The purpose of this dissertation is to model biosocial determinants of group-directed altruistic behavior – exploring the nomological net around it. To do this a study will be presented to determine existing associations among various biological and social predictors and test a life-history-derived causal cascade using a partially exploratory and partially confirmatory statistical technique called Sequential Canonical Analysis to ultimately predict living-donor, non-directed kidney donations (NDKD). Toward that end, some important methodological considerations first need to be discussed. The first consideration revolves around the level of analysis and how this frames the cascade model and its interpretation. Following a general discussion, an exercise in some of the general principles is provided – investigating the higher-order factor structure of the Big-5 personality constructs across two levels of analysis. The second consideration is the use of unit-weighted factor scores and their appropriateness. Following the theoretical discussion, a demonstration is provided – deriving an estimate of genetic relatedness from a set of heterogeneous data sets. Once the methodological considerations have been discussed, the primary cascade model is presented in two parts: 1) the measurement model – operationalizing the measures incorporated into 2) the structural model – testing the proposed causal cascade using Sequential Canonical Analysis. A discussion follows in which the results are summarized, limitations are articulated, and further research directions are explored.

Online communities provide a fertile ground for analyzing people\'s behavior and improving our understanding of social processes. For instance, when modeling social interaction online, it is important to understand when people are reacting to each other. Also, since both people and communities change over time, we argue that analyses of online communities that take time into account will lead to deeper and more accurate results. In many cases, however, users behavior can be easily missed: users react to content in many more ways than observed by explicit indicators (such as likes on Facebook or replies on Twitter) and poorly aggregated temporal data might hide, misrepresent and even lead to wrong conclusions about how users are evolving. In order to address the problem of detecting non-explicit responses, we present a new approach that uses tf-idf similarity between a user\'s own tweets and recent tweets by people they follow. Based on a month\'s worth of posting data from 449 ego networks in Twitter, this method demonstrates that it is likely that at least 11% of reactions are not captured by the explicit reply and retweet mechanisms. Further, these uncaptured reactions are not evenly distributed between users: some users, who create replies and retweets without using the official interface mechanisms, are much more responsive to followees than they appear. This suggests that detecting non-explicit responses is an important consideration in mitigating biases and building more accurate models when using these markers to study social interaction and information diffusion. We also address the problem of users evolution in Reddit based on comment and submission data from 2007 to 2014. Even using one of the simplest temporal differences between usersyearly cohortswe find wide differences in people\'s behavior, including comment activity, effort, and survival. Furthermore, not accounting for time can lead us to misinterpret important phenomena. For instance, we observe that average comment length decreases over any fixed period of time, but comment length in each cohort of users steadily increases during the same period after an abrupt initial drop, an example of Simpson\'s Paradox. Dividing cohorts into sub-cohorts based on the survival time in the community provides further insights; in particular, longer-lived users start at a higher activity level and make more and shorter comments than those who leave earlier. These findings both give more insight into user evolution in Reddit in particular, and raise a number of interesting questions around studying online behavior going forward. / Comunidades online proporcionam um ambiente fértil para análise do comportamento de indivíduos e processos sociais. Por exemplo, ao modelarmos interações sociais online, é importante compreendemos quando indivíduos estão reagindo a outros indivíduos. Além disso, pessoas e comunidades mudam com o passar do tempo, e levar em consideração sua evolução temporal nos leva a resultados mais precisos. Entretanto, em muitos casos, o comportamento dos usuários pode ser perdido: suas reações ao conteúdo ao qual são expostos não são capturadas por indicadores explícitos (likes no Facebook, replies no Twitter). Agregações temporais de dados pouco criteriosas podem ocultar, enviesar ou até levar a conclusões equivocadas sobre como usuários evoluem. Apresentamos uma nova abordagem para o problema de detectar respostas não-explicitas que utiliza similaridade tf-idf entre tweets de um usuário e tweets recentes que este usuário recebeu de quem segue. Com base em dados de postagens de um mês para 449 redes egocêntricas do Twitter, este método evidencia que temos um volume de ao menos 11% de reações não capturadas pelos mecanismos explicitos de reply e retweet. Além disso, essas reações não capturadas não estão uniformemente distribuídas entre os usuários: alguns usuários que criam replies e retweets sem utilizar os mecanismos formais da interface são muito mais responsivos a quem eles seguem do que aparentam. Isso sugere que detectar respostas não-explicitas é importante para mitigar viéses e construir modelos mais precisos a fim de estudar interações sociais e difusão de informação. Abordamos o problema de evolução de usuários no Reddit com base em dados entre o período de 2007 a 2014. Utilizando métodos simples de diferenciação temporal dos usuários -- cohorts anuais -- encontramos amplas diferenças entre o comportamento, que incluem criação de comentários, métricas de esforço e sobrevivência. Desconsiderar a evolução temporal pode levar a equívocos a respeito de fenômenos importantes. Por exemplo, o tamanho médio dos comentários na rede decresce ao longo de qualquer intervalo de tempo, mas este tamanho é crescente em cada uma das cohorts de usuários no mesmo período, salvo de uma queda inicial. Esta é uma observação do Paradoxo de Simpson. Dividir as cohorts de usuários em sub-cohorts baseadas em anos de sobrevivência na rede nos fornece uma perspectiva melhor; usuários que sobrevivem por mais tempo apresentam um maior nível de atividade inicial, com comentários mais curtos do que aqueles que sobrevivem menos. Com isto, compreendemos melhor como usuários evoluem no Reddit e levantamos uma série de questões a respeito de futuros desdobramentos do estudo de comportamento online.

