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PubMed2006

Pseudo-second order models for the adsorption of safranin onto activated carbon: comparison of linear and non-linear regression methods.

Kinetic experiments were carried out for the sorption of safranin onto activated carbon particles. The kinetic data were fitted to pseudo-second order model of Ho, Sobkowsk and Czerwinski, Blanchard et al. and Ritchie by linear and non-linear regression methods. Non-linear method was found to be a better way of obtaining the parameters involved in the second order rate kinetic expressions. Both linear and non-linear regression showed that the Sobkowsk and Czerwinski and Ritchie's pseudo-second order models were the same. Non-linear regression analysis showed that both Blanchard et al. and Ho have similar ideas on the pseudo-second order model but with different assumptions. The best fit of experimental data in Ho's pseudo-second order expression by linear and non-linear regression method showed that Ho pseudo-second order model was a better kinetic expression when compared to other pseudo-second order kinetic expressions.

Adsorption↗

PubMed1998

Identification of persistent trophoblastic diseases based on a human chorionic gonadotropin regression curve by means of a stepwise piecewise linear regression analysis after the evacuation of uneventful moles.

Among the 191 patients with complete hydatidiform moles who were diagnosed and treated at Kyushu University Hospital from 1982 until 1996, 167 patients were diagnosed with uneventful moles retrospectively. The serial beta human chorionic gonadotropin (hCG) values in the 167 patients with uneventful moles were analyzed by a stepwise piecewise linear regression analysis in order to establish a normal regression curve of a human chorionic gonadotropin after a molar pregnancy. This normal regression curve is considered to be excellent regarding sensitivity (24/24-100%) and to be equivalent to the identification based on a plateau or a rise regarding specificity (156/167-93.4%). To distinguish patients with persistent trophoblastic disease (PTD) from uneventful moles, this normal curve is thus considered to be accurate since the accuracy was 180/191 (94. 2%). The weeks exceeding the normal regression curve in 24 PTD patients were 5.04 +/- 3.85 weeks and were also earlier than the weeks based on a plateau or a rise (P = 0.01). Within 7 weeks after evacuation, in 21/24 (87.5%) of the PTD cases, the beta-hCG values exceeded the normal range, while in only 14/24 (58.3%) the beta-hCG showed a change in the shape of a plateau or a rise. In addition, in 19/24 (79.2%) of the PTD patients, the time exceeding the normal range was shorter than the time exhibiting a plateau or a rise in the beta-hCG change. The above findings thus led us to conclude that this normal regression curve was useful for discriminating PTD from uneventful moles more precisely and more quickly than by identification based on a plateau or a rise.

Chorionic Gonadotropin↗

PubMed1997

Accuracy of international conversions of elite sires and cows when conversion equations are based on linear regression.

Conversion equations that are based on linear regression are used widely to transform estimated breeding values (EBV) of sires for production, type, health, and management traits from the genetic base, scale, and units of measurement of an exporting country to that of an importing country. One of the major deficiencies of these regression equations is that the accuracy of converted EBV of elite sires and cows, which are of primary interest in genetic selection programs, is lower than that for average animals. In this study, it is shown both mathematically and in practical examples that the standard error (SE) of prediction of elite dairy sires can be much larger than for average sires. When more than 100 sires are used to develop conversion equations, the SE of prediction for elite AI sires is up to 10% larger than for an average sire, and, when 50 sires are used to develop conversion equations, the SE of prediction is up to 25% larger than for an average sire. When fewer than 50 sires are used to develop conversion equations, the accuracy of converted EBV of elite sires is very poor, and SE can be 30 to 60% larger than for an average sire. Based on this study, it is recommended that international sire evaluations based on BLUP methodology (rather than linear regression) be made available as soon as possible for nonproduction traits in all countries and for production traits in countries that currently do not participate in routine INTERBULL (International Bull Evaluation Service) analyses.

Animals↗

PubMed2006

A simple linear regression method for quantitative trait loci linkage analysis with censored observations.

