Ordinary least squares regression is indicated for studies of allometry

When it comes to fitting simple allometric slopes through measurement data, evolutionary biologists have been torn between regression methods. On the one hand, there is the ordinary least squares (OLS) regression, which is commonly used across many disciplines of biology to fit lines through data, but which has a reputation for underestimating slopes when measurement error is present. On the other hand, there is the reduced major axis (RMA) regression, which is often recommended as a substitute for OLS regression in studies of allometry, but which has several weaknesses of its own. Here, we review statistical theory as it applies to evolutionary biology and studies of allometry. We point out that the concerns that arise from measurement error for OLS regression are small and straightforward to deal with, whereas RMA has several key properties that make it unfit for use in the field of allometry. The recommended approach for researchers interested in allometry is to use OLS regression on measurements taken with low (but realistically achievable) measurement error. If measurement error is unavoidable and relatively large, it is preferable to correct for slope attenuation rather than to turn to RMA regression, or to take the expected amount of attenuation into account when interpreting the data.     

Why, when, and how biochemists should use least squares.

One of the most commonly used methods for the analysis of experimental data in the biochemical literature is nonlinear least squares (regression). This group of methods are also commonly misused. The purpose of this article is to review the assumptions inherent in the use of least-squares techniques and how these assumptions govern the ways that least-squares techniques can and should be used. Since these assumptions pertain to the nature of the experimental data to be analyzed they also dictate many aspects of the data collection protocol. The examination of these assumptions includes a discussion of questions like: Why would a biochemist want to use nonlinear least-squares techniques? When is it appropriate for a biochemist to use nonlinear least-squares techniques? What confidence can be assigned to the results of a nonlinear least-squares analysis?     

A rapid generalized least squares model for a genome-wide quantitative trait association analysis in families

Genome-wide association studies (GWAS) using family data involve association analyses between hundreds of thousands of markers and a trait for a large number of related individuals. The correlations among relatives bring statistical and computational challenges when performing these large-scale association analyses. Recently, several rapid methods accounting for both within- and between-family variation have been proposed. However, these techniques mostly model the phenotypic similarities in terms of genetic relatedness. The familial resemblances in many family-based studies such as twin studies are not only due to the genetic relatedness, but also derive from shared environmental effects and assortative mating. In this paper, we propose 2 generalized least squares (GLS) models for rapid association analysis of family-based GWAS, which accommodate both genetic and environmental contributions to familial resemblance. In our first model, we estimated the joint genetic and environmental variations. In our second model, we estimated the genetic and environmental components separately. Through simulation studies, we demonstrated that our proposed approaches are more powerful and computationally efficient than a number of existing methods are. We show that estimating the residual variance-covariance matrix in the GLS models without SNP effects does not lead to an appreciable bias in the p values as long as the SNP effect is small (i.e. accounting for no more than 1% of trait variance).     

A program for least squares analysis of reassociation and hybridization data.

A computer program is described for the rapid calculation of least squares solutions for data fitted to different functions normally used in reassociation and hybridization kinetic measurements. The equations for the fraction not reacted as a function of Cot follow: First order, exp(-kCot); second order, (1+kCot)-1; variable order, (1+kCot)-n; approximate fraction of DNA sequence remaining single stranded, (1+kCot)-.44; and a function describing the pairing of tracer when the rate constant for the tracer (k) is distinct from the driver rate constant (kd): (formula: see text). Several components may be used for most of these functional forms. The standard deviations of the individual parameters at the solutions are calculated.     

A weighted least squares method for inverse dynamic analysis

Internal forces in the human body can be estimated from measured movements and external forces using inverse dynamic analysis. Here we present a general method of analysis which makes optimal use of all available data, and allows the use of inverse dynamic analysis in cases where external force data is incomplete. The method was evaluated for the analysis of running on a partially instrumented treadmill. It was found that results correlate well with those of a conventional analysis where all external forces are known.     

Methods of least squares and SIRT in reconstruction.

