On the effect of confounding in linear regression models: an approach based on the theory of quadratic forms

The main topic of this thesis is confounding in linear regression models. It arises when a relationship between an observed process, the covariate, and an outcome process, the response, is influenced by an unmeasured process, the confounder, associated with both. Consequently, the estimators for the regression coefficients of the measured covariates might be severely biased, less efficient and characterized by misleading interpretations. Confounding is an issue when the primary target of the work is the estimation of the regression parameters. The central point of the dissertation is the evaluation of the sampling properties of parameter estimators. This work aims to extend the spatial confounding framework to general structured settings and to understand the behaviour of confounding as a function of the data generating process structure parameters in several scenarios focusing on the joint covariate-confounder structure. In line with the spatial statistics literature, our purpose is to quantify the sampling properties of the regression coefficient estimators and, in turn, to identify the most prominent quantities depending on the generative mechanism impacting confounding. Once the sampling properties of the estimator conditionally on the covariate process are derived as ratios of dependent quadratic forms in Gaussian random variables, we provide an analytic expression of the marginal sampling properties of the estimator using Carlson’s R function. Additionally, we propose a representative quantity for the magnitude of confounding as a proxy of the bias, its first-order Laplace approximation. To conclude, we work under several frameworks considering spatial and temporal data with specific assumptions regarding the covariance and cross-covariance functions used to generate the processes involved. This study allows us to claim that the variability of the confounder-covariate interaction and of the covariate plays the most relevant role in determining the principal marker of the magnitude of confounding.

Spatial variability of flow statistics within regular building arrays

Turbulence statistics obtained by direct numerical simulations are analysed to investigate spatial heterogeneity within regular arrays of building-like cubical obstacles. Two different array layouts are studied, staggered and square, both at a packing density of λp=0.25 . The flow statistics analysed are mean streamwise velocity ( u− ), shear stress ( u′w′−−−− ), turbulent kinetic energy (k) and dispersive stress fraction ( u˜w˜ ). The spatial flow patterns and spatial distribution of these statistics in the two arrays are found to be very different. Local regions of high spatial variability are identified. The overall spatial variances of the statistics are shown to be generally very significant in comparison with their spatial averages within the arrays. Above the arrays the spatial variances as well as dispersive stresses decay rapidly to zero. The heterogeneity is explored further by separately considering six different flow regimes identified within the arrays, described here as: channelling region, constricted region, intersection region, building wake region, canyon region and front-recirculation region. It is found that the flow in the first three regions is relatively homogeneous, but that spatial variances in the latter three regions are large, especially in the building wake and canyon regions. The implication is that, in general, the flow immediately behind (and, to a lesser extent, in front of) a building is much more heterogeneous than elsewhere, even in the relatively dense arrays considered here. Most of the dispersive stress is concentrated in these regions. Considering the experimental difficulties of obtaining enough point measurements to form a representative spatial average, the error incurred by degrading the sampling resolution is investigated. It is found that a good estimate for both area and line averages can be obtained using a relatively small number of strategically located sampling points.

Measurement of genetic structure within populations using Moran's spatial autocorrelation statistics.

Spatial structure of genetic variation within populations, an important interacting influence on evolutionary and ecological processes, can be analyzed in detail by using spatial autocorrelation statistics. This paper characterizes the statistical properties of spatial autocorrelation statistics in this context and develops estimators of gene dispersal based on data on standing patterns of genetic variation. Large numbers of Monte Carlo simulations and a wide variety of sampling strategies are utilized. The results show that spatial autocorrelation statistics are highly predictable and informative. Thus, strong hypothesis tests for neutral theory can be formulated. Most strikingly, robust estimators of gene dispersal can be obtained with practical sample sizes. Details about optimal sampling strategies are also described.

Disaggregation of polygons of surficial geology and soil maps using spatial modelling and legacy data

Examples from the Murray-Darling basin in Australia are used to illustrate different methods of disaggregation of reconnaissance-scale maps. One approach for disaggregation revolves around the de-convolution of the soil-landscape paradigm elaborated during a soil survey. The descriptions of soil ma units and block diagrams in a soil survey report detail soil-landscape relationships or soil toposequences that can be used to disaggregate map units into component landscape elements. Toposequences can be visualised on a computer by combining soil maps with digital elevation data. Expert knowledge or statistics can be used to implement the disaggregation. Use of a restructuring element and k-means clustering are illustrated. Another approach to disaggregation uses training areas to develop rules to extrapolate detailed mapping into other, larger areas where detailed mapping is unavailable. A two-level decision tree example is presented. At one level, the decision tree method is used to capture mapping rules from the training area; at another level, it is used to define the domain over which those rules can be extrapolated. (C) 2001 Elsevier Science B.V. All rights reserved.

