Levene-type Test Research Articles

Preliminary Tests of Homogeneity - Type I Error Rates under Non-Normality

Many statistical procedures utilize preliminary tests to enhance the accuracy of the final inferences. Preliminary tests like Goldfeld-Quandt (GQ) and Levene-type tests are used to assess the assumption of equality of population variances with normality as the underlying distributional assumption. Such tests must be used with care as the final inferences are conditional on the performance of these tests at first stage. This study explores the size distortions of GQ and Levene-type tests under non-normality. The results do not warrant the use of GQ & Levene test under non-normality as the size distortions are as high as 88 & 48% for the respective statistics. However, the modified form of Levene test (BF-test) retains its size properties except for the multi-model alternatives with relatively big outliers.

A Comparison of Five Bootstrap and Non-Bootstrap Levene-Type Tests of Homogeneity of Variances

The test of homogeneity of variances arises in various statistical problems. In this paper, we compared five Levene-type tests including Levene’s test, Welch’s test, James’s test, Brown and Forsythe’s test and Alexander–Govern’s test in terms of power and size. The simulation results show that the bootstrapping after applying Noguchi and Gel (J Nonparametr Stat 22:897–913, 2010) correction and removing structural zeros significantly improve the performance of Type I error rates. We identify the best in terms of controlling the Type I error rate and power for each number of groups.

In vitro evaluation of accuracy and precision of automated robotic tooth preparation system for porcelain laminate veneers

Statement of problemControlling tooth reduction for porcelain laminate veneers (PLVs) in fractions of millimeters is challenging. PurposeThe purpose of this study was to assess an automated robotic tooth preparation system for PLVs for accuracy and precision compared with conventional freehand tooth preparation. Material and methodsTwenty maxillary central incisor tooth models were divided into 2 groups. Ten were assigned to a veneer preparation with a robotic arm according to preoperative preparation design-specific guidelines (experimental group). Ten were assigned to conventional tooth preparation by a clinician (control group). Initially, all tooth models were scanned with a 3- dimensional (3D) laser scanner, and a tooth preparation for PLVs was designed on a 3D image. Each tooth model was attached to a typodont. For the experimental group, an electric high-speed handpiece with a 0.9-mm-diameter round diamond rotary cutting instrument was mounted on the robotic arm. The teeth were prepared automatically according to the designed image. For the control group, several diamond rotary cutting instruments were used to prepare the tooth models according to preoperative preparation design guidelines. All prepared tooth models were scanned. The preoperative preparation design image and scanned postoperative preparation images were superimposed. The dimensional difference between those 2 images was measured on the facial aspect, finish line, and incisal edge. Differences between the experimental and the control groups from the 3D design image were computed. Accuracy and precision were compared for all sites and separately for each tooth surface (facial, finish line, incisal). Statistical analyses were conducted with a permutation test for accuracy and with a modified robust Brown-Forsythe Levene-type test for precision (α=.05). ResultsFor accuracy for all sites, the mean absolute deviation was 0.112 mm in the control group and 0.133 mm in the experimental group. No significant difference was found between the 2 (P=.15). For precision of all sites, the standard deviation was 0.141 mm in the control group and 0.185 mm in the experimental group. The standard deviation in the control group was significantly lower (P=.030). In terms of accuracy for the finish line, the control group was significantly less accurate (P=.038). For precision, the standard deviation in the control group was significantly higher at the finish line (P=.034). ConclusionsFor the data from all sites, the experimental procedure was able to prepare the tooth model as accurately as the control, and the control procedure was able to prepare the tooth model with better precision. The experimental group showed better accuracy and precision at the finish line.

A Levene-type test of homogeneity of variances against ordered alternatives

Investigations focusing on differences in scale parameters across multiple samples appear in various scientific fields, e.g., when assessing measurement precision. In such cases a natural order of ...

Palaeoclimatological perspective on river basin hydrometeorology: case of the Mekong Basin

