Box-Cox Transformation Parameter Research Articles

Likelihood-based inference for semi-parametric transformation cure models with interval censored data

A simple yet effective way of modeling survival data with cure fraction is by considering Box-Cox transformation cure model (BCTM) that unifies mixture and promotion time cure models. In this article, we numerically study the statistical properties of the BCTM when applied to interval censored data. Time-to-events associated with susceptible subjects are modeled through proportional hazards structure that allows for non-homogeneity across subjects, where the baseline hazard function is estimated by distribution-free piecewise linear function with varied degrees of non-parametricity. Due to missing cured statuses for right censored subjects, maximum likelihood estimates of model parameters are obtained by developing an expectation-maximization (EM) algorithm. Under the EM framework, the conditional expectation of the complete data log-likelihood function is maximized by considering all parameters (including the Box-Cox transformation parameter α) simultaneously, in contrast to conventional profile-likelihood technique of estimating α. The robustness and accuracy of the model and estimation method are established through a detailed simulation study under various parameter settings, and an analysis of real-life data obtained from a smoking cessation study.

Functional coefficient cointegration models with Box–Cox transformation

This paper considers a functional coefficient cointegration model where the dependent variable is subject to a Box–Cox transformation. A profile method is proposed to estimate the Box–Cox transformation parameter β0. We first approximate the functional coefficients by sieve method assuming that β0 is known. Then an extremum estimator of β0 is proposed based on a loss function which measures the relative variation of the regression residual compared to the variation in the transformed dependent variable. Finally, a plug-in estimator of the functional coefficients is obtained. Asymptotic properties of the proposed estimators are developed. Numerical results demonstrate the nice performance of the estimators and corroborate the theoretical development.

Changing Box–Cox transformation parameter as an early warning signal for abrupt climate change

There are some critical thresholds in many complex dynamic systems. When a system approaches its critical threshold, the probability density distribution of the system might be changed. In view of this, we presented a new early warning indicator of an abrupt climate change based on the Box–Cox transformation, namely, the changing Box–Cox transformation parameter (BCTP). A significant increasing or decreasing trend in BCTP can always be found before the onset of an abrupt change in different folding models. Based on the Greenland paleotemperature data reconstructed through Greenland Ice Sheet Project II, we found that BCTP showed an increasing trend about 2800 years before the onset of Bølling–Allerød warming. The trends of BCTP are statistically significant at a significant level of of α = 0.05 for both artificial abrupt changes and Greenland paleotemperature. Compared with kurtosis coefficient and skewness coefficient, BCTP can present an effectively early warning in some cases of abrupt change where kurtosis coefficient or skewness coefficient showed a poor warning performance. In general, the variation magnitudes of BCTP are greater than that of kurtosis coefficient and skewness coefficient, and BCTP shows a better warning performance in most of cases. These results indicate that changing BCTP could be a good new indicator for warning upcoming abrupt changes.

A novel method for determining the depth of kickoff points in cluster wells based on probability analysis

The purpose of this study is to determine the depth range of kickoff points of cluster wells. A probability distribution method is proposed to analyze the control deviation of the closure distance of vertical well sections. A Box-Cox transformation was performed on the positively skewed raw data, and a function of the closure distance probability distribution with the Box-Cox transformation parameter λ and the measured depth was established. The closure distance range was nonlinearly related to the measured depth and the distribution probability. The factors that should be considered in the function when determining the depth of the kickoff points were discussed. An engineering example was used to verify the applicability of the method. The proposed function can be used for assessing the collision risk during the project planning phase and optimizing the number of cluster wells and the wellhead spacing.

Parametric and non-parametric estimation of reference intervals for routine laboratory tests: an analysis of health check-up data for 260 889 young men in the South Korean military

ObjectivesDetermination of reference intervals (RIs) using big data faces several obstacles due to heterogeneity in analysers, period and ethnicity. The present study aimed to establish the RIs for routine common...

Efficient approach for dynamic reliability analysis based on uniform design method and Box-Cox transformation

This paper proposes a novel approach for dynamic reliability analysis of uncertain structures subjected to non-stationary stochastic excitations. This approach calculates the first-passage probabilities by evaluating the transformed extreme value distribution (TEVD) instead of the original extreme value distribution (EVD). The Box-Cox transformation is first used to symmetrize the extreme value, and then the generalized Gaussian distribution (GGD) is suggested to approximate the main body and tails of the TEVD. An absolute skewness minimization criterion is proposed to find the optimal Box-Cox transformation parameter. Furthermore, the random-function spectral representation method (RFSRM) and uniform design method (UDM) are successfully combined to estimate the first four central moments of the transformed extreme value with a small sample size. Two numerical examples involving linear and nonlinear shear-frame structures under nonstationary ground motions are investigated. The results indicate that the proposed method can be used for dynamic reliability analysis with high accuracy and efficiency.

