Multiplicative Error Term Research Articles

An alternative approximation to consumer durables expenditures

Consumer durables expenditures are normally assumed to be of linear form with additive error term. Hansen and Singleton (1983) derive the log linear form for consumption, but they do not employ a depreciation rate. In this article, we use a multiplicative error term to obtain a log linear AR(1) with a unit root as an approximate process that drives durables expenditure. The Box–Cox test rejects the linear form in favour of the log linear one. The Box–Jenkins model selection procedure and the augmented Dickey–Fuller test support an AR(1) in log linear form.

Statistical estimation of gene expression using multiple laser scans of microarrays

We propose a statistical model for estimating gene expression using data from multiple laser scans at different settings of hybridized microarrays. A functional regression model is used, based on a non-linear relationship with both additive and multiplicative error terms. The function is derived as the expected value of a pixel, given that values are censored at 65 535, the maximum detectable intensity for double precision scanning software. Maximum likelihood estimation based on a Cauchy distribution is used to fit the model, which is able to estimate gene expressions taking account of outliers and the systematic bias caused by signal censoring of highly expressed genes. We have applied the method to experimental data. Simulation studies suggest that the model can estimate the true gene expression with negligible bias. FORTRAN 90 code for implementing the method can be obtained from the authors.

Statistical inference for calibration points in nonlinear mixed effects models

Many disciplines conduct studies in which the primary objectives depend on inference based on a nonlinear relationship between the treatment and response. In particular, interest often focuses on calibration—that is, estimating the best treatment level to achieve a particular result. Often, data for such calibration come from experiments with split-plots or other features that result in multiple error terms or other nontrivial error structures. One such example is the time-of-weed-removal study in weed science, designed to estimate the critical period of weed control. Calibration, or inverse prediction, is not a trivial problem with simple linear regression, and the complexities of experiments such as the time-of-weed-removal study further complicate the procedure. In this article, we extend existing calibration techniques to nonlinear mixed effects models, and illustrate the procedure using data from a time-of-weed-removal study.

Mixed models: getting the best use of parasitological data.

Statistical analysis of parasitological data provides a powerful method for understanding the biological processes underlying parasite infection. However, robust and reliable analysis of parasitological data from natural and experimental infections is often difficult where: (1) the distribution of parasites between hosts is aggregated; (2) multiple measurements are made on the same individual host in longitudinal studies; or (3) data are from 'noisy' natural systems. Mixed models, which allow multiple error terms, provide an excellent opportunity to overcome these problems, and their application to the analysis of various types of parasitological data are reviewed here.

USING PROC NLMIXED TO ANALYZE A TIME OF WEED REMOVAL STUDY

Many studies in weed science involve fitting a nonlinear model to experimental data. Examples of such studies include dose-response experiments and studies to determine the critical period of weed control. The experiments typically use block designs and often have additional complexity such as split-plot features. However, nonlinear models are typically fit using software such as SAS PROC NLIN that are limited to a single error term and whose ability to account for blocking is either awkward or lacking entirely. For example, Seefeldt et al. (1995) only proceeded in fitting the nonlinear model after establishing that the block effect was negligible. Issues such as multiple error terms in split-plot designs are simply not dealt with at all. In this paper, we examine a weed removal study carried out as a split-plot design with blocks and illustrate the use of SAS PROC NLMIXED to account for blocks and the two-level error structure.

Modelling health-related performance indices

Primary objective: The purpose of the present review is to critically examine the various models used to describe health-related performance indices (e.g. grip-strength, lung function, arterial blood pressure, anaerobic and aerobic power, etc.). Main outcomes and results: The majority of health-related performance indices are known to increase proportionally with body size and decline with age, although some, for example arterial blood pressure and cholesterol, increase with both age and excessive body weight. Historically, the confounding effects of body size and age have been controlled using multiple linear or polynomial regression models assuming a constant additive error variance. Based on maximum likelihood goodness-of-fit criterion, an alternative class of allometric models with a multiplicative error term are shown to describe such proportional relationships more adequately in both cross-sectional and longitudinal studies. Conclusions: When modelling health-related performance indices, multiplicative, allometric models incorporate the concept of a proportional association as an integral part of its model form. Such models guarantee realistic values of health-related response variables both within and beyond the range of observations, ensuring biologically sound estimates of the response variables for subjects of all ages. After a logarithmic transformation, the models can be fitted using ordinary linear regression that will naturally help to overcome the heteroscedasticity observed in such data. However, if the logarithmic transformation fails to provide a constant error variance, an alternative approach is to model the error variance itself using log-linear regression. For longitudinal data, the use of multilevel modelling provides an alternative method of modelling error variation observed in health-related performance indices.

