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Linear mixed models are frequently used to analyze data with random effects and/or repeated measures. A common approach to such analyses requires choosing a covariance structure. Information criteria, such as AIC and BIC, are often used by statisticians to help with this task. However, these criteria do not always point to the true covariance structure and therefore the wrong covariance structure is sometimes chosen. Once this step is complete, Wald statistics are used to test fixed effects. Degrees of freedom for these statistics are not known. However, there are approximation methods, such as Kenward and Roger (KR) and Satterthwaite (SW) that have been shown to work well in some situations. Schaalje et al. (2002) concluded that the KR method would perform at least as well as or better than the SW method in many cases assuming that the covariance structure was known. On the other hand, Keselman et al. (1999) concluded that the performance of the SW method when the covariance structure was selected using AIC was poor for negative pairings of treatment sizes and covariance matrices and small sample sizes. Our study used simulations to investigate Type I error rates in test of fixed effects using Wald statistics with the KR adjustment method, incorporating the selection of the covariance structure using AIC and BIC. Performance of the AIC and BIC criteria in selecting the true covariance structure was also studied. The MIXED procedure (SAS v. 9) was used to analyze each simulated data set. Type I error rates from the best AIC and BIC models were always higher than target values. However, Type I error rates obtained by using the BIC criterion were better than those obtained by using the AIC criterion. Type I error rates for the correct models were often adequate depending on the sample size and complexity of covariance structure. Performance of AIC and BIC was poor. This could be a consequence of small sample sizes and the high number of covariance structures these criteria had to choose from.

Generalized linear mixed models (GLMMs) have become extremely popular in recent years. The main computational problem in parameter estimation for GLMMs is that, in contrast to linear mixed models, closed analytical expressions for the likelihood are not available. To overcome this problem, several approaches have been proposed in the literature. For this study we have used one quasi-likelihood approach, penalized quasi-likelihood (PQL), and two integral approaches: Laplace and adaptive Gauss-Hermite quadrature (AGHQ) approximation. Our primary objective was to measure the performances of each estimation method. AGHQ approximation is more accurate than Laplace approximation, but slower. So the question is when Laplace approximation is adequate, versus when AGHQ approximation provides a significantly more accurate result. We have run two simulations using PQL, Laplace and AGHQ approximations with different quadrature points for varying random effect standard deviation (Ɵ) and number of replications per cluster. The performances of these three methods were measured base on the root mean square error (RMSE) and bias. Based on the simulated data, we have found that for both smaller values of Ɵ and small number of replications and for larger values of and for larger values of Ɵ and lager number of replications, the RMSE of PQL method is much higher than Laplace and AGHQ approximations. However, for intermediate values of Ɵ (random effect standard deviation) ranging from 0.63 to 3.98, regardless of number of replications per cluster, both Laplace and AGHQ approximations gave similar estimates. But when both number of replications and Ɵ became small, increasing quadrature points increases RMSE values indicating that Laplace approximation perform better than the AGHQ method. When random effect standard deviation is large, e.g. Ɵ=10, and number of replications is small the Laplace RMSE value is larger than that of AGHQ approximation. Increasing quadrature points decreases the RMSE values. This indicates that AGHQ performs better in this situation. The difference in RMSE between PQL vs Laplace and AGHQ vs Laplace is approximately 12% and 10% respectively. In addition, we have tested the relative performance and the accuracy between two different packages of R (lme4, glmmML) and SAS (PROC GLIMMIX) based on real data. Our results suggested that all of them perform well in terms of accuracy, precision and convergence rates. In most cases, glmmML was found to be much faster than lme4 package and SAS. The only difference was found in the Contraception data where the required computational time for both R packages was exactly the same. The difference in required computational times for these two platforms decreases as the number of quadrature points increases. / Thesis / Master of Science (MSc)

