Observed Correlation Structure Research Articles

Spatial variability of atrazine sorption parameters and other soil properties in a podzoluvisol

The spatial variability of the K f and n parameters of the Freundlich sorption isotherm for atrazine and their correlation with soil textural variables, cation exchange capacity and organic carbon content were studied in a stagnic podzoluvisol. Ninety-three sample points were organized on an irregular three-dimensional grid to a depth of 3.2 m. A trend in the vertical direction explains, for most variables, about 85% of the observed variance. This trend also significantly influences the observed correlation structure between the variables. The horizontal and vertical trends were removed from the data set with the median polish algorithm. The residuals resulting from this technique obey the intrinsic hypothesis. Organic carbon content, cation exchange capacity and n revealed spatial structure. The estimated correlation length scales in the vertical direction were between 0.63–0.81 m for n and the organic carbon content, and between 0.25–0.40 m for the cation exchange capacity. The variograms of sand, loam, clay and K f exhibited pure nugget. The correlation structures between the variables differ for different spatial increments. Variables appeared correlated at small spatial increments whereas they are not correlated if the spatial location of the sample points is neglected.

Smoothing Regression Coefficients in an Overspecified Regression Model with Interrelated Explanatory Variables

Standard rank reducing methods such as principal components regression and partial least squares take account of the x-variables being interrelated only through the observed correlation structure in the data. Alternatively, such interrelationships can be taken into account by using generalized ridge regression, motivated as a form of penalized likelihood estimation. When the form of the penalty function is chosen appropriately, it has the effect of smoothing the regression coefficients corresponding to successive x-variables in a similar way to the use of spline functions for interpolation using a single x-variable. Such a smoothing of the regression coefficients is demonstrated in an example in which the response variable is a single measure of the growth rate of Sitka spruce trees at each of 363 sites in Scotland and the 12 x-variables available are 30-year mean temperatures for each calendar month at each site. The smoothing used penalizes the sum of squared third differences of the regression coefficients and leads to a reduction in the average variance of the fitted values of two-thirds when compared with the unsmoothed regression. Furthermore, the smoothed regression coefficients suggest a bimodal relationship between growth rate and monthly mean temperature which would probably be missed by using standard rank reducing techniques.

Effects of observed correlation structure among tumor types on experimentwise false positive rate in animal carcinogenicity studies.

Probabilities P1, P2, ..., Pn of n events E1, E2, ..., En can impose constraints on the probability of a further event E. deFinetti's fundamental theorem of probability characterizes the interval of coherent probability of E. Lad, Dickey, and Rahman (5) extended deFinetti's theorem in a few directions. In this work we will show some applications of these results to the calculation of experimentwise false positive error rate in multiple hypotheses testing procedure.

Methods for the Comparative Analysis of Variation Patterns

-Although comparisons of variation patterns with theoretical expectations and across species are playing an increasingly important role in systematics, there has been a lack of appropriate procedures for statistically testing the proposed hypotheses. We present a series of statistical tests for hypotheses of morphological integration and for interspecific comparison, along with examples of their application. These tests are based on various randomization and resampling procedures, such as Mantel's test with its recent extensions and bootstrapping. They have the advantage of avoiding the specific and strict distributional assumptions invoked by analytically-based statistics. The statistical procedures described include one for testing the fit of observed correlation matrices to hypotheses of morphological integration and a related test for significant differences in the fit of two alternative hypotheses of morphological integration to the observed correlation structure. Tests for significant similarity in the patterns and magnitudes of variance and correlation among species are also provided. [Morphometrics; comparative analysis; morphological integration; quadratic assignment procedures; Mantel's test; bootstrap.] Comparing observed patterns of morphometric variation to theories of morphological integration (Olson and Miller, 1958; Cheverud, 1982) and among species, or subspecific populations (Arnold, 1981; Riska, 1985), has been a largely ad hoc procedure. Previously, a large body of methods has been used to analyze variation patterns, including various forms of cluster analysis, factor analysis, principal components, multi-dimensional scaling, matrix correlations, and visual inspection. The results of such analyses were then discussed relative to some theory of variation patterns or compared between species or populations. These comparisons might either be verbal or quantitative, but tests of statistical significance were rarely employed. More recently, there has been an increase in statistical rigor in the field, particularly involving the use of quadratic assignment procedures (QAP; sometimes referred to as Mantel's test) (Mantel, 1967; Deitz, 1983; Dow and Cheverud, 1985; Smouse et al., 1986; Dow et al., 1987a, b; Hubert, 1987) for testing the statistical significance of matrix comparisons (Cheverud and Leamy, 1985; Lofsvold, 1986; Kohn and Atchley, 1988; Cheverud, 1989a; Wagner, 1989) and the use of confirmatory factor analysis (Zelditch, 1987, 1988) for testing hypotheses concerning levels and patterns of variation. These new methods allow statistical inference for hypotheses of morphological integration and for comparisons across species. We will describe the use of several of these newer methods, especially those using randomization, for testing hypotheses of morphological integration and interspecific comparison and provide brief examples of their use. The procedures described below can be used to rigorously test hypotheses concerning the causes of morphological variation and covariation patterns. A closely related set of procedures can be directed towards comparative, cross-taxon, analyses of variation and correlation patterns. The systematic study of distinction among group means is well known and extensively represented in the literature. However, systematic studies of variation patterns (as measured by a multivariate variance/covariance or correlation matrix) have been relatively rare. This has been due, in part, to a lack of relevant theory and appropriate systematic methodology. Important theoretical advances over the

A spatial-temporal model of storm rainfall

This paper proposes a model to describe the spatial and temporal distribution of small-scale rainfall patterns. The model is characterised by a multiple autoregressive scheme which has the property that its correlation structure is factorised into space-time components. Such a factorisation explains the observed correlation structure of 2-min. rainfall of storms recorded by the Cardington and Winchcombe, England, networks.