Modeling the influence of body size on V(O2) peak: effects of model choice and body composition.

This study examined the bivariate relationship between peak oxygen uptake (V(O2) peak); l/min) and body size in adult men (n = 1,314, age 17-66 yr), using both "simple" and "full" iterative nonlinear allometric models. The simple model was described by V(O2) peak = M(b) (or FFM(b)) exp(c SR-PA) exp(a + d age) epsilon (where M is body mass in kg; FFM is fat-free mass in kg; SR-PA is self-reported physical activity; epsilon is a multiplicative error term; and exp indicates natural antilogarithms). The full model was described by V(O2) peak = M(b) (or FFM(b)) exp(c SR-PA) exp(a + d age) + e (epsilon), where e is a permitted Y-intercept term. The M exponent obtained from simple allometry was 0.65 [95% confidence interval (CI), 0.59-0.71], suggestive of a curvilinear relationship constrained to pass through the origin. This "zero Y-intercept" assumption was examined via the full allometric model, which revealed an M exponent of 1.00 (95% CI, 0.7-1.31), together with a positive Y-intercept term (e) of 1.13 (95% CI, 0.54-1.73). The FFM exponents were not significantly different from unity in either the simple or full allometric models. It appears that the curvilinearity of the simple allometric model (using total M) is fictitious and is due to the inappropriate forcing of the regression line through the origin. Utilizing FFM as the body-size variable revealed a linear relationship between body size and V(O2) peak, irrespective of model choice. We conclude that the population mass exponent for V(O2) peak is close to unity.

Correct and Incorrect Error Specifications in Statistical Cost Models

A functional form often employed in cost-estimating relationships is Y = otYP, where Y and X are the cost and predictor variables, respectively, and a and (3 are parameters to be estimated from empirical data. Logarithmic transformation of either side of this intrinsically linear relationship leads to 7* a* + fL¥*, with the asterisks denoting logs. What typically happens in practice is that ordinary least squares (OLS) regression is applied to a set of observations on Y* and X*, generating estimates of a* and p. From a statistical point of view, however, justification of this process requires the presence of a multiplicative random error term in the untransformed model. In addition, the multiplicative structure gives rise to variance in cost that changes systematically with X. If the error term is additive, the variance in cost will be constant, but the parameters in that model must be estimated by nonlinear least squares (NLS) applied to the untransformed variables. In general, OLS and NLS lead to different parameter estimates and different cost predictions, and frequently analysts are unsure as to which method is correct. This paper reports on a Monte Carlo study of the accuracy of each estimation method under correct and incorrect error specifications. The multi-predictor model that serves as the basis of the study is representative of models with which cost analysts frequently deal. Results suggest that OLS, applied to the logs of the variables, may be the preferred method under either specification, provided the estimate of a is properly developed. This finding, however, is applicable strictly to the issue of parameter estimation. Accuracy of cost prediction requires further analysis.

Losses from effluent taxes and quotas under uncertainty

Recent theoretical papers by Adar and Griffin (J. Environ. Econ. Manag.3, 178–188 (1976)), Fishelson (J. Environ. Econ. Manag.3, 189–197 (1976)), and Weitzman (Rev. Econ. Studies41, 477–491 (1974)) show that,different expected social losses arise from using effluent taxes and quotas as alternative control instruments when marginal control costs are uncertain. Key assumptions in these analyses are linear marginal cost and benefit functions and an additive error for the marginal cost function (to reflect uncertainty). In this paper, empirically derived nonlinear functions and more realistic multiplicative error terms are used to estimate expected control and damage costs and to identify (empirically) the mix of control instruments that minimizes expected losses.

Comparison of the Logistic and the Gompertz Growth Functions Considering Additive and Multiplicative Error Terms

Both the logistic growth function and the Gompertz growth function are used to describe growth processes. In this paper it is argued that, although the logistic has been more studied, the Gompertz growth function may better meet the features of some growth processes. In order to make possible the comparison between both growth functions when an additive error is assumed, a method of fitting the Gompertz growth function under homoscedasticity is described herein. Two numerical examples and a brief discussion of the results are also provided.