Quantifying how diagnostic test accuracy depends on threshold in a meta-analysis.

Tests for disease often produce a continuous measure, such as the concentration of some biomarker in a blood sample. In clinical practice, a threshold C is selected such that results, say, greater than C are declared positive and those less than C negative. Measures of test accuracy such as sensitivity and specificity depend crucially on C, and the optimal value of this threshold is usually a key question for clinical practice. Standard methods for meta‐analysis of test accuracy (i) do not provide summary estimates of accuracy at each threshold, precluding selection of the optimal threshold, and furthermore, (ii) do not make use of all available data. We describe a multinomial meta‐analysis model that can take any number of pairs of sensitivity and specificity from each study and explicitly quantifies how accuracy depends on C. Our model assumes that some prespecified or Box‐Cox transformation of test results in the diseased and disease‐free populations has a logistic distribution. The Box‐Cox transformation parameter can be estimated from the data, allowing for a flexible range of underlying distributions. We parameterise in terms of the means and scale parameters of the two logistic distributions. In addition to credible intervals for the pooled sensitivity and specificity across all thresholds, we produce prediction intervals, allowing for between‐study heterogeneity in all parameters. We demonstrate the model using two case study meta‐analyses, examining the accuracy of tests for acute heart failure and preeclampsia. We show how the model can be extended to explore reasons for heterogeneity using study‐level covariates.

Optimal transformations leading to normal distributions of positron emission tomography standardized uptake values

The statistical analysis of positron emission tomography (PET) standardized uptake value (SUV) measurements is challenging due to the skewed nature of SUV distributions. This limits utilization of powerful parametric statistical models for analyzing SUV measurements. An ad-hoc approach, which is frequently used in practice, is to blindly use a log transformation, which may or may not result in normal SUV distributions. This study sought to identify optimal transformations leading to normally distributed PET SUVs extracted from tumors and assess the effects of therapy on the optimal transformations. Methods. The optimal transformation for producing normal distributions of tumor SUVs was identified by iterating the Box–Cox transformation parameter (λ) and selecting the parameter that maximized the Shapiro–Wilk P-value. Optimal transformations were identified for tumor SUVmax distributions at both pre and post treatment. This study included 57 patients that underwent 18F-fluorodeoxyglucose (18F-FDG) PET scans (publically available dataset). In addition, to test the generality of our transformation methodology, we included analysis of 27 patients that underwent 18F-Fluorothymidine (18F-FLT) PET scans at our institution. Results. After applying the optimal Box–Cox transformations, neither the pre nor the post treatment 18F-FDG SUV distributions deviated significantly from normality (P > 0.10). Similar results were found for 18F-FLT PET SUV distributions (P > 0.10). For both 18F-FDG and 18F-FLT SUV distributions, the skewness and kurtosis increased from pre to post treatment, leading to a decrease in the optimal Box–Cox transformation parameter from pre to post treatment. There were types of distributions encountered for both 18F-FDG and 18F-FLT where a log transformation was not optimal for providing normal SUV distributions. Conclusion. Optimization of the Box–Cox transformation, offers a solution for identifying normal SUV transformations for when the log transformation is insufficient. The log transformation is not always the appropriate transformation for producing normally distributed PET SUVs.

Multivariate correlation among resilient modulus and cone penetration test parameters of cohesive subgrade soils