Standard quantitative trait loci (QTL) mapping techniques commonly assume that the trait is both fully observed and normally distributed. When considering survival or age-at-onset traits these assumptions are often incorrect. Methods have been developed to map QTL for survival traits; however, they are both computationally intensive and not available in standard genome analysis software packages. We propose a grouped linear regression method for the analysis of continuous survival data. Using simulation we compare this method to both the Cox and Weibull proportional hazards models and a standard linear regression method that ignores censoring. The grouped linear regression method is of equivalent power to both the Cox and Weibull proportional hazards methods and is significantly better than the standard linear regression method when censored observations are present. The method is also robust to the proportion of censored individuals and the underlying distribution of the trait. On the basis of linear regression methodology, the grouped linear regression model is computationally simple and fast and can be implemented readily in freely available statistical software.

Chromosome Mapping↗

PubMed2004

Reliability of calculating the cepstral peak without linear regression analysis.

Measures of cepstral peak prominence, using the smoothing algorithm and linear regression analysis software developed by Hillenbrand, have been shown to be reliable predictors of dysphonia in voice samples.(1-4) Recently, the Computerized Speech Laboratory [(CSL) Kay Elemetrics, Pinebrook, New Jersey] has introduced cepstral analysis as a component of that software package. The cepstral peak, in this instance, is calculated by the voice clinician analyzing the phonatory sample by subtracting the value of the peak from the apparent baseline signal. This study compares the ability of cepstral peak values calculated from the CSL software to predict dysphonia reliably with that of the values produced by the smoothing algorithm and linear regression analysis of Hillenbrand. The results of this study show that linear regression analysis is an important step in calculating the cepstral peak prominence, thus limiting the usefulness of software programs that do not employ this step.

Algorithms↗

PubMed1989

Comparison of linear regression methods when both variables contain error: relation to clinical studies.

Five common linear regression methods were evaluated for their ability to determine the correct values of slope and intercept of a known function after random errors were added to x and y. The error variances were controlled to simulate research problems commonly studied by linear regression. The total error of each method was assessed by the absolute value of the bias in the estimate of slope. Whenever differences among methods were observed, the mean of the slope determined by two reciprocal techniques performed as well as or better than orthogonal regression, regression of y upon x, or x upon y. All the methods studied appeared to perform equally well when x and y errors were heteroscedastic or when the data set was small (n = 7). Regression of y upon x was equal or superior to other methods when n = 7 or n = 20 and y and x errors were homoscedastic. When the data set was large (n = 50) and the error in x greater than that in y, the standard method (regression of y upon x) was inferior to all other methods. It is suggested that linear regression by the traditional method of y upon x (a method present in many hand-held calculators) is appropriate in the majority of clinical situations, but when n is large and errors in x are much larger than those in y, orthogonal regression or the averaging method may be preferable.

Monte Carlo Method↗

PubMed1999

[On the problems of fitting linear regression models for hierarchically structured data in medical research].

There are a large number of the hierarchically structured data in the field of medical sciences, which have been analyzed usually by conventional linear regression models. The objective of this paper is to explore the problems and the relationship of parameter estimates of the three common linear regression models in fitting the hierarchically structured date, and the correction of the precision of parameter estimates. It is shown that the estimate of parameter and it's precision of linear regression models is related to the variation of independent variable between and within level 2 units, and the difference of residual estimates is associated with the difference of parameter estimates. The three common linear regression models are all inappropriate for the hierarchically structured data, but the standard error of the level 1 combined model can be corrected by variance inflation factor in conditions.

Analysis of Variance↗

PubMed1996

A note on fitting a marginal model to mixed effects log-linear regression data via GEE.

Marginal generalized linear models for correlated data are often fit by generalized estimating equations (Liang, K.Y. and Zeger, S.L., 1986, Biometrika 73, 13-22), which requires the marginal expectation to be correctly specified while the covariance structure is allowed to be misspecified. This note is concerned with the correct specification of the marginal mean for data from a mixed log-linear regression model and the relationship between the true subject-specific parameters and the true marginal parameters.

