Likelihood-based instrumental variable analysis in the presence of an unobserved latent confounder

Instrumental variable analysis (IVA) is used to control unobserved confounders and estimate average causal effects in observational studies. Classical IVA involves a two-stage procedure with two ordinary linear models. The first stage relates the treatment or intervention to the instrument, and the second relates the outcome to the expected treatment predicted by the first stage. The average causal effect can be estimated using the difference in outcomes between the strata of the instrumental variable. D.B. Rubin in a series of papers (summarized in Angrist, Imbens, and Rubin, 1996) re-framed IVA in terms of a causal model which can be applied to binary outcome variables when the instrumental variable and treatment status are also binary. However, the average causal effect is typically expressed as a difference. When causal effects expressed as rate ratios or odds ratios are desired in nonlinear models, it is problematic to obtain the unbiased estimators for these parameters. We propose a two-stage likelihood-based IVA model. In both stages, the estimates of parameters of interest are obtained using maximum likelihood functions. In the first stage, patient compliance with the instrumental variable is estimated. Treatment effect is then imputed in the second stage with the adjustment of compliance. Essentially, the likelihood function is formulated using the joint distribution of outcome and instrumental variables by integrating out the treatment and unknown confounder, assuming the distribution of the confounder is known, and the associations between the confounder and treatment, and confounder and outcome are also known or can be estimated. This likelihood function is maximized to obtain an estimator of the coefficient of the treatment variable. The variance of this maximum likelihood estimation (MLE) of treatment effect can be estimated using average Fisher’s information matrix. We illustrate this two-stage likelihood-based IVA model using data from a study of primary androgen deprivation therapy (PADT) in men with localized prostate cancer (Lu-Yao, Albertsen, Moore, et al. 2008). We also examine the optimal minimum sample size needed for each health service area in order to reduce the misclassifications, and obtain unbiased estimates of the average causal effect.Ph. D.Includes bibliographical referencesby Anjun Ca