The factors affecting tree establishment in boreal forests after fire will help determine the community composition of the regenerating forest. These may have large consequences on the community dynamics for years after the fire disturbance. Factors such as burn severity and soil moisture among others have been shown to play a key role in influencing several facets of establishment. However, tree establishment after megafire in boreal forest in Europe has not yet been fully understood. Here I capitalise on a megafire in Sweden in 2014 to investigate the relative impact of different abiotic factors and preconditions on tree establishment six years after the fire. This study used a systematic survey of tree saplings (height >30cm) at 625 locations inside the nature reserve set up within the burnt area. Tested factors were: The number of dead trees lying down, slope and slope aspect, elevation, soil wetness, pre-fire standing volume, distance to fire perimeter, forest stand age, stand productivity index, previous stand dominant tree species, humus thickness after fire and depth of burn. Generalized Linear Mixed Models (GLMMs) were used to estimate the effect of these factors on specific tree species abundance. Strong influences from previous wood volume, soil wetness, elevation, and dead wood lying down had an effective influence on sapling abundance but were typically species-specific. Only elevation and wood volume had a consistent effect on all species’ abundances. Habitat context was important on a landscape scale. These results support the pattern of increasing boreal deciduousness caused by high burn severity and shorter disturbance intervals, in turn, caused by hotter, drier weather, which will have implications on the composition of boreal forests of tomorrow.

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Self-thinning quantifies the reduction in tree numbers due to density-dependent mortality. Maximum size-density relationships (MSDRs) are a component of self-thinning that describe the maximum tree density per unit area obtainable for a given average tree size, often quadratic mean diameter (D). An MSDR species boundary line has been defined as a static upper limit of maximum tree density -- D relationships that applies to all stands of a certain species within a particular geographical area. MSDR dynamic thinning lines have been defined as the maximum tree density obtainable within an individual stand for a particular D which have been shown to vary relative to planting density. Results from this study show that differences in boundary levels of individual stands cause the MSDR species boundary line slope estimate to be sensitive to the range of planting densities within the model fitting dataset. Thus, a second MSDR species boundary line was defined whose slope is the average slope of all MSDR dynamic thinning lines. Mixed-models are presented as a statistical method to obtain an estimate of the population average MSDR dynamic thinning line slope. A common problem when modeling self-thinning is to determine what observations are within generally accepted stages of stand development. Segmented regression is presented as a statistical and less subjective method to determine what observations are within various stages of stand development. Estimates of D and trees per acre (N) where MSDR dynamic thinning lines begin and end on the logarithmic scale were used as response variables and predicted as a function of planting density. Predictions of MSDR dynamic thinning line beginning and ending D and N are used in an alternative MSDR dynamic thinning line slope estimation method. These models show that the maximum value of Reineke's Stand Density Index (SDI) varies relative to planting density. By relating planting density specific Zone of Imminent Competition Mortality boundaries to a MSDR species boundary line, self-thinning was found not to begin at a constant relative SDI. Thus, planting density specific Density Management Diagrams (DMD) showed that self-thinning began at 40 to 72% for planting densities of 605 and 2722 seedlings per acre, respectively. / Ph. D.