Extensive research has been conducted to establish empirical equations between the resilient modulus (Mr) and other testing parameters. Despite an increase in the correlation studies, less effort has been made in developing reliable correlations to predict the resilient modulus using cone penetration testing indices. In this research, an attempt has been made to propose new correlations between Mr and the piezocone penetration test (CPTU) indices of clayey soils by using a multivariate normal distribution approach. A database collected from 16 different sites in Jiangsu province, China, was presented to achieve this study. The database contains 124 sets of resilient modulus (Mr) values at the in-situ stress condition, cone tip resistance (qc), sleeve frictional resistance (fs), moisture (w) and dry density (γd). Four major procedures including the data transformation, correlation analysis, Bayesian updating, and back transformation were applied to derive the correlations among Mr and other indices in the framework of the multivariate normal distribution model. First, each individual parameter was converted to a standard normal variable using the Box-Cox transformation. Later, the correlation matrix of the multivariate normal distribution model was estimated using the product-moment (Pearson) correlation coefficients between all pairwise data. The uncertainties associated with the Box-Cox transformation parameters and the correlation coefficients were evaluated using a bootstrapping technique. Furthermore, the formulas for predicting the posterior mean, coefficient of variation (COV), median, and 95% confidence interval (CI) of Mr conditional on different indices were derived using Bayesian updating combined with back transformation. A new approach based on the Taylor-series expansion was proposed to approximate these statistics in this study. Comparisons between the correlations derived from the multivariate model, the database of Jiangsu clay, and the data collected from the literature demonstrated that the model should be capable of the correlations among the five indices, but it might be slightly biased when applied in the global dataset. This research highlighted a new method to establish the reliable correlations to update Mr using different testing indices.

Power-transformed linear regression on quantile residual life for censored competing risks data

ABSTRACTThis paper proposes a power-transformed linear quantile regression model for the residual lifetime of competing risks data. The proposed model can describe the association between any quantile of a time-to-event distribution among survivors beyond a specific time point and the covariates. Under covariate-dependent censoring, we develop an estimation procedure with two steps, including an unbiased monotone estimating equation for regression parameters and cumulative sum processes for the Box–Cox transformation parameter. The asymptotic properties of the estimators are also derived. We employ an efficient bootstrap method for the estimation of the variance–covariance matrix. The finite-sample performance of the proposed approaches are evaluated through simulation studies and a real example.

A supply-and-demand based price model for financial assets

This paper proposes a price model for financial assets using the supply-demand relationship, referred to as a supply and demand based price (SDP) model for financial assets. The model demonstrates that stock price volatility is characterized by the downward sloping demand curve and volume fluctuation, not by another stochastic volatility process often employed in security markets. In particular, it is found that the flatter demand curve using inverse Box-Cox transformation with a positive parameter causes the asymmetric volatility, i.e., negative price return-volatility correlation, often observed in security markets. The model can also classify the asymmetric volatility into leverage effect and feedback effect examining the relationship between expected price return and volatility. We then characterize the time varying market price of risk based on the SDP model. Empirical studies using a recent decade data for many asset classes show that stock market indices, freight indices, and carbon assets possess positive inverse Box-Cox transformation parameters, resulting in asymmetric volatility while commodity-related assets tend to have negative parameters, resulting in inverse leverage effects. In contrast, using the long run data from 1950 to 2009, we illustrate the inverse leverage effect in security markets. Finally, we show that the positive risk and return relationship holds if the expected return is positive, regardless of the asset class.

Selection of the optimal Box–Cox transformation parameter for modelling and forecasting age-specific fertility

The Box-Cox transformation can sometimes yield noticeable improvements in model simplicity, variance homogeneity and precision of estimation, such as in modelling and forecasting age-specific fertility. Despite its importance, there have been few studies focusing on the optimal selection of Box-Cox transformation parameters in demographic forecasting. A simple method is proposed for selecting the optimal Box-Cox transformation parameter, along with an algorithm based on an in-sample forecast error measure. Illustrated by Australian age-specific fertility, the out-of-sample accuracy of a forecasting method can be improved with the selected Box-Cox transformation parameter. Furthermore, the log transformation is not adequate for modelling and forecasting age-specific fertility. The Box-Cox transformation parameter should be embedded in statistical analysis of age-specific demographic data, in order to fully capture forecast uncertainties.

Box-Cox transformation of firm size data in statistical analysis

Firm size data usually do not show the normality that is often assumed in statistical analysis such as regression analysis. In this study we focus on two firm size data: the number of employees and sale. Those data deviate considerably from a normal distribution. To improve the normality of those data we transform them by the Box-Cox transformation with appropriate parameters. The Box-Cox transformation parameters are determined so that the transformed data best show the kurtosis of a normal distribution. It is found that the two firm size data transformed by the Box-Cox transformation show strong linearity. This indicates that the number of employees and sale have the similar property as a firm size indicator. The Box-Cox parameters obtained for the firm size data are found to be very close to zero. In this case the Box-Cox transformations are approximately a log-transformation. This suggests that the firm size data we used are approximately log-normal distributions.

