Box-Cox Power Transformation Research Articles

The British, American NCHS, and Dutch weight standards compared using the LMS method.

AbstractThe LMS method (Cole, 1988) is a method for fitting and summarising growth standards. The standard is represented by three curves plotted against age: The Box–Cox power transformation needed to bring the distribution close to normality (the L Curve), the mean (the M curve), and the coefficient of variation (the S curve), respectively. Individual centiles of the weight distribution at any age can be obtained from the age‐specific L, M, and S values by assuming that transformed weight is normally distributed. The British (Tanner et al.,1966 ), American NCHS (Hamill et al., 1977), and Ducth (Roede and Van Wieringen, 1985) national weight standards are compared by this method, and the centiles closely. The fit is better for the British and Dutch standards, where the centiles were originally drawn by eye, than for the American NCHS standard, which used cubic splines in its construction. The L curves from the three standards show a consistent pattern of changing skewness during childhood, particularly in puberty, which has not previously been noted. The values of the L, M, and S curves in each standard are tabulated by age and sex, and they allow any reference weight centile to be calculated. Equally, observed weights including those in the tails of the weight distribution can be expressed as centiles or SD scores to full accuracy.

Transformations and Influential Observations in Minimum Sum of Absolute Errors Regression

The Box-Cox power transformation model is considered for minimum sum of absolute errors (MSAE) regression. A long-tailed error distribution, specifically the Laplace distribution, is included and could accommodate observations that in the normal case are outlying or unduly influencing a choice of transformation. The log-likelihood procedure of Box and Cox (1964) for obtaining the optimal transformation parameter is adapted for Laplace errors and MSAE regression. Graphical methods for detecting the influence of individual observations on the choice of transformation are described. Application to examples illustrates that this approach can provide valuable additional information to the data analyst.

Transformations and Influential Observations in Minimum Sum of Absolute Errors Regression

The Box-Cox power transformation model is considered for minimum sum of absolute errors (MSAE) regression. A long-tailed error distribution, specifically the Laplace distribution, is included and could accommodate observations that in the normal case are outlying or unduly influencing a choice of transformation. The log-likelihood procedure of Box and Cox (1964) for obtaining the optimal transformation parameter is adapted for Laplace errors and MSAE regression. Graphical methods for detecting the influence of individual observations on the choice of transformation are described. Application to examples illustrates that this approach can provide valuable additional information to the data analyst.

The Retransformed Mean after a Fitted Power Transformation

Abstract An approximate method is given to estimate the mean of a dependent variable after a linear model is fitted to the Box—Cox power transformation of this variable. The estimate is accurate except when the transformation power parameter is near zero. The properties of the estimate in the one-sample and regression cases are considered, by both asymptotic calculations and Monte Carlo simulations, and comparisons are made with the smearing estimate (Duan 1983). It is shown that there can be some cost due to estimating the power transformation, as opposed to assuming it is known; however, this cost is not severe.

Box-Cox power transformations and the demand for money functions in France, Germany, Italy and the Netherlands

Box-Cox power transformations and the demand for money functions in France, Germany, Italy and the Netherlands

Some considerations concerning the functional form of earnings

The problem of choosing the correct functional form of earnings has plagued studies of social inequality and labor market segmentation. Although this choice has often been made on superficial grounds, the results presented in the current paper demonstrate that this choice has very significant implications for the empirical findings that emerge from the study of earnings determination. Four functional forms of earnings are compared: dollar earnings, earnings rank based on social perceptions as calibrated by P. Coleman and L. Rainwater (1978, Social Standing in America, Basic Books, New York) the natural logarithm of earnings, and the best-fitting power function transformation of earnings as estimated by the Box-Cox technique. Evaluation of a generalized earnings determination model on a sample of private sector employees produces very similar results across these four functional forms of earnings. More divergent results are obtained when the models are compared across subsamples based on class, economic sector, and gender. Significant contrasts between earnings coefficients across groups are much more likely when utilizing log earnings or the Box-Cox power transformation than when using either dollar earnings or perceived earnings. In addition, the contrasts observed for log earnings and Box-Cox earnings are frequently in the opposite direction of those observed for dollar earnings or perceived earnings. The paper concludes with a discussion of the implications of these findings for theoretical developments in the study of social inequality and labor market segmentation.

Measures of Location of Skew Distributions Obtained through Box-Cox Transformations

Abstract Box—Cox power transformations are used to transform a set of observations to symmetry. A measure of location of the original data is obtained by applying the inverse transformation to the center of the transformed data. Its properties are discussed, and it is compared with other measures of location.

Transformations in Regression: A Robust Analysis

We consider two approaches to robust estimation for the Box–Cox power-transformation model. One approach maximizes weighted, modified likelihoods. A second approach bounds a measure of gross-error sensitivity. Among our primary concerns is the performance of these estimators on actual data. In examples that we study, there seem to be only minor differences between these two robust estimators, but they behave rather differently than the maximum likelihood estimator or estimators that bound only the influence of the residuals. These examples show that model selection, determination of the transformation parameter, and outlier identification are fundamentally interconnected.

Transformations in Regression: A Robust Analysis

We consider two approaches to robust estimation for the Box–Cox power-transformation model. One approach maximizes weighted, modified likelihoods. A second approach bounds a measure of gross-error sensitivity. Among our primary concerns is the performance of these estimators on actual data. In examples that we study, there seem to be only minor differences between these two robust estimators, but they behave rather differently than the maximum likelihood estimator or estimators that bound only the influence of the residuals. These examples show that model selection, determination of the transformation parameter, and outlier identification are fundamentally interconnected.

Box-Cox Power Transformations and the Demand for Money Function

Box-Cox Power Transformations and the Demand for Money Function

The estimation of economic depreciation using vintage asset prices: An application of the Box-Cox power transformation

This paper applies the Box–Cox power transformation to the problem of estimating the rate and form of economic depreciation from data on used asset prices. The use of the Box–Cox model to statistically discriminate between geometric, linear and ‘one-hoss-shay’ depreciation patterns (which are special cases of the Box–Cox form) is analogous to the use of the CES production function to discriminate between Cobb–Douglas and fixed proportion technologies. We apply the Box–Cox model to a sample of used building prices, and find that the appropriate depreciation pattern is approximately geometric.

Determining Statistical Characteristics of a Vehicle Emissions Audit Procedure

The Environmental Protection Agency currently audits automobiles for emissions compliance at assembly plants with multiple attribute multiple staged sampling plans. This paper summarizes the auditing procedure and develops methods for determining its statistical characteristics. Specifically, we consider how to evaluate a multiple attribute multiple staged sampling plan, give a procedure that efficiently estimates attribute-type probabilities from multivariate variables-type data, and quantify the effect of tightening the acceptable quality level of the sampling plan. A key technique for this analysis is the multivariate version of the Box-Cox power transformation. All procedures are illustrated with data from a typical vehicle configuration.