Cox Power Transformation Research Articles

Serum Analytes of American Mink (Neovison Vison) Challenged with Aleutian Mink Disease Virus.

Simple SummaryAleutian mink disease virus (AMDV) causes major health problems in the mink industry worldwide. The disease caused by AMDV has no cure or effective vaccine, and long-term viral eradication programs have failed in many countries. Some AMDV infected mink are genetically capable of tolerating the infection and living healthy and productive lives. Genetic selection for tolerance is, thus, a practical strategy to combat this virus. Accurate identification of tolerant animals is the fundamental issue in selection programs. The concentrations of some blood analytes, which are widely used as indicators of the presence and severity of diseases in humans and animals, are known to increase the accuracy of identifying tolerant mink. The objective of this study was to evaluate the merits of 14 serum analytes as biomarkers of tolerance to AMDV infection. Blood samples from 493 AMDV inoculated mink collected between 120 and 1211 days post-inoculation were analyzed. Total serum protein and globulin were found to be the most useful biomarkers of tolerance, whereas the relationships of other serum analytes to tolerance were weak or negligible.Black American mink (Neovison vison), which had been selected for tolerance to Aleutian mink disease virus (AMDV) for more than 20 years (TG100) or were from herds that have been free of AMDV (TG0), along with their progeny and crosses with 50% and 75% tolerance ancestry, were inoculated with a local isolate of AMDV. Blood samples were collected from 493 mink between 120 and 1211 days post-inoculation, and concentrations of 14 serum analytes were measured. Distributions of all analytes significantly deviated from normality, and data were analyzed after Box–Cox power transformation. Significant differences were observed among tolerant groups in the concentrations of globulin (GLO), total protein (TP), alkaline phosphatase, urea nitrogen, and calcium. Concentrations of GLO and TP linearly and significantly decreased with an increasing percentage of tolerance ancestry. Eleven analytes had the smallest values in the tolerant groups (TG100 or TG75), and eight analytes had the greatest values in the non-selected groups (TG0 or TG50). Antibody titer had the greatest correlation coefficients with GLO (0.62), TP (0.53), and creatinine (0.36). It was concluded that selection for tolerance decreased the concentrations of most serum analytes, and TP and GLO were the most accurate biomarkers of tolerance to AMDV infection. Males had significantly greater values than females for phosphorus and total bilirubin concentrations, but females had significantly greater amylase, cholesterol, and BUN concentrations than males.

Variable selection in the Box–Cox power transformation model

High dimensional data are frequently collected across research fields such as genomics, health sciences, economics, and social sciences. Recently, variable selection in the high dimensional setting has drawn great attention, with many effective methods developed to reduce the dimensionality of the data. However, most of these methods apply only to normally or near normally distributed outcomes in a linear regression model, while few studies focus on variable selection for skewed data. Simulation studies show that ignoring an appropriate transformation for the outcome can lead to biased inferences (e.g., missing important covariates). In this paper, we develop a variable selection procedure for the Box–Cox power transformation model by developing a penalized maximum likelihood estimate and deriving the consistency, oracle property, and asymptotic distribution of this estimate. Simulation studies demonstrate that the proposed method can yield higher sensitivity, while the naive method that without doing transformation can lead to lower sensitivity. We apply the proposed method to a gene expression study.

The Box–Cox Transformation: Review and Extensions

The Box–Cox power transformation family for nonnegative responses in linear models has a long and interesting history in both statistical practice and theory, which we summarize. The relationship between generalized linear models and log transformed data is illustrated. Extensions investigated include the transform both sides model and the Yeo–Johnson transformation for observations that can be positive or negative. The paper also describes an extended Yeo–Johnson transformation that allows positive and negative responses to have different power transformations. Analyses of data show this to be necessary. Robustness enters in the fan plot for which the forward search provides an ordering of the data. Plausible transformations are checked with an extended fan plot. These procedures are used to compare parametric power transformations with nonparametric transformations produced by smoothing.