In this paper we show that a particular version of the Simultaneous Iterative Reconstruction Technique (SIRT) proposed by Gilbert in 1972 strongly resembles the Richardson least-squares algorithm.By adopting the adjustable parameters of the general Richardson algorithm, we have been able to produce generalized SIRT algorithms with improved convergence.A particular generalization of the SIRT algorithm, GSIRT, has an adjustable parameter σ and the starting picture ρ⁰ as input. A value $\text{1}\text{2}$ for σ and a weighted back-projection for ρ⁰ produce a stable algorithm.We call the SIRT-like algorithms for the solution of the weighted leastsquares problems LSIRT and present two such algorithms, LSIRT1 and LSIRT2, which have definite computational advantages over SIRT and GSIRT.We have tested these methods on mathematically simulated phantoms and find that the new SIRT methods converge faster than Gilbert's SIRT but are more sensitive to noise present in the data. However, the faster convergence rates allow termination before the noise contribution degrades the reconstructed image excessively.     

A Pascal program for weighted least squares regression on a microcomputer

Weighted least-squares regression has been programmed in Pascal for a microcomputer. A double precision Pascal compiler and the Motorola 6809 assembler produce a fast machine-code program occupying 22,000 bytes of memory when appended to the Pascal run-time module. Large data sets fit in the remaining memory. A regression with 72 observations and 24 parameters runs in 7 min, excluding optional print out of large matrices. The maximum dimensions of the design matrix, X, can be altered by modifying two Pascal constants. Minor changes to the Pascal source program will make it compatible with other Pascal compilers. The program optionally orthogonalises the X matrix to detect linearly-dependent columns in X, and/or generate orthogonal parameter estimates. After orthogonalizing X and fitting the model, the parameter estimates for the original X can be retrieved by the program. Regressions on a repeatedly reduced model are performed through elimination of columns in X until the minimum adequate model is obtained.     

A new regularized least squares support vector regression for gene selection

Background  

Selection of influential genes with microarray data often faces the difficulties of a large number of genes and a relatively small group of subjects. In addition to the curse of dimensionality, many gene selection methods weight the contribution from each individual subject equally. This equal-contribution assumption cannot account for the possible dependence among subjects who associate similarly to the disease, and may restrict the selection of influential genes.     

A hybrid least squares and principal component analysis algorithm for Raman spectroscopy

Raman spectroscopy is a powerful technique for detecting and quantifying analytes in chemical mixtures. A critical part of Raman spectroscopy is the use of a computer algorithm to analyze the measured Raman spectra. The most commonly used algorithm is the classical least squares method, which is popular due to its speed and ease of implementation. However, it is sensitive to inaccuracies or variations in the reference spectra of the analytes (compounds of interest) and the background. Many algorithms, primarily multivariate calibration methods, have been proposed that increase robustness to such variations. In this study, we propose a novel method that improves robustness even further by explicitly modeling variations in both the background and analyte signals. More specifically, it extends the classical least squares model by allowing the declared reference spectra to vary in accordance with the principal components obtained from training sets of spectra measured in prior characterization experiments. The amount of variation allowed is constrained by the eigenvalues of this principal component analysis. We compare the novel algorithm to the least squares method with a low-order polynomial residual model, as well as a state-of-the-art hybrid linear analysis method. The latter is a multivariate calibration method designed specifically to improve robustness to background variability in cases where training spectra of the background, as well as the mean spectrum of the analyte, are available. We demonstrate the novel algorithm's superior performance by comparing quantitative error metrics generated by each method. The experiments consider both simulated data and experimental data acquired from in vitro solutions of Raman-enhanced gold-silica nanoparticles.     

Surrogate variable analysis using partial least squares (SVA-PLS) in gene expression studies