Bayesian inference for multivariate meta‐analysis Box–Cox transformation models for individual patient data with applications to evaluation of cholesterol‐lowering drugs

In this paper, we propose a class of Box-Cox transformation regression models with multidimensional random effects for analyzing multivariate responses for individual patient data in meta-analysis. Our modeling formulation uses a multivariate normal response meta-analysis model with multivariate random effects, in which each response is allowed to have its own Box-Cox transformation. Prior distributions are specified for the Box-Cox transformation parameters as well as the regression coefficients in this complex model, and the deviance information criterion is used to select the best transformation model. Because the model is quite complex, we develop a novel Monte Carlo Markov chain sampling scheme to sample from the joint posterior of the parameters. This model is motivated by a very rich dataset comprising 26 clinical trials involving cholesterol-lowering drugs where the goal is to jointly model the three-dimensional response consisting of low density lipoprotein cholesterol (LDL-C), high density lipoprotein cholesterol (HDL-C), and triglycerides (TG) (LDL-C, HDL-C, TG). Because the joint distribution of (LDL-C, HDL-C, TG) is not multivariate normal and in fact quite skewed, a Box-Cox transformation is needed to achieve normality. In the clinical literature, these three variables are usually analyzed univariately; however, a multivariate approach would be more appropriate because these variables are correlated with each other. We carry out a detailed analysis of these data by using the proposed methodology.

The power normal distribution - modelling, simulation and applications

In the following work we model a random sample having support on the positive orthant using the power normal (PN) distribution. The PN distribution is unimodal, obtained from inverse Box-Cox (BC) transformation and there exists no close form expression for moment estimation. Hence, we simulate from the distribution to estimate its moments and also implement a naive bootstrap, on the random sample prior to the modelling, to account for the uncertainty in estimating the BC transformation parameters. In the real world examples we consider, we find the distribution efficient in modelling the tails of the underlying unknown process. Due to the minimalistic assumptions with PN distribution it can be used to model data observed both on the nominal scale and interval scale. In our simulation studies, wherein we consider samples from a known distribution with support defined on the positive orthant only, we observe that the PN distribution performs better than Kernel density estimation methods based on Gaussian assumptions for sampling and density estimation.

Variable selection in joint mean and variance models of Box–Cox transformation

In many applications, a single Box–Cox transformation cannot necessarily produce the normality, constancy of variance and linearity of systematic effects. In this paper, by establishing a heterogeneous linear regression model for the Box–Cox transformed response, we propose a hybrid strategy, in which variable selection is employed to reduce the dimension of the explanatory variables in joint mean and variance models, and Box–Cox transformation is made to remedy the response. We propose a unified procedure which can simultaneously select significant variables in the joint mean and variance models of Box–Cox transformation which provide a useful extension of the ordinary normal linear regression models. With appropriate choice of the tuning parameters, we establish the consistency of this procedure and the oracle property of the obtained estimators. Moreover, we also consider the maximum profile likelihood estimator of the Box–Cox transformation parameter. Simulation studies and a real example are used to illustrate the application of the proposed methods.

The Box-Cox power transformation on nursing sensitive indicators: Does it matter if structural effects are omitted during the estimation of the transformation parameter?

BackgroundMany nursing and health related research studies have continuous outcome measures that are inherently non-normal in distribution. The Box-Cox transformation provides a powerful tool for developing a parsimonious model for data representation and interpretation when the distribution of the dependent variable, or outcome measure, of interest deviates from the normal distribution. The objectives of this study was to contrast the effect of obtaining the Box-Cox power transformation parameter and subsequent analysis of variance with or without a priori knowledge of predictor variables under the classic linear or linear mixed model settings.MethodsSimulation data from a 3 × 4 factorial treatments design, along with the Patient Falls and Patient Injury Falls from the National Database of Nursing Quality Indicators (NDNQI®) for the 3rd quarter of 2007 from a convenience sample of over one thousand US hospitals were analyzed. The effect of the nonlinear monotonic transformation was contrasted in two ways: a) estimating the transformation parameter along with factors with potential structural effects, and b) estimating the transformation parameter first and then conducting analysis of variance for the structural effect.ResultsLinear model ANOVA with Monte Carlo simulation and mixed models with correlated error terms with NDNQI examples showed no substantial differences on statistical tests for structural effects if the factors with structural effects were omitted during the estimation of the transformation parameter.ConclusionsThe Box-Cox power transformation can still be an effective tool for validating statistical inferences with large observational, cross-sectional, and hierarchical or repeated measure studies under the linear or the mixed model settings without prior knowledge of all the factors with potential structural effects.