Box–Cox power transformation unconditional quantile regressions with an application on wage inequality

This study proposes a semi-parametric estimation method, Box–Cox power transformation unconditional quantile regression, to estimate the impact of changes in the distribution of the explanatory variables on the unconditional quantile of the outcome variable. The proposed method consists of running a nonlinear regression of the recentered influence function (RIF) of the outcome variable on the explanatory variables. We also show the asymptotic properties of the proposed estimator and apply the estimation method to address an existing puzzle in labor economics–why the 50th/10th percentile wage gap has been falling in the USA since the late 1980s. Our results show that declining unionization can explain approximately 10% of the decline in the 50/10 wage gap in 1990–2000 and 23% in 2000–2010.

Effects of life expectancy on economic growth: new results using the flexible Box–Cox power transformation model

ABSTRACT This article revisits the effects of life expectancy (LE) on economic growth reported in previous research. Maximum likelihhod estimates are obtained based on the monotonic nonlinear flexible Box–Cox power transformation regression model. This variance stabilizing model, unlike the a priori restrictive nested log and linear forms, is a parametrically richer flexible functional form for transforming the dependent and independent variables separately with different power parameters. The conditional LE elasticity of income estimates are found to vary across fitted models and the more robust flexible Box–Cox power model outperforms the restrictive linear and log specifications fitted in past work. Contrary to past findings indicating large negative and highly elastic (statistically significant) LE elasticity of income, new results from the preferred fully flexible Box–Cox model incorporating the nonlinear effect yield a much smaller negative (statistically significant) and inelastic LE elasticity of income estimate. The study explores some policy options for enhancing economic growth during periods of high or rising LE.

An algorithm for estimating Box–Cox transformation parameter in ANOVA

ABSTRACTIn this study, we construct a feasible region, in which we maximize the likelihood function, by using Shapiro–Wilk and Bartlett's test statistics to obtain Box–Cox power transformation parameter for solving the issues of non-normality and/or heterogeneity of variances in analysis of variance (ANOVA). Simulation studies illustrate that the proposed approach is more successful in attaining normality and variance stabilization, and is at least as good as the usual maximum likelihood estimation (MLE) in estimating the transformation parameter for different conditions. Our proposed method is illustrated on two real-life datasets. Moreover, the proposed algorithm is released under R package AID under the name of “boxcoxfr” for implementation.

Derivation of gender and age-specific reference intervals from fully normal Japanese individuals and the implications for health screening

BackgroundWith nationwide standardization of laboratory tests among institutions for health screening in Japan, common reference intervals (RIs) were derived from records of 1,500,000 health check attendees. MethodsTargets were 20 basic laboratory tests including body mass index (BMI) and systolic and diastolic blood pressures (SBP, DBP). Individuals fulfilling the following strict criteria were chosen: SBP<130, DBP<85mmHg, BMI<25kg/m2, non-smoking, ethanol consumption<20g/day and under no mediation with no remarkable current/past illnesses. The latent abnormal values exclusion (LAVE) method was applied to ensure fully normal results. RIs were derived by parametric method using modified Box–Cox power transformation. ResultsAmong all attendees, 23% fulfilled the criteria. Application of the LAVE method further reduced the dataset by 40%–50%. Age-related charts of test results differed greatly between genders in almost all tests. Comparison of derived RIs with clinical decision limits (CDLs) revealed that the upper limits of RIs differed from CDLs according to gender and age. ConclusionsImplementation of gender and age-specific RIs derived from individuals with fully normal health attributes will (1) enable appropriate interpretation of test results in health screening and (2) promote judicious application of CDLs for therapeutic intervention, taking into account gender, age and other health attributes.

Estimating Box-Cox power transformation parameter via goodness-of-fit tests

ABSTRACTBox–Cox power transformation is a commonly used methodology to transform the distribution of the data into a normal distribution. The methodology relies on a single transformation parameter. In this study, we focus on the estimation of this parameter. For this purpose, we employ seven popular goodness-of-fit tests for normality, namely Shapiro–Wilk, Anderson–Darling, Cramer-von Mises, Pearson Chi-square, Shapiro-Francia, Lilliefors and Jarque–Bera tests, together with a searching algorithm. The searching algorithm is based on finding the argument of the minimum or maximum depending on the test, i.e., maximum for the Shapiro–Wilk and Shapiro–Francia, minimum for the rest. The artificial covariate method of Dag et al. (2014) is also included for comparison purposes. Simulation studies are implemented to compare the performances of the methods. Results show that Shapiro–Wilk and the artificial covariate method are more effective than the others and Pearson Chi-square is the worst performing method. The methods are also applied to two real-life datasets. The R package AID is proposed for implementation of the aforementioned methods.

