Test For Homoscedasticity Research Articles

Application of modern tests for stationarity to single-trial MEG data

Stationarity is a crucial yet rarely questioned assumption in the analysis of time series of magneto- (MEG) or electroencephalography (EEG). One key drawback of the commonly used tests for stationarity of encephalographic time series is the fact that conclusions on stationarity are only indirectly inferred either from the Gaussianity (e.g. the Shapiro-Wilk test or Kolmogorov-Smirnov test) or the randomness of the time series and the absence of trend using very simple time-series models (e.g. the sign and trend tests by Bendat and Piersol). We present a novel approach to the analysis of the stationarity of MEG and EEG time series by applying modern statistical methods which were specifically developed in econometrics to verify the hypothesis that a time series is stationary. We report our findings of the application of three different tests of stationarity--the Kwiatkowski-Phillips-Schmidt-Schin (KPSS) test for trend or mean stationarity, the Phillips-Perron (PP) test for the presence of a unit root and the White test for homoscedasticity--on an illustrative set of MEG data. For five stimulation sessions, we found already for short epochs of duration of 250 and 500 ms that, although the majority of the studied epochs of single MEG trials were usually mean-stationary (KPSS test and PP test), they were classified as nonstationary due to their heteroscedasticity (White test). We also observed that the presence of external auditory stimulation did not significantly affect the findings regarding the stationarity of the data. We conclude that the combination of these tests allows a refined analysis of the stationarity of MEG and EEG time series.

Tests of Homoscedasticity, Normality, and Missing Completely at Random for Incomplete Multivariate Data

Test of homogeneity of covariances (or homoscedasticity) among several groups has many applications in statistical analysis. In the context of incomplete data analysis, tests of homoscedasticity among groups of cases with identical missing data patterns have been proposed to test whether data are missing completely at random (MCAR). These tests of MCAR require large sample sizes n and/or large group sample sizes n(i), and they usually fail when applied to non-normal data. Hawkins (1981) proposed a test of multivariate normality and homoscedasticity that is an exact test for complete data when n(i) are small. This paper proposes a modification of this test for complete data to improve its performance, and extends its application to test of homoscedasticity and MCAR when data are multivariate normal and incomplete. Moreover, it is shown that the statistic used in the Hawkins test in conjunction with a nonparametric k-sample test can be used to obtain a nonparametric test of homoscedasticity that works well for both normal and non-normal data. It is explained how a combination of the proposed normal-theory Hawkins test and the nonparametric test can be employed to test for homoscedasticity, MCAR, and multivariate normality. Simulation studies show that the newly proposed tests generally outperform their existing competitors in terms of Type I error rejection rates. Also, a power study of the proposed tests indicates good power. The proposed methods use appropriate missing data imputations to impute missing data. Methods of multiple imputation are described and one of the methods is employed to confirm the result of our single imputation methods. Examples are provided where multiple imputation enables one to identify a group or groups whose covariance matrices differ from the majority of other groups.

A robust test for homoscedasticity in nonparametric regression

We consider a nonparametric location scale model and propose a new test for homoscedasticity (constant scale function). The test is based on an estimate of a deterministic function that vanishes if and only if the hypothesis of a constant scale function is satisfied and an empirical process estimating this function is investigated. Weak convergence to a scaled Brownian bridge is established, which allows a simple calculation of critical values. The new test can detect alternatives converging to the null hypothesis at a rate n −1/2 and is robust with respect to the presence of outliers. The finite sample properties are investigated by means of a simulation study, and the test is compared with some nonrobust tests for a constant scale function, which have recently been proposed in the literature.

Inconsistency of the MLE and inference based on weighted LS for LARCH models

This paper considers a class of finite-order autoregressive linear ARCH models. The model captures the leverage effect, allows the volatility to be arbitrarily close to zero and to reach its minimum for non-zero innovations, and is appropriate for long memory modeling when infinite orders are allowed. However, the (quasi-)maximum likelihood estimator is, in general, inconsistent. A self-weighted least-squares estimator is proposed and is shown to be asymptotically normal. A score test for conditional homoscedasticity and diagnostic portmanteau tests are developed. Their performance is illustrated via simulation experiments. It is also investigated whether stock market returns exhibit some of the characteristic features of the linear ARCH model.

