A versatile digital computer program for non-linear regression analysis

1.

1. A general purpose digital computer program is described for application to biological experiments that require a non-linear regression analysis. The mathematical function, or model, to be fitted to a given set of experimental data is written as a section within the program. Given initial estimates for the parameters of the function, the program uses an iterative procedure to adjust the parameters until the sum of squares of residuals has converged to a minimum.     

OutlierD: an R package for outlier detection using quantile regression on mass spectrometry data

It is important to preprocess high-throughput data generated from mass spectrometry experiments in order to obtain a successful proteomics analysis. Outlier detection is an important preprocessing step. A naive outlier detection approach may miss many true outliers and instead select many non-outliers because of the heterogeneity of the variability observed commonly in high-throughput data. Because of this issue, we developed a outlier detection software program accounting for the heterogeneous variability by utilizing linear, non-linear and non-parametric quantile regression techniques. Our program was developed using the R computer language. As a consequence, it can be used interactively and conveniently in the R environment. AVAILABILITY: An R package, OutlierD, is available at the Bioconductor project at http://www.bioconductor.org     

A computer program suitable for fitting linear models when the dependent variable is dichotomous, polichotomous or censored survival and non-linear models when the dependent variable is quantitative.

Given a set of measurements of s explanatory variables corresponding to each experimental unit, a computer program, whose methodological background can be found in [2] has been written in FORTRAN IV language in order to perform regression analyses when the dependent variable is: (i) dichotomous; (ii) polichotomous; (iii) censored survival. In the two former the Cox's [6] linear logistic models are used while in the third one it has been resorted to the models suggested by Feigl and Zelen [8]. The statistical estimation procedure is maximum likelihood and among the different algorithms developed to reach this goal, the one published by Van der Voort and Dorpema [3], has been utilized. Furthermore, when the dependent variable is quantitative, the program is suitable to fit any function non-linear in the parameters; the pertinent function and its first and second derivatives must be provided by the user. In the present version, implemented on a Univac 1106 machine, the program fits directly the Gompertz function.     

Kinetic studies of chloride inhibition in aspartate aminotransferase activity

The inhibitive effects of chloride anion on the activity of mitochondrial aspartate aminotransferase (L-aspartate: 2-oxoglutarate-aminotransferase EC. 2.6.1.1.) from chicken (Gallus domesticus) and turkey (Maleagris gallopavo) were studied. Steady-state velocities were obtained from a wide range of chloride concentrations. The data were fitted by rational functions of 0:2 and 1:2 for chloride, using a non-linear regression program which guaranteed the fit. The goodness of fit was improved by the use of a computer program that combined model discrimination, parameter refinement and sequential design. It was concluded that chloride aspartate aminotransferase inhibition requires a minimum velocity equation of 1:2 with regard to chloride, and a plausible kinetic mechanism for this experimental result was proposed.     

The effect of pH on stomatal sensitivity to abscisic acid

Abstract. The sensitivity of stomata of Commelina communis L. to abscisic acid (ABA) was evaluated by analysing the initial rates of response to the compound at different hormone concentrations. This was carried out at pH 6.8 and pH 5.5. The data were modelled and statistically analyzed by means of a computer program employing non-linear regression techniques and step-down analysis of variance. The response kinetics as quantified in terms of three sensitivity parameters were found to differ significantly between the two pH values. This finding is discussed in relation to previous research on purified ABA-binding proteins.     

The appropriate use of regression equations for the estimation of segmental inertia parameters

Linear regression equations are commonly used in conjunction with experimental data to provide linear relationships between quantities which are dimensionally distinct. In many cases theoretical relationships between such quantities are known and can be used as a basis for non-linear regression equations. This study compares linear and non-linear approaches for estimating the segmental moments of inertia from anthropometric measurements using the data of Chandler et al. [Chandler et al. (1975) Investigation of inertial properties of the human body. AMRL Technical Report 74-137, Wright Patterson Air Force Base. OH.] Right limb data were used to derive the equations while left limb data were used as a cross-validation sample to evaluate the inertia estimates calculated from the equations. For the limb segments the standard error estimates had average values of 21% for the linear equations and 13% for the non-linear equations. Data on a 10 yr-old boy was used to compare the two approaches outside the sample range. The mean percentage residuals were 286% for the linear equations and 20% for the non-linear equations. A set of non-linear equations is provided.     