Analysis of Variance↗

PubMed2005

Mathematical algorithms applied to the multi-linear regression functions for the multicomponent determination of pharmaceutical dosage form containing three-component mixtures.

In the presence of closely overlapping spectra, the quantitative multiresolution of ternary mixtures of three active compounds paracetamol (PAR), caffeine (CAF) and acetylsalycilic acid (ASP) in tablets, without using pretreatment such as separation step and graphical procedure of spectra was accomplished by the multivariate spectral calibration models, tri-linear regression calibration (TLRC), multi-linear regression calibration (MLRC) and Cramer's rule solution (CRS) of three linear equation functions in the matrix form. In the first two models, TLRC and MLRC are based on the use of the linear regression functions at selected wavelength sets in the spectral region of 210-300 nm. In the case of CRS model, A1(1) (1%, 1 cm) were used to obtain three linear equation functions and this linear equation system was resolved by the Cramer's rule for the prediction of PAR, CAF and ASP in samples. In the TLRC and CRS models, the selection of the appropriate wavelength set was performed by the Kaiser's technique. The algorithms of these mathematical calibration models were briefly described. The validation of TLRC, MLRC and CRS models was carried out by analyzing various synthetic ternary mixtures and by using the standard addition technique. These three calibration approaches were applied to the analysis of the real pharmaceutical tablets containing PAR, CAF and ASP. The obtained results were statistically compared with each other by using experimental and statistical tests. In the comparison of TLRC and MLRC models to the classical approach, CRS technique, the successful assay results were observed for the quantitative multiresolution of ternary mixture of the subject active compounds.

Acetaminophen↗

PubMed2001

Monitoring of nonlinear respiratory elastance using a multiple linear regression analysis.

The elastic pressure/volume (P/V) curve obtained by the multiple linear regression (MLR) technique using a new model, was compared with the quasi-static P/V points obtained by the rapid airway occlusion technique. Seven infants were studied during mechanical ventilation using a pressure controlled mode. The resistive pressure was subtracted from airway opening pressure, thus determining the elastance related pressure, which was then plotted against the volume to make an MLR-elastance curve. Quasi-static P/V curves of the rapid occlusion technique were constructed by plotting the different inspiratory and expiratory volumes against the corresponding values of the quasi-static airway pressure. The calculated MLR-elastance curves closely fit the experimental quasi-static P/V points obtained by the occlusion technique. There were, however, some discrepancies due to the viscoelastic behaviour of the respiratory system. Although slightly altered by these discrepancies, the multiple linear regression-elastance curves did fit the observed quasi-static pressure/volume characteristics for use in clinical practice. The multiple linear regression technique may prove to be clinically useful by continuous monitoring of respiratory system mechanics during mechanical ventilation.

Airway Resistance↗

PubMed2002

Linear regression for calibration lines revisited: weighting schemes for bioanalytical methods.

When the assumption of homoscedasticity is not met for analytical data, a simple and effective way to counteract the greater influence of the greater concentrations on the fitted regression line is to use weighted least squares linear regression (WLSLR). The purpose of the present paper is to stress the relevance of weighting schemes for linear regression analysis and to show how this approach can be useful in the bioanalytical field. The steps to be taken in the study of the linear calibration approach are described. The application of weighting schemes was shown by using a high-performance liquid chromatography method for the determination of lamotrigine in biological fluids as a practical example. By using the WLSLR, the accuracy of the analytical method was improved at the lower end of the calibration curve. Bioanalytical methods data analysis was improved by using the WLSLR procedure.

Calibration↗

PubMed2004

Linear regression approach to study amino acid digestibility in broiler chickens.

