The purpose of this study, based on the third year of a three-year research study, was to examine Grade 6 students’ previously developed abilities to integrate their understanding of geometric growing patterns with graphic representations as a means of further developing their conception of linear relationships. In addition, I included an investigation to determine whether the students’ understanding of linear relationships of positive values could be extended to support their understanding of negative numbers. The theoretical approach to the microgenetic analyses I conducted is based on Noss & Hoyles’ notion of situated abstractions, which can be defined as the development of successive approximation of formal mathematical knowledge in individuals. I also looked to Roschelle’s work on collaborative conceptual change, which allowed me to examine and document successive mathematical abstractions at a whole-class level. I documented in detail the development of ten grade 6 students’ understanding of linear relationships as they engaged in seven experimental lessons. The results show that these learners were all able to grasp the connections among multiple representations of linear relationships. The students were also able to use their grasp of pattern sequences, graphs and tables of value to work out how to operate with negative numbers, both as the multiplier and as the additive constant. As a contribution to research methodology, the use of two analytical frameworks provides a model of how frameworks can be used to make sense of data and in particular to pinpoint the interplay between individual and collective actions and understanding.

The purpose of this study, based on the third year of a three-year research study, was to examine Grade 6 students’ previously developed abilities to integrate their understanding of geometric growing patterns with graphic representations as a means of further developing their conception of linear relationships. In addition, I included an investigation to determine whether the students’ understanding of linear relationships of positive values could be extended to support their understanding of negative numbers. The theoretical approach to the microgenetic analyses I conducted is based on Noss & Hoyles’ notion of situated abstractions, which can be defined as the development of successive approximation of formal mathematical knowledge in individuals. I also looked to Roschelle’s work on collaborative conceptual change, which allowed me to examine and document successive mathematical abstractions at a whole-class level. I documented in detail the development of ten grade 6 students’ understanding of linear relationships as they engaged in seven experimental lessons. The results show that these learners were all able to grasp the connections among multiple representations of linear relationships. The students were also able to use their grasp of pattern sequences, graphs and tables of value to work out how to operate with negative numbers, both as the multiplier and as the additive constant. As a contribution to research methodology, the use of two analytical frameworks provides a model of how frameworks can be used to make sense of data and in particular to pinpoint the interplay between individual and collective actions and understanding.

This thesis was motivated by data from the Emerging Risk Factors Collaboration (ERFC), a large individual participant data (IPD) meta-analysis of risk factors for coronary heart disease(CHD). Cardiovascular disease is the largest cause of death in almost all countries in the world, therefore it is important to be able to characterise the shape of risk factor–CHD relationships. Many of the risk factors for CHD considered by the ERFC are subject to substantial measurement error, and their relationship with CHD non-linear. We firstly consider issues associated with modelling the risk factor–disease relationship in a single study, before using meta-analysis to combine relationships across studies. It is well known that classical measurement error generally attenuates linear exposure–disease relationships, however its precise effect on non-linear relationships is less well understood. We investigate the effect of classical measurement error on the shape of exposure–disease relationships that are commonly encountered in epidemiological studies, and then consider methods for correcting for classical measurement error. We propose the application of a widely used correction method, regression calibration, to fractional polynomial models. We also consider the effects of non-classical error on the observed exposure–disease relationship, and the impact on our correction methods when we erroneously assume classical measurement error. Analyses performed using categorised continuous exposures are common in epidemiology. We show that MacMahon’s method for correcting for measurement error in analyses that use categorised continuous exposures, although simple, does not provide the correct shape for nonlinear exposure–disease relationships. We perform a simulation study to compare alternative methods for categorised continuous exposures. Meta-analysis is the statistical synthesis of results from a number of studies addressing similar research hypotheses. The use of IPD is the gold standard approach because it allows for consistent analysis of the exposure–disease relationship across studies. Methods have recently been proposed for combining non-linear relationships across studies. We discuss these methods, extend them to P-spline models, and consider alternative methods of combining relationships across studies. We apply the methods developed to the relationships of fasting blood glucose and lipoprotein(a) with CHD, using data from the ERFC.