MOTIVATION: In a typical gene expression profiling study, our prime objective is to identify the genes that are differentially expressed between the samples from two different tissue types. Commonly, standard analysis of variance (ANOVA)/regression is implemented to identify the relative effects of these genes over the two types of samples from their respective arrays of expression levels. But, this technique becomes fundamentally flawed when there are unaccounted sources of variability in these arrays (latent variables attributable to different biological, environmental or other factors relevant in the context). These factors distort the true picture of differential gene expression between the two tissue types and introduce spurious signals of expression heterogeneity. As a result, many genes which are actually differentially expressed are not detected, whereas many others are falsely identified as positives. Moreover, these distortions can be different for different genes. Thus, it is also not possible to get rid of these variations by simple array normalizations. This both-way error can lead to a serious loss in sensitivity and specificity, thereby causing a severe inefficiency in the underlying multiple testing problem. In this work, we attempt to identify the hidden effects of the underlying latent factors in a gene expression profiling study by partial least squares (PLS) and apply ANCOVA technique with the PLS-identified signatures of these hidden effects as covariates, in order to identify the genes that are truly differentially expressed between the two concerned tissue types. RESULTS: We compare the performance of our method SVA-PLS with standard ANOVA and a relatively recent technique of surrogate variable analysis (SVA), on a wide variety of simulation settings (incorporating different effects of the hidden variable, under situations with varying signal intensities and gene groupings). In all settings, our method yields the highest sensitivity while maintaining relatively reasonable values for the specificity, false discovery rate and false non-discovery rate. Application of our method to gene expression profiling for acute megakaryoblastic leukemia shows that our method detects an additional six genes, that are missed by both the standard ANOVA method as well as SVA, but may be relevant to this disease, as can be seen from mining the existing literature.     

A shared response model for clustered binary data in developmental toxicity studies

Existing distributions for modeling fetal response data in developmental toxicology such as the beta-binomial distribution have a tendency of inflating the probability of no malformed fetuses, and hence understating the risk of having at least one malformed fetus within a litter. As opposed to a shared probability extra-binomial model, we advocate a shared response model that allows a random number of fetuses within the same litter to share a common response. An explicit formula is given for the probability function and graphical plots suggest that it does not suffer from the problem of assigning too much probability to the event of no malformed fetuses. The EM algorithm can be used to estimate the model parameters. Results of a simulation study show that the EM estimates are nearly unbiased and the associated confidence intervals based on the usual standard error estimates have coverage close to the nominal level. Simulation results also suggest that the shared response model estimates of the marginal malformation probabilities are robust to misspecification of the distributional form, but not so for the estimates of intralitter correlation and the litter-level probability of having at least one malformed fetus. The proposed model is fitted to a set of data from the U.S. National Toxicology Program. For the same dose-response relationship, the fit based on the shared response distribution is superior to that based on the beta-binomial, and comparable to that based on the recently proposed q-power distribution (Kuk, 2004, Applied Statistics53, 369-386). An advantage of the shared response model over the q-power distribution is that it is more interpretable and can be extended more easily to the multivariate case. To illustrate this, a bivariate shared response model is fitted to fetal response data involving visceral and skeletal malformation.     

Using nonlinear least squares to assess relative expression and its uncertainty in real-time qPCR studies

Relative expression ratios are commonly estimated in real-time qPCR studies by comparing the quantification cycle for the target gene with that for a reference gene in the treatment samples, normalized to the same quantities determined for a control sample. For the “standard curve” design, where data are obtained for all four of these at several dilutions, nonlinear least squares can be used to assess the amplification efficiencies (AE) and the adjusted ΔΔC_q and its uncertainty, with automatic inclusion of the effect of uncertainty in the AEs. An algorithm is illustrated for the KaleidaGraph program.     

Application of partial least squares discriminant analysis to two-dimensional difference gel studies in expression proteomics

Two-dimensional difference gel electrophoresis (DIGE) is a tool for measuring changes in protein expression between samples involving pre-electrophoretic labeling ith cyanine dyes. In multi-gel experiments, univariate statistical tests have been used to identify differential expression between sample types by looking for significant changes in spot volume. Multivariate statistical tests, which look for correlated changes between sample types, provide an alternate approach for identifying spots with differential expression. Partial least squares-discriminant analysis (PLS-DA), a multivariate statistical approach, was combined with an iterative threshold process to identify which protein spots had the greatest contribution to the model, and compared to univariate test for three datasets. This included one dataset where no biological difference was expected. The novel multivariate approach, detailed here, represents a method to complement the univariate approach in identification of differentially expressed protein spots. This new approach has the advantages of reduced risk of false-positives and the identification of spots that are significantly altered in terms of correlated expression rather than absolute expression values.