One of the main problems in statistical climatology is to construct a parsimonious model of natural climate variability. Such a model serves for instance as a null hypothesis for detection of human induced climate changes and of periodic climate signals. Fitting thismodel to various climatic time series also helps to infer the origins of underlying temporal variability and to cross validate it between different data sets. We consider the use of a spectral power-law model in this role for the surface temperature, for the free atmospheric air temperature of the troposphere and stratosphere, and for the total ozone. First, we lay down a methodological foundation for our work. We compare two variants of five different power-law fitting methods by means of Monte-Carlo simulations and their application to observed air temperature. Then using the best two methods we fit the power-law model to several observational products and climate model simulations. We make use of specialized atmospheric general circulation model simulations and of the simulations of the Coupled Model Intercomparison Project 3 (CMIP3). The specialized simulations allow us to explain the power-law exponent spatial distribution and to account for discrepancies in scaling behaviour between different observational products. We find that most of the pre-industrial control and 20th century model simulations capture many aspects of the observed horizontal and vertical distribution of the power-law exponents. At the surface, regions with robust power-law exponents—the North Atlantic, the North Pacific, and the Southern Ocean — coincide with regions with strong inter-decadal variability. In the free atmosphere, the large power-law exponents are detected on annual to decadal time scales in the tropical and subtropical troposphere and stratosphere. The spectral steepness in the former is explained by its strong coupling to the surface and in the latter by its sensitivity to volcanic aerosols. However power-law behaviour in the tropics and in the free atmosphere saturates on multi-decadal timescales. We propose a novel diagnostic to evaluate the relative goodness-of-fit of the autoregressive model of the first order (AR1) and the power-law model. The collective behaviour of CMIP3 simulations appears to fall between the two statistical models. Our results suggest that the power-law model should serve as an upper bound and the AR1 model should serve as a lower bound for climate persistence on monthly to decadal time scales. On the applied side we find that the presence of power-law like natural variability increases the uncertainty on the long-term total ozone trend in the Northern Hemisphere high latitudes attributable to anthropogenic chlorine by about a factor of 1.5, and lengthens the expected time to detect ozone recovery by a similar amount.

Structural equation models are systems of simultaneous linear equations that are generalizations of linear regression, and have many applications in the social, behavioural and biological sciences. A serious barrier to applications is that it is easy to specify models for which the parameter vector is not identifiable from the distribution of the observable data, and it is often difficult to tell whether a model is identified or not. In this thesis, we study the most straightforward method to check for identification – solving a system of simultaneous equations. However, the calculations can easily get very complex. Grobner basis is introduced to simplify the process. The main idea of checking identification is to solve a set of finitely many simultaneous equations, called identifying equations, which can be transformed into polynomials. If a unique solution is found, the model is identified. Grobner basis reduces the polynomials into simpler forms making them easier to solve. Also, it allows us to investigate the model-induced constraints on the covariances, even when the model is not identified. With the explicit solution to the identifying equations, including the constraints on the covariances, we can (1) locate points in the parameter space where the model is not identified, (2) find the maximum likelihood estimators, (3) study the effects of mis-specified models, (4) obtain a set of method of moments estimators, and (5) build customized parametric and distribution free tests, including inference for non-identified models.

One of the main problems in statistical climatology is to construct a parsimonious model of natural climate variability. Such a model serves for instance as a null hypothesis for detection of human induced climate changes and of periodic climate signals. Fitting thismodel to various climatic time series also helps to infer the origins of underlying temporal variability and to cross validate it between different data sets. We consider the use of a spectral power-law model in this role for the surface temperature, for the free atmospheric air temperature of the troposphere and stratosphere, and for the total ozone. First, we lay down a methodological foundation for our work. We compare two variants of five different power-law fitting methods by means of Monte-Carlo simulations and their application to observed air temperature. Then using the best two methods we fit the power-law model to several observational products and climate model simulations. We make use of specialized atmospheric general circulation model simulations and of the simulations of the Coupled Model Intercomparison Project 3 (CMIP3). The specialized simulations allow us to explain the power-law exponent spatial distribution and to account for discrepancies in scaling behaviour between different observational products. We find that most of the pre-industrial control and 20th century model simulations capture many aspects of the observed horizontal and vertical distribution of the power-law exponents. At the surface, regions with robust power-law exponents—the North Atlantic, the North Pacific, and the Southern Ocean — coincide with regions with strong inter-decadal variability. In the free atmosphere, the large power-law exponents are detected on annual to decadal time scales in the tropical and subtropical troposphere and stratosphere. The spectral steepness in the former is explained by its strong coupling to the surface and in the latter by its sensitivity to volcanic aerosols. However power-law behaviour in the tropics and in the free atmosphere saturates on multi-decadal timescales. We propose a novel diagnostic to evaluate the relative goodness-of-fit of the autoregressive model of the first order (AR1) and the power-law model. The collective behaviour of CMIP3 simulations appears to fall between the two statistical models. Our results suggest that the power-law model should serve as an upper bound and the AR1 model should serve as a lower bound for climate persistence on monthly to decadal time scales. On the applied side we find that the presence of power-law like natural variability increases the uncertainty on the long-term total ozone trend in the Northern Hemisphere high latitudes attributable to anthropogenic chlorine by about a factor of 1.5, and lengthens the expected time to detect ozone recovery by a similar amount.