1. An experiment was conducted to investigate whether a linear regression approach is a suitable tool for determining the amino acid (AA) digestibility up to the terminal ileum of broiler chickens. Solvent-extracted rapeseed meal (RSM) was used as the model ingredient. 2. Ten diets with 5 different inclusion rates of RSM (60, 120, 180, 240 and 300 g/kg, corresponding to crude protein concentrations from 170 to 250 g/kg in the diet), each without or with a supplementation of phytase (500 U/kg), were fed ad libitum to broiler chickens between 14 and 21 d of age. Seven pens of 12 chickens were allocated to each treatment. Digesta were sampled on a pen basis from the section of the gastrointestinal tract between Meckel's diverticulum and 2 cm anterior to the ileo-caeco-colonic junction. Titanium dioxide was included as an indigestible marker. 3. The amounts of crude protein and AAs digested up to the terminal ileum constantly increased with increasing AA intake over the entire range of intakes. When the amount of an AA digested at the terminal ileum is linearly regressed against its intake, the deviation of the slope from 1 is caused by both the unabsorbed AA from RSM and from specific endogenous losses related to RSM. These slopes varied between 0.68 and 0.88 for individual AAs, and the slopes were unaffected by phytase supplementation. 4. It is suggested that a linear regression approach be adopted to study the AA digestibility of raw materials in chickens. Digestibility determined this way does not need any correction for basal endogenous loss.

6-Phytase↗

PubMed1997

A software program based on robust linear regression methods for instrumental calibration in biochemical analysis.

A microcomputer program written in GWBASIC for conventional least squares regression (LS) and robust linear regression methods (RRMs) such as single median (SM) and repeated median (RM) is described. The main advantages of RRMs is the ability to detect outliers which can cause an inaccurate estimation of the analyzed samples. It requires MS DOS PC version 3.3 or higher and at least 256 kbyte of RAM. The joint application of RRMs and residuals analysis by the studentized method (STRM) allows a more precise biochemical analysis.

Biochemistry↗

PubMed2006

Pseudo second order kinetics and pseudo isotherms for malachite green onto activated carbon: comparison of linear and non-linear regression methods.

Pseudo second order kinetic expressions of Ho, Sobkowsk and Czerwinski, Blanachard et al. and Ritchie were fitted to the experimental kinetic data of malachite green onto activated carbon by non-linear and linear method. Non-linear method was found to be a better way of obtaining the parameters involved in the second order rate kinetic expressions. Both linear and non-linear regression showed that the Sobkowsk and Czerwinski and Ritchie's pseudo second order model were the same. Non-linear regression analysis showed that both Blanachard et al. and Ho have similar ideas on the pseudo second order model but with different assumptions. The best fit of experimental data in Ho's pseudo second order expression by linear and non-linear regression method showed that Ho pseudo second order model was a better kinetic expression when compared to other pseudo second order kinetic expressions. The amount of dye adsorbed at equilibrium, q(e), was predicted from Ho pseudo second order expression and were fitted to the Langmuir, Freundlich and Redlich Peterson expressions by both linear and non-linear method to obtain the pseudo isotherms. The best fitting pseudo isotherm was found to be the Langmuir and Redlich Peterson isotherm. Redlich Peterson is a special case of Langmuir when the constant g equals unity.

Adsorption↗

PubMed1983

Prediction of body volume by a stepwise linear regression technique.

Body volume and 35 anthropometric measurements were obtained from 88 active soldiers using standard techniques. These anthropometric measurements were examined for their possible relationships to body volume using stepwise linear regression analysis. Four measurements (Body weight, anterior thigh skinfold thickness, subscapular skinfold thickness and suprailiac skinfold thickness) accounted for 99.7% of the variation in body volume and the introduction of each of these measurements in the equation was significant. The regression equation for predicting body volume from these 4 anthropometric measurements had a multiple correlation coefficient of 0.9987 (P less than 0.001). Body weight alone was correlated with body volume to the extent of 0.9966. An attempt has therefore been made to develop a multiple linear regression equation without incorporation of body weight in the regression analysis. Nine measurements were selected by stepwise linear regression analysis for predicting body volume. These nine measurements accounted for 97.1% of the variation in body volume. These equations have been validated on another small sample of 22 soldiers. The analysis has also revealed that a direct regression of body density from the anthropometric variables gives more accurate results than when estimated body volumes are utilized for calculating body density.

