Degree Polynomial Regression Model Research Articles

Modeling the recovery patterns from solid phase extraction purification of secoisolariciresinol diglucoside, p-coumaric acid glucoside, and ferulic acid glucoside from microwave-assisted flaxseed extracts

Flaxseed extracts obtained by microwave-assisted extraction (MAE) were cleaned of leftover salts from hydrolysis by solid phase extraction (SPE). The SPE set up was affordable, non-automated and vacuum-driven. The recovery patterns of secoisolariciresinol diglucoside (SDG), p-coumaric acid glucoside (PCouAG), and ferulic acid glucoside (FerAG) were modeled in two stages using regression procedures (p<0.05). At stage one, the recovery patterns were predicted as a function of SPE eluent concentration in ethanol (10–50%). At stage two, the accuracy of the predictions was increased by enlarging the SPE eluent regressor's range (0–100% ethanol in water) and arranging the solvent system into three practical elution groups. The groups 1, 2 and 3 reflected the major loss, the major recovery and the minor loss of SDG, respectively. Second degree polynomial regression models were fitted for accurately predicting the recoveries of compounds. Microwave-assisted extracts obtained from 0.6 and 1.5g defatted flaxseed meal were purified; the total SDG recoveries from the SPE funnel were 97.8 and 99.8%; and the SDG amounts obtained were 8.54 and 20mg, respectively. The HPLC analysis of eluates pooled into practical groups allows for significant reductions in HPLC analysis time and solvent consumption which could have a positive impact on future purification studies. The results of this study allow for designing simplified, efficient and economical pilot scale studies for the purification of SDG from flaxseed extracts.

A note on the de la Garza phenomenon for locally optimal designs

The celebrated de la Garza phenomenon states that for a polynomial regression model of degree $p-1$ any optimal design can be based on at most $p$ design points. In a remarkable paper, Yang [Ann. Statist. 38 (2010) 2499--2524] showed that this phenomenon exists in many locally optimal design problems for nonlinear models. In the present note, we present a different view point on these findings using results about moment theory and Chebyshev systems. In particular, we show that this phenomenon occurs in an even larger class of models than considered so far.

Nomogram of fetal upper arm volume by three-dimensional ultrasound using extended imaging virtual organ computer-aided analysis (XI VOCAL)

To define the reference range of the fetal upper arm volume by three-dimensional (3D) ultrasound using the eXtended Imaging Virtual Organ Computer-aided AnaLysis (XI VOCAL) method. This prospective, cross-sectional study enrolled 425 healthy pregnant women with 20-40 weeks of gestation. The XI VOCAL technique was used for the volumetric calculations, considering 10 sequential sectional plane areas, performed along the axial plan of the fetal upper arm. The proximal and distal epiphyses were the beginning and final reference. Second degree polynomial regression models were created to evaluate the correlation between the volume of the fetal arm and the gestational age (GA), determining the 5th, 10th, 25th, 50th, 75th, 90th and 95th corresponding percentiles. The intraclass correlation coefficient (ICC) was used to evaluate the intra- and interobserver reproducibility. The mean fetal upper arm volume ranged from 4.59-1.18 (3.10-7.40 cm(3)) to 53.87-10.72 cm(3) (40.30-76.60 cm(3)). The fetal upper arm volume and GA were highly associated (R(2)=0.913). The intra- and interobserver reproducibility were reliable, with ICC=0.997 (95% CI 0.995-0.999) and 0.996 (95% CI 0.993-0.998), respectively. The reference range for the fetal upper arm volume was determined by 3D-ultrasound using the XI VOCAL method, and was found to be highly reproducible.

D-optimal designs for weighted polynomial regression

By utilizing the equivalence theorem and Descartes's rule of signs, we construct D-optimal designs for a weighted polynomial regression model of degree k, with specific weight function w( x)=1/( a 2− x 2) δ , on the compact interval [−1,1]. The main result shows that in most cases, the number of support points of the D-optimal design is k+1, while in other cases, the D-optimal design has k+2 support points.

Constrained $D$- and $D_1$-optimal designs for polynomial regression

In the common polynomial regression model of degree m we consider the problem of determining the $D$- and $D_1$-optimal designs subject to certain constraints for the $D_1$-efficiencies in the models of degree $m - j, m - j + 1,\dots, m + k(m > j \geq 0, k \geq 0 \text{given})$.We present a complete solution of these problems, which on the one hand allow a fast computation of the constrained optimal designs and, on the other hand, give an answer to the question of the existence of a design satisfying all constraints. Our approach is based on a combination of general equivalence theory with the theory of canonical moments. In the case of equal bounds for the $D_1$-efficiencies the constrained optimal designs can be found explicitly by an application of recent results for associated orthogonal polynomials.

Regression models for the determination of the absorbed dose rate with an extrapolation chamber for flat ophthalmic applicators.