The effect of a low dose of clenbuterol on rat soleus muscle submitted to joint immobilization

The aim of the present study was to evaluate the effect of joint immobilization on morphometric parameters and glycogen content of soleus muscle treated with clenbuterol. Male Wistar (3-4 months old) rats were divided into 4 groups (N = 6 for each group): control, clenbuterol, immobilized, and immobilized treated with clenbuterol. Immobilization was performed with acrylic resin orthoses and 10 microg/kg body weight clenbuterol was administered subcutaneously for 7 days. The following parameters were measured the next day on soleus muscle: weight, glycogen content, cross-sectional area, and connective tissue content. The clenbuterol group showed an increase in glycogen (81.6%, 0.38 +/- 0.09 vs 0.69 +/- 0.06 mg/100 g; P < 0.05) without alteration in weight, cross-sectional area or connective tissue compared with the control group. The immobilized group showed a reduction in muscle weight (34.2%, 123.5 +/- 5.3 vs 81.3 +/- 4.6 mg; P < 0.05), glycogen content (31.6%, 0.38 +/- 0.09 vs 0.26 +/- 0.05 mg/100 mg; P < 0.05) and cross-sectional area (44.1%, 2574.9 +/- 560.2 vs 1438.1 +/- 352.2 microm(2); P < 0.05) and an increase in connective tissue (216.5%, 8.82 +/- 3.55 vs 27.92 +/- 5.36%; P < 0.05). However, the immobilized + clenbuterol group showed an increase in weight (15.9%; 81.3 +/- 4.6 vs 94.2 +/- 4.3 mg; P < 0.05), glycogen content (92.3%, 0.26 +/- 0.05 vs 0.50 +/- 0.17 mg/100 mg; P < 0.05), and cross-sectional area (19.9%, 1438.1 +/- 352.2 vs 1724.8 +/- 365.5 microm(2); P < 0.05) and a reduction in connective tissue (52.2%, 27.92 +/- 5.36 vs 13.34 +/- 6.86%; P < 0.05). Statistical analysis was performed using Kolmogorov-Smirnov and homoscedasticity tests. For the muscle weight and muscle glycogen content, two-way ANOVA and the Tukey test were used. For the cross-sectional area and connective tissue content, Kruskal-Wallis and Tukey tests were used. This study emphasizes the importance of anabolic pharmacological protection during immobilization to minimize skeletal muscle alterations resulting from disuse.

Heterogeneity of neuroendocrine stress responses in aging rat strains

Hyperactivity of the hypothalamo-pituitary-adrenocortical (HPA) axis is linked with age-related decrements in cognition and neuronal survival. However, the nature and extent of age-related HPA axis deficits vary considerably across and indeed, within strains. The current study was designed to assess variance in HPA axis function using two rodent models commonly used in aging studies: Fischer 344 (F344) and F344/Brown–Norway F1 hybrid rats (F344/BN). We examined both basal and stress-induced ACTH and corticosterone (CORT) release in two stress contexts thought to differ in intensity: novel environment (‘mild’) and restraint (‘intense’). Variability of the data was tested with a modification of the Brown–Forsythe test of homoscedasticity. The results indicated that F344 rats exhibit greater peak HPA responses. Furthermore, in most cases variability was increased in aged rats relative to young and middle-aged rats of the same strain, indicative of the emergence of individual differences in stress responsivity amongst older rats. The results suggest that these older rat strains may be useful models to further assess individual differences in neuroendocrine aging.

Bayesian modeling and significant features exploration in wavelet power spectra

Abstract. This study proposes and justifies a Bayesian approach to modeling wavelet coefficients and finding statistically significant features in wavelet power spectra. The approach utilizes ideas elaborated in scale-space smoothing methods and wavelet data analysis. We treat each scale of the discrete wavelet decomposition as a sequence of independent random variables and then apply Bayes' rule for constructing the posterior distribution of the smoothed wavelet coefficients. Samples drawn from the posterior are subsequently used for finding the estimate of the true wavelet spectrum at each scale. The method offers two different significance testing procedures for wavelet spectra. A traditional approach assesses the statistical significance against a red noise background. The second procedure tests for homoscedasticity of the wavelet power assessing whether the spectrum derivative significantly differs from zero at each particular point of the spectrum. Case studies with simulated data and climatic time-series prove the method to be a potentially useful tool in data analysis.