A Supply and Demand Based Price Model for Financial Assets

This paper proposes a price model for financial assets using the supply-demand relationship, referred to as a supply and demand based price (SDP) model for financial assets. The model demonstrates that stock price volatility is characterized by the downward sloping demand curve and volume fluctuation, not by another stochastic volatility process often employed in security markets. In particular, it is found that the flatter demand curve using inverse Box-Cox transformation with a positive parameter causes the asymmetric volatility, i.e., negative price return-volatility correlation, often observed in security markets. The model can also classify the asymmetric volatility into leverage effect and feedback effect examining the relationship between expected price return and volatility. We then characterize the time varying market price of risk based on the SDP model. Empirical studies using a recent decade data for many asset classes show that stock market indices, freight indices, and carbon assets possess positive inverse Box-Cox transformation parameters, resulting in asymmetric volatility while commodity-related assets tend to have negative parameters, resulting in inverse leverage effects. In contrast, using the long run data from 1950 to 2009, we illustrate the inverse leverage effect in security markets. Finally, we show that the positive risk and return relationship holds if the expected return is positive, regardless of the asset class.

Comparing four methods to estimate usual intake distributions

The aim of this paper was to compare methods to estimate usual intake distributions of nutrients and foods. As 'true' usual intake distributions are not known in practice, the comparison was carried out through a simulation study, as well as empirically, by application to data from the European Food Consumption Validation (EFCOVAL) Study in which two 24-h dietary recalls (24-HDRs) and food frequency data were collected. The methods being compared were the Iowa State University Method (ISU), National Cancer Institute Method (NCI), Multiple Source Method (MSM) and Statistical Program for Age-adjusted Dietary Assessment (SPADE). Simulation data were constructed with varying numbers of subjects (n), different values for the Box-Cox transformation parameter (λ(BC)) and different values for the ratio of the within- and between-person variance (r(var)). All data were analyzed with the four different methods and the estimated usual mean intake and selected percentiles were obtained. Moreover, the 2-day within-person mean was estimated as an additional 'method'. These five methods were compared in terms of the mean bias, which was calculated as the mean of the differences between the estimated value and the known true value. The application of data from the EFCOVAL Project included calculations of nutrients (that is, protein, potassium, protein density) and foods (that is, vegetables, fruit and fish). Overall, the mean bias of the ISU, NCI, MSM and SPADE Methods was small. However, for all methods, the mean bias and the variation of the bias increased with smaller sample size, higher variance ratios and with more pronounced departures from normality. Serious mean bias (especially in the 95th percentile) was seen using the NCI Method when r(var) = 9, λ(BC) = 0 and n = 1000. The ISU Method and MSM showed a somewhat higher s.d. of the bias compared with NCI and SPADE Methods, indicating a larger method uncertainty. Furthermore, whereas the ISU, NCI and SPADE Methods produced unimodal density functions by definition, MSM produced distributions with 'peaks', when sample size was small, because of the fact that the population's usual intake distribution was based on estimated individual usual intakes. The application to the EFCOVAL data showed that all estimates of the percentiles and mean were within 5% of each other for the three nutrients analyzed. For vegetables, fruit and fish, the differences were larger than that for nutrients, but overall the sample mean was estimated reasonably. The four methods that were compared seem to provide good estimates of the usual intake distribution of nutrients. Nevertheless, care needs to be taken when a nutrient has a high within-person variation or has a highly skewed distribution, and when the sample size is small. As the methods offer different features, practical reasons may exist to prefer one method over the other.

Bayesowska analiza transformacji Boxa i Coxa dla ilorazu cen w modelach FCSV

The continuously compounded rates of return (or logarithmic returns) and the simple rates of return are commonly used in econometric analyses of financial data. These two types of data transformation are applied arbitrarily. However, both are variants of the well-known Box- Cox transformation of the xt/xt-1 ratio (where xt denotes the asset price at time t) with parameter 0 and 1, respectively. In the paper, we consider the Box-Cox transformation of financial time series in Stochastic Volatility (SV) models. Bayesian approach is applied to make inference about the Box-Cox transformation parameter (λ). As parameter λ is estimated along with other unknown parameters, information in the data is used to determine which transformation is appropriate for the data. Using daily data (quotations of stock indices), we show that in the Stochastic Volatility models with fat tails and correlated errors (FCSV), the posterior distribution of parameter λ strongly depends on the prior assumption about this parameter. It seems that in more cases the values of λ close to 0 are more probable a posteriori than the one close to 1.