Online communities provide a fertile ground for analyzing people\'s behavior and improving our understanding of social processes. For instance, when modeling social interaction online, it is important to understand when people are reacting to each other. Also, since both people and communities change over time, we argue that analyses of online communities that take time into account will lead to deeper and more accurate results. In many cases, however, users behavior can be easily missed: users react to content in many more ways than observed by explicit indicators (such as likes on Facebook or replies on Twitter) and poorly aggregated temporal data might hide, misrepresent and even lead to wrong conclusions about how users are evolving. In order to address the problem of detecting non-explicit responses, we present a new approach that uses tf-idf similarity between a user\'s own tweets and recent tweets by people they follow. Based on a month\'s worth of posting data from 449 ego networks in Twitter, this method demonstrates that it is likely that at least 11% of reactions are not captured by the explicit reply and retweet mechanisms. Further, these uncaptured reactions are not evenly distributed between users: some users, who create replies and retweets without using the official interface mechanisms, are much more responsive to followees than they appear. This suggests that detecting non-explicit responses is an important consideration in mitigating biases and building more accurate models when using these markers to study social interaction and information diffusion. We also address the problem of users evolution in Reddit based on comment and submission data from 2007 to 2014. Even using one of the simplest temporal differences between usersyearly cohortswe find wide differences in people\'s behavior, including comment activity, effort, and survival. Furthermore, not accounting for time can lead us to misinterpret important phenomena. For instance, we observe that average comment length decreases over any fixed period of time, but comment length in each cohort of users steadily increases during the same period after an abrupt initial drop, an example of Simpson\'s Paradox. Dividing cohorts into sub-cohorts based on the survival time in the community provides further insights; in particular, longer-lived users start at a higher activity level and make more and shorter comments than those who leave earlier. These findings both give more insight into user evolution in Reddit in particular, and raise a number of interesting questions around studying online behavior going forward. / Comunidades online proporcionam um ambiente fértil para análise do comportamento de indivíduos e processos sociais. Por exemplo, ao modelarmos interações sociais online, é importante compreendemos quando indivíduos estão reagindo a outros indivíduos. Além disso, pessoas e comunidades mudam com o passar do tempo, e levar em consideração sua evolução temporal nos leva a resultados mais precisos. Entretanto, em muitos casos, o comportamento dos usuários pode ser perdido: suas reações ao conteúdo ao qual são expostos não são capturadas por indicadores explícitos (likes no Facebook, replies no Twitter). Agregações temporais de dados pouco criteriosas podem ocultar, enviesar ou até levar a conclusões equivocadas sobre como usuários evoluem. Apresentamos uma nova abordagem para o problema de detectar respostas não-explicitas que utiliza similaridade tf-idf entre tweets de um usuário e tweets recentes que este usuário recebeu de quem segue. Com base em dados de postagens de um mês para 449 redes egocêntricas do Twitter, este método evidencia que temos um volume de ao menos 11% de reações não capturadas pelos mecanismos explicitos de reply e retweet. Além disso, essas reações não capturadas não estão uniformemente distribuídas entre os usuários: alguns usuários que criam replies e retweets sem utilizar os mecanismos formais da interface são muito mais responsivos a quem eles seguem do que aparentam. Isso sugere que detectar respostas não-explicitas é importante para mitigar viéses e construir modelos mais precisos a fim de estudar interações sociais e difusão de informação. Abordamos o problema de evolução de usuários no Reddit com base em dados entre o período de 2007 a 2014. Utilizando métodos simples de diferenciação temporal dos usuários -- cohorts anuais -- encontramos amplas diferenças entre o comportamento, que incluem criação de comentários, métricas de esforço e sobrevivência. Desconsiderar a evolução temporal pode levar a equívocos a respeito de fenômenos importantes. Por exemplo, o tamanho médio dos comentários na rede decresce ao longo de qualquer intervalo de tempo, mas este tamanho é crescente em cada uma das cohorts de usuários no mesmo período, salvo de uma queda inicial. Esta é uma observação do Paradoxo de Simpson. Dividir as cohorts de usuários em sub-cohorts baseadas em anos de sobrevivência na rede nos fornece uma perspectiva melhor; usuários que sobrevivem por mais tempo apresentam um maior nível de atividade inicial, com comentários mais curtos do que aqueles que sobrevivem menos. Com isto, compreendemos melhor como usuários evoluem no Reddit e levantamos uma série de questões a respeito de futuros desdobramentos do estudo de comportamento online.