Blue Catfish were introduced in the tidal tributaries of the Chesapeake Bay in the 1970s and 1980s to establish new fisheries during a time period when many fisheries were in decline due to pollution, habitat alteration, disease, overfishing, and environmental catastrophes. Having expanded their range to most Bay tributaries, the species has drawn concern from many stakeholders and scientists for its effects on at-risk and economically important native and naturalized species. My study focused on understanding the dynamics of this species based on multiple long-term monitoring data and evaluating potential management strategies to meet stakeholder needs. I sought to understand how is growth variability was partitioned over time and space, how Blue Catfish populations changed from 1994 to 2016, and how predation on native species and fishery-based performance measures may respond to management intervention. As Blue Catfish length-at-age is exceptionally variable in Virginia tributaries of the Chesapeake Bay, I evaluated the variability in growth using candidate non-linear mixed effects models that described variability in growth over time and space. Linear trend tests supported declines in growth over time within river systems, but did not support the presence of synchronous growth responses among river systems. To better understand population dynamics of Blue Catfish in the Chesapeake Bay watershed, I developed a statistical catch-at-length model for the James River to estimate population size, instantaneous fishing mortality, and size structure over time. The statistical catch-at-length model estimated that Blue Catfish abundance increased slowly and peaked in the mid-2000s before undergoing a recent decline. The model estimated a large spike in abundance due to an estimated large recruitment event in 2011, but may be an artifact of missing data in 2012 in both relative abundance indices examined. The newly developed statistical catch-at-length model provides most detailed information on population dynamics of Blue Catfish in the James River and can be expanded and updated as new data become available. Based on results of the statistical catch-at-length model, I examined population responses to unregulated, maximum length limit (60 cm), and harvest slot limit regulations (harvest allowed 25 –60 cm) in a management strategy evaluation framework. The management strategy evaluation supported that the James River Blue Catfish population could be reduced with increased harvest, but trophy-size fish would decline. Consequently, fishery managers tasked with invasive species management must consider this tradeoff of fishery economic benefits and predation on native populations, especially those prey in which population sizes are unknown. / Doctor of Philosophy / Blue Catfish are non-native to the Chesapeake Bay watershed, but were stocked in the 1970s and 1980s to provide fishing opportunities to the region. Unknowingly, Blue Catfish expanded downstream and beyond the boundaries of the rivers to which they were originally stocked and now exist in extremely dense populations in places. This expansion in population size and distribution has generated concern for the health of the Chesapeake Bay and calls for population control. I wanted to learn more about Blue Catfish in Virginia, specifically Blue Catfish growth rates, population dynamics, and how they might respond to control efforts. I examined Blue Catfish growth rates and found growth rates differed over time and across river systems. Blue Catfish tended to grow more slowly over time as their populations matured. As growth rates declined, population size increased with maximum population sizes in the late 2000s in the James River with a subsequent decline in abundance. Many invasive species exhibit this sort of phenomenon, where population sizes increase and reach a maximum before declining. Finally, I looked at Blue Catfish responses to different fishing regulations and harvest levels, finding that increased harvest could help control Blue Catfish population sizes. However, Blue Catfish management objectives are in conflict as regulations that limit predation of native species of interest also reduce the proportion of large fish in populations. Blue Catfish management will require stakeholder-driven approaches to ensure buy-in and reduce user conflicts.

Product reliability is an important characteristic for all manufacturers, engineers and consumers. Industrial statisticians have been planning experiments for years to improve product quality and reliability. However, rarely do experts in the field of reliability have expertise in design of experiments (DOE) and the implications that experimental protocol have on data analysis. Additionally, statisticians who focus on DOE rarely work with reliability data. As a result, analysis methods for lifetime data for experimental designs that are more complex than a completely randomized design are extremely limited. This dissertation provides two new analysis methods for reliability data from life tests. We focus on data from a sub-sampling experimental design. The new analysis methods are illustrated on a popular reliability data set, which contains sub-sampling. Monte Carlo simulation studies evaluate the capabilities of the new modeling methods. Additionally, Monte Carlo simulation studies highlight the principles of experimental design in a reliability context. The dissertation provides multiple methods for statistical inference for the new analysis methods. Finally, implications for the reliability field are discussed, especially in future applications of the new analysis methods. / Ph. D.

In public health, biomedical, epidemiological, and other applications, data collected are often measured with error. When mismeasured data is used in a regression analysis, not accounting for the measurement error can lead to incorrect inference about the relationships between the covariates and the response. We investigate measurement error in the covariates of two types of regression models.  For each we propose a fully Bayesian approach that treats the variable measured with error as a latent variable to be integrated over, and a semi-Bayesian approach which uses a first order Laplace approximation to marginalize the variable measured with error out of the likelihood. The first model is the matched case-control study for analyzing clustered binary outcomes. We develop low-rank thin plate splines for the case where a variable measured with error has an unknown, nonlinear relationship with the response. In addition to the semi- and fully Bayesian approaches, we propose another using expectation-maximization to detect both parametric and nonparametric relationships between the covariates and the binary outcome. We assess the performance of each method via simulation terms of mean squared error and mean bias. We illustrate each method on a perturbed example of 1--4 matched case-control study. The second regression model is the generalized linear model (GLM) with unknown link function. Usually, the link function is chosen by the user based on the distribution of the response variable, often to be the canonical link. However, when covariates are measured with error, incorrect inference as a result of the error can be compounded by incorrect choice of link function. We assess performance via simulation of the semi- and fully Bayesian methods in terms of mean squared error. We illustrate each method on the Framingham Heart Study dataset. The simulation results for both regression models support that the fully Bayesian approach is at least as good as the semi-Bayesian approach for adjusting for measurement error, particularly when the distribution of the variable of measure with error and the distribution of the measurement error are misspecified. / Ph. D.

