Estimating relative risk of disease from outputs of logistic regression when the disease is not rare

Abstract

Many epidemiologic studies in the veterinary field aim to quantify the relationships between risk factors and the occurrence of diseases. The strength of the association between a factor and a disease can be measured by (i) a relative risk (RR), or (ii) an odds ratio (OR) which is widely used because it is directly derived from the estimates of logistic regression. RR directly provides the relative increase in the probability of disease occurrence in case of exposure. OR is often interpretated as a multiplicative factor of the risk of disease occurrence when exposed, although it is not a good approximation of RR when the disease is not rare. The objective of this paper is to propose a method to estimate RR of disease from adjusted odds ratios derived from logistic regression when the disease is not rare. The method of estimation is developed for three different cases: (i) the factor and the outcome are dichotomous; (ii) the factor has more than two classes, and the outcome is dichotomous; and (iii) the factor and the outcome both have more than two classes. In all cases, the principles of estimation are the same: in a subpopulation including individuals diseased at level j (Dj) and not diseased (Do), when exposed to level i (Fi) or not exposed to the factor (Fo), RRij can be calculated with adjusted ORij, and the frequencies of individuals exposed to level i (ni&j), of those not exposed (no&j) and of those diseased (ni&j) among the individuals exposed to level i and not exposed. RRij is the positive solution of the formula: [formula: see text] Simulations were done to assess the relative weight of the exposure rate, disease risk and value of ORij in the difference between RRij and ORij. The difference between ORij and RRij depends upon (i), first of all, the disease risk in the population, but also (ii) the value of ORij, and to a less extent (iii) the exposure rate. Simulations also showed that ranking of the risk-factor levels according to their effect cannot always rely on OR.