Normality testing: two new tests using L-moments

Establishing that there is no compelling evidence that some population is not normally distributed is fundamental to many statistical inferences, and numerous approaches to testing the null hypothesis of normality have been proposed. Fundamentally, the power of a test depends on which specific deviation from normality may be presented in a distribution. Knowledge of the potential nature of deviation from normality should reasonably guide the researcher's selection of testing for non-normality. In most settings, little is known aside from the data available for analysis, so that selection of a test based on general applicability is typically necessary. This research proposes and reports the power of two new tests of normality. One of the new tests is a version of the R-test that uses the L-moments, respectively, L-skewness and L-kurtosis and the other test is based on normalizing transformations of L-skewness and L-kurtosis. Both tests have high power relative to alternatives. The test based on normalized transformations, in particular, shows consistently high power and outperforms other normality tests against a variety of distributions.     

A note on the correlation structure of transformed Gaussian random fields

Transformed Gaussian random fields can be used to model continuous time series and spatial data when the Gaussian assumption is not appropriate. The main features of these random fields are specified in a transformed scale, while for modelling and parameter interpretation it is useful to establish connections between these features and those of the random field in the original scale. This paper provides evidence that for many ‘normalizing’ transformations the correlation function of a transformed Gaussian random field is not very dependent on the transformation that is used. Hence many commonly used transformations of correlated data have little effect on the original correlation structure. The property is shown to hold for some kinds of transformed Gaussian random fields, and a statistical explanation based on the concept of parameter orthogonality is provided. The property is also illustrated using two spatial datasets and several ‘normalizing’ transformations. Some consequences of this property for modelling and inference are also discussed.     

Sample size calculations for single-arm survival studies using transformations of the Kaplan–Meier estimator

In single-arm clinical trials with survival outcomes, the Kaplan–Meier estimator and its confidence interval are widely used to assess survival probability and median survival time. Since the asymptotic normality of the Kaplan–Meier estimator is a common result, the sample size calculation methods have not been studied in depth. An existing sample size calculation method is founded on the asymptotic normality of the Kaplan–Meier estimator using the log transformation. However, the small sample properties of the log transformed estimator are quite poor in small sample sizes (which are typical situations in single-arm trials), and the existing method uses an inappropriate standard normal approximation to calculate sample sizes. These issues can seriously influence the accuracy of results. In this paper, we propose alternative methods to determine sample sizes based on a valid standard normal approximation with several transformations that may give an accurate normal approximation even with small sample sizes. In numerical evaluations via simulations, some of the proposed methods provided more accurate results, and the empirical power of the proposed method with the arcsine square-root transformation tended to be closer to a prescribed power than the other transformations. These results were supported when methods were applied to data from three clinical trials.     

Finite sample correction factors for several simple robust estimators of normal standard deviation

Estimation of the standard deviation of a normal population is an important practical problem that in industrial practice must often be done from small and possibly contaminated data sets. Using multiple estimators is useful, as differences in the estimates may indicate whether the data set is contaminated and the form of the contamination. In this paper, finite sample correction factors have been estimated by simulation for several simple robust estimators of the standard deviation of a normal population. The estimators are the median absolute deviation, interquartile range, shortest half interval (Shorth), and median moving range. Finite sample correction factors have also been estimated for the commonly used non-robust estimators: mean absolute deviation and mean moving range. The simulation has been benchmarked against finite sample correction factors for the sample standard deviation and the sample range.     

A Simple Derivation of r* for Curved Exponential Families

For curved exponential families we consider modified likelihood ratio statistics of the form r_L=r+ log( u/r)/r , where r is the signed root of the likelihood ratio statistic. We are testing a one-dimensional hypothesis, but in order to specify approximate ancillary statistics we consider the test as one in a series of tests. By requiring asymptotic independence and asymptotic normality of the test statistics in a large deviation region there is a particular choice of the statistic u which suggests itself. The derivation of this result is quite simple, only involving a standard saddlepoint approximation followed by a transformation. We give explicit formulas for the statistic u , and include a discussion of the case where some coordinates of the underlying variable are lattice.     

On robustness of usual confidence region under transformation misspecification

Robustness of confidence region for linear model parameters following a misspecified transformation of dependent variable is studied. It is shown that when error standard deviation is moderate to large the usual confidence region is robust against transformation misspecification. When error standard deviation is small the usual confidence region could be very conservative for structured models and slightly liberal for unstructured models. However, the conservativeness in structured case can be controlled if the transformation is selected with the help of data rather than prior information since this is the case when data is able to provide a very accurate estimate of transformation.     

Statistical inference for multiple change-point models

In this article, we propose a new technique for constructing confidence intervals for the mean of a noisy sequence with multiple change-points. We use the weighted bootstrap to generalize the bootstrap aggregating or bagging estimator. A standard deviation formula for the bagging estimator is introduced, based on which smoothed confidence intervals are constructed. To further improve the performance of the smoothed interval for weak signals, we suggest a strategy of adaptively choosing between the percentile intervals and the smoothed intervals. A new intensity plot is proposed to visualize the pattern of the change-points. We also propose a new change-point estimator based on the intensity plot, which has superior performance in comparison with the state-of-the-art segmentation methods. The finite sample performance of the confidence intervals and the change-point estimator are evaluated through Monte Carlo studies and illustrated with a real data example.     