Spatial and temporal patterns in Australian wheat yield and their relationship with ENSO

Spatial and temporal variability in wheat production in Australia is dominated by rainfall occurrence. The length of historical production records is inadequate, however, to analyse spatial and temporal patterns conclusively. In this study we used modelling and simulation to identify key spatial patterns in Australian wheat yield, identify groups of years in the historical record in which spatial patterns were similar, and examine association of those wheat yield year groups with indicators of the El Nino Southern Oscillation (ENSO). A simple stress index model was trained on 19 years of Australian Bureau of Statistics shire yield data (1975-93). The model was then used to simulate shire yield from 1901 to 1999 for all wheat-producing shires. Principal components analysis was used to determine the dominating spatial relationships in wheat yield among shires. Six major components of spatial variability were found. Five of these represented near spatially independent zones across the Australian wheatbelt that demonstrated coherent temporal (annual) variability in wheat yield. A second orthogonal component was required to explain the temporal variation in New South Wales. The principal component scores were used to identify high- and low-yielding years in each zone. Year type groupings identified in this way were tested for association with indicators of ENSO. Significant associations were found for all zones in the Australian wheatbelt. Associations were as strong or stronger when ENSO indicators preceding the wheat season (April-May phases of the Southern Oscillation Index) were used rather than indicators based on classification during the wheat season. Although this association suggests an obvious role for seasonal climate forecasting in national wheat crop forecasting, the discriminatory power of the ENSO indicators, although significant, was not strong. By examining the historical years forming the wheat yield analog sets within each zone, it may be possible to identify novel climate system or ocean-atmosphere features that may be causal and, hence, most useful in improving seasonal forecasting schemes.

Spatial study of homicide rates in the state of Bahia, Brazil, 1996-2010

OBJECTIVE To analyze the spatial distribution of homicide mortality in the state of Bahia, Northeastern Brazil. METHODS Ecological study of the 15 to 39-year old male population in the state of Bahia in the period 1996-2010. Data from the Mortality Information System, relating to homicide (X85-Y09) and population estimates from the Brazilian Institute of Geography and Statistics were used. The existence of spatial correlation, the presence of clusters and critical areas of the event studied were analyzed using Moran’s I Global and Local indices. RESULTS A non-random spatial pattern was observed in the distribution of rates, as was the presence of three clusters, the first in the north health district, the second in the eastern region, and the third cluster included townships in the south and the far south of Bahia. CONCLUSIONS The homicide mortality in the three different critical areas requires further studies that consider the socioeconomic, cultural and environmental characteristics in order to guide specific preventive and interventionist practices.

Temporal trends and spatial distribution of unsafe abortion in Brazil, 1996-2012

OBJECTIVE To analyze temporal trends and distribution patterns of unsafe abortion in Brazil. METHODS Ecological study based on records of hospital admissions of women due to abortion in Brazil between 1996 and 2012, obtained from the Hospital Information System of the Ministry of Health. We estimated the number of unsafe abortions stratified by place of residence, using indirect estimate techniques. The following indicators were calculated: ratio of unsafe abortions/100 live births and rate of unsafe abortion/1,000 women of childbearing age. We analyzed temporal trends through polynomial regression and spatial distribution using municipalities as the unit of analysis. RESULTS In the study period, a total of 4,007,327 hospital admissions due to abortions were recorded in Brazil. We estimated a total of 16,905,911 unsafe abortions in the country, with an annual mean of 994,465 abortions (mean unsafe abortion rate: 17.0 abortions/1,000 women of childbearing age; ratio of unsafe abortions: 33.2/100 live births). Unsafe abortion presented a declining trend at national level (R2: 94.0%, p < 0.001), with unequal patterns between regions. There was a significant reduction of unsafe abortion in the Northeast (R2: 93.0%, p < 0.001), Southeast (R2: 92.0%, p < 0.001) and Central-West regions (R2: 64.0%, p < 0.001), whereas the North (R2: 39.0%, p = 0.030) presented an increase, and the South (R2: 22.0%, p = 0.340) remained stable. Spatial analysis identified the presence of clusters of municipalities with high values for unsafe abortion, located mainly in states of the North, Northeast and Southeast Regions. CONCLUSIONS Unsafe abortion remains a public health problem in Brazil, with marked regional differences, mainly concentrated in the socioeconomically disadvantaged regions of the country. Qualification of attention to women’s health, especially to reproductive aspects and attention to pre- and post-abortion processes, are necessary and urgent strategies to be implemented in the country.