Structural equation models are systems of simultaneous linear equations that are generalizations of linear regression, and have many applications in the social, behavioural and biological sciences. A serious barrier to applications is that it is easy to specify models for which the parameter vector is not identifiable from the distribution of the observable data, and it is often difficult to tell whether a model is identified or not. In this thesis, we study the most straightforward method to check for identification – solving a system of simultaneous equations. However, the calculations can easily get very complex. Grobner basis is introduced to simplify the process. The main idea of checking identification is to solve a set of finitely many simultaneous equations, called identifying equations, which can be transformed into polynomials. If a unique solution is found, the model is identified. Grobner basis reduces the polynomials into simpler forms making them easier to solve. Also, it allows us to investigate the model-induced constraints on the covariances, even when the model is not identified. With the explicit solution to the identifying equations, including the constraints on the covariances, we can (1) locate points in the parameter space where the model is not identified, (2) find the maximum likelihood estimators, (3) study the effects of mis-specified models, (4) obtain a set of method of moments estimators, and (5) build customized parametric and distribution free tests, including inference for non-identified models.

Bioassessment can be performed through several methods and with different bioindicators. In Canadian Areas of Concern (AOC), fishes are used as a proxy for site condition. The Index of Biotic Integrity (IBI), a multimetric index for biological assessment, has been applied to fish data across Canadian AOCs to detect recovery. Previous studies, however, have indicated the IBI is not sensitive to assemblage changes characteristic of later stages of recovery. In this study, the IBI and multivariate methods were applied to data from two AOCs, the Detroit and St. Clair rivers. The results revealed that the IBI is susceptible to species substitutions within metric categories. The substitutions produced high variability within narrative ranks and rendered the IBI insensitive to changes, detected by multivariate methods, in the fish assemblage. In the absence of reference sites, the multivariate analyses were supplemented with the development of a reference condition based on best professional judgment.

The primary aim of this thesis is the elucidation of covariate effects on the dependence structure of random variables in bivariate or multivariate models. We develop a unified approach via a conditional copula model in which the copula is parametric and its parameter varies as the covariate. We propose a nonparametric procedure based on local likelihood to estimate the functional relationship between the copula parameter and the covariate, derive the asymptotic properties of the proposed estimator and outline the construction of pointwise confidence intervals. We also contribute a novel conditional copula selection method based on cross-validated prediction errors and a generalized likelihood ratio-type test to determine if the copula parameter varies significantly. We derive the asymptotic null distribution of the formal test. Using subsets of the Matched Multiple Birth and Framingham Heart Study datasets, we demonstrate the performance of these procedures via analyses of gestational age-specific twin birth weights and the impact of change in body mass index on the dependence between two consequent pulse pressures taken from the same subject.

Bioassessment can be performed through several methods and with different bioindicators. In Canadian Areas of Concern (AOC), fishes are used as a proxy for site condition. The Index of Biotic Integrity (IBI), a multimetric index for biological assessment, has been applied to fish data across Canadian AOCs to detect recovery. Previous studies, however, have indicated the IBI is not sensitive to assemblage changes characteristic of later stages of recovery. In this study, the IBI and multivariate methods were applied to data from two AOCs, the Detroit and St. Clair rivers. The results revealed that the IBI is susceptible to species substitutions within metric categories. The substitutions produced high variability within narrative ranks and rendered the IBI insensitive to changes, detected by multivariate methods, in the fish assemblage. In the absence of reference sites, the multivariate analyses were supplemented with the development of a reference condition based on best professional judgment.

Master of Science / Department of Statistics / Weixing Song / The relationship between a random variable and a random vector is often investigated through the regression modeling. Because of its relative simplicity and ease of interpretation, a particular parametric form is often assumed for the regression function. If the pre-specified function form truly reflects the truth, then the resulting estimators and inference procedures would be reliable and efficient. But if the regression function does not represent the true relationship between the response and the predictors, then the inference results might be very misleading. Therefore, lack-of-fit test should be an indispensable part in regression modeling. This report compares the finite sample performance of several classical lack-of-fit tests in regression models via simulation studies. It has three chapters. The conception of the lack-of-fit test, together with its basic setup, is briefly introduced in Chapter 1; then several classical lack-of-fit test procedures are discussed in Chapter 2; finally, thorough simulation studies are conducted in Chapter 3 to assess the finite sample performance of each procedure introduced in Chapter 2. Some conclusions are also summarized in Chapter 3. A list of MATLAB codes that are used for the simulation studies is given in the appendix.

Master of Science / Department of Statistics / Thomas M. Loughin / Traditionally, major-league scouts have evaluated young “position players,” those who are not pitchers, using the “Five Tools”: hitting for average, hitting for power, running, throwing, and fielding. However, “sabermetricians,” those who study the science of baseball, e.g. Bill James, have been trying to evaluate position players using quantifiable measures of performance. In this study, a factor analysis was used to determine underlying characteristics of minor-league hitters. The underlying factors were determined to be slugging ability, lead-off hitting ability, “patience” at the plate, and pure-hitting ability. Additionally, an ordinal response was created from the number of at-bats and on-base plus slugging percentage in the majors during the 2002-05 seasons. The underlying characteristics along with other variables such as a player’s age, position, and level in the minors are used in a cumulative logit logistic regression model to predict a player’s probability of notable success in the majors. The model is built upon data from the 2002 minor-league season and data from the 2002, 2003, 2004, and 2005 major-league seasons.

Master of Science / Department of Statistics / John E. Boyer Jr / It has been shown that the most straightforward approach to testing for the difference of two independent population proportions, called the Wald procedure, tends to declare differences too often. Because of this poor performance, various researchers have proposed simple adjustments to the Wald approach that tend to provide significance levels closer to the nominal. Additionally, several tests that take advantage of different methodologies have been proposed. This paper extends the work of Tebbs and Roths (2008), who wrote an R program to compare confidence interval coverage for a variety of these procedures when used to estimate a contrast in two or more binomial parameters. Their program has been adapted to generate exact significance levels and power for the two parameter hypothesis testing situation. Several combinations of binomial parameters and sample sizes are considered. Recommendations for a choice of procedure are made for practical situations.