Classification trees aided mixed regression model

This paper introduces a novel hybrid regression method (MixReg) combining two linear regression methods, ordinary least square (OLS) and least squares ratio (LSR) regression. LSR regression is a method to find the regression coefficients minimizing the sum of squared error rate while OLS minimizes the sum of squared error itself. The goal of this study is to combine two methods in a way that the proposed method superior both OLS and LSR regression methods in terms of R² statistics and relative error rate. Applications of MixReg, on both simulated and real data, show that MixReg method outperforms both OLS and LSR regression.     

Measurement error in regression analysis

Consider the linear regression model Y = Xθ+ ε where Y denotes a vector of n observations on the dependent variable, X is a known matrix, θ is a vector of parameters to be estimated and e is a random vector of uncorrelated errors. If X'X is nearly singular, that is if the smallest characteristic root of X'X s small then a small perurbation in the elements of X, such as due to measurement errors, induces considerable variation in the least squares estimate of θ. In this paper we examine for the asymptotic case when n is large the effect of perturbation with regard to the bias and mean squared error of the estimate.     

Geometric characterization of planar regression

The geometric characterization of linear regression in terms of the ‘concentration ellipse’ by Galton [Galton, F., 1886, Family likeness in stature (with Appendix by Dickson, J.D.H.). Proceedings of the Royal Society of London, 40, 42–73.] and Pearson [Pearson, K., 1901, On lines and planes of closest fit to systems of points in space. Philosophical Magazine, 2, 559–572.] was extended to the case of unequal variances of the presumably uncorrelated errors in the experimental data [McCartin, B.J., 2003, A geometric characterization of linear regression. Statistics, 37(2), 101–117.]. In this paper, this geometric characterization is further extended to planar (and also linear) regression in three dimensions where a beautiful interpretation in terms of the concentration ellipsoid is developed.     

The peculiar shrinkage properties of partial least squares regression

Partial least squares regression has been widely adopted within some areas as a useful alternative to ordinary least squares regression in the manner of other shrinkage methods such as principal components regression and ridge regression. In this paper we examine the nature of this shrinkage and demonstrate that partial least squares regression exhibits some undesirable properties.     

Tests of regression coefficients under ridge regression models

This paper investigates two “non-exact” t-type tests, t( k₂) and t(k₂), of the individual coefficients of a linear regression model, based on two ordinary ridge estimators. The reported results are built on a simulation study covering 84 different models. For models with large standard errors, the ridge-based t-tests have correct levels with considerable gain in powers over those of the least squares t-test, t(0). For models with small standard errors, t(k₁) is found to be liberal and is not safe to use while, t(k₂) is found to slightly exceed the nominal level in few cases. When tie two ridge tests art: not winners, the results indicate that they don't loose much against t(0).     

An algorithm for weighted bilinear regression

Bilinear models in which the expectation of a two-way array is the sum of products of parameters are widely used in spectroscopy. In this paper we present an algorithm called combined-vector successive overrelaxation (COV-SOR) for bilinear models, and compare it with methods like alternating least squares, singular value decomposition, and the Marquardt procedure. Comparisons are done for missing data also.     

Quantile regression via iterative least squares computations

We present an estimating framework for quantile regression where the usual L ₁-norm objective function is replaced by its smooth parametric approximation. An exact path-following algorithm is derived, leading to the well-known ‘basic’ solutions interpolating exactly a number of observations equal to the number of parameters being estimated. We discuss briefly possible practical implications of the proposed approach, such as early stopping for large data sets, confidence intervals, and additional topics for future research.     

A Markov-chain-based regression model with random effects for the analysis of 18O-labelled mass spectra

The enzymatic ¹⁸O-labelling is a useful technique for reducing the influence of the between-spectra variability on the results of mass-spectrometry experiments. A difficulty in applying the technique lies in the quantification of the corresponding peptides due to the possibility of an incomplete labelling, which may result in biased estimates of the relative peptide abundance. To address the problem, Zhu et al. [A Markov-chain-based heteroscedastic regression model for the analysis of high-resolution enzymatically ¹⁸O-labeled mass spectra, J. Proteome Res. 9(5) (2010), pp. 2669–2677] proposed a Markov-chain-based regression model with heteroscedastic residual variance, which corrects for the possible bias. In this paper, we extend the model by allowing for the estimation of the technical and/or biological variability for the mass spectra data. To this aim, we use a mixed-effects version of the model. The performance of the model is evaluated based on results of an application to real-life mass spectra data and a simulation study.     

Penalized regression with model‐based penalties

Nonparametric regression techniques such as spline smoothing and local fitting depend implicitly on a parametric model. For instance, the cubic smoothing spline estimate of a regression function ∫ μ based on observations ti, Yi is the minimizer of Σ{Yi ‐ μ(ti)}² + λ∫(μ′′)². Since ∫(μ″)² is zero when μ is a line, the cubic smoothing spline estimate favors the parametric model μ(t) = α_o + α₁t. Here the authors consider replacing ∫(μ″)² with the more general expression ∫(Lμ)² where L is a linear differential operator with possibly nonconstant coefficients. The resulting estimate of μ performs well, particularly if Lμ is small. They present an O(n) algorithm for the computation of μ. This algorithm is applicable to a wide class of L's. They also suggest a method for the estimation of L. They study their estimates via simulation and apply them to several data sets.     

