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Journal ArticleDOI

Systems of frequency curves generated by methods of translation.

01 Jun 1949-Biometrika (Oxford University Press)-Vol. 36, pp 149-176
About: This article is published in Biometrika.The article was published on 1949-06-01. It has received 2085 citations till now.
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Journal ArticleDOI
S. S. Shapiro1, M. B. Wilk1
TL;DR: In this article, a new statistical procedure for testing a complete sample for normality is introduced, which is obtained by dividing the square of an appropriate linear combination of the sample order statistics by the usual symmetric estimate of variance.
Abstract: The main intent of this paper is to introduce a new statistical procedure for testing a complete sample for normality. The test statistic is obtained by dividing the square of an appropriate linear combination of the sample order statistics by the usual symmetric estimate of variance. This ratio is both scale and origin invariant and hence the statistic is appropriate for a test of the composite hypothesis of normality. Testing for distributional assumptions in general and for normality in particular has been a major area of continuing statistical research-both theoretically and practically. A possible cause of such sustained interest is that many statistical procedures have been derived based on particular distributional assumptions-especially that of normality. Although in many cases the techniques are more robust than the assumptions underlying them, still a knowledge that the underlying assumption is incorrect may temper the use and application of the methods. Moreover, the study of a body of data with the stimulus of a distributional test may encourage consideration of, for example, normalizing transformations and the use of alternate methods such as distribution-free techniques, as well as detection of gross peculiarities such as outliers or errors. The test procedure developed in this paper is defined and some of its analytical properties described in ? 2. Operational information and tables useful in employing the test are detailed in ? 3 (which may be read independently of the rest of the paper). Some examples are given in ? 4. Section 5 consists of an extract from an empirical sampling study of the comparison of the effectiveness of various alternative tests. Discussion and concluding remarks are given in ?6. 2. THE W TEST FOR NORMALITY (COMPLETE SAMPLES) 2 1. Motivation and early work This study was initiated, in part, in an attempt to summarize formally certain indications of probability plots. In particular, could one condense departures from statistical linearity of probability plots into one or a few 'degrees of freedom' in the manner of the application of analysis of variance in regression analysis? In a probability plot, one can consider the regression of the ordered observations on the expected values of the order statistics from a standardized version of the hypothesized distribution-the plot tending to be linear if the hypothesis is true. Hence a possible method of testing the distributional assumptionis by means of an analysis of variance type procedure. Using generalized least squares (the ordered variates are correlated) linear and higher-order

16,906 citations

Journal ArticleDOI
TL;DR: The principles of the method and how to impute categorical and quantitative variables, including skewed variables, are described and shown and the practical analysis of multiply imputed data is described, including model building and model checking.
Abstract: Multiple imputation by chained equations is a flexible and practical approach to handling missing data. We describe the principles of the method and show how to impute categorical and quantitative variables, including skewed variables. We give guidance on how to specify the imputation model and how many imputations are needed. We describe the practical analysis of multiply imputed data, including model building and model checking. We stress the limitations of the method and discuss the possible pitfalls. We illustrate the ideas using a data set in mental health, giving Stata code fragments. Copyright © 2010 John Wiley & Sons, Ltd.

6,349 citations


Cites background from "Systems of frequency curves generat..."

  • ...These include the Johnson SU family [20] and the modulus-exponential-Normal (MEN) and modulus-power-Normal (MPN) transformations [21]....

