Path Analysis for Binary Random Variables

The decomposition of the overall effect of a treatment into direct and indirect effects is here investigated with reference to a recursive system of binary random variables. We show how, for the single mediator context, the marginal effect measured on the log odds scale can be written as the sum of the indirect and direct effects plus a residual term that vanishes under some specific conditions. We then extend our definitions to situations involving multiple mediators and address research questions concerning the decomposition of the total effect when some mediators on the pathway from the treatment to the outcome are marginalized over. Connections to the counterfactual definitions of the effects are also made. Data coming from an encouragement design on students’ attitude to visit museums in Florence, Italy, are reanalyzed. The estimates of the defined quantities are reported together with their standard errors to compute p values and form confidence intervals.

A hidden Markov model for lymphatic tumor progression in the head and neck

Currently, elective clinical target volume (CTV-N) definition for head and neck squamous cell carcinoma (HNSCC) is mostly based on the prevalence of nodal involvement for a given tumor location. In this work, we propose a probabilistic model for lymphatic metastatic spread that can quantify the risk of microscopic involvement in lymph node levels (LNL) given the location of macroscopic metastases and T-category. This may allow for further personalized CTV-N definition based on an individual patient’s state of disease. We model the patient's state of metastatic lymphatic progression as a collection of hidden binary random variables that indicate the involvement of LNLs. In addition, each LNL is associated with observed binary random variables that indicate whether macroscopic metastases are detected. A hidden Markov model (HMM) is used to compute the probabilities of transitions between states over time. The underlying graph of the HMM represents the anatomy of the lymphatic drainage system. Learning of the transition probabilities is done via Markov chain Monte Carlo sampling and is based on a dataset of HNSCC patients in whom involvement of individual LNLs was reported. The model is demonstrated for ipsilateral metastatic spread in oropharyngeal HNSCC patients. We demonstrate the model's capability to quantify the risk of microscopic involvement in levels III and IV, depending on whether macroscopic metastases are observed in the upstream levels II and III, and depending on T-category. In conclusion, the statistical model of lymphatic progression may inform future, more personalized, guidelines on which LNL to include in the elective CTV. However, larger multi-institutional datasets for model parameter learning are required for that.

Mediation analysis for binary random variables : parametric decomposition of the total effect

This thesis is centered on the evaluation of direct and indirect effects via mediation analysis. A researcher is usually interested in assessing to what extent an exposure variable affects an outcome. However, identifying the overall effect does not answer questions concerning how and why such an effect arises. Single mediation analysis decomposes the overall effect of the exposure on the outcome into an indirect and a direct effect. The former refers to the to the effect of the exposure on the outcome due to a third variable, the mediator, which is supposed to fall in the pathway. The latter effect is the effect of the exposure on the outcome after keeping the mediator to whatever value might be of interest. Specifically, we derived novel exact parametric decompositions of the total effect into direct and indirect effect for binary random variables, both in the counterfactual and path-analysis frameworks. In the single mediation context, we derive parametric expressions of the counterfactual entities and their relationships with the associational definitions coming from the path analysis context. We apply these methodological results on a dataset coming from a randomly allocated microcredit program in Bosnia-Herzegovina to evaluate the effect on client’s bankability. We re-analyse the data, in order to build a mediation scheme that allows a better understanding of the main effect of the study, by assuming business ownership as a possible mediator. We also implement a simulation study to compare the proposed estimator to several competing ones. When multiple mediators are involved, we found alternative definitions for the decomposition of the total effect. These new definitions are more appropriate for variables modelled as a recursive system of univariate logistic regressions. Thus, by making use of graphical models, the overall effect was defined as the sum of the direct, indirect effects and a residual term that is null under certain hypotheses. In general, these expressions are written such that one can maintain the link between effects and their corresponding coefficients in logistic regression models assumed in the system.

On marginal and conditional parameters in logistic regression models

Summary We derive the exact formula linking the parameters of marginal and conditional logistic regression models with binary mediators when no conditional independence assumptions can be made. The formula has the appealing property of being the sum of terms that vanish whenever parameters of the conditional models vanish, thereby recovering well-known results as particular cases. It also permits the disentangling of direct and indirect effects as well as quantifying the distortion induced by the omission of relevant covariates, opening the way to sensitivity analysis. As the parameters of the conditional models are multiplied by terms that are always bounded, the derivations may also be used to construct reasonable bounds on the parameters of interest when relevant intermediate variables are unobserved. We assume that, conditionally on a set of covariates, the data-generating process can be represented by a directed acyclic graph. We also show how the results presented here lead to the extension of path analysis to a system of binary random variables.

Mixtures and products in two graphical models

We compare two statistical models of three binary random variables. One is a mixture model and the other is a product of mixtures model called a restricted Boltzmann machine. Although the two models we study look different from their parametrizations, we show that they represent the same set of distributions on the interior of the probability simplex, and are equal up to closure. We give a semi-algebraic description of the model in terms of six binomial inequalities and obtain closed form expressions for the maximum likelihood estimates. We briefly discuss extensions to larger models.