Walking Through the Revolution: A Spatial Reading of Literary Echoes

This paper presents an embryo of a literary guide on the Carnation Revolution to be explored for educational historical excursions other than leisure and tourism. We propose a historical trail through the centre of Lisbon, city of the Carnation Revolution, called Walk through the Revolution. The trail aims to reinforce collective memory about the major events that occurred in the early moments leading to the coup. The trail is made up by nine places of rememberance, for which literary excerpts are suggested and which are supported by a digital research procedure. A set of seven fixed and observer-independent categories are used to analyse the literary contents of 23 literary works published up to 2013. These literary works refer to events that happened between the eve of April 25 and May 1, 1974. At the same time, literary descriptions are explored using a spatial approach in order to define the literary geography of the most iconic military actions and popular demonstrations that occurred in Lisbon and the surroundings. The literary geography and the cartography of the historical events are then compared. Data analysis and visualization benefit from the use of standardised and quantitative methods, including basic statistics and geographic information systems.

Spatial and spatiotemporal occurrence of human visceral leishmaniasis in Adamantina, State of São Paulo, Brazil

Abstract: INTRODUCTION : Several municipalities of the Western region of the State of São Paulo have been affected by human visceral leishmaniasis (HVL), including the City of Adamantina, where the first autochthonous cases occurred in 2004. Therefore, this study aimed to describe the spatial and spatiotemporal occurrence of HVL in Adamantina. METHODS : Secondary data regarding the occurrence of HVL in Adamantina between 2004 and 2011 were used. Incidence, mortality, and case fatality rates were calculated. We used local empirical Bayesian incidence rates to represent the occurrence of the disease in the census sector of the city. The existence of spatial and spatiotemporal clusters of cases was evaluated using scan statistics. In situ observation was performed to assess the socioeconomic and environmental characteristics of the areas with medium and high incidences. RESULTS : Adamantina reported cases in 70% of its census sectors. No differences were observed between sexes. The group aged 0-4 years presented the highest incidence and mortality rates, and the group aged 40-59 years presented the highest fatality rate. We detected a spatiotemporal cluster, which coincided with the commencement of the endemic in the city. CONCLUSIONS : The individuals most affected by the disease were children. The disease was present in areas with better and worse socioeconomic conditions. The use of spatial analysis techniques was important to achieve the study objectives.

Scan statistics analysis of forest fire clusters

Spatio-temporal clusters in 1997?2003 fire sequences of Tuscany region (central Italy) have been identified and analysed by using the scan statistic, a method which was devised to evidence clusters in epidemiology. Results showed that the method is reliable to find clusters of events and to evaluate their significance via Monte Carlo replication. The evaluation of the presence of spatial and temporal patterns in fire occurrence and their significance could have a great impact in forthcoming studies on fire occurrences prediction.

Hiding the evidence: The State and spatial inequalities in health in Ireland

The Irish State has consistently reduced its production of publicly accessible disease / mortality maps over the last fifty years. State health statistics, and the small number of disease / mortality maps that have been produced in official publications, show a declining level of detail and are routinely out of date. Following a review of the production of disease / mortality maps in Ireland by the State and allied health agencies, two reasons are suggested for this decline. The first explanation relates to spatial inequalities in healthcare provision and to the absence of a health funding formulae in Ireland. The second explanation focuses on the potential politicisation of spatial inequalities in health status. Researchers in these fields are urged to disseminate information widely on spatial inequalities in healthcare provision, healthcare access and health status, both within and outside of the academic literature. Researchers in these fields are also urged to adopt an advocacy role on these issues, or to develop strategic alliances with such advocates.����

