Statistical error in isothermal titration calorimetry: variance function estimation from generalized least squares

The method of generalized least squares (GLS) is used to assess the variance function for isothermal titration calorimetry (ITC) data collected for the 1:1 complexation of Ba(2+) with 18-crown-6 ether. In the GLS method, the least squares (LS) residuals from the data fit are themselves fitted to a variance function, with iterative adjustment of the weighting function in the data analysis to produce consistency. The data are treated in a pooled fashion, providing 321 fitted residuals from 35 data sets in the final analysis. Heteroscedasticity (nonconstant variance) is clearly indicated. Data error terms proportional to q(i) and q(i)/v are well defined statistically, where q(i) is the heat from the ith injection of titrant and v is the injected volume. The statistical significance of the variance function parameters is confirmed through Monte Carlo calculations that mimic the actual data set. For the data in question, which fall mostly in the range of q(i)=100-2000 microcal, the contributions to the data variance from the terms in q(i)(2) typically exceed the background constant term for q(i)>300 microcal and v<10 microl. Conversely, this means that in reactions with q(i) much less than this, heteroscedasticity is not a significant problem. Accordingly, in such cases the standard unweighted fitting procedures provide reliable results for the key parameters, K and DeltaH(degrees) and their statistical errors. These results also support an important earlier finding: in most ITC work on 1:1 binding processes, the optimal number of injections is 7-10, which is a factor of 3 smaller than the current norm. For high-q reactions, where weighting is needed for optimal LS analysis, tips are given for using the weighting option in the commercial software commonly employed to process ITC data.     

A comparison of simultaneous autoregressive and generalized least squares models for dealing with spatial autocorrelation

Aim  In their recent paper, Kissling & Carl (2008 ) recommended the spatial error simultaneous autoregressive model (SAR_err) over ordinary least squares (OLS) for modelling species distribution. We compared these models with the generalized least squares model (GLS) and a variant of SAR (SAR_vario). GLS and SAR_vario are superior to standard implementations of SAR because the spatial covariance structure is described by a semivariogram model.
Innovation  We used the complete datasets employed by Kissling & Carl (2008 ), with strong spatial autocorrelation, and two datasets in which the spatial structure was degraded by sample reduction and grid coarsening. GLS performed consistently better than OLS, SAR_err and SAR_vario in all datasets, especially in terms of goodness of fit. SAR_vario was marginally better than SAR_err in the degraded datasets.
Main conclusions  GLS was more reliable than SAR-based models, so its use is recommended when dealing with spatially autocorrelated data.     

On the mean square error associated with adaptive generalized ridge regression

Some numerical results are presented for generalized ridge regression where the additive constants are based on the data. The adaptive estimator so obtained is compared with the least-squares estimator on the basis of mean square error (MSE). It is shown that the MSE of each component of the vector of ridge estimators may be as low as 47.1% of the variance of the corresponding component of the least squares vector or as high as 125.2%.     

On the difference between gauss-markov and least squares estimates

For the usual full rank univariate least squares regression model y = XB + e, E(e) = 0, E(ee) = A, the equality of the estimates occurs when B-B^* = (XA^?1X)^?1XA^-1y-(XX)^?1Xy = 0. A necessary and sufficient condition for this equality is that A has some N - k + 1 roots equal where N is the rank of A and k is the rank of X.     

Iterative partial least squares with right-censored data analysis: a comparison to other dimension reduction techniques

In the linear model with right-censored responses and many potential explanatory variables, regression parameter estimates may be unstable or, when the covariates outnumber the uncensored observations, not estimable. We propose an iterative algorithm for partial least squares, based on the Buckley-James estimating equation, to estimate the covariate effect and predict the response for a future subject with a given set of covariates. We use a leave-two-out cross-validation method for empirically selecting the number of components in the partial least-squares fit that approximately minimizes the error in estimating the covariate effect of a future observation. Simulation studies compare the methods discussed here with other dimension reduction techniques. Data from the AIDS Clinical Trials Group protocol 333 are used to motivate the methodology.     

A Bayesian extension of phylogenetic generalized least squares: Incorporating uncertainty in the comparative study of trait relationships and evolutionary rates

Phylogenetic comparative methods use tree topology, branch lengths, and models of phenotypic change to take into account nonindependence in statistical analysis. However, these methods normally assume that trees and models are known without error. Approaches relying on evolutionary regimes also assume specific distributions of character states across a tree, which often result from ancestral state reconstructions that are subject to uncertainty. Several methods have been proposed to deal with some of these sources of uncertainty, but approaches accounting for all of them are less common. Here, we show how Bayesian statistics facilitates this task while relaxing the homogeneous rate assumption of the well-known phylogenetic generalized least squares (PGLS) framework. This Bayesian formulation allows uncertainty about phylogeny, evolutionary regimes, or other statistical parameters to be taken into account for studies as simple as testing for coevolution in two traits or as complex as testing whether bursts of phenotypic change are associated with evolutionary shifts in intertrait correlations. A mixture of validation approaches indicates that the approach has good inferential properties and predictive performance. We provide suggestions for implementation and show its usefulness by exploring the coevolution of ankle posture and forefoot proportions in Carnivora.     

On the asymptotics of marginal regression splines with longitudinal data

There have been studies on how the asymptotic efficiency ofa nonparametric function estimator depends on the handling ofthe within-cluster correlation when nonparametric regressionmodels are used on longitudinal or cluster data. In particular,methods based on smoothing splines and local polynomial kernelsexhibit different behaviour. We show that the generalized estimationequations based on weighted least squares regression splinesfor the nonparametric function have an interesting property:the asymptotic bias of the estimator does not depend on theworking correlation matrix, but the asymptotic variance, andtherefore the mean squared error, is minimized when the truecorrelation structure is specified. This property of the asymptoticbias distinguishes regression splines from smoothing splines.     

A semi-parametric approach for comparing the estimated spectra of two stationary point processes

In this paper, we provide a semi-parametric test for the hypothesis that the spectra of two stationary point processes (SPPs) are the same. The estimates of the second-order spectral density functions of the SPPs are obtained by using two different approaches: (a) by smoothing the modified periodogram statistics using a moving average weighting scheme, (b) by employing the Welch's method on the modified periodogram statistics. The test is based on the likelihood ratio function under two alternative hypotheses. The first alternative hypothesis suggests that the ratio of the spectra is a constant (there is a shift to the power) while the second one adopts a quadratic model for the logarithmic ratio of the spectra. A comparison with a log-linear model indicates coincidence in the results. This is explained by using an illustrative example from the field of neurophysiology. It is shown that the information transferred to the spinal cord by the sensory axons, closely related with the complex physiological system called muscle spindle, under the influence of two different stimuli can be separated in two parts. The first part corresponds to the range of frequencies 0-19 Hz while the second to the range of frequencies 19-100 Hz. However, the dependence on the frequency in both parts is of a quadratic form.