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Point and interval estimation of exposure effects and interaction between the exposures based on logistic model for observational studies

In observational studies with dichotomous outcome of a population, researchers need to present the effects of exposures and interaction between the exposures jointly in order to learn the relationship between the exposure effects and the interaction. In this article we study point and interval estimation of exposure effects and the interaction based on logistic model, where the exposure effects are measured by risk differences while the interaction is measured by difference between risk differences. Using approximate normal distribution of the maximum-likelihood (ML) estimate of the model parameters, we obtain approximate non-normal distribution of the ML estimate of the exposure effects and the interaction. Using the obtained distribution, we obtain point estimate and confidence region of (exposure effect, interaction) as well as point estimate and confidence interval of the interaction when the ML estimate of an exposure effect falls into specified range. Our maximum-likelihood-based approach provides a simple but reliable method of interval estimation of exposure effects and the interaction.