On Aspects of Mathematical Reasoning : Affect and Gender

This thesis explores two aspects of mathematical reasoning: affect and gender. I started by looking at the reasoning of upper secondary students when solving tasks. This work revealed that when not guided by an interviewer, algorithmic reasoning, based on memorising algorithms which may or may not be appropriate for the task, was predominant in the students reasoning. Given this lack of mathematical grounding in students reasoning I looked in a second study at what grounds they had for different strategy choices and conclusions. This qualitative study suggested that beliefs about safety, expectation and motivation were important in the central decisions made during task solving.  But are reasoning and beliefs gendered? The third study explored upper secondary school teachers conceptions about gender and students mathematical reasoning. In this study I found that upper secondary school teachers attributed gender symbols including insecurity, use of standard methods and imitative reasoning to girls and symbols such as multiple strategies especially on the calculator, guessing and chance-taking were assigned to boys. In the fourth and final study I found that students, both male and female, shared their teachers view of rather traditional feminities and masculinities. Remarkably however, this result did not repeat itself when students were asked to reflect on their own behaviour: there were some discrepancies between the traits the students ascribed as gender different and the traits they ascribed to themselves. Taken together the thesis suggests that, contrary to conceptions, girls and boys share many of the same core beliefs about mathematics, but much work is still needed if we should create learning environments that provide better opportunities for students to develop beliefs that guide them towards well-grounded mathematical reasoning. 

Teachers' conceptions about students' mathematical reasoning : Gendered or not?

This study looks at how upper secondary school teachers gender stereotype aspects of students' mathematical reasoning. Girls were attributed gender symbols including insecurity, use of standard methods and imitative reasoning. Boys were assigned the symbols such as multiple strategies especially on the calculator, guessing and chance-taking. 

Statistical bound of genetic solutions to quadratic assignment problems

Quadratic assignment problems (QAPs) are commonly solved by heuristic methods, where the optimum is sought iteratively. Heuristics are known to provide good solutions but the quality of the solutions, i.e., the confidence interval of the solution is unknown. This paper uses statistical optimum estimation techniques (SOETs) to assess the quality of Genetic algorithm solutions for QAPs. We examine the functioning of different SOETs regarding biasness, coverage rate and length of interval, and then we compare the SOET lower bound with deterministic ones. The commonly used deterministic bounds are confined to only a few algorithms. We show that, the Jackknife estimators have better performance than Weibull estimators, and when the number of heuristic solutions is as large as 100, higher order JK-estimators perform better than lower order ones. Compared with the deterministic bounds, the SOET lower bound performs significantly better than most deterministic lower bounds and is comparable with the best deterministic ones. 

On statistical bounds of heuristic solutions to location problems

Solutions to combinatorial optimization problems, such as problems of locating facilities, frequently rely on heuristics to minimize the objective function. The optimum is sought iteratively and a criterion is needed to decide when the procedure (almost) attains it. Pre-setting the number of iterations dominates in OR applications, which implies that the quality of the solution cannot be ascertained. A small, almost dormant, branch of the literature suggests using statistical principles to estimate the minimum and its bounds as a tool to decide upon stopping and evaluating the quality of the solution. In this paper we examine the functioning of statistical bounds obtained from four different estimators by using simulated annealing on p-median test problems taken from Beasley’s OR-library. We find the Weibull estimator and the 2nd order Jackknife estimator preferable and the requirement of sample size to be about 10 being much less than the current recommendation. However, reliable statistical bounds are found to depend critically on a sample of heuristic solutions of high quality and we give a simple statistic useful for checking the quality. We end the paper with an illustration on using statistical bounds in a problem of locating some 70 distribution centers of the Swedish Post in one Swedish region. 

Novel Statistical Methods in Quantitative Genetics : Modeling Genetic Variance for Quantitative Trait Loci Mapping and Genomic Evaluation

This thesis develops and evaluates statistical methods for different types of genetic analyses, including quantitative trait loci (QTL) analysis, genome-wide association study (GWAS), and genomic evaluation. The main contribution of the thesis is to provide novel insights in modeling genetic variance, especially via random effects models. In variance component QTL analysis, a full likelihood model accounting for uncertainty in the identity-by-descent (IBD) matrix was developed. It was found to be able to correctly adjust the bias in genetic variance component estimation and gain power in QTL mapping in terms of precision.  Double hierarchical generalized linear models, and a non-iterative simplified version, were implemented and applied to fit data of an entire genome. These whole genome models were shown to have good performance in both QTL mapping and genomic prediction. A re-analysis of a publicly available GWAS data set identified significant loci in Arabidopsis that control phenotypic variance instead of mean, which validated the idea of variance-controlling genes.  The works in the thesis are accompanied by R packages available online, including a general statistical tool for fitting random effects models (hglm), an efficient generalized ridge regression for high-dimensional data (bigRR), a double-layer mixed model for genomic data analysis (iQTL), a stochastic IBD matrix calculator (MCIBD), a computational interface for QTL mapping (qtl.outbred), and a GWAS analysis tool for mapping variance-controlling loci (vGWAS).