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Journal ArticleDOI
TL;DR: The Box‐Cox power exponential (BCPE) method, with curve smoothing by cubic splines, was used to construct the curves and the concordance between smoothed percentile curves and empirical percentiles was excellent and free of bias.
Abstract: Aim: To describe the methods used to construct the WHO Child Growth Standards based on length/height, weight and age, and to present resulting growth charts. Methods: The WHO Child Growth Standards were derived from an international sample of healthy breastfed infants and young children raised in environments that do not constrain growth. Rigorous methods of data collection and standardized procedures across study sites yielded very high-quality data. The generation of the standards followed methodical, state-of-the-art statistical methodologies. The Box-Cox power exponential (BCPE) method, with curve smoothing by cubic splines, was used to construct the curves. The BCPE accommodates various kinds of distributions, from normal to skewed or kurtotic, as necessary. A set of diagnostic tools was used to detect possible biases in estimated percentiles or z-score curves. Results: There was wide variability in the degrees of freedom required for the cubic splines to achieve the best model. Except for length/height-for-age, which followed a normal distribution, all other standards needed to model skewness but not kurtosis. Length-for-age and height-for-age standards were constructed by fitting a unique model that reflected the 0.7-cm average difference between these two measurements. The concordance between smoothed percentile curves and empirical percentiles was excellent and free of bias. Percentiles and z-score curves for boys and girls aged 0–60 mo were generated for weight-for-age, length/height-for-age, weight-for-length/height (45 to 110 cm and 65 to 120 cm, respectively) and body mass index-for-age. Conclusion: The WHO Child Growth Standards depict normal growth under optimal environmental conditions and can be used to assess children everywhere, regardless of ethnicity, socio-economic status and type of feeding.

2,850 citations


Cites methods from "Systems of frequency curves generat..."

  • ...Five distributions were identified for detailed testing: the Box-Cox power exponential [22], the Box-Cox t [23], the Box-Cox normal [24], the Johnson’s SU [25] and the modulus-exponentialnormal [26]....

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Journal ArticleDOI
TL;DR: The generalized additive model for location, scale and shape (GAMLSS) as mentioned in this paper is a general class of statistical models for a univariate response variable, which assumes independent observations of the response variable y given the parameters, the explanatory variables and the values of the random effects.
Abstract: Summary. A general class of statistical models for a univariate response variable is presented which we call the generalized additive model for location, scale and shape (GAMLSS). The model assumes independent observations of the response variable y given the parameters, the explanatory variables and the values of the random effects. The distribution for the response variable in the GAMLSS can be selected from a very general family of distributions including highly skew or kurtotic continuous and discrete distributions. The systematic part of the model is expanded to allow modelling not only of the mean (or location) but also of the other parameters of the distribution of y, as parametric and/or additive nonparametric (smooth) functions of explanatory variables and/or random-effects terms. Maximum (penalized) likelihood estimation is used to fit the (non)parametric models. A Newton–Raphson or Fisher scoring algorithm is used to maximize the (penalized) likelihood. The additive terms in the model are fitted by using a backfitting algorithm. Censored data are easily incorporated into the framework. Five data sets from different fields of application are analysed to emphasize the generality of the GAMLSS class of models.

2,386 citations


Additional excerpts

  • ...μ, σ, ν, τ / (Johnson, 1949), is defined by assuming that z=ν + τ sinh−1{....

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  • ...The Johnson–Su family for −∞ < y < ∞, denoted by JSU0.µ, σ, ν, τ / (Johnson, 1949), is defined by assuming that z=ν + τ sinh−1{.y −µ/=σ} has a standard normal distribution....

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References
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Book
01 Jan 1902
TL;DR: The volume now gives a somewhat exhaustive account of the various ramifications of the subject, which are set out in an attractive manner and should become indispensable, not only as a textbook for advanced students, but as a work of reference to those whose aim is to extend the knowledge of analysis.
Abstract: This classic work has been a unique resource for thousands of mathematicians, scientists and engineers since its first appearance in 1902 Never out of print, its continuing value lies in its thorough and exhaustive treatment of special functions of mathematical physics and the analysis of differential equations from which they emerge The book also is of historical value as it was the first book in English to introduce the then modern methods of complex analysis This fifth edition preserves the style and content of the original, but it has been supplemented with more recent results and references where appropriate All the formulas have been checked and many corrections made A complete bibliographical search has been conducted to present the references in modern form for ease of use A new foreword by Professor SJ Patterson sketches the circumstances of the book's genesis and explains the reasons for its longevity A welcome addition to any mathematician's bookshelf, this will allow a whole new generation to experience the beauty contained in this text

8,965 citations