Adaptive robust estimation and testing

Testing the normality of heart associated variables having neutrosophic numbers

In this paper, tests of skewness and kurtosis are introduced under neutrosophic statistics. The necessary measures and neutrosophic forms of these estimators are introduced. The application of the proposed tests is given using the data associated with heart diseases. From the real example analysis, the proposed tests are quite flexible and informative than the existing tests under classical statistics. In addition, it is concluded from the analysis that the proposed tests give information about the measure of indeterminacy in the presence of uncertainty.

Bias, Precision, and Accuracy of Skewness and Kurtosis Estimators for Frequently Used Continuous Distributions

Several measures of skewness and kurtosis were proposed by Hogg (1974) in order to reduce the bias of conventional estimators when the distribution is non-normal. Here we conducted a Monte Carlo simulation study to compare the performance of conventional and Hogg’s estimators, considering the most frequent continuous distributions used in health, education, and social sciences (gamma, lognormal and exponential distributions). In order to determine the bias, precision and accuracy of the skewness and kurtosis estimators for each distribution we calculated the relative bias, the coefficient of variation, and the scaled root mean square error. The effect of sample size on the estimators is also analyzed. In addition, a SAS program for calculating both conventional and Hogg’s estimators is presented. The results indicated that for the non-normal distributions investigated, the estimators of skewness and kurtosis which best reflect the shape of the distribution are Hogg’s estimators. It should also be noted that Hogg’s estimators are not as affected by sample size as are conventional estimators.

Location Tests for Biomarker Studies: A Comparison Using Simulations for the Two-sample Case

Background: Gene, protein, or metabolite expression levels are often non-normally distributed, heavy tailed and contain outliers. Standard statistical approaches may fail as location tests in this situation.

Objectives: In three Monte-Carlo simulation studies, we aimed at comparing the type I error levels and empirical power of standard location tests and three adaptive tests [O’Gorman, Can J Stat 1997; 25: 269 –279; Keselman et al., Brit J Math Stat Psychol 2007; 60: 267– 293; Szymczak et al., Stat Med 2013; 32: 524 – 537] for a wide range of distributions.

Methods: We simulated two-sample scena -rios using the g-and-k-distribution family to systematically vary tail length and skewness with identical and varying variability between groups.

Results: All tests kept the type I error level when groups did not vary in their variability. The standard non-parametric U-test per -formed well in all simulated scenarios. It was outperformed by the two non-parametric adaptive methods in case of heavy tails or large skewness. Most tests did not keep the type I error level for skewed data in the case of heterogeneous variances.

Conclusions: The standard U-test was a powerful and robust location test for most of the simulated scenarios except for very heavy tailed or heavy skewed data, and it is thus to be recommended except for these cases. The non-parametric adaptive tests were powerful for both normal and non-normal distributions under sample variance homogeneity. But when sample variances differed, they did not keep the type I error level. The parametric adaptive test lacks power for skewed and heavy tailed distributions.

Evaluating clinical significance: incorporating robust statistics with normative comparison tests

The purpose of this study was to evaluate a modified test of equivalence for conducting normative comparisons when distribution shapes are non-normal and variances are unequal. A Monte Carlo study was used to compare the empirical Type I error rates and power of the proposed Schuirmann-Yuen test of equivalence, which utilizes trimmed means, with that of the previously recommended Schuirmann and Schuirmann-Welch tests of equivalence when the assumptions of normality and variance homogeneity are satisfied, as well as when they are not satisfied. The empirical Type I error rates of the Schuirmann-Yuen were much closer to the nominal α level than those of the Schuirmann or Schuirmann-Welch tests, and the power of the Schuirmann-Yuen was substantially greater than that of the Schuirmann or Schuirmann-Welch tests when distributions were skewed or outliers were present. The Schuirmann-Yuen test is recommended for assessing clinical significance with normative comparisons.