Cluster recognition in spatial-temporal sequences: The case of forest fires

Forest fire sequences can be modelled as a stochastic point process where events are characterized by their spatial locations and occurrence in time. Cluster analysis permits the detection of the space/time pattern distribution of forest fires. These analyses are useful to assist fire-managers in identifying risk areas, implementing preventive measures and conducting strategies for an efficient distribution of the firefighting resources. This paper aims to identify hot spots in forest fire sequences by means of the space-time scan statistics permutation model (STSSP) and a geographical information system (GIS) for data and results visualization. The scan statistical methodology uses a scanning window, which moves across space and time, detecting local excesses of events in specific areas over a certain period of time. Finally, the statistical significance of each cluster is evaluated through Monte Carlo hypothesis testing. The case study is the forest fires registered by the Forest Service in Canton Ticino (Switzerland) from 1969 to 2008. This dataset consists of geo-referenced single events including the location of the ignition points and additional information. The data were aggregated into three sub-periods (considering important preventive legal dispositions) and two main ignition-causes (lightning and anthropogenic causes). Results revealed that forest fire events in Ticino are mainly clustered in the southern region where most of the population is settled. Our analysis uncovered local hot spots arising from extemporaneous arson activities. Results regarding the naturally-caused fires (lightning fires) disclosed two clusters detected in the northern mountainous area.

Spatial analysis of the relationship between mortality from cardiovascular and cerebrovascular disease and drinking water hardness.

Previously published scientific papers have reported a negative correlation between drinking water hardness and cardiovascular mortality. Some ecologic and case-control studies suggest the protective effect of calcium and magnesium concentration in drinking water. In this article we present an analysis of this protective relationship in 538 municipalities of Comunidad Valenciana (Spain) from 1991-1998. We used the Spanish version of the Rapid Inquiry Facility (RIF) developed under the European Environment and Health Information System (EUROHEIS) research project. The strategy of analysis used in our study conforms to the exploratory nature of the RIF that is used as a tool to obtain quick and flexible insight into epidemiologic surveillance problems. This article describes the use of the RIF to explore possible associations between disease indicators and environmental factors. We used exposure analysis to assess the effect of both protective factors--calcium and magnesium--on mortality from cerebrovascular (ICD-9 430-438) and ischemic heart (ICD-9 410-414) diseases. This study provides statistical evidence of the relationship between mortality from cardiovascular diseases and hardness of drinking water. This relationship is stronger in cerebrovascular disease than in ischemic heart disease, is more pronounced for women than for men, and is more apparent with magnesium than with calcium concentration levels. Nevertheless, the protective nature of these two factors is not clearly established. Our results suggest the possibility of protectiveness but cannot be claimed as conclusive. The weak effects of these covariates make it difficult to separate them from the influence of socioeconomic and environmental factors. We have also performed disease mapping of standardized mortality ratios to detect clusters of municipalities with high risk. Further standardization by levels of calcium and magnesium in drinking water shows changes in the maps when we remove the effect of these covariates.

Quantifying rock fabrics: a test of autocorrelation of the spatial distribution of cristals

A novel test of spatial independence of the distribution of crystals or phases in rocksbased on compositional statistics is introduced. It improves and generalizes the commonjoins-count statistics known from map analysis in geographic information systems.Assigning phases independently to objects in RD is modelled by a single-trial multinomialrandom function Z(x), where the probabilities of phases add to one and areexplicitly modelled as compositions in the K-part simplex SK. Thus, apparent inconsistenciesof the tests based on the conventional joins{count statistics and their possiblycontradictory interpretations are avoided. In practical applications we assume that theprobabilities of phases do not depend on the location but are identical everywhere inthe domain of de nition. Thus, the model involves the sum of r independent identicalmultinomial distributed 1-trial random variables which is an r-trial multinomialdistributed random variable. The probabilities of the distribution of the r counts canbe considered as a composition in the Q-part simplex SQ. They span the so calledHardy-Weinberg manifold H that is proved to be a K-1-affine subspace of SQ. This isa generalisation of the well-known Hardy-Weinberg law of genetics. If the assignmentof phases accounts for some kind of spatial dependence, then the r-trial probabilitiesdo not remain on H. This suggests the use of the Aitchison distance between observedprobabilities to H to test dependence. Moreover, when there is a spatial uctuation ofthe multinomial probabilities, the observed r-trial probabilities move on H. This shiftcan be used as to check for these uctuations. A practical procedure and an algorithmto perform the test have been developed. Some cases applied to simulated and realdata are presented.Key words: Spatial distribution of crystals in rocks, spatial distribution of phases,joins-count statistics, multinomial distribution, Hardy-Weinberg law, Hardy-Weinbergmanifold, Aitchison geometry