Instrumental variables analysis using genetic markers as instruments
is now a widely used technique in epidemiology and biostatistics. As
single markers tend to explain only a small proportion of phenotypic variation,
there is increasing interest in using multiple genetic markers to obtain
more precise estimates of causal parameters. Structural mean models
(SMMs) are semiparametric models that use instrumental variables to identify
causal parameters. Recently, interest has started to focus on using these
models with multiple instruments, particularly for multiplicative and logistic
SMMs. In this paper we show how additive, multiplicative and logistic
SMMs with multiple orthogonal binary instrumental variables can be estimated
efficiently in models with no further (continuous) covariates, using
the generalised method of moments (GMM) estimator. We discuss how the
Hansen J-test can be used to test for model misspecification, and how standard
GMM software routines can be used to fit SMMs. We further show that
multiplicative SMMs, like the additive SMM, identify a weighted average of
local causal effects if selection is monotonic. We use these methods to reanalyse
a study of the relationship between adiposity and hypertension using
SMMs with two genetic markers as instruments for adiposity. We find strong
effects of adiposity on hypertension.