Nancy E. Heckman

A quantitative genetic model for growth, shape, reaction norms, and other infinite-dimensional characters

Infinite-dimensional characters are those in which the phenotype of an individual is described by a function, rather than by a finite set of measurements. Examples include growth trajectories, morphological shapes, and norms of reaction. Methods are presented here that allow individual phenotypes, population means, and patterns of variance and covariance to be quantified for infinite-dimensional characters. A quantitative-genetic model is developed, and the recursion equation for the evolution of the population mean phenotype of an infinite-dimensional character is derived. The infinite-dimensional method offers three advantages over conventional finite-dimensional methods when applied to this kind of trait: (1) it describes the trait at all points rather than at a finite number of landmarks, (2) it eliminates errors in predicting the evolutionary response to selection made by conventional methods because they neglect the effects of selection on some parts of the trait, and (3) it estimates parameters of interest more efficiently.

Local Polynomial Kernel Regression for Generalized Linear Models and Quasi-Likelihood Functions

Abstract We investigate the extension of the nonparametric regression technique of local polynomial fitting with a kernel weight to generalized linear models and quasi-likelihood contexts. In the ordinary regression case, local polynomial fitting has been seen to have several appealing features in terms of intuitive and mathematical simplicity. One noteworthy feature is the better performance near the boundaries compared to the traditional kernel regression estimators. These properties are shown to carry over to generalized linear model and quasi-likelihood settings. We also derive the asymptotic distributions of the proposed class of estimators that allow for straightforward interpretation and extensions of state-of-the-art bandwidth selection methods.

Hypothesis Testing in Comparative and Experimental Studies of Function-Valued Traits

Abstract Many traits of evolutionary interest, when placed in their developmental, physiological, or environmental contexts, are function-valued. For instance, gene expression during development is typically a function of the age of an organism and physiological processes are often a function of environment. In comparative and experimental studies, a fundamental question is whether the function-valued trait of one group is different from another. To address this question, evolutionary biologists have several statistical methods available. These methods can be classified into one of two types: multivariate and functional. Multivariate methods, including univariate repeated-measures analysis of variance (ANOVA), treat each trait as a finite list of data. Functional methods, such as repeated-measures regression, view the data as a sample of points drawn from an underlying function. A key difference between multivariate and functional methods is that functional methods retain information about the ordering and spacing of a set of data values, information that is discarded by multivariate methods. In this study, we evaluated the importance of that discarded information in statistical analyses of function-valued traits. Our results indicate that functional methods tend to have substantially greater statistical power than multivariate approaches to detect differences in a function-valued trait between groups.

Nonparametric testing for a monotone hazard function via normalized spacings

We study the problem of testing whether a hazard function is monotonic or not. The proposed test statistics, a global test and four localized tests, are all based on normalized spacings. The global test is in fact just the test statistic [Proschan, F. and Pyke, R. (1967). Tests for monotone failure rate. Fifth Berkeley Symposium, 3, 293–313], introduced for testing a constant hazard function versus a nondecreasing nonconstant hazard function. This global test is powerful for detecting global departures of the null hypothesis, but lacks power when there are local departures from the null hypothesis. By localizing the global test, we obtain tests that respond to this drawback. We also show how the testing procedures can be used when dealing with Type II censored data. We evaluate the performance of the test statistics via simulation studies and illustrate them on some data sets. E-mail: gijbels@stat.ucl.ac.be

Minimax Estimates in a Semiparametric Model

Abstract The statistical analysis of multidimensional data has been greatly influenced by the widespread use of computer-intensive smoothing techniques. These techniques allow one to look at the dependency of a response on several covariates without actually specifying the exact form of that dependency. Nevertheless, when the responses dependency on some subset of the covariates can be assumed to be, say, linear, common sense dictates incorporating this assumption into the estimation technique. In the semiparametric model with covariates X and t, the response, Y, is modeled as the sum of three components: g(t) with g (unknown) lying in an infinite-dimensional space, a linear component Xβ, and random error. If β were the parameter of interest, with the function g treated as a nuisance parameter, the conservative statistician would be led to a minimax approach, with the estimate of β minimizing the worst that could happen over a specified class of functions. Of course, the class of functions should be one ...

Spline smoothing with model-based penalties

Nonparametric regression techniques, which estimate functions directly from noisy data rather than relying on specific parametric models, now play a central role in statistical analysis. We can improve the efficiency and other aspects of a nonparametric curve estimate by using prior knowledge about general features of the curve in the smoothing process. Spline smoothing is extended in this paper to express this prior knowledge in the form of a linear differential operator that annihilates a specified parametric model for the data. Roughness in the fitted function is defined in terms of the integrated square of this operator applied to the fitted function. A fastO(n) algorithm is outlined for this smart smoothing process. Illustrations are provided of where this technique proves useful.

Rapid adaptive evolution of colour vision in the threespine stickleback radiation.

Vision is a sensory modality of fundamental importance for many animals, aiding in foraging, detection of predators and mate choice. Adaptation to local ambient light conditions is thought to be commonplace, and a match between spectral sensitivity and light spectrum is predicted. We use opsin gene expression to test for local adaptation and matching of spectral sensitivity in multiple independent lake populations of threespine stickleback populations derived since the last ice age from an ancestral marine form. We show that sensitivity across the visual spectrum is shifted repeatedly towards longer wavelengths in freshwater compared with the ancestral marine form. Laboratory rearing suggests that this shift is largely genetically based. Using a new metric, we found that the magnitude of shift in spectral sensitivity in each population corresponds strongly to the transition in the availability of different wavelengths of light between the marine and lake environments. We also found evidence of local adaptation by sympatric benthic and limnetic ecotypes to different light environments within lakes. Our findings indicate rapid parallel evolution of the visual system to altered light conditions. The changes have not, however, yielded a close matching of spectrum-wide sensitivity to wavelength availability, for reasons we discuss.

CriSP: A tool for bump hunting

We propose a test of multimodality of regression functions and their derivatives. The test statistic is a critical smoothing parameter (CriSP), giving the minimum amount of smoothing necessary to force the regression function to satisfy the null hypothesis. The p values are computed via bootstrapping. Our idea is motivated by Silvermans test concerning the number of modes in the density function. Simulation studies indicate that the test works well, even when testing for bumps in the derivative. We apply CriSP to childrens growth data, to study the number of spurts of growth.

A note on generalized cross-validation with replicates

Generalized cross-validation (GCV) is a popular method for choosing the smoothing parameter in generalized spline smoothing when there are independent errors with common unknown variance. When data points are replicated, one can choose the smoothing parameter by minimizing one of three functions: the GCV score computed from the averaged observations, the GCV score computed from the original data, or an unbiased estimate of the risk using an independent estimate of the unknown variance [sigma]2. In this note we show how these three methods are related.

Bump hunting in regression analysis

Computer techniques for scatter-plot smoothing are valuable exploratory analysis tools. However, one must exercise caution when making inferences based on features of the smoothed data. Here, the size and number of observed modes are studied: the asymptotic distributions are determined under the assumption of constancy of the true regression function, and sufficient conditions for consistency of the estimated number of modes and their locations are given in the general case.