Estimation of relative risk using a log-binomial model with constraints

Abstract

The relative risk/prevalence ratio and odds ratio are very popular in medical research and epidemiological studies. The odds ratio is overused in practice due to its direct relation with the logistic regression. Interpreting the odds ratio in terms of “relative risk” may lead to incorrect inference on the prevalence of certain event. In this paper, we propose to estimate the relative risk using the log-binomial model by maximizing the likelihood with a linear constraint, which can be easily implemented by an existing function, such as “constrOptim” in R. Furthermore, we review other existing methods and compare their performance under a variety of settings using simulated data. We suggest the proposed and COPY methods in practice, based on its performance in the simulations. For illustration, we investigate the vigorous physical activity effects on obesity using data from the National Health and Nutrition Examination Survey, and the treatment effects on lung cancer mortality using data in R.

Appendix

1.1 R code

1.1.1 Constraint function in R

For a purpose of illustration, let data$Y denote the binary outcome $Y$, data$x denote the possible variable $\mathbf x$ with dimension $n\times (p+1)$, “loglike” denote the negative logarithm of likelihood function (3), and $\hat{\beta }$ denote the estimator of $\beta$. We illustrate the code for $\hat{\beta }$ by “constrOptim” function. First, we set the starting value in a feasible region, such as $(-1, 0, \ldots , 0)$ since $\exp (-1)<1$. Second, we need to specify the restriction region (4). In the “constrOptim” function, it can be realized by ui %*%$\beta$ $-$ ci $\ge$ 0 with ui = $-$data$x and ci = ($0, 0,\ldots , 0$). We can obtain estimates of the log-binomial model by

Covariance matrix

1.1.2 Generate simulated dataset

1.1.3 COPY method

1.1.4 IPTW method

1.1.5 Poisson method

1.1.6 NLS method