灵芝胞外多糖分批发酵非结构动力学模型

在２Ｌ搅拌发酵罐上提出了描述了灵芝胞外多糖分批发酵过程中菌球生长、底物消耗和胞外多糖形成的非结构动力学模型。首先研究了灵芝分批发酵特性,结果表明该发酵过程属菌体生长和产物形成相偶联型。然后在总结文献的基础上,运用动力学模型,经过非线性回归,得到了模型中的参数值。通过计算机模拟,证明模型预测值与实际实验值具有良好的拟合性。     

Progress-curve equations for reversible enzyme-catalysed reactions inhibited by tight-binding inhibitors.

The rate equation for a tight-binding inhibitor of an enzyme-catalysed first-order reversible reaction was used to derive two integrated equations. One of them covers the situations in which competitive, uncompetitive or non-competitive inhibition occurs and the other refers to the special non-competitive case where the two inhibition constants are equal. For these equations, graphical and non-linear regression methods are proposed for distinguishing between types of inhibition and for calculating inhibition constants from progress-curve data. The application of the non-linear regression to the analysis of stimulated progress curves in the presence of a tight-binding inhibitor is also presented. The results obtained are valid for any type of 'dead-end'-complex-forming inhibitor and can be used to characterize an unknown inhibitor on the basis of progress curves.     

Simulation of biochemical systems

A computer program has been developed which solves sets of first-order non-linear differential equations which are typical of those encountered in biochemistry. The program includes its own algebraic compiler, which writes the subroutines needed by the integration routine, which is a fourth-order Runge-Kutta process incorporating variable step-length. The routine is simple to use and very fast in operation.     

Kinetic studies of chicken and turkey liver mitochondrial aspartate aminotransferase.

The kinetic behaviour of chicken liver and turkey liver aspartate aminotransferases (L-aspartate:2-oxoglutarate aminotransferase, EC 2.6.1.1) was studied. Steady-state data were obtained from a wide range of concentrations of substrates and product L-glutamate. The data were fitted by rational functions of degree 1:1, 1:2 and 2:2 with respect to substrates and 0:1, 1:1, 0:2 and 1:2 with regard to product (L-glutamate), by using a non-linear regression program that guarantees the fit. The goodness of fit was improved by the use of a computer program that combines model discrimination parameter refinement and sequential experimental design. It was concluded that aspartate aminotransferase requires a minimum velocity equation of degree 2:2 for L-aspartate, 2:2 for 2-oxoglutarate and 1:2 for L-glutamate. Finally, a plausible kinetic mechanism that justifies these experimental results is proposed.     

Merging Spline Approximations with Analysis of Nonlinear Regression Models

The paper outlines a method of analysis of intricically non-linear regression functions. The nonlinear model is first approximated by a cubic spline function. Thereafter, the standard methods of analysis in linear models are applied to obtain estimates and test statistics.     

Determination of photoperiod response characteristics using a greenhouse

The photoperiod response characteristics of two cultivars of seven long-day species of crop plants were determined by seeding at 33 biweekly intervals in a greenhouse at 49? 43′ N Latitude. Parameters of a segmented linear regression were estimated using a non-linear computer program. This gave good estimates of the length of the basic vegetative and reproductive phases, the photoperiod sensitivity and minimum optimal photoperiods. It was apparent that the photoperiod of importance was that of the day of floral initiation. If the plant is old enough and the photoperiod of the day is not inhibitory, initiation will occur. At certain times of the year, the photoperiod changes so rapidly that flower initiation is prevented. Using this method of analysis, comparisons of large numbers of genotypes can be made.     

Comparison of non-linear and linear models for estimating haemoglobin adduct stability

According to the kinetic theory for the build-up and elimination of haemoglobin (Hb) adducts, unstable Hb adducts are simultaneously eliminated by zero-order Hb turnover and first-order chemical instability. Thus, the elimination of unstable Hb adducts is non-linear with respect to time. Nonetheless, many studies of Hb adduct stability have characterized the elimination of Hb adducts using linear zero-order or linear first-order models. This paper demonstrates the use of non-linear regression to estimate the first-order rate constant of Hb adduct instability (k) using data on the elimination of Hb adducts in rats dosed with benzene or ortho -toluidine. Results obtained using non-linear regression models are compared with results from the more commonly employed zero- and first-order linear models. It is shown that exposure estimates based on measured levels of unstable Hb adducts can be severely biased if zero-order turnover is assumed. Furthermore, based on published data, estimates of k are subject to estimated relative biases in the range of -4% to 96% when first-order linear models are used to characterize Hb adduct instability.     