Parameter estimation of Poisson generalized linear mixed models based on three different statistical principles : a simulation study

Generalized linear mixed models are flexible tools for modeling non-normal data and are useful for accommodating overdispersion in Poisson regression models with random effects. Their main difficulty resides in the parameter estimation because there is no analytic solution for the maximization of the marginal likelihood. Many methods have been proposed for this purpose and many of them are implemented in software packages. The purpose of this study is to compare the performance of three different statistical principles - marginal likelihood, extended likelihood, Bayesian analysis-via simulation studies. Real data on contact wrestling are used for illustration.

PokerMapper: : mapping executive functions, poker playing ability and responsible gambling in online environments

Research objectives Poker and responsible gambling both entail the use of the executive functions (EF), which are higher-level cognitive abilities. The main objective of this work was to assess if online poker players of different ability show different performances in their EF and if so, which functions are the most discriminating ones. The secondary objective was to assess if the EF performance can predict the quality of gambling, according to the Gambling Related Cognition Scale (GRCS), the South Oaks Gambling Screen (SOGS) and the Problem Gambling Severity Index (PGSI). Sample and methods The study design consisted of two stages: 46 Italian active players (41m, 5f; age 32±7,1ys; education 14,8±3ys) fulfilled the PGSI in a secure IT web system and uploaded their own hand history files, which were anonymized and then evaluated by two poker experts. 36 of these players (31m, 5f; age 33±7,3ys; education 15±3ys) accepted to take part in the second stage: the administration of an extensive neuropsychological test battery by a blinded trained professional. To answer the main research question we collected all final and intermediate scores of the EF tests on each player together with the scoring on the playing ability. To answer the secondary research question, we referred to GRCS, PGSI and SOGS scores.  We determined which variables that are good predictors of the playing ability score using statistical techniques able to deal with many regressors and few observations (LASSO, best subset algorithms and CART). In this context information criteria and cross-validation errors play a key role for the selection of the relevant regressors, while significance testing and goodness-of-fit measures can lead to wrong conclusions.   Preliminary findings We found significant predictors of the poker ability score in various tests. In particular, there are good predictors 1) in some Wisconsin Card Sorting Test items that measure flexibility in choosing strategy of problem-solving, strategic planning, modulating impulsive responding, goal setting and self-monitoring, 2) in those Cognitive Estimates Test variables related to deductive reasoning, problem solving, development of an appropriate strategy and self-monitoring, 3) in the Emotional Quotient Inventory Short (EQ-i:S) Stress Management score, composed by the Stress Tolerance and Impulse Control scores, and in the Interpersonal score (Empathy, Social Responsibility, Interpersonal Relationship). As for the quality of gambling, some EQ-i:S scales scores provide the best predictors: General Mood for the PGSI; Intrapersonal (Self-Regard; Emotional Self-Awareness, Assertiveness, Independence, Self-Actualization) and Adaptability  (Reality Testing, Flexibility, Problem Solving) for the SOGS, Adaptability for the GRCS. Implications for the field Through PokerMapper we gathered knowledge and evaluated the feasibility of the construction of short tasks/card games in online poker environments for profiling users’ executive functions. These card games will be part of an IT system able to dynamically profile EF and provide players with a feedback on their expected performance and ability to gamble responsibly in that particular moment. The implementation of such system in existing gambling platforms could lead to an effective proactive tool for supporting responsible gambling. 

Recent developments in statistical methods for detecting genetic loci affecting phenotypic variability

A number of recent works have introduced statistical methods for detecting genetic loci that affect phenotypic variability, which we refer to as variability-controlling quantitative trait loci (vQTL). These are genetic variants whose allelic state predicts how much phenotype values will vary about their expected means. Such loci are of great potential interest in both human and non-human genetic studies, one reason being that a detected vQTL could represent a previously undetected interaction with other genes or environmental factors. The simultaneous publication of these new methods in different journals has in many cases precluded opportunity for comparison. We survey some of these methods, the respective trade-offs they imply, and the connections between them. The methods fall into three main groups: classical non-parametric, fully parametric, and semi-parametric two-stage approximations. Choosing between alternatives involves balancing the need for robustness, flexibility, and speed. For each method, we identify important assumptions and limitations, including those of practical importance, such as their scope for including covariates and random effects. We show in simulations that both parametric methods and their semi-parametric approximations can give elevated false positive rates when they ignore mean-variance relationships intrinsic to the data generation process. We conclude that choice of method depends on the trait distribution, the need to include non-genetic covariates, and the population size and structure, coupled with a critical evaluation of how these fit with the assumptions of the statistical model.