that there are generally valid relationships between f, Wright's coefficient of inbreeding, and sex ratio, z(*), the generality being with respect to population structure. To focus the discussion, we concentrate on malaria and show that the previously derived result, f = 1 - 2z(*), does not depend on the artificial assumptions about population structure that were previously made. Not only does this justify the use of sex ratio as an indirect measure of f, but also we argue that it may actually be preferable to measure f by measuring sex ratios, rather than by measuring departures from Hardy-Weinberg genotypic proportions both in malaria and parasites more generally.     

Adaptive sex ratios and parent-offspring conflict

Studies testing the theoretical prediction that birds would adaptively vary the sex ratio of their offspring either supported theoretical predictions or simply found a 1:1 sex ratio. Four recent papers, in particular one by Kate Oddie, of Great Tit nestling sex ratios, however, found that, when conditions are poor, the sex ratio is male biased, opposite of what was predicted by theory. The development of molecular markers to sex birds using minute amounts of blood has allowed experiments that help us to explain this apparent anomaly.     

Achieving women and declining sex ratios

Professionally written biographies of 353 women selected for inclusion in two volumes of a dictionary of biography were analyzed separately for data on family structure, occupation, and personality characteristics. Findings from Volume 4 (1921-1940) were similar to those from Volume 5 (1941-1960). Results showed that 55.2% of women biographees had no children. They were either unmarried or, if married, childless. Of those who did have children, biographees had a significantly high ratio of sons to daughters compared with the norm (0.514) (Vol. 4: males = 106, females = 76; Vol. 5: males = 115, females = 80; totals: males = 221, females = 156; chi2 = 7.87; p = 0.005). These data, interpreted according to the maternal dominance hypothesis (Grant 1998) and set in the context of increased workforce participation by women and declining fertility, suggest another possible explanation for the contemporary decline in the secondary sex ratio.     

Activity ratios of ribulose-1,5-bisphosphate carboxylase accurately reflect carbamylation ratios

Activity ratios and carbamylation ratios of ribulose-1,5-bisphosphate carboxylase (RuBPCase) were determined for leaves of Phaseolus vulgaris and Spinacia oleracea exposed to a variety of partial pressures of CO₂ and O₂ and photon flux densities (PFD). It was found that activity ratios accurately predicted carbamylation ratios except in extracts from leaves held in low PFD. In particular, it was confirmed that the loss of RuBPCase activity in low partial pressure of O₂ and high PFD results from reduced carbamylation. Activity ratios of RuBPCase were lower than carbamylation ratios for Phaseolus leaves sampled in low PFD, presumably because of the presence of 2-carboxyarabinitol 1-phosphate. Spinacia leaves sampled in darkness also exhibited lower activity ratios than carbamylation ratios indicating that this species may also have an RuBPCase inhibitor even though carboxyarabinitol 1-phosphate has not been detected in this species in the past.