On Symmetrizing Transformation of the Sample Coefficient of Variation from a Normal Population

Variance-stabilizing transformation (VST) for the sample coefficient of variation is often used as a normalizing transformation and may be used for inference on the population coefficient of variation. However, for small samples, the VST may not be symmetric and hence there is a scope of improvement in its performance by seeking a symmetrizing transformation. This article investigates such a transformation that has been obtained by solving a differential equation. The solution may be complex; hence, a numerical strategy is employed in order to make the approximation practically useful. This transformation has been compared with explicitly available VST. The approach has been illustrated on real data from an agricultural experiment concentrating on inference on single samples; however, the method may be generally applicable to multiple samples when testing the homogeneity of coefficients of variation for many populations by following usual normal-theory-based methods applied on transformed statistics.     

GRAPHICAL SENSITIVITY ANALYSIS WITH DIFFERENT METHODS OF IMPUTATION FOR A TRIAL WITH PROBABLE NON‐IGNORABLE MISSING DATA

Graphical sensitivity analyses have recently been recommended for clinical trials with non‐ignorable missing outcome. We demonstrate an adaptation of this methodology for a continuous outcome of a trial of three cognitive‐behavioural therapies for mild depression in primary care, in which one arm had unexpectedly high levels of missing data. Fixed‐value and multiple imputations from a normal distribution (assuming either varying mean and fixed standard deviation, or fixed mean and varying standard deviation) were used to obtain contour plots of the contrast estimates with their P‐values superimposed, their confidence intervals, and the root mean square errors. Imputation was based either on the outcome value alone, or on change from baseline. The plots showed fixed‐value imputation to be more sensitive than imputing from a normal distribution, but the normally distributed imputations were subject to sampling noise. The contours of the sensitivity plots were close to linear in appearance, with the slope approximately equal to the ratio of the proportions of subjects with missing data in each trial arm.     

On Sensitivity of Inverse Response Plot Estimation and the Benefits of a Robust Estimation Approach

Abstract. Inverse response plots are a useful tool in determining a response transformation function for response linearization in regression. Under some mild conditions it is possible to seek such transformations by plotting ordinary least squares fits versus the responses. A common approach is then to use nonlinear least squares to estimate a transformation by modelling the fits on the transformed response where the transformation function depends on an unknown parameter to be estimated. We provide insight into this approach by considering sensitivity of the estimation via the influence function. For example, estimation is insensitive to the method chosen to estimate the fits in the initial step. Additionally, the inverse response plot does not provide direct information on how well the transformation parameter is being estimated and poor inverse response plots may still result in good estimates. We also introduce a simple robustified process that can vastly improve estimation.     

The null distribution of the likelihood ratio test for a mixture of two normals after a restricted box-cox transformation

Maclean et al. (1976) applied a specific Box-Cox transformation to test for mixtures of distributions against a single distribution. Their null hypothesis is that a sample of n observations is from a normal distribution with unknown mean and variance after a restricted Box-Cox transformation. The alternative is that the sample is from a mixture of two normal distributions, each with unknown mean and unknown, but equal, variance after another restricted Box-Cox transformation. We developed a computer program that calculated the maximum likelihood estimates (MLEs) and likelihood ratio test (LRT) statistic for the above. Our algorithm for the calculation of the MLEs of the unknown parameters used multiple starting points to protect against convergence to a local rather than global maximum. We then simulated the distribution of the LRT for samples drawn from a normal distribution and five Box-Cox transformations of a normal distribution. The null distribution appeared to be the same for the Box-Cox transformations studied and appeared to be distributed as a chi-square random variable for samples of 25 or more. The degrees of freedom parameter appeared to be a monotonically decreasing function of the sample size. The null distribution of this LRT appeared to converge to a chi-square distribution with 2.5 degrees of freedom. We estimated the critical values for the 0.10, 0.05, and 0.01 levels of significance.     

Elliptical safety region plots for C pk

The process capability index C _pk is widely used when measuring the capability of a manufacturing process. A process is defined to be capable if the capability index exceeds a stated threshold value, e.g. C _pk >4/3. This inequality can be expressed graphically using a process capability plot, which is a plot in the plane defined by the process mean and the process standard deviation, showing the region for a capable process. In the process capability plot, a safety region can be plotted to obtain a simple graphical decision rule to assess process capability at a given significance level. We consider safety regions to be used for the index C _pk . Under the assumption of normality, we derive elliptical safety regions so that, using a random sample, conclusions about the process capability can be drawn at a given significance level. This simple graphical tool is helpful when trying to understand whether it is the variability, the deviation from target, or both that need to be reduced to improve the capability. Furthermore, using safety regions, several characteristics with different specification limits and different sample sizes can be monitored in the same plot. The proposed graphical decision rule is also investigated with respect to power.     

