Counterfactual conditional

Summary

Counterfactual conditionals (also subjunctive or X-marked) are conditional sentences which discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be here." Counterfactuals are contrasted with indicatives, which are generally restricted to discussing open possibilities. Counterfactuals are characterized grammatically by their use of fake tense morphology, which some languages use in combination with other kinds of morphology including aspect and mood. Counterfactuals are one of the most studied phenomena in philosophical logic, formal semantics, and philosophy of language. They were first discussed as a problem for the material conditional analysis of conditionals, which treats them all as trivially true. Starting in the 1960s, philosophers and linguists developed the now-classic possible world approach, in which a counterfactual's truth hinges on its consequent holding at certain possible worlds where its antecedent h

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https://en.wikipedia.org/wiki/Counterfactual_conditional

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This thesis is a contribution to multivariate extreme value statistics. The tail of a multivariate distribution function is characterized by its spectral distribution, for which we propose a new semi-parametric model based on mixtures of Dirichlet distributions. To estimate the components of this model, reversible jump Monte Carlo Markov chain and EM algorithms are developed. Their performances are illustrated on real and simulated data, obtained using new representations of the extremal logistic and Dirichlet models. In parallel with the estimation of the spectral distribution, extreme value statistic machinery requires the selection of a threshold in order to classify data as extreme or not. This selection is achieved by a new method based on heuristic arguments. It allows a selection independent of the dimension of the data. Its performance is illustrated on real and simulated data. Primal scientific interests behind a multivariate extreme value analysis reside in the estimation of quantiles of rare events and in the exploration of the dependence structure, for which the estimation of the spectral measure is a means rather than an end. These two issues are addressed. For the first, a Monte Carlo method is developed based on simulation of extremes. It is compared with classical and new methods of the literature. For the second one, an original conditional dependence analysis is proposed, which enlightens various aspects of the dependence structure of the data. Examples using real data sets are given. In the last part, the semi-parametric model and the presented methods are extended to spatial extremes. It is made possible by considering the spectral distribution as the distribution of a random probability, an original viewpoint adopted throughout this thesis. Classical multivariate extremes are extended to extremes of random measures. The application is illustrated on rainfall data in China.

EPFL2004

Effect heterogeneity and variable selection for standardizing causal effects to a target population

Mats Julius Stensrud

The participants in randomized trials and other studies used for causal inference are often not representative of the populations seen by clinical decision-makers. To account for differences between populations, researchers may consider standardizing results to a target population. We discuss several different types of homogeneity conditions that are relevant for standardization: Homogeneity of effect measures, homogeneity of counterfactual outcome state transition parameters, and homogeneity of counterfactual distributions. Each of these conditions can be used to show that a particular standardization procedure will result in an unbiased estimate of the effect in the target population, given assumptions about the relevant scientific context. We compare and contrast the homogeneity conditions, in particular their implications for selection of covariates for standardization and their implications for how to compute the standardized causal effect in the target population. While some of the recently developed counterfactual approaches to generalizability rely upon homogeneity conditions that avoid many of the problems associated with traditional approaches, they often require adjustment for a large (and possibly unfeasible) set of covariates.

2019

On the collapsibility of measures of effect in the counterfactual causal framework

Mats Julius Stensrud

The relationship between collapsibility and confounding has been subject to an extensive and ongoing discussion in the methodological literature. We discuss two subtly different definitions of collapsibility, and show that by considering causal effect measures based on counterfactual variables (rather than measures of association based on observed variables) it is possible to separate out the component of non-collapsibility which is due to the mathematical properties of the effect measure, from the components that are due to structural bias such as confounding. We provide new weights such that the causal risk ratio is collapsible over arbitrary baseline covariates. In the absence of confounding, these weights may be used for standardization of the risk ratio.

2019