Least squares linear regression is a common tool in ecological research. One of the central assumptions of least squares linear regression is that the independent variable is measured without error. But this variable is measured with error whenever it is a sample mean. The significance of such contraventions is not regularly assessed in ecological studies. A simulation program was made to provide such an assessment. The program requires a hypothetical data set, and using estimates of S$ sp2$ it scatters the hypothetical data to simulate the effect of sampling error. A regression line is drawn through the scattered data, and SSE and r$ sp2$ are measured. This is repeated numerous times (e.g. 1000) to generate probability distributions for r$ sp2$ and SSE. From these distributions it is possible to assess the likelihood of the hypothetical data resulting in a given SSE or r$ sp2$. The method was applied to survey data used in a published TP-CHLa regression (Pace 1984). Beginning with a hypothetical, linear data set (r$ sp2$ = 1), simulated scatter due to sampling exceeded the SSE from the regression through the survey data about 30% of the time. Thus chances are 3 out of 10 that the level of uncertainty found in the surveyed TP-CHLa relationship would be observed if the true relationship were perfectly linear. If this is so, more precise and more comprehensive models will only be possible when better estimates of the means are available. This simulation approach should apply to all least squares regression studies that use sampled means, and should be especially relevant to studies that use log-transformed values.

In this thesis we develop a theoretical framework for the identification of situations where the equal frequency (EF) or equal variance (EV) subclassification may produce lower bias and/or variance of the estimator. We conduct simulation studies to examine the EF and EV approaches under different types of model misspecification. We apply two weighting schemes in our simulations: equal weights (EW) and inverse variance (IV) weights. Our simulation results indicate that under the quadratic term misspecification, the EF-IV estimator provides the lowest bias and root mean square error as compared to the ordinary least square estimator and other propensity score estimators. Our theorem development demonstrates that if higher variation occurs with larger bias for within subclass treatment effect estimates then the EF-IV estimator has a smaller overall bias than the EF-EW estimator. We show that the EF-IV estimator always has a smaller variance than the EF-EW estimator. We also propose a novel method of subclassification that focuses on creating homogeneous propensity score subclasses to produce an estimator with reduced biased in some circumstances. We feel our research contributes to the field of propensity score adjustments by providing new theorems to compare the overall bias and variance between different propensity score estimators. / Graduation date: 2012

Algorithms based on Markov chains are ubiquitous across scientific disciplines, as they provide a method for extracting statistical information about large, complicated systems. Although these algorithms may be applied to arbitrary graphs, many physical applications are more naturally studied under the restriction to regular lattices. We study several local Markov chains on lattices, exploring how small changes to some parameters can greatly influence efficiency of the algorithms. We begin by examining a natural Markov Chain that arises in the context of "monotonic surfaces", where some point on a surface is sightly raised or lowered each step, but with a greater rate of raising than lowering. We show that this chain is rapidly mixing (converges quickly to the equilibrium) using a coupling argument; the novelty of our proof is that it requires defining an exponentially increasing distance function on pairs of surfaces, allowing us to derive near optimal results in many settings. Next, we present new methods for lower bounding the time local chains may take to converge to equilibrium. For many models that we study, there seems to be a phase transition as a parameter is changed, so that the chain is rapidly mixing above a critical point and slow mixing below it. Unfortunately, it is not always possible to make this intuition rigorous. We present the first proofs of slow mixing for three sampling problems motivated by statistical physics and nanotechnology: independent sets on the triangular lattice (the hard-core lattice gas model), weighted even orientations of the two-dimensional Cartesian lattice (the 8-vertex model), and non-saturated Ising (tile-based self-assembly). Previous proofs of slow mixing for other models have been based on contour arguments that allow us prove that a bottleneck in the state space constricts the mixing. The standard contour arguments do not seem to apply to these problems, so we modify this approach by introducing the notion of "fat contours" that can have nontrivial area. We use these to prove that the local chains defined for these models are slow mixing. Finally, we study another important issue that arises in the context of phase transitions in physical systems, namely how the boundary of a lattice can affect the efficiency of the Markov chain. We examine a local chain on the perfect and near-perfect matchings of the square-octagon lattice, and show for one boundary condition the chain will mix in polynomial time, while for another it will mix exponentially slowly. Strikingly, the two boundary conditions only differ at four vertices. These are the first rigorous proofs of such a phenomenon on lattice graphs.