Conselho Nacional de Desenvolvimento Científico e Tecnológico / The objective of this study was to determine the sample size necessary to estimate the average, the coefficient of variation, the Pearson linear correlation coefficient and the direct effects of explanatory variables on grain yield in maize. In 361, 373 and 416 plants, respectively, of the simple, triple and double hybrids of the 2008/09 crop and, in 1,777, 1,693 and 1,720 plants, respectively, of the simple, triple and double hybrids of the 2009/10 crop, were measured eleven explanatory variables: plant height at harvest (AP), ear height (AIE), ear weight (PE), number of grain rows per ear (NF), ear length (CE), ear diameter (DE), cob weight (PS), cob diameter (DS), weight of hundred grains (MCG), number of grains per ear (NGR), grain length (CGR) and the main variable, grain yield (PROD). For each hybrid and crop, descriptives statistics for each variable were calculated and the correlation coefficients and direct effects of explanatory variables on PROD were estimated, in nine scenarios of traditional and ridge path analysis. Then, the sample size necessary to estimate the average, the coefficients of variation and of correlation and the direct effects of each explanatory variable on PROD were determined, for each type of hybrid, crop, scenario and type of path analysis, by resampling with replacement. The sample size necessary to estimate the mean and the coefficients of variation and of correlation ranges among hybrids, crops and variables or pairs of variables. The sample size necessary to estimate the direct effects ranges among hybrids, crops, scenarios, types of path analysis and explanatory variables. Independently of hybrid, crop and variable, 375 plants are enough to estimate the mean and the coefficient of variation with amplitude of the confidence interval of 95% (AIC95%) maximum of 10% and for the estimation of the correlation coefficients with a AIC95% maximum of 0.25. For the estimation of direct effects, with AIC95% maximum of 0.25, are required from 10 to 530 plants, depending of the type of hybrid, crop, scenario, type of path analysis and explanatory variable. The measurement of 120 plants is sufficient to estimate the average with AIC95% maximum of 20%, for the estimation of the coefficient of variation with AIC95% maximum of 15% and for the estimation of correlation coefficients with AIC95% maximum of 0.45, independently of the hybrid, crop and variable. The measurement of 120 plants is also sufficient for the estimation of the direct effects of AIE, CE and DE on PROD in the ninth scenario, with AIC95% maximum of 0.25, and in the ninth scenario, CE and DE have greater direct effects on PROD, independent of the type of hybrid, the crop and the type of path analysis. / O objetivo deste trabalho foi determinar o tamanho de amostra necessário para a estimação da média, do coeficiente de variação, do coeficiente de correlação linear de Pearson e dos efeitos diretos de variáveis explicativas sobre a produtividade de grãos em milho. Em 361, 373 e 416 plantas, respectivamente, dos híbridos simples, triplo e duplo da safra 2008/09 e, em 1.777, 1.693 e 1.720 plantas, respectivamente, dos híbridos simples, triplo e duplo da safra 2009/10, foram mensuradas onze variáveis explicativas: altura de planta na colheita (AP), altura de inserção de espiga (AIE), peso de espiga (PE), número de fileiras de grãos por espiga (NF), comprimento de espiga (CE), diâmetro de espiga (DE), peso de sabugo (PS), diâmetro de sabugo (DS), massa de cem grãos (MCG), número de grãos por espiga (NGR), comprimento de grãos (CGR) e, a variável principal produtividade de grãos (PROD). A seguir, em cada híbrido e safra, foram calculadas estatísticas descritivas para cada variável e estimados os coeficientes de correlação e os efeitos diretos de variáveis explicativas sobre a PROD, para nove cenários de análises de trilha tradicional e em crista. Após, determinou-se o tamanho de amostra necessário para a estimação da média, dos coeficientes de variação e de correlação e dos efeitos diretos de cada variável explicativa sobre a PROD, em cada tipo de híbrido, safra, cenário e tipo de análise de trilha, por meio de reamostragem com reposição. O tamanho de amostra necessário para a estimação da média e dos coeficientes de variação e de correlação varia entre híbridos, safras e variáveis ou pares de variáveis. O tamanho de amostra necessário para a estimação dos efeitos diretos varia entre híbridos, safras, cenários, tipos de análises de trilha e variáveis explicativas. Independentemente do híbrido, da safra e da variável, 375 plantas são suficientes para a estimação da média e do coeficiente de variação com amplitude do intervalo de confiança de 95% (AIC95%) máxima de 10% e, para a estimação de coeficientes de correlação com AIC95% máxima de 0,25. Para a estimação de efeitos diretos com AIC95% máxima de 0,25, são necessárias de 10 a 530 plantas, dependendo do tipo de híbrido, da safra, do cenário, do tipo de análise de trilha e da variável explicativa. A mensuração de 120 plantas é suficiente para a estimação da média com AIC95% máxima de 20%, para a estimação do coeficiente de variação com AIC95% máxima de 15% e, para a estimação de coeficientes de correlação com AIC95% máxima de 0,45, independentemente do híbrido, da safra e da variável. A mensuração de 120 plantas também é suficiente para a estimação dos efeitos diretos de AIE, CE e DE sobre PROD no nono cenário, com AIC95% máxima de 0,25, sendo que nesse cenário, CE e DE possuem maiores efeitos diretos sobre PROD, independentemente do tipo de híbrido, da safra e do tipo de análise de trilha.