The average surface absorbed dose rate, given by flat ophthalmic applicators (90Sr/90Y, 925 MBq) is determined in equivalent soft tissue using an extrapolation chamber with two flat parallel electrodes of variable separation; the input electrode is fixed in relation to the collector electrode of constant area. When estimating the extrapolation curve slope using a linear regression model, it has been observed that average surface dose rate values were underestimated by up to 19%, as compared to estimations of these values by means of a second degree polynomial regression model, while an improvement of up to 37% is observed in the standard error of the slope in the quadratic model, as compared to that of the linear model. With the aim of validating the results of these models, goodness of fit tests to a Normal (the Shapiro-Wilk test) as well as homogeneity tests on treatment variance (the Bartlett test) were applied. The analysis of variance (ANOVA) tables of fit and residual error breakdown are given: table 3a and 3b for linear fit; 7a and 7b for quadratic fit, and table 10 to error breakdown. Also presented is the global uncertainty of the average dose rate, taking into account the reproducibility of the experimental set-up. It may be inferred that by using this type of measurement for the extrapolation curve slope, quadratic regression models allow for a greater degree of accuracy and precision in determining surface dose rate values. The effective area of the collector electrode and the effective electrode separation in the chamber are also determined by measuring the chamber's electric capacity. Finally, there is an attempt to relate the use of the regression models to the experimental conditions during the measurement of ionization currents (diameter of collector electrode, electrical field gradient, radiation field uniformity, radiation field intensity, etc.). In this particular case, deviations in the distance inverse square law and the "screening" effects during the collection of negative charges (both for primary radiation and the ionization generated by it), are presented as necessary, but insufficient, conditions to explain thoroughly the quadratic behavior of ionizing currents.

Comparing robust properties of A, D, E and G-optimal designs

We examine the A, D, E and G-efficiencies of using the optimal design for the polynomial regression model of degree k when the hypothesized model is of degree j and 1⩽ j⩽ k⩽8. The robustness properties of each of these optimal designs with respect to the other optimality criteria are also investigated. Relationships among these efficiencies are noted and practical implications of the results are discussed. In particular, our numerical results show E-optimal designs possess several properties not shared by the A, D and G-optimal designs.

Discrimination Designs for Polynomial Regression on Compact Intervals

In the polynomial regression model of degree $m \in \mathbb{N}$ we consider the problem of determining a design for the identification of the correct degree of the underlying regression. We propose a new optimality criterion which minimizes a weighted $p$-mean of the variances of the least squares estimators for the coefficients of $x^l$ in the polynomial regression models of degree $l = 1,\cdots, m$. The theory of canonical moments is used to determine the optimal designs with respect to the proposed criterion. It is shown that the canonical moments of the optimal measure satisfy a (nonlinear) equation and that the support points are given by the zeros of an orthogonal polynomial. An explicit solution is given for the discrimination problem between polynomial regression models of degree $m - 2, m - 1$ and $m$ and the results are used to calculate the discrimination designs in the sense of Atkinson and Cox for polynomial regression models of degree $1,\cdots,m$.

Robust designs for multivariate polynomial regression on the d-cube

For the multivariate polynomial regression model of degree n ∈ N we determine the model robust designs in the sense of Läuter ( Math. Operat. Statist. , 1974) within the class of product measures on the d -cube. The optimal product design is completely described by its canonical moments. The results hold for all d isin; N , n ∈ N and can be generalized to the case that the exponents of the components x 1 ,…, x d in the multivariate polynomial are either of odd or even order. As special cases we give explicit representations of the support points and the weights of the D- and D 1 -optimal product designs for the polynomial which components x 1 ,…, x d appear either with odd or even exponents.

On a mixture of the D- and D1-optimality criterion in polynomial regression

We consider a mixture of the D1-optimality criterion (minimizing the variance of the estimate for the highest coefficient) and the D-optimality criterion (minimizing the volume of the ellipsoid of concentration for the unknown parameter vector) in the polynomial regression model of degree n ⊆ N. The mixture is defined as a weighted product of both optimality criteria and explicit solutions are given for the proposed criterion. The derived designs have excellent efficiencies compared to the G-, D-, and D1-optimal design. This is illustrated in some examples for polynomial regression of lower degree. The optimal designs are calculated using the theory of canonical moments. Further applications are given in the field of model robust designs for polynomial regression models where only an upper bound for the degree of the polynomial regression model is known by the experimenter before the experiments are carried out.

The Standard Physical Growth Chart for Saudi Arabian Preschool Children

This study presents growth charts of preschool based on cross-sectional data of the population in Riyadh, Saudi Arabia. Charts on weights and heights per age were designed separately for boys and girls. Boys consistently weighed more than girls for this age group at every age. Also, the boys were taller than the girls after the age of one year. Since the growth deficits in children from Saudi Arabia compared with internationally recommended reference populations may be attributed to genetic and environmental factors, the charts presented will suffice as a standard in monitoring the growth of Saudi children in particular and those in the Gulf region in general.

A Generalization of $D$- and $D_1$-Optimal Designs in Polynomial Regression

In the class of polynomial regression models up to degree $n$ we determine the design on $\lbrack -1, 1\rbrack$ that maximizes a product of $n + 1$ determinants of information matrices weighted with a prior $\beta$, where the $l$-th information matrix corresponds to a polynomial regression model of degree $l$, for $l = 0, 1, \cdots, n$. The designs are calculated using canonical moments. We identify a special class of priors $\beta(z)$ depending on one real parameter $z$ so that analogous results are obtained as in the classical $D$- and $D_1$-optimal design problems. The interior support of the optimal design with respect to the prior $\beta(z)$ is given by the zeros of a Jacobi polynomial and all the interior support points have the same masses. The masses at the boundary points $-1$ and $1$ are $(z + 1)/2$ times bigger than the masses of the interior points. The results found in one dimension are generalized to the problem of determining optimal product designs in the case of multivariate polynomial regression on the $q$-cube $\lbrack -1,1\rbrack^q$. Explicit solutions are obtained for the $D$- and $D_1$-optimal product designs in the polynomial model of degree $n$ for all $n \in \mathbb{N}$ and $q \in \mathbb{N}$.