Box‐Cox transformations in linear models: Large sample theory and tests of normality

The authors provide a rigorous large sample theory for linear models whose response variable has been subjected to the Box‐Cox transformation. They provide a continuous asymptotic approximation to the distribution of estimators of natural parameters of the model. They show, in particular, that the maximum likelihood estimator of the ratio of slope to residual standard deviation is consistent and relatively stable. The authors further show the importance for inference of normality of the errors and give tests for normality based on the estimated residuals. For non‐normal errors, they give adjustments to the log‐likelihood and to asymptotic standard errors.     

The Effectiveness of Risk Scores: the Logit Rank Plot

A risk score s for event E is a function of covariates with the property that P ( E | s ) is an increasing function of s . Motivated by applications in medicine and in criminology, we suggest the logit rank plot as a good way of summarizing the effectiveness of such a score. Explicitly, plot logit{ P ( E | s )} against logit( r ), where r is the proportional rank of s in a sample or population. The slope of this plot gives an overall measure of effectiveness, and the logit rank transformation provides a common basis on which different risk scores can be compared. Some practical and theoretical aspects are discussed.     

Methods to Distinguish Between Polynomial and Exponential Tails

Two methods to distinguish between polynomial and exponential tails are introduced. The methods are based on the properties of the residual coefficient of variation for the exponential and non‐exponential distributions. A graphical method, called a CV‐plot, shows departures from exponentiality in the tails. The plot is applied to the daily log‐returns of exchange rates of US dollar and Japanese yen. New statistics are introduced for testing the exponentiality of tails using multiple thresholds. They give better control of the significance level than previous tests. The powers of the new tests are compared with those of some others for various sample sizes.     

Bounds on Sample Variation Measures Based on Majorization

Using majorization theory, upper and lower bounds are derived for different measures of variation as progressively more items of information are available about the sample data. As a convenient starting point, bounds are first established for a one-parameter family of variation measures, which is a generalized mean difference measure of which Gini's mean difference, the standard deviation, and the range are particular cases. While, as pointed out, some of the derived bounds are well known, others do not appear to have been published and are tighter than established bounds. Some 40 different bounds are derived, besides any number of bounds given for the generalized family of variation measures. A number of interesting inequalities are also derived on the basis of some of the bounds. While the bounds have been developed in terms of real-valued sample data generally, the paper concludes with a brief discussion of the bounds for categorical data when the sample data consists of frequencies (counts).     

Robust weighted one-way ANOVA: Improved approximation and efficiency

A robust test for the one-way ANOVA model under heteroscedasticity is developed in this paper. The data are assumed to be symmetrically distributed, apart from some outliers, although the assumption of normality may be violated. The test statistic to be used is a weighted sum of squares similar to the Welch [1951. On the comparison of several mean values: an alternative approach. Biometrika 38, 330-336.] test statistic, but any of a variety of robust measures of location and scale for the populations of interest may be used instead of the usual mean and standard deviation. Under the commonly occurring condition that the robust measures of location and scale are asymptotically normal, we derive approximations to the distribution of the test statistic under the null hypothesis and to its distribution under alternative hypotheses. An expression for relative efficiency is derived, thus allowing comparison of the efficiency of the test as a function of the choice of the location and scale estimators used in the test statistic. As an illustration of the theory presented here, we apply it to three commonly used robust location–scale estimator pairs: the trimmed mean with the Winsorized standard deviation; the Huber Proposal 2 estimator pair; and the Hampel robust location estimator with the median absolute deviation.     

An alternative approximation to the variance of transformation score

An alternative approximation to the variance of transformation score is given, based on an asymptotic expansion of the transformation estimator. It is then compared with the variance approximation given by Lawrance (1987) in terms of standardized scores. Simulations show that the two standardized scores behave very similarly when model error standard deviation is small. However,when error standard deviation is not small, the new standardized score outperforms that of Lawrance,especially in the structured models.     

Estimation of the population mean when the coefficient of variation is known

Theory has been developed to provide an optimum estimator of the population mean based on a “mean per unit” estimator and the estimated standard deviation, assuming that the form of the distribution as well as its coefficient of variation (c.v.) are known. Theory has been extended to the case when an estimate of c.v. is available from an independent sample drawn in the past; the case when the form of the distribution is not known is also discussed. It is shown that the relative efficiency of the estimator with respect to “mean per unit estimator” is generally high for normal or near normal populations. For log-normal populations, an increase in efficiency of about 17 percent can be achieved. The results have been illustrated with data from biological populations.     

A New Kernel Distribution Function Estimator Based on a Non-parametric Transformation of the Data

Abstract.  A new kernel distribution function (df) estimator based on a non-parametric transformation of the data is proposed. It is shown that the asymptotic bias and mean squared error of the estimator are considerably smaller than that of the standard kernel df estimator. For the practical implementation of the new estimator a data-based choice of the bandwidth is proposed. Two possible areas of application are the non-parametric smoothed bootstrap and survival analysis. In the latter case new estimators for the survival function and the mean residual life function are derived.