Inference for causal effects can benefit from the availability of an instrumental variable (IV) which, by definition, is associated with the given exposure, but not with the outcome of interest other than through a causal exposure effect. Estimation methods for instrumental variables are now well established for continuous outcomes, but much less so for dichotomous outcomes. In this article we review IV estimation of so-called conditional causal odds ratios which express the effect of an arbitrary exposure on a dichotomous outcome conditional on the exposure level, instrumental variable and measured covariates. In addition, we propose IV estimators of so-called marginal causal odds ratios which express the effect of an arbitrary exposure on a dichotomous outcome at the population level, and are therefore of greater public health relevance. We explore interconnections between the different estimators and support the results with extensive simulation studies and three applications.Comment: Published in at http://dx.doi.org/10.1214/11-STS360 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Primeval forests in the temperate zone exist only as a few remnants, but theses serve as important reference areas for conservation. As key habitats, tree-related microhabitats (TreMs) are of intense interest to forest ecologists, but little is known about their natural composition and dynamics in different tree species. Beech forms a major part of the temperate forests that extend from Europe, home to European beech Fagus sylvatica L. (Fs), eastward to Iran, where Oriental beech Fagus orientalis Lipsky (Fo) is the dominant species. In this study, we compared TreMs in primeval forests of both species, using data from Fo growing in 25 inventory plots throughout the Hyrcanian forest belt in Iran and from Fs growing in a 9 ha permanent plot in the Uholka Forest of Ukraine. TreMs based on 47 types and 11 subgroups were recorded. Beech trees in the Hyrcanian forest had a higher mean diameter at breast height (dbh) than beech trees in Uholka and contained twice as many TreMs per hectare. Although the mean richness of TreMs per TreM bearing tree was similar in the two species, on the basis of the comparison single trees in two groups (n = 405 vs. 2251), the composition of the TreMs clearly differed, as the proportions of rot holes, root-buttress concavities, and crown deadwood were higher in the Hyrcanian Forest, and those of bark losses, exposed heartwood, and burrs and cankers higher in Uholka Forest. Estimates of TreMs dynamics based on dbh and using Weibull models showed a significantly faster cumulative increase of TreMs in Fo, in which saturation occurred already in trees with a dbh of 70–80 cm. By contrast, the increase in TreMs in Fs was continuous. In both species, the probability density was highest at a dbh of about 30 cm, but was twice as high in Fo. Because of limitations of our study design, the reason behind observed differences of TreM formation and composition between regions remains unclear, as it could be either result of the tree species or the environment, or their interaction. However, the observed differences were more likely the result of differences in the environment than in the two tree species. Nevertheless, our findings demonstrate that the Hyrcanian Forest, recently designated as a natural heritage site in Iran, is unique, not only as a tertiary relict or due to its endemic trees, herbs and arthropods, but also because of its TreMs, which form a distinct and rich habitat for associated taxa, including endemic saproxylic species.

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