Effects of outliers on the identification and estimation of GARCH models

Abstract. This paper analyses how outliers affect the identification of conditional heteroscedasticity and the estimation of generalized autoregressive conditionally heteroscedastic (GARCH) models. First, we derive the asymptotic biases of the sample autocorrelations of squared observations generated by stationary processes and show that the properties of some conditional homoscedasticity tests can be distorted. Second, we obtain the asymptotic and finite sample biases of the ordinary least squares (OLS) estimator of ARCH(p) models. The finite sample results are extended to generalized least squares (GLS), maximum likelihood (ML) and quasi‐maximum likelihood (QML) estimators of ARCH(p) and GARCH(1,1) models. Finally, we show that the estimated asymptotic standard deviations are biased estimates of the sample standard deviations.

Classification of Gastric Tumors using Shape Features of Gland

Recently in Japan, pathologists have been in short supply, while each pathological diagnosis requires a substantial amount of time because each analyte must be inspected by multiple pathologists for adequate diagnosis. This paper deals with the classification method of gastric cancer and gastric adenoma, using image processing and pattern analysis. We first select the R component and G component from the RGB basis of the digital image, and the Y component from the YIQ basis for our system. After pre-processing, we automatically extracted the shape of the nucleus and cytoplasm. After many inspections, we selected 40 features for shape of the nucleus and cytoplasm and 14 features for texture within the cytoplasm for assessment of tumors. Principal component analysis, F test of homoscedasticity, t test of difference of average, stepwise method for selecting the smaller number of features, and discriminant method using Mahalanobis distance were all performed. Total ratio of diagnosis reached 96.9%, showing the validity of our proposed method.

Properties of the Sample Autocorrelations of Nonlinear Transformations in Long-Memory Stochastic Volatility Models

The autocorrelations of log-squared, squared, and absolute financial returns are often used to infer the dynamic properties of the underlying volatility. This article shows that, in the context of long-memory stochastic volatility models, these autocorrelations are smaller than the autocorrelations of the log volatility and so is the rate of decay for squared and absolute returns. Furthermore, the corresponding sample autocorrelations could have severe negative biases, making the identification of conditional heteroscedasticity and long memory a difficult task. Finally, we show that the power of some popular tests for homoscedasticity is larger when they are applied to absolute returns.

On a test for a parametric form of volatility in continuous time financial models

Abstract. A new specification test for the parametric form of the variance function in diffusion processes is proposed, which does not require specific knowledge of the functional form of the model. The corresponding test statistic has an asymptotic normal distribution under the null hypothesis and diverges at an appropriate rate under the alternative. In contrast to recent work the approach of the present paper does not require the specification of particular time points at which the hypothesis of a parametric form is checked. As a by-product we obtain a very simple test for homoscedasticity in diffusion processes. Moreover, the new test does not use nonparametric estimation techniques for estimating the variance function and is therefore independent of the specification of a particular smoothing parameter. The results are illustrated by a small simulation study and a data example is analyzed.

Liquid chromatography–UV determination and liquid chromatography–atmospheric pressure chemical ionization mass spectrometric characterization of sitosterol and stigmasterol in soybean oil

A narrow-bore HPLC–UV method was developed for the analysis of two of the more abundant naturally occurring phytosterols in vegetable oils: sitosterol and stigmasterol. The method enabled detection of the compounds at a concentration of 0.42 μg/ml and quantitation at concentrations of 0.52 and 0.54 μg/ml for sitosterol and stigmasterol, respectively. An excellent linearity was determined over two orders of concentration magnitude ( r 2 0.999–1.000) and verified by applying the Mandel fitting test ( p>0.099) and the lack-of-fit test ( p>0.057) performed at the 95% confidence level. A good intra-day precision ranging from 0.15 to 1.16% was calculated at two concentration levels (2 and 100 μg/ml). The inter-day reproducibility was verified on 3 different days by performing an homoscedasticity test and analysis of variance. A solid-phase extraction method was developed on silica cartridges for the isolation of phytosterols from soybean oil providing recovery values of 101±9 and 106±7% for sitosterol and stigmasterol, respectively. Good accuracy of the method was statistically demonstrated since no matrix effect was found for both the analytes. The developed method was applied to the quantitative assay of phytosterols in a soybean oil sample (61±5 mg/100 g of stigmasterol and 118±4 mg/100 g sitosterol). The HPLC–atmospheric pressure chemical ionization MS technique enabled the identification of stigmasterol, sitosterol and campesterol in the oil sample.