Improving TAIEX forecasting using fuzzy time series with Box–Cox power transformation

Box–Cox together with our newly proposed transformation were implemented in three different real world empirical problems to alleviate noisy and the volatility effect of them. Consequently, a new domain was constructed. Subsequently, universe of discourse for transformed data was established and an approach for calculating effective length of the intervals was then proposed. Considering the steps above, the initial forecasts were performed using frequently used fuzzy time series (FTS) methods on transformed data. Final forecasts were retrieved from initial forecasted values by proper inverse operation. Comparisons of the results demonstrate that the proposed method produced more accurate forecasts compared with existing FTS on original data.

Better Articulating Normal Curve Theory for Introductory Mathematical Statistics Students: Power Transformations and Their Back-Transformations

This article addresses a gap in many, if not all, introductory mathematical statistics textbooks, namely, transforming a random variable so that it better mimics a normal distribution. Virtually all such textbooks treat the subject of variable transformations, which furnishes a nice opportunity to introduce and study this transformation-to-normality topic, a topic students frequently encounter in subsequent applied statistics courses. Accordingly, this article reviews variable power transformations of the Box–Cox type within the context of normal curve theory, as well as addresses their corresponding back-transformations. It presents four theorems and a conjecture that furnish the basics needed to derive equivalent results for all nonnegative values of the Box–Cox power transformation exponent. Results are illustrated with the exponential random variable. This article also includes selected pedagogic tools created with R code.

Effect of individual observations on the Box–Cox transformation

In this paper, we consider the influence of individual observations on inferences about the Box–Cox power transformation parameter from a Bayesian point of view. We compare Bayesian diagnostic measures with the ‘forward’ method of analysis due to Riani and Atkinson. In particular, we look at the effect of omitting observations on the inference by comparing particular choices of transformation using the conditional predictive ordinate and the k d measure of Pettit and Young. We illustrate the methods using a designed experiment. We show that a group of masked outliers can be detected using these single deletion diagnostics. Also, we show that Bayesian diagnostic measures are simpler to use to investigate the effect of observations on transformations than the forward search method.

Vanadium removal from roasted LD converter slag: Optimization of parameters by response surface methodology (RSM)

Alkaline roasting-sulfuric acid leaching process was used to remove vanadium from LD (Linz Donawitz) converter slag. Leaching parameters were investigated and optimized in order to maximize the recovery of vanadium as the vanadium content in the slag is low. The independent leaching factors investigated were liquid to solid ratio (L/S) (10–20mL/g), temperature (40–70°C), sulfuric acid concentration (1.5–3.0M), stirring rate (500–700rpm) and time (60–120min). Response surface methodology (RSM) was used to optimize the process parameters. To choose the best model for describing the leaching process, Box–Cox power transformation technique was used which in turn was used to normalize the data. The most influencing parameter was found to be L/S and the less effective was stirring rate as well. Based on the results, optimum removal condition (approx. 96%) according to process parameters was obtained with L/S ratio of (19.99), temperature of (70°C), sulfuric acid concentration of 2.05M, rpm of (558) and leaching time of (60min), respectively.

Estimating the prevalence of low-lumbar spine bone mineral density in older men with or at risk for HIV infection using normal mixture models

Bone mineral density decreases naturally as we age because existing bone tissue is reabsorbed by the body faster than new bone tissue is synthesized. When this occurs, bones lose calcium and other minerals. What is normal bone mineral density for men 50 years and older? Suitable diagnostic cutoff values for men are less well defined than for women. In this paper, we propose using normal mixture models to estimate the prevalence of low-lumbar spine bone mineral density in men 50 years and older with or at risk for human immunodeficiency virus infection when normal values of bone mineral density are not generally known. The Box–Cox power transformation is used to determine which transformation best suits normal mixture distributions. Parametric bootstrap tests are used to determine the number of mixture components and to determine whether the mixture components are homoscedastic or heteroscedastic.