Comparison of non-linear and linear models for estimating haemoglobin adduct stability

According to the kinetic theory for the build-up and elimination of haemoglobin (Hb) adducts, unstable Hb adducts are simultaneously eliminated by zero-order Hb turnover and first-order chemical instability. Thus, the elimination of unstable Hb adducts is non-linear with respect to time. Nonetheless, many studies of Hb adduct stability have characterized the elimination of Hb adducts using linear zero-order or linear first-order models. This paper demonstrates the use of non-linear regression to estimate the first-order rate constant of Hb adduct instability (k) using data on the elimination of Hb adducts in rats dosed with benzene or ortho -toluidine. Results obtained using non-linear regression models are compared with results from the more commonly employed zero- and first-order linear models. It is shown that exposure estimates based on measured levels of unstable Hb adducts can be severely biased if zero-order turnover is assumed. Furthermore, based on published data, estimates of k are subject to estimated relative biases in the range of -4% to 96% when first-order linear models are used to characterize Hb adduct instability.     

Statistical testing of equality of two break-points in experimental data.

Two examples in quantitative biology are examined to emphasize the need for two-phase regression models: the osmotic behaviour of cells and the non-linear temperature kinetics of membrane-bound enzyme systems. Existing statistical techniques are inadequate to test the equality of break-points of two data sets for specific reasons. We suggest here a pragmatic solution by way of a computer programme useful in applying two-phase regression models to such data sets wherein a decision needs to be made whether the critical transition differs or not.     

Genome-enabled prediction of genetic values using radial basis function neural networks

The availability of high density panels of molecular markers has prompted the adoption of genomic selection (GS) methods in animal and plant breeding. In GS, parametric, semi-parametric and non-parametric regressions models are used for predicting quantitative traits. This article shows how to use neural networks with radial basis functions (RBFs) for prediction with dense molecular markers. We illustrate the use of the linear Bayesian LASSO regression model and of two non-linear regression models, reproducing kernel Hilbert spaces (RKHS) regression and radial basis function neural networks (RBFNN) on simulated data and real maize lines genotyped with 55,000 markers and evaluated for several trait-environment combinations. The empirical results of this study indicated that the three models showed similar overall prediction accuracy, with a slight and consistent superiority of RKHS and RBFNN over the additive Bayesian LASSO model. Results from the simulated data indicate that RKHS and RBFNN models captured epistatic effects; however, adding non-signal (redundant) predictors (interaction between markers) can adversely affect the predictive accuracy of the non-linear regression models.     

A microcomputer program for analysis of nucleic acid hybridization data

The study of nucleic acid hybridization is facilitated by computer mediated fitting of theoretical models to experimental data. This paper describes a non-linear curve fitting program, using the `Patternsearch' algorithm, written in BASIC for the Apple II microcomputer. The advantages and disadvantages of using a microcomputer for local data processing are discussed.     

A structured growth model for Cynara cardunculus cell suspension

In this study a model was developed to describe the growth of Cynara cardunculus L. suspended cells as a function of the availability of two substrates, sucrose as the carbon and energy source and phosphate. It was assumed that the maintenance energy need was fulfilled by the consumption of extracellular carbohydrates, in non-limiting conditions, or by the consumption of structural biomass when sucrose is depleted. A production of secondary metabolites was also assumed. This model was developed based on a structured model previously described by Van Gulik et al. (1993). The model was applied to the experimental results of C. cardunculus suspended cells grown in a Gamborg B₅ medium supplemented with 2% sucrose, using a non-linear regression program.     

A comparison of methods for estimating the kinetic parameters of two simple types of transport process

Sets of experimental data, with known characteristics and error structures, have been simulated for the Michaelis-Menten equation plus a second term, either for linear transport or for competitive inhibition. The Michaelis-Menten equation plus linear term was fitted by several methods and the accuracy and the precision of the parameter estimates from the several methods were compared. The model-fitting methods were: three for least-squares non-linear regression, computer versions of two graphical methods and of two non-parametric methods. The most precise and accurate method was that of D.W. Marquardt (J. Soc. Ind. Appl. Math. 11 (1963) 431–441). The Michaelis-Menten equation with competitive inhibition was also fitted by several methods, viz., two for least-squared non-linear regression, a non-parametric method and four variants of the Preston-Schaeffer-Curran plot (Preston, R.L. et al. (1974) J. Gen. Physiol. 64, 443–467). The most precise and accurate of these was the non-linear regression method of W.W. Cleland (Adv. Enzymol. 29 (1967) 1–32). For both these models, the various graphical methods and non-parametric methods gave poor results and are not recommended.     

A computer program for the implicit regression of the Hill equation]

A computer program for implicit regression of the Hill equation. The fundamentals for the regression of implicit functions following the Gauss method are dealt with. Basing on these fundamentals a program for the regression of the Hill equation is described. The binding constant, the Hill coefficient and end extinction can be estimated. The mean errors of parameters were calculated in the linear model. The program was written in ALGOL 60 for the computer Robotron 300. The program is available on request.