  The Metropolis algorithm is a Markov chain Monte Carlo (MCMC) algorithm used
to simulate from parameter distributions of interest, such as generalized
linear model parameters. The "Metropolis step" is a keystone concept that
underlies classical and modern MCMC methods and facilitates simple analysis of
complex statistical models. Beyond Bayesian analysis, MCMC is useful for
generating uncertainty intervals, even under the common scenario in causal
inference in which the target parameter is not directly estimated by a single,
fitted statistical model. We demonstrate, with a worked example, pseudo-code,
and R code, the basic mechanics of the Metropolis algorithm. We use the
Metropolis algorithm to estimate the odds ratio and risk difference contrasting
the risk of childhood leukemia among those exposed to high versus low level
magnetic fields. This approach can be used for inference from Bayesian and
frequentist paradigms and, in small samples, offers advantages over
large-sample methods like the bootstrap.