Does the Box–Cox transformation help in forecasting macroeconomic time series?

The paper investigates whether transforming a time series leads to an improvement in forecasting accuracy. The class of transformations that is considered is the Box–Cox power transformation, which applies to series measured on a ratio scale. We propose a nonparametric approach for estimating the optimal transformation parameter based on the frequency domain estimation of the prediction error variance, and also conduct an extensive recursive forecast experiment on a large set of seasonal monthly macroeconomic time series related to industrial production and retail turnover. In about a fifth of the series considered, the Box–Cox transformation produces forecasts which are significantly better than the untransformed data at the one-step-ahead horizon; in most cases, the logarithmic transformation is the relevant one. As the forecast horizon increases, the evidence in favour of a transformation becomes less strong. Typically, the naive predictor that just reverses the transformation leads to a lower mean square error than the optimal predictor at short forecast lead times. We also discuss whether the preliminary in-sample frequency domain assessment conducted here provides reliable guidance as to which series should be transformed in order to improve the predictive performance significantly.

New Techniques for the Detection and Adjustment of Shifts in Daily Precipitation Data Series

Abstract This study integrates a Box–Cox power transformation procedure into a common trend two-phase regression-model-based test (the extended version of the penalized maximal F test, or “PMFred,” algorithm) for detecting changepoints to make the test applicable to non-Gaussian data series, such as nonzero daily precipitation amounts or wind speeds. The detection-power aspects of the transformed method (transPMFred) are assessed by a simulation study that shows that this new algorithm is much better than the corresponding untransformed method for non-Gaussian data; the transformation procedure can increase the hit rate by up to ∼70%. Examples of application of this new transPMFred algorithm to detect shifts in real daily precipitation series are provided using nonzero daily precipitation series recorded at a few stations across Canada that represent very different precipitation regimes. The detected changepoints are in good agreement with documented times of changes for all of the example series. This study clarifies that it is essential for homogenization of daily precipitation data series to test the nonzero precipitation amount series and the frequency series of precipitation occurrence (or nonoccurrence), separately. The new transPMFred can be used to test the series of nonzero daily precipitation (which are non Gaussian and positive), and the existing PMFred algorithm can be used to test the frequency series. A software package for using the transPMFred algorithm to detect shifts in nonzero daily precipitation amounts has been developed and made freely available online, along with a quantile-matching (QM) algorithm for adjusting shifts in nonzero daily precipitation series, which is applicable to all positive data. In addition, a similar QM algorithm has also been developed for adjusting Gaussian data such as temperatures. It is noticed that frequency discontinuities are often inevitable because of changes in the measuring precision of precipitation, and that they could complicate the detection of shifts in nonzero daily precipitation data series and void any attempt to homogenize the series. In this case, one must account for all frequency discontinuities before attempting to adjust the measured amounts. This study also proposes approaches to account for detected frequency discontinuities, for example, to fill in the missed measurements of small precipitation or the missed reports of trace precipitation. It stresses the importance of testing the homogeneity of the frequency series of reported zero precipitation and of various small precipitation events, along with testing the series of daily precipitation amounts that are larger than a small threshold value, varying the threshold over a set of small values that reflect changes in measuring precision over time.

Neutral Zone Classifiers Using a Decision-Theoretic Approach With Application to DNA Array Analyses

Two-class neutral zone classifiers were recently proposed for use in microbial community profiling applications. These classifiers allow a region of neutrality for cases where probe hybridization outcomes are too ambiguous to have adequate confidence in assigning a "binding" or "no binding" result. In this paper, we generalize the idea of neutral zone classifiers to an arbitrary number of classes and apply it to improve the process of microbial community profiling by considering a third class for the outcome of probe hybridization experiments, "partial binding." We introduce a family of class distributions that uses a mixture of Gaussian distributions as a model for a Box-Cox power transformation of the raw intensity measurements. Stratified cross-validation analyses are used to assess the efficacy of the proposed three-class neutral zone classifier. This article has supplementary material online.

