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8 D. Madigan et al. interactions and a vector of time-varying covariates z id can be written as λ id = e φi + β′ x id + ∑ r≠s γrs x idr x ids + α ′ z id where γ denotes a two-way interaction between drugs r and s. Remark 1. In practice, many adverse effects can occur at most once in a given day suggesting a binary rather than Poisson model. One can show that adopting a logistic model yields an identical conditional likelihood to (1). This equivalence allows shifting to a logistic model with follow-up truncated at the outcome event, when that event is the onset of an enduring condition that permanently changes exposure propensity (see Discussion below.) Remark 2. It is straightforward to show that the conditional likelihood in (1) is log-concave. 3.3. Bayesian Self-Controlled Case Series We have now set up the full conditional likelihood for multiple drugs, so one could proceed by finding conditional maximum likelihood estimates of the drug parameter vector β. However in the problem of drug safety surveillance in LODs there are millions of potential drug exposure predictors (tens of thousands of drug main effects along with drug interactions). This high dimensionality leads to potential overfitting under the usual maximum likelihood approach, so regularization is necessary. We take a Bayesian approach by putting a prior over the drug effect parameter vector and performing inference based on posterior mode estimates. There are many choices of prior distributions that shrink the parameter estimates toward zero and address overfitting. In particular, we focus on the (1) Normal prior and (2) Laplacian prior. (i) Normal prior. Here we shrink the estimates toward zero by putting an independent Normal prior on each of the parameter components. Taking the posterior mode estimates would be analogous to a ridge Poisson regression, placing a constraint on the L 2 -norm of the parameter vector. (ii) Laplace prior. Under this choice of prior a portion of the posterior mode estimates will shrink all the way to zero, and their corresponding predictors will effectively be selected out of the model. This is equivalent to a lasso Poisson regression, where there is a constraint on the L 1 -norm of the parameter vector estimate. Efficient algorithms exist for finding posterior modes, rendering our approach tractable even in the large-scale setting. In particular, we have adapted the cyclic-coordinate descent algorithm of Genkin et al. (2007) to the SCCS context. An open-source implementation is available at http://omop.fnih.org.
Self-Controlled Case Series 9 4. EXTENSIONS TO THE BASIC SCCS MODEL 4.1. Relaxing the Independence Assumptions I: Events . Farrington and Hocine (2010) present an approach that extends SCCS to allow for within-individual event dependence. This method treats the vector of observed event times t i = (t i1 , . . . , t ini ) ′ for each individual i as a single point in an n i -dimensional region, where n i denotes the number of events experienced by individual i. This region is restricted to Q i (n i ) = {t i ∈ (a i , b i ] ni : t i1 < · · · < t ini } (where (a i , b i ] denotes the observation period for individual i) since the components of t i are ordered by time, and no event times can occur outside of the observation window (a i , b i ]. Standard SCCS assumes that events are realizations of a one-dimensional Poisson process and conditions upon the observed number of events n i . Under Farrington and Hocine’s model, however, the event time vector t i is treated as a single point arising from an n i -dimensional Poisson process. In this framework, conditioning on n i is equivalent to conditioning on the occurrence of a single point in the region Q i (n i ). If λ i (t 1 , . . . , t ni | x i ) is the intensity of the n i -dimensional Poisson process on Q i (n i ), the conditional likelihood of t i given the occurrence of one such point in Q i (n i ) is L ni i = ∫ Q i(n i) λ i (t i1 , . . . , t ini | x i ) (2) λ i (u 1 , . . . , u ni | x i )du 1 · · · du ni Farrington and Hocine assume that the n i -dimensional Poisson intensity can be written in the form ∏n i λ i (t 1 , . . . , t ni | x i ) = λ i (t j | x i ) × H ni (t 1 , . . . , t ni ) (3) j=1 where the product term is made up of independent univariate intensities λ i (t | x i ), and the H ni (.) function determines the dependence between events. From (2) and (3) we can see that terms of λ i (t | x i ) that are fixed in time will cancel out of the conditional likelihood, as they do in the original SCCS model. Similarly, fixed terms of H ni (.) will also drop out of the conditional likelihood. Farrington and Hoccine explore different possible choices for H. 4.2. Relaxing the Independence Assumptions II: The PD Model . The PD-SCCS model (Simpson, 2011) extends SCCS to allow positive dependence between events, meaning that the occurrence of an event can
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