This diploma thesis is focused on the analysis of small samples where it is not possible to obtain more data. It can be especially due to the capital intensity or time demandingness. Where the production have not a wherewithall for the realization more data or absence of the financial resources. Of course, analysis of small samples is very uncertain, because inferences are always encumbered with the level of uncertainty.

In this thesis, we present some new results in distribution theory for both discrete and continuous random variables, together with their motivating applications. We start with some results about the Multivariate Gaussian Distribution and its characterization as a maximizer of the Strichartz Estimates. Then, we present some characterizations of discrete and continuous distributions through ideas coming from optimal transportation. After this, we pass to the Simpson's Paradox and see that it is ubiquitous and it appears in Quantum Mechanics as well. We conclude with a group of results about discrete and continuous distributions invariant under symmetries, in particular invariant under the groups $A_1$, an elliptical version of $O(n)$ and $\mathbb{T}^n$. As mentioned, all the results proved in this thesis are motivated by their applications in different research areas. The applications will be thoroughly discussed. We have tried to keep each chapter self-contained and recalled results from other chapters when needed. The following is a more precise summary of the results discussed in each chapter. In chapter \ref{chapter 2}, we discuss a variational characterization of the Multivariate Normal distribution (MVN) as a maximizer of the Strichartz Estimates. Strichartz Estimates appear as a fundamental tool in the proof of wellposedness results for dispersive PDEs. With respect to the characterization of the MVN distribution as a maximizer of the entropy functional, the characterization as a maximizer of the Strichartz Estimate does not require the constraint of fixed variance. In this chapter, we compute the precise optimal constant for the whole range of Strichartz admissible exponents, discuss the connection of this problem to Restriction Theorems in Fourier analysis and give some statistical properties of the family of Gaussian Distributions which maximize the Strichartz estimates, such as Fisher Information, Index of Dispersion and Stochastic Ordering. We conclude this chapter presenting an optimization algorithm to compute numerically the maximizers. Chapter \ref{chapter 3} is devoted to the characterization of distributions by means of techniques from Optimal Transportation and the Monge-Amp\`{e}re equation. We give emphasis to methods to do statistical inference for distributions that do not possess good regularity, decay or integrability properties. For example, distributions which do not admit a finite expected value, such as the Cauchy distribution. The main tool used here is a modified version of the characteristic function (a particular case of the Fourier Transform). An important motivation to develop these tools come from Big Data analysis and in particular the Consensus Monte Carlo Algorithm. In chapter \ref{chapter 4}, we study the \emph{Simpson's Paradox}. The \emph{Simpson's Paradox} is the phenomenon that appears in some datasets, where subgroups with a common trend (say, all negative trend) show the reverse trend when they are aggregated (say, positive trend). Even if this issue has an elementary mathematical explanation, the statistical implications are deep. Basic examples appear in arithmetic, geometry, linear algebra, statistics, game theory, sociology (e.g. gender bias in the graduate school admission process) and so on and so forth. In our new results, we prove the occurrence of the \emph{Simpson's Paradox} in Quantum Mechanics. In particular, we prove that the \emph{Simpson's Paradox} occurs for solutions of the \emph{Quantum Harmonic Oscillator} both in the stationary case and in the non-stationary case. We prove that the phenomenon is not isolated and that it appears (asymptotically) in the context of the \emph{Nonlinear Schr\"{o}dinger Equation} as well. The likelihood of the \emph{Simpson's Paradox} in Quantum Mechanics and the physical implications are also discussed. Chapter \ref{chapter 5} contains some new results about distributions with symmetries. We first discuss a result on symmetric order statistics. We prove that the symmetry of any of the order statistics is equivalent to the symmetry of the underlying distribution. Then, we characterize elliptical distributions through group invariance and give some properties. Finally, we study geometric probability distributions on the torus with applications to molecular biology. In particular, we introduce a new family of distributions generated through stereographic projection, give several properties of them and compare them with the Von-Mises distribution and its multivariate extensions. / Thesis / Doctor of Philosophy (PhD)