The impact of study design and diagnostic approach in a large multi-centre ADHD study. Part 1: ADHD symptom patterns

BACKGROUND

The International Multi-centre ADHD Genetics (IMAGE) project with 11 participating centres from 7 European countries and Israel has collected a large behavioural and genetic database for present and future research. Behavioural data were collected from 1068 probands with the combined type of attention deficit/hyperactivity disorder (ADHD-CT) and 1446 'unselected' siblings. The aim was to analyse the IMAGE sample with respect to demographic features (gender, age, family status, and recruiting centres) and psychopathological characteristics (diagnostic subtype, symptom frequencies, age at symptom detection, and comorbidities). A particular focus was on the effects of the study design and the diagnostic procedure on the homogeneity of the sample in terms of symptom-based behavioural data, and potential consequences for further analyses based on these data.

METHODS

Diagnosis was based on the Parental Account of Childhood Symptoms (PACS) interview and the DSM-IV items of the Conners' teacher questionnaire. Demographics of the full sample and the homogeneity of a subsample (all probands) were analysed by using robust statistical procedures which were adjusted for unequal sample sizes and skewed distributions. These procedures included multi-way analyses based on trimmed means and winsorised variances as well as bootstrapping.

RESULTS

Age and proband/sibling ratios differed between participating centres. There was no significant difference in the distribution of gender between centres. There was a significant interaction between age and centre for number of inattentive, but not number of hyperactive symptoms. Higher ADHD symptom frequencies were reported by parents than teachers. The diagnostic symptoms differed from each other in their frequencies. The face-to-face interview was more sensitive than the questionnaire. The differentiation between ADHD-CT probands and unaffected siblings was mainly due to differences in hyperactive/impulsive symptoms.

CONCLUSIONS

Despite a symptom-based standardized inclusion procedure according to DSM-IV criteria with defined symptom thresholds, centres may differ markedly in probands' ADHD symptom frequencies. Both the diagnostic procedure and the multi-centre design influence the behavioural characteristics of a sample and, thus, may bias statistical analyses, particularly in genetic or neurobehavioral studies.

A comparative study of robust tests for spread: asymmetric trimming strategies

We examined 633 procedures that can be used to compare the variability of scores across independent groups. The procedures, except for one, were modifications of the procedures suggested by Levene (1960) and O'Brien (1981). We modified their procedures by substituting robust measures of the typical score and variability, rather than relying on classical estimators. The robust measures that we utilized were either based on a priori or empirically determined symmetric or asymmetric trimming strategies. The Levene-type and O'Brien-type transformed scores were used with either the ANOVA F test, a robust test due to Lee and Fung (1985), or the Welch (1951) test. Based on four measures of robustness, we recommend a Levene-type transformation based upon empirically determined 20% asymmetric trimmed means, involving a particular adaptive estimator, where the transformed scores are then used with the ANOVA F test.

Robust step-down tests for multivariate independent group designs

A composite step-down procedure, in which a set of step-down tests are summarized collectively with Fisher's combination statistic, was considered to test for multivariate mean equality in two-group designs. An approximate degrees of freedom (ADF) composite procedure based on trimmed/Winsorized estimators and a non-pooled estimate of error variance is proposed, and compared to a composite procedure based on trimmed/Winsorized estimators and a pooled estimate of error variance. The step-down procedures were also compared to Hotelling's T (2) and Johansen's ADF global procedure based on trimmed estimators in a simulation study. Type I error rates of the pooled step-down procedure were sensitive to covariance heterogeneity in unbalanced designs; error rates were similar to those of Hotelling's T (2) across all of the investigated conditions. Type I error rates of the ADF composite step-down procedure were insensitive to covariance heterogeneity and less sensitive to the number of dependent variables when sample size was small than error rates of Johansen's test. The ADF composite step-down procedure is recommended for testing hypotheses of mean equality in two-group designs except when the data are sampled from populations with different degrees of multivariate skewness.