Risk Comparison of Inequality Constrained Estimators in the Heteroscedastic Linear Model

There exist many studies which treat the inequality and/or interval constraints on coefficients in the homoscedastic linear regression model. However, the sampling performance of the inequality constrained estimators in the heteroscedastic linear model has not been examined. This paper considers the inequality constrained estimators in the heteroscedastic linear regression model and derives their risks under a quadratic loss function. Furthermore, using the inequality constrained estimators, we introduce a pre‐test estimator which might be employed after the test for homoscedasticity and derive its risk. In addition, the risk performance of these estimators is evaluated numerically.

Grain size along two gravel-bed rivers: statistical variation, spatial pattern and sedimentary links

A new set of field data facilitates a detailed analysis of variations in bed material grain size within two confluent gravel-bed rivers in northeastern British Columbia, Canada. A preliminary assessment of grain-size variability establishes a basis for examination of the spatial pattern of grain-size change. Standard ANOVA techniques are inappropriate because individual samples have unequal variances and are not normally distributed. Alternative tests for homoscedasticity and comparison of means are therefore utilized. Within-site, between-sample variability is not significant. The grain-size distributions that were obtained at individual sites are therefore representative of the depositional environments that were sampled. In both rivers mean grain size does vary significantly between sites and there is therefore a basis for examining the data for spatial patterns such as downstream fining. Textural variations along the two rivers studied here are complex and show negligible overall fining (in over 100 km). This is the consequence of a large number of tributary inputs and non-alluvial sediment sources which are the legacy of Late Pleistocene glaciation. The identification of lateral sources like these is fundamental for understanding textural changes within rivers. The sedimentary link (a channel reach between significant lateral sediment inputs) provides a means of isolating fluvial maturation processes (abrasion and sorting) from contingent lateral inputs. Strong fining trends are apparent in most links and classification of grain-size measurements according to their location within particular links greatly improves the statistical explanation of textural variation. Identification of sedimentary links provides a means of applying models of fluvial fining processes, so isolation of link networks will aid the development of basin-scale models of textural variation. © 1998 John Wiley & Sons, Ltd.

Analysis of Levene’s Test Under Design Imbalance

By means of three analyses, the operating characteristics of Levene‘s test of homoscedasticity in a 1 -way layout are shown to be systematically affected by design imbalance when the error terms are normal. We consider a formal general linear hypothesis analysis, an empirical analysis, and an analysis of expected mean squares, employing the heuristic justification for an ordinary ANOVA. We propose a simple correction involving leverage and analyze its effect, as well as that of another adjustment, on power and level.

Distribution-Free Tests for One-Way Homoscedasticity in a Two-Way Classification

Homoscedasticity is an important assumption for the analysis of variance of data from a two-way table. A preliminary test for equality of row (column) variances is often in order. Moreover, the row (column) variance may be a parameter characterizing physical or biological properties of the treatment associated with a row (column), e.g., in the analysis of measurement errors or in the analysis of phenotypic stability. In this case, a test for homoscedasticity is central to the statistical analysis. Unfortunately, parametric procedures available for this problem are quite sensitive to the underlying assumption of normality. This paper examines two rank tests given by Nassar and Hiihn (1987, Biometrics 43, 45-53) and Hiihn and Nassar (1989, Biometrics 45, 977-1000), which are applicable for detecting homoscedasticity in a two-way classification. These tests are approximately valid for normally distributed errors, but they may be anticonservative for nonnormal distributions. A modification is suggested that leads to a distribution-free test for homoscedasticity in a two-way layout. Simulations indicate that under normality the modified test has low power compared to parametric tests.

A Robust Test for Homoscedasticity in a Two‐Way Layout

AbstractThis paper discusses tests for homogeneity of row‐variances in a two‐way classification. Parametric tests for this problem are shown to be highly sensitive to departures from the normality assumption. Levene's robust tests tend to be rather liberal when the number of rows exceeds the number of columns. A modification is suggested which improves the approximation for this situation, while retaining the desirable robustness property.

C410. a distribution-free test for homoscedasticity in a two-way layout

C410. a distribution-free test for homoscedasticity in a two-way layout

C409. A randomization test for homoscedasticity in a balanced one-way layout

C409. A randomization test for homoscedasticity in a balanced one-way layout

Testing Homoscedasticity in a Two-Way Table

Shukla (1972, Biometrics 28, 1063-1072) proposed a test of homoscedasticity for a two-way table, which he illustrated by comparing column variances in a 13 X 4 table. In this note it is argued that his method is appropriate only if the number of columns is less than or equal to the number of rows. We offer an alternative that is obtained by modifying Shukla's approach and that can be applied irrespective of the dimensions of the table.