Abstract. Globally, there have been many extreme weather events in recent decades. A challenge has been to determine whether these extreme weather events have increased in number and intensity compared to the past. This challenge is made more difficult due to the lack of long-term instrumental data, particularly in terms of river discharge, in many regions including Southeast Asia. Thus our main aim in this paper is to develop a river basin scale approach for assessing interannual hydrometeorological and discharge variability on long, palaeological, time scales. For the development of the basin-wide approach, we used the Mekong River basin as a case study area, although the approach is also intended to be applicable to other basins. Firstly, we derived a basin-wide Palmer Drought Severity Index (PDSI) from the Monsoon Asia Drought Atlas (MADA). Secondly, we compared the basin-wide PDSI with measured discharge to validate our approach. Thirdly, we used basin-wide PDSI to analyse the hydrometeorology and discharge of the case study area over the study period of 1300–2005. For the discharge-MADA comparison and hydrometeorological analyses, we used methods such as linear correlations, smoothing, moving window variances, Levene type tests for variances, and wavelet analyses. We found that the developed basin-wide approach based on MADA can be used for assessing long-term average conditions and interannual variability for river basin hydrometeorology and discharge. It provides a tool for studying interannual discharge variability on a palaeological time scale, and therefore the approach contributes to a better understanding of discharge variability during the most recent decades. Our case study revealed that the Mekong has experienced exceptional levels of interannual variability during the post-1950 period, which could not be observed in any other part of the study period. The increased variability was found to be at least partly associated with increased El Niño Southern Oscillation (ENSO) activity.

Crossing survival curves: alternatives to the log-rank test

Background: It is not uncommon for clinical trials to present results on survival time as Kaplan-Meier survival curves that cross, indicating non-proportional hazards. A recent example was given in a pivotal trial in advanced non-small cell lung cancer (The ‘IPASS study’ [1]). Trials such as these present a hazard ratio and log-rank test for treatment comparison as this is their planned primary analysis. However, the validity of such analysis is questionable and has received published criticism. This paper reviews the use of the log-rank test with crossing curves and considers alternatives that have been proposed. Methods: The review of the alternative approaches includes weighted log-rank tests (Wilcoxon, Tarone-Ware, Peto-Prentice and Fleming-Harrington), supremum versions of the log-rank test (modified Kolmogorov-Smirnov and Renyi-type tests) which are based on the maximum difference between estimates of two survivor functions and modified log-rank tests (Lin and Wang test using squared differences at each time point, and Levene-type test focusing on variance differences). In addition, methods based on splitting the analysis at the crossing point have also been proposed. Methods are compared and evaluated using both real and simulated datasets using Weibull and Weibull-Cox distributions representing realistic situations. Results: Crossing survival curves is generally a result of the survival times having greater variance in one treatment group than another. The performance of the log-rank test and alternatives depend on the type of crossing (early, mid or late) but in general the probability of a Type II error is increased for log-rank and weighted log-rank tests but performance is improved with the alternatives. The choice of time-point for the split-analysis is problematic. Standard software such as sts test (Stata), proc lifetest (SAS) and survfit (R) and routines-on-demand support some but not all the tests considered. Conclusions: There is a need in the clinical community to clarify methods that are appropriate when survival curves cross. Statistical analysis plans for clinical trials with survival as primary outcome measure should specify an analysis dependent on the proportionality of hazard rates and explicitly consider non-proportionality issues, powering the analyses based on log-rank alternatives. Modelling the survival data may be more appropriate than simple univariate hypothesis tests when hazards are not proportional. Finally, there are some feasibility issues regarding software for such analysis that remain to be tackled.

Levene type tests for the ratio of two scales

Tests for the equality of variances are of interest in many areas such as quality control, agricultural production systems, experimental education, pharmacology, biology, as well as a preliminary to the analysis of variance, dose–response modelling or discriminant analysis. The literature is vast. Traditional non-parametric tests are due to Mood, Miller and Ansari–Bradley. A test which usually stands out in terms of power and robustness against non-normality is the W50 Brown and Forsythe [Robust tests for the equality of variances, J. Am. Stat. Assoc. 69 (1974), pp. 364–367] modification of the Levene test [Robust tests for equality of variances, in Contributions to Probability and Statistics, I. Olkin, ed., Stanford University Press, Stanford, 1960, pp. 278–292]. This paper deals with the two-sample scale problem and in particular with Levene type tests. We consider 10 Levene type tests: the W50, the M50 and L50 tests [G. Pan, On a Levene type test for equality of two variances, J. Stat. Comput. Simul. 63 (1999), pp. 59–71], the R-test [R.G. O'Brien, A general ANOVA method for robust tests of additive models for variances, J. Am. Stat. Assoc. 74 (1979), pp. 877–880], as well as the bootstrap and permutation versions of the W50, L50 and R tests. We consider also the F-test, the modified Fligner and Killeen [Distribution-free two-sample tests for scale, J. Am. Stat. Assoc. 71 (1976), pp. 210–213] test, an adaptive test due to Hall and Padmanabhan [Adaptive inference for the two-sample scale problem, Technometrics 23 (1997), pp. 351–361] and the two tests due to Shoemaker [Tests for differences in dispersion based on quantiles, Am. Stat. 49(2) (1995), pp. 179–182; Interquantile tests for dispersion in skewed distributions, Commun. Stat. Simul. Comput. 28 (1999), pp. 189–205]. The aim is to identify the effective methods for detecting scale differences. Our study is different with respect to the other ones since it is focused on resampling versions of the Levene type tests, and many tests considered here have not ever been proposed and/or compared. The computationally simplest test found robust is W50. Higher power, while preserving robustness, is achieved by considering the resampling version of Levene type tests like the permutation R-test (recommended for normal- and light-tailed distributions) and the bootstrap L50 test (recommended for heavy-tailed and skewed distributions). Among non-Levene type tests, the best one is the adaptive test due to Hall and Padmanabhan.

