Nova Acta Leopoldina Band 110 Nummer 377

based cohort of 37,842 new NSAID users, among Medicare beneficiaries eligible for a state- run pharmaceutical benefit plan in Pennsylvania. They wished specifically to infer the effect of prescribing a COX-2 versus non-selective NSAID on gastrointestinal (GI) bleeding within 60 days of initiating an NSAID. Let therefore – Zi indicate the physician’s prescribing preference for COX-2 inhibitors (1) versus non-se- lective NSAIDs (0) for subject i; – Yi stand for an observed gastrointestinal (GI) bleeding in subject i within 60 days of initiating an NSAID (1), or not (0); – Xi indicate subject i receiving the prescription for COX-2 inhibitors (1) versus non-selective NSAIDs (0). RASSEN et al. (2008) applied linear and log linear causal models to define and estimate an ex- pected causal effect of x for given Z and X = x values. Both models have the drawback that they do not automatically respect the constraint that a risk lies between 0 and 100 %. Logistic regression models do obey the constraint but lead to more complicated causal model estima- tors. Recent evolutions in this field allow nevertheless to develop a causal or structural logistic regression model here. More details on this development can be found in VANSTEELANDT et al. (2010). The approach is outlined below. 5.3 The Logistic Structural Model Let the odds of a bleed under observed treatment relative to the potential treatment free odds be modeled within subsets of given instrument and exposure value (X, Z) as = exp(Ψ* X ) [8] or equivalently, assume: logit E(Yi| Xi,Zi) – logit E(Yi0| Xi,Zi) = Ψ* Xi, [9] with logit(p)=log {p/(1–p)}. To derive an exact estimating equation for Ψ in logistic structural or causal model [9] based on the IV assumptions, VANSTEELANDT and GOETGHEBEUR (2003) added to structural model [9] an association model linking observed variables: logit {P(Yi = 1| Xi, Zi)} = β0 + β1 Xi + β2 Zi + β3 Xi Zi [10] or more generally, of the form: logit {P (Yi=1| Xi,Zi)} = m(Xi,Zi; β* ) where m(Xi,Zi; β* ) is a known function, smooth in β, with β* an unknown finite-dimensional parameter. An estimate β ^ can be obtained using standard methods, e.g. Maximum Likelihood Estimation, so that causal model [9]+ association model [10] yield predicted Yi0 for given Ψ: Y ^ i0 (Ψ, β ^ ) = expit {m(Xi,Zi ; β ^ ) – Ψ Xi} [11] With expit(a) = exp(a)/{1+exp(a)}. Since the IV-assumptions imply: E(Yi0|Zi)= E (Yi0) we let the estimator Ψ´` be the Ψ–value that makes this happen, after Yi0 has been replaced by Y ^ i0 (Ψ, β ^ ). Nova Acta Leopoldina NF 110, Nr. 377, 47–64 (2011) Els Goetghebeur 58 ! odds Yi = 1Xi ,Zi( ) odds Yi0 = 1Xi ,Zi( )