Spatial heterogeneity of the environment, as measured by floral diversity, composition and structure, is known to influence the distribution and diversity of invertebrates. Heterogeneity brought about by anthropogenic disturbance may be a threat to invertebrate diversity. This thesis investigates the impacts of vegetation heterogeneity at a range of scales on the diversity of invertebrate populations in modified high-altitude indigenous grasslands on the Rock and Pillar Range, Central Otago. Invertebrates were sampled in and on the edges of snow tussock fragments to assess whether species richness increased systematically with fragment area. Invertebrate composition was poorly related to fragment area, plant composition and environmental variables. Taxon richness, abundance and/or diversity for three invertebrate groups increased as fragment area decreased, perhaps reflecting an influx of species from the surrounding matrix. For snow tussock leaf invertebrates in autumn, richness and abundance were at least two times lower in tussocks exposed to the wind than those in the centre of fragments, suggesting selection of habitat may be based on microclimatic characteristics. Invertebrates were sampled from the bases of tussocks after they were clipped to simulate three levels of vertebrate grazing. Invertebrate community composition differed between sites and sampling dates but was unaffected by clipping treatment. At the higher altitude site invertebrate abundance was 1.45 times greater and Shannon-Wiener diversity (H�) 1.22 times lower than at the lower altitude site. The latter sampling date had higher abundance (2.12 times) and taxon richness (1.14 times) than the earlier date. Pitfall-trapped invertebrates in cushionfield, herbfield and snow tussock differed in community composition and often by taxon richness, abundance and diversity. Across habitats, plant composition, plant diversity and some environmental variables were correlated with invertebrate variables, but could not be separated from vegetation type. The invertebrates collected in the course of the study are listed. Four Phyla, eight Classes, 24 orders and over 300 taxa were recorded. Almost all taxa are endemic and many have limited distributions and/or are undescribed. A species list is provided with collection altitude, method and habitat type. Invertebrate assemblages from sites differing in altitude, vegetation type and level of habitat modification on the Rock and Pillar Range are compared. Sites differed in species composition and rank orders of abundance and richness. At lower elevations, invertebrate richness was at least 25% less, and standardised trap abundance at least 44% less, than that at the highest elevation. Richness and abundance of exotic invertebrates decreased with increasing altitude. This thesis highlights several points concerning the study of grassland invertebrates and heterogeneity on the Rock and Pillar Range. First, there are differences in invertebrate assemblages at a range of scales. Conserving invertebrate diversity will therefore require altitudinal sequences and different habitat types, including disturbed areas. At high elevations, tussock habitat may be disproportionately important due to its relative rarity. Second, the effects of disturbance on invertebrates were only visible at large spatial scales. Third, there is a paucity of research on New Zealand invertebrates, especially in regard to terrestrial disturbance, which has resulted in a shortfall of biological, distributional, taxonomic and ecological knowledge.

Thesis (Ph. D.)--Carleton University, 2004. / Includes bibliographical references (p. 175-179). Also available in electronic format on the Internet.

Thesis (Ph.D.)--Baylor University, 2008. / Includes bibliographical references (p. 99-101).

Thesis (M.S.)--University of North Carolina Wilmington, 2009. / Title from PDF title page (January 19, 2010) Includes bibliographical references (p. 32-33)

Thesis (Ph.D.) - Carleton University, 2007. / Includes bibliographical references (p. 207-211). Also available in electronic format on the Internet.

Thesis (Ph.D.)--University of Melbourne, Dept. of Geomatics, 2009. / Typescript. Includes bibliographical references (leaves 127-134)

Thesis (M.S.)--Humboldt State University, 2009. / Includes bibliographical references (leaves 46-48). Also available via Humboldt Digital Scholar.