Combination of Levene-type tests and a finite-intersection method for testing equality of variances against ordered alternatives

The problem of detecting monotonic trends in variances from k samples is widely met in many applications, e.g. finance, economics, medicine, biopharmaceutical, and environmental studies. However, most of the tests for equality of variances against ordered alternatives rely on the assumption of normality and are often non-robust to its violation, which eventually leads to unreliable conclusions. In this paper, we propose a new distribution-free test against trends in variances which is based on a combination of a robust Levene-type approach and a finite-intersection method. The new test can be viewed as a piecewise linear approximation to possibly non-linear dynamics of variances, and hence is applicable to a broad range of alternatives. The new combined procedure yields a more accurate estimate of size and provides a competitive power for a variety of distributions and alternatives. In addition, we develop a modification of the proposed test for unbalanced designs with small sample sizes. We discuss asymptotic properties of the new test and illustrate its applications with simulations and case studies from soil pollution analysis, real estate markets, engineering, and epidemiology.

The Impact of Levene’s Test of Equality of Variances on Statistical Theory and Practice

In many applications, the underlying scientific question concerns whether the variances of $k$ samples are equal. There are a substantial number of tests for this problem. Many of them rely on the assumption of normality and are not robust to its violation. In 1960 Professor Howard Levene proposed a new approach to this problem by applying the $F$-test to the absolute deviations of the observations from their group means. Levene's approach is powerful and robust to nonnormality and became a very popular tool for checking the homogeneity of variances. This paper reviews the original method proposed by Levene and subsequent robust modifications. A modification of Levene-type tests to increase their power to detect monotonic trends in variances is discussed. This procedure is useful when one is concerned with an alternative of increasing or decreasing variability, for example, increasing volatility of stocks prices or "open or closed gramophones" in regression residual analysis. A major section of the paper is devoted to discussion of various scientific problems where Levene-type tests have been used, for example, economic anthropology, accuracy of medical measurements, volatility of the price of oil, studies of the consistency of jury awards in legal cases and the effect of hurricanes on ecological systems.

Lawstat: AnRPackage for Law, Public Policy and Biostatistics

We present a new R software package lawstat that contains statistical tests and procedures that are utilized in various litigations on securities law, antitrust law, equal employment and discrimination as well as in public policy and biostatistics. Along with the well known tests such as the Bartels test, runs test, tests of homogeneity of several sample proportions, the Brunner-Munzel tests, the Lorenz curve, the Cochran-Mantel-Haenszel test and others, the package contains new distribution-free robust tests for symmetry, robust tests for normality that are more sensitive to heavy-tailed departures, measures of relative variability, Levene-type tests against trends in variances etc. All implemented tests and methods are illustrated by simulations and real-life examples from legal cases, economics and biostatistics. Although the package is called lawstat, it presents implementation and discussion of statistical procedures and tests that are also employed in a variety of other applications, e.g., biostatistics, environmental studies, social sciences and others, in other words, all applications utilizing statistical data analysis. Hence, name of the package should not be considered as a restriction to legal statistics. The package will be useful to applied statisticians and

On a levene type test for equality of two variances

Among the many tests for the equality of two population variances, a modified Levene test W50 stands out as very robust and reasonably efficient, and thus recommended in several published comparison studies. This paper points out a hidden structural problem of this test. More specifically, the power of test W50 is bounded above by a value that can be considerably smaller than 1 for some underlying distributions, no matter how different the two population variances are. In terms of the confidence set corresponding to test W50, noninformative intervals such as (0, + ∞) may occur with non-negligible probabilities for some distributions. The above problem exists for normal distributions, but is especially severe for contaminated normal distributions. Several intuitive remedies are investigated in an attempt to correct the flaw of test W50, but retain its good features such as robustness, efficiency, simplicity and asymptotic distribution-freeness. It is found that a proposed confidence interval and its corresponding test have satisfactory robustness and are more efficient than test W50. The proposed procedures are asymptotically distribution-free and have completely eliminated the hidden problem of test W50.