Este trabalho objetivou a avaliação da presença de plasticidade fenotípica e de adaptação local de três procedências de palmiteiro: Ombrófila Densa, Estacional Semidecidual e Restinga, em três locais no Estado de São Paulo: Parque Estadual da Ilha do Cardoso, Parque Estadual de Carlos Botelho e Estação Ecológica dos Caetetus, em ensaios de adaptação no estabelecimento (ou de semeadura) e de adaptação em juvenis (ou de crescimento). Os conjuntos de dados foram analisados utilizando estruturas de grupos de experimentos, com efeitos cruzados e aninhados. As variáveis relacionadas com a massa de matéria seca das plantas, nos dois ensaios, foram analisadas usando a abordagem de modelos lineares de efeitos mistos, por meio da incorporação de fatores de efeito aleatório, e fazendo uso do método da máxima verossimilhança restrita (REML) para estimação dos componentes de variância associados a tais fatores com um menor viés. Por outro lado, para a proporção de sementes germinadas, no ensaio de adaptação no estabelecimento, a análise estatística foi realizada a partir da abordagem dos modelos lineares generalizados mistos, sob a pressuposição de que a variável segue uma distribuição binomial, com função de ligação logito. O método da pseudo-verossimilhança foi empregado para obtenção da solução das equações de verossimilhança. Os resultados mostraram que as plantas originadas de sementes dos três biomas avaliados apresentaram um comportamento plástico, para todos os caracteres avaliados no ensaio de adaptação no estabelecimento. Com relação ao ensaio de adaptação em juvenis, a característica de plasticidade foi verificada somente para a massa de matéria seca da folha em plantas provenientes do bioma Estacional Semidecidual. A característica de adaptação local, apresentou-se de forma evidente no ensaio de adaptação no estabelecimento. Estes resultados evidenciaram que em cada local avaliado, as plantas originadas das sementes de diferentes procedências apresentaram um comportamento diferenciado nos caracteres relacionados à massa de matéria seca, podendo em alguns casos, tratar-se de adaptação local. Concluiu-se que os locais Carlos Botelho e Ilha do Cardoso são os mais favoráveis para a germinação das sementes de sua mesma procedência. / The aim of this work was to evaluate the presence of phenotypic plasticity and local adaptation of three provenances of the palm specie Euterpe edulis: Atlantic Rainforest, Seasonally Dry Forest and Restinga Forest, in permanent parcels inserted in three forest types of the São Paulo State (Brazil): Parque Estadual da Ilha do Cardoso, Parque Estadual de Carlos Botelho e Estação Ecológica dos Caetetus, in experiments of seedling establishment and juveniles plants growth. The data sets were analyzed using structures of groups of experiments, with crossed and nested effects. The variables related to dry matter content of plants in both assays were analyzed using linear mixed models (LMM) approach, through the incorporation of random effect factors, and using the restricted maximum likelihood method (REML) for estimation of variance components associated with these factors with a minor bias. On the other hand, germination proportion of the seeds at seedling establishment assay was analyzed using the generalized linear mixed models (GLMM) approach, under the assumption that the variable follows a binomial distribution, with logit link function. The pseudo-likelihood (PL) method was used to obtain the numerical solution of the likelihood equations. The results showed that, plants from seeds of the three biomes evaluated presented a plastic behavior for all characters assessed in the seedling establishment assay. In respect to juveniles adaptation assay, the phenotypic plasticity characteristic was observed only to the leaf dry matter content of plants from Seasonally Dry Forest biome. The local adaptation characteristic was clearly observed in the seedling establishment assay. These results showed that at each site evaluated, plants originating from seeds of different provenances exhibited different behavior on characters related to the dry matter content and may in some cases be local adaptation. It was concluded that locations Carlos Botelho and Ilha do Cardoso are the most favorable for seed germination of its same provenance.