Adolescent↗

PubMed2007

Multiple linear regression with some correlated errors: classical and robust methods.

In this paper we consider classical and robust methods of estimation and diagnostics for the multiple linear regression model when some of the errors are correlated. This work was motivated by the analysis of a medical data set, from an observational study aimed at identifying factors affecting the outcome of a surgical method for the correction of scoliosis (abnormal lateral spinal curvature). There are 392 observations but some of them are on the same patient (double curves). It seems adequate to consider a multiple linear regression model but, since it is not desirable to discard the double curves, the assumption of non-correlated errors is clearly violated, and this is indeed confirmed by related diagnostics on the residuals (Durbin-Watson test). A more appropriate model retains the linear structure but allows for non-null correlation between the errors on the same patient. We propose two different procedures for the estimation of the parameters of the linear model and the correlation parameters: maximum likelihood assuming normal errors and a robustified version obtained by plugging-in results from robust linear regression. The latter procedure is designed to be resistant to outlying observations or error distributions with heavy tails and has produced the most satisfactory results for the analysed data set.

Data Interpretation, Statistical↗

PubMed1979

Methodological problems arising from the choice of an independent variable in linear regression, with application to an air pollution epidemiological study.

In epidemiological studies using linear regression, it is often necessary for reasons of economy or unavailability of data to use as the independent variable not the variable ideally demanded by the hypothesis under study but some convenient practical approximation to it. We show that if the correlation coefficient between the "practical" and "ideal" variables can be obtained, then a range of uncertainty can be obtained within which the desired regression coefficient of dependent on "ideal" variable may lie. This range can be quite wide, even if the practical and ideal variables are fairly well correlated. These points are illustrated with data on observed regression coefficients from an air pollution epidemiological study, in which pollution measured at one station in a large metropolitan area (containing 40 aerometric stations) was used as the practical approximation to the city-wide average pollution. The uncertainties in the regression coefficients were found to exceed the regression coefficients themselves by large factors. The problem is one that may afflict application of linear regression in general, and suggests caution when selecting independent variables for regression analysis on the basis of convenience, rather than relevance to the hypotheses tested.

Air Pollution↗

PubMed1997

Linear regression estimation of minimal detectable concentration. Thyrotropin as an example.

BACKGROUND: Minimal detectable concentration is an important analytic feature of certain clinical immunoassays. We believe that accuracy is an important component of the minimal detectable concentration; for a given observed concentration to be meaningful, it should reflect a consistent linear relationship with the amount of analyte actually present. METHODS: To evaluate the minimal detectable concentration, we developed a linearity regression protocol based on accuracy and also accounting for between-run variability. Using serial twofold dilutions of serum samples, we regressed the log of concentration (x) and of dilution (y) with linear, second-, and third-order polynomials. Initially, we evaluated two elements to find the linear region of the dataset, establishing the statistical significance of the beta coefficients with a t test and the reduction of the sum of square of the residuals between the linear regression and the higher-order regressions by means of an F test. As needed, we successively eliminated the lowest point until the linear regression was the best fit. Once we found the best fit, we added the most recently removed point back and calculated the difference between the value predicted by the first-order regression and the observed value. If the difference was not analytically significant, then we considered the point to be part of the linear set; otherwise, it was not included. In either case, the lowest included point was considered to be the minimal detectable concentration. RESULTS: We applied the technique in evaluating two automated systems for serum thyrotropin. One system appeared linear and accurate down to 0.02 mU/L, or better, approximately 77% of the time, and to 0.01 mU/L 68% of the time. The second system was linear infrequently and appeared to be useful down to 0.02 mU/L, or better, only about 20% of the time. CONCLUSIONS: This accuracy-based approach to determining the minimal detectable concentration is an attractive alternative to current empiric approaches, which are based only on interassay variability.

Humans↗