A simulation study on SPSS ridge regression and ordinary least squares regression procedures for multicollinearity data

This study compares the SPSS ordinary least squares (OLS) regression and ridge regression procedures in dealing with multicollinearity data. The LS regression method is one of the most frequently applied statistical procedures in application. It is well documented that the LS method is extremely unreliable in parameter estimation while the independent variables are dependent (multicollinearity problem). The Ridge Regression procedure deals with the multicollinearity problem by introducing a small bias in the parameter estimation. The application of Ridge Regression involves the selection of a bias parameter and it is not clear if it works better in applications. This study uses a Monte Carlo method to compare the results of OLS procedure with the Ridge Regression procedure in SPSS.     

On the efficiency of regression analysis with AR(p) errors

In this paper we will consider a linear regression model with the sequence of error terms following an autoregressive stationary process. The statistical properties of the maximum likelihood and least squares estimators of the regression parameters will be summarized. Then, it will be proved that, for some typical cases of the design matrix, both methods produce asymptotically equivalent estimators. These estimators are also asymptotically efficient. Such cases include the most commonly used models to describe trend and seasonality like polynomial trends, dummy variables and trigonometric polynomials. Further, a very convenient asymptotic formula for the covariance matrix will be derived. It will be illustrated through a brief simulation study that, for the simple linear trend model, the result applies even for sample sizes as small as 20.     

Study of partial least squares and ridge regression methods

This article considers both Partial Least Squares (PLS) and Ridge Regression (RR) methods to combat multicollinearity problem. A simulation study has been conducted to compare their performances with respect to Ordinary Least Squares (OLS). With varying degrees of multicollinearity, it is found that both, PLS and RR, estimators produce significant reductions in the Mean Square Error (MSE) and Prediction Mean Square Error (PMSE) over OLS. However, from the simulation study it is evident that the RR performs better when the error variance is large and the PLS estimator achieves its best results when the model includes more variables. However, the advantage of the ridge regression method over PLS is that it can provide the 95% confidence interval for the regression coefficients while PLS cannot.     

Estimation of the linear-linear segmented regression model in the presence of measurement error

A simple segmented regression model in which the independent variable is measured with error is considered. The method of moments is used to obtain parameter estimates and the joint asymptotic distribution of the estimators is presented. The small sample properties of the inference procedures based on the asymptotic distribution of the estimators are studied numerically.     

Trimmed least squares estimator as best trimmed linear conditional estimator for linear regression model

A class of trimmed linear conditional estimators based on regression quantiles for the linear regression model is introduced. This class serves as a robust analogue of non-robust linear unbiased estimators. Asymptotic analysis then shows that the trimmed least squares estimator based on regression quantiles ( Koenker and Bassett ( 1978 ) ) is the best in this estimator class in terms of asymptotic covariance matrices. The class of trimmed linear conditional estimators contains the Mallows-type bounded influence trimmed means ( see De Jongh et al ( 1988 ) ) and trimmed instrumental variables estimators. A large sample methodology based on trimmed instrumental variables estimator for confidence ellipsoids and hypothesis testing is also provided.     

An iterative procedure for the estimation of parameters in a dose-response model

The least squares estimates of the parameters in the multistage dose-response model are unduly affected by outliers in a data set whereas the minimum sum of absolute errors, MSAE estimates are more resistant to outliers. Algorithms to compute the MSAE estimates can be tedious and computationally burdensome. We propose a linear approximation for the dose-response model that can be used to find the MSAE estimates by a simple and computationally less intensive algorithm. A few illustrative ex-amples and a Monte Carlo study show that we get comparable values of the MSAE estimates of the parameters in a dose-response model using the exact model and the linear approximation.     

Large sample inference in random coefficient regression models

Random coefficient regression models have been used t odescribe repeated measures on members of a sample of n in dividuals . Previous researchers have proposed methods of estimating the mean parameters of such models. Their methods require that eachindividual be observed under the same settings of independent variablesor , lesss stringently , that the number of observations ,r , on each individual be the same. Under the latter restriction ,estimators of mean regression parameters exist which are consist ent as both r^→∞and n^→∞ and efficient as r^→∞, and large sample ( r large ) tests of mean parameters are available . These results are easily extended to the case where not a11 individuals are observed an equal number of times provided limit are taken as min(r) → ∞. Existing methods of inference , however, are not justified by the current literature when n is large and r is small, as is the case i n many bio-medical applications . The primary con tribution of the current paper is a derivation of the asymptotic properties of modifications of existing estimators as n alone tends to infinity, r fixed. From these properties it is shown that existing methods of inference, which are currently justified only when min(r) is large, are also justifiable when n is large and min(r) is small. A secondary contribution is the definition of a positive definite estimator of the covariance matrix for the random coefficients in these models. Use of this estimator avoids computational problems that can otherwise arise.     

A small sample estimator for a polynomial regression with errors in the variables

An adjusted least squares estimator, introduced by Cheng and Schneeweiss for consistently estimating a polynomial regression of any degree with errors in the variables, is modified such that it shows good results in small samples without losing its asymptotic properties for large samples. Simulation studies corroborate the theoretical findings.     

Comparisons among regression estimators under the generalized mean square error criterion

It is not always prossible to establish a preference ordering among regression estimators in terms of the generalized mean square error criterion. In the paper, we determine when it is feasible to use this criteion to couduct comparisons among ordinary least squares, principal components, ridge regression, and shrunken least squares estimators.     

Bandwidth selection for kernel regression with correlated errors

In this paper, we propose bandwidth selectors for nonparametric regression with dependent errors. The methods are based on criteria that approximate the average squared error. We show that these approximations are uniform over the bandwidth sequence. The criteria involve some constants that depend on the unknown error correlations. We propose a novel way of estimating these constants. Our numerical study shows that the method is quite efficient in a variety of error models.