We propose to estimate average exposure (or treatment) effects from observational data for multiple exposure groups by fitting an approximation of the marginal sample distribution of the response variable in each exposure group to the data. The marginal sample distribution is a function of the true distribution of the response variable in the population and the assignment rule governing the allocation of the subjects to different exposure groups. The assignment rule can depend on the response variable in addition to measured covariates, and hence the method is appropriate even when the assumption of ignorable treatment assignment is not justified. We estimate the exposure effects are estimated based on the population expectation (PE) of the outcome variable. We compare the PE approach with an instrumental variable method and with several other methods including propensity score based approaches that assume ignorable assignment mechanisms. We evaluate the robustness of the PE method under model misspecifications and illustrate it using data from a study of the impact of soy consumption on urinary concentrations of estrogen and estrogen metabolites in Asian American women. Published 2012. This article is a US Government work and is in the public domain in the USA.