Accounting for choice of measurement scale in extreme value modeling

We investigate the effect that the choice of measurement scale has upon inference and extrapolation in extreme value analysis. Separate analyses of variables from a single process on scales which are linked by a nonlinear transformation may lead to discrepant conclusions concerning the tail behavior of the process. We propose the use of a Box--Cox power transformation incorporated as part of the inference procedure to account parametrically for the uncertainty surrounding the scale of extrapolation. This has the additional feature of increasing the rate of convergence of the distribution tails to an extreme value form in certain cases and thus reducing bias in the model estimation. Inference without reparameterization is practicably infeasible, so we explore a reparameterization which exploits the asymptotic theory of normalizing constants required for nondegenerate limit distributions. Inference is carried out in a Bayesian setting, an advantage of this being the availability of posterior predictive return levels. The methodology is illustrated on both simulated data and significant wave height data from the North Sea.

Box and Cox power-transformation to additivity and homoscedasticity in regression

We describe a Box and Cox power-transformation to simultaneously provide additivity and homoscedasticity in regression. The two methods developed here are extensions of the power-additive transformation (PAT) discussed by Goto (1992, 1995) and Hamasaki and Goto (2005). The PAT aims to improve the additivity or linearity of some simple model represented by linear predicators. We then consider combinations of the PAT with the weighting and transform-both-sides methods. We discuss the procedures to find the maximum likelihood estimates of parameters and then consider the relationship between the methods. Also, we compare the performances of the methods through a simulation study.

Statistical analyses of geochemical variables in soils of Ireland

Geochemical variables in soils are affected by multiple factors, and different variables have different responses to these factors, causing complicated multivariate relationships between them. In this study, the relationships between a total of 45 geochemical variables in soils and in particular mineral soils of Ireland were investigated using multivariate analyses following an investigation into their probability features and data transformation based on a total of 1310 surface soil samples. There was strong variation in the values of geochemical variables. Multi-modal features in the histograms and multi-kink features in the normal quantile–quantile (Q–Q) plots were observed for many variables implying the existence of multiple populations. Obvious outliers were identified using the normal Q–Q plots. Most of the variables did not pass a Kolmogorov–Smirnov test for either normal or lognormal distribution, and the Box–Cox power transformation was effective in transposing data to a form suited to further parametric statistical analyses. There were generally good correlations between most metals, but relatively poor correlations between extractable nutrients (available P, K, and Mg) and metals. The extractable nutrients were more affected by agricultural activities that were more spatially variable compared with natural factors such as geology and soil type. Cluster analysis classified the variables into groups of metals of mainly natural origin, mobile elements, and variables that are affected by agricultural activities and pollution. Among the three factors of rock type, soil type and land use investigated, land use was found to be the least important influencing factor for most of the geochemical variables studied.

Application of a regionalized Clark IUH model with limited hydrologic data availability

AbstractTo aid prediction of the flow hydrograph in a basin with limited data, a practical approach to determining a regionalized Clark instantaneous unit hydrograph (IUH) model is presented. The proposed model is described in terms of the synthetic time–area concentration curve, the concentration time, and a special regional similarity value that is valid in the whole basin. The latter was estimated from a Monte Carlo testing procedure based on the normal probability distribution of transformed regional similarity values composed of the time of concentration and the storage coefficient in gauged basins. The time–area concentration curve and the concentration time were calculated from a rational equation as in conventional methods. The method of transformation adopted was the Box–Cox power transformation, which is known to make non‐normal values resemble normal data. By introducing the regional similarity value into a Clark IUH, a statistically best estimate of IUH for given data conditions and its quantified degree of uncertainty were realized. The Wi River basin in Korea was used to test the applicability of the regionalized Clark IUH. The performance of the suggested methodology was evaluated by assuming an ungauged sub‐basin at the site. The results showed that the IUH model developed in this work was an effective tool, predicting a reliable hydrograph within the study area even though only limited data were available. Copyright © 2008 John Wiley &amp; Sons, Ltd.