Heteroscedastic Error Variances Research Articles

Robust Hausman Pretest for Panel Data Model in the Presence of Heteroscedasticity and Influential Observations

The classical Hausman pretest (HT) is used to specified the right model between random and fixed effect panel data models. However, in the presence of heteroscedastic error variances and influential observations (IOs) in the data set, it may not correctly identify the right model. Therefore, this motivated us to proposed a new method termed Robust Hausman Test (RHTFIID) which employed residuals from weighted least square (WLS) instead of OLS in the construction of heteroscedasticity consistent covariance matrix (HCCM) estimator. The weighting method is based on an efficient High Leverage Points (HLPs) detection method called Fast Improvised Influential Distance (FIID) which down weight only vertical outliers and bad HLPs. The good HLPs were allowed in the estimation as they might contribute to the precision of the estimate. The result indicates that the new proposed RHTFIID outperformed the existing classical Hausman pretest by identifying the right model with and without heteroscedasticity and influential observations.

Optimality VIS-À-VIS robustness in mixture models with heteroscedastic error

Mixture models and designs are used in situations where the response depends on the proportions of the factors (components). Optimum designs were derived for mixture models with fixed regression parameters under homoscedastic error variance by several authors. In this paper, an attempt has been made to find D- and A-optimum designs for the estimation of model parameters with heteroscedastic error variance. It is assumed that the error variance has constant value for all points equidistant from the center of the design. Equivalence theorem plays an important role in this investigation. Robustness of the standard optimum designs in the homoscedastic case under changes in the error variances has also been studied.

Using rheological monitoring to determine the gelation kinetics of chitosan-based systems.

The modeling of polymeric reactions is a topic of large interest. The gelation reactions that may result from self-crosslinking or hybrid (agent based-) crosslinking are examples with interest specially in biomaterials applications. The composition of polymer entities during the reaction is hard to follow, and their concentration is not a good measure of the system dynamics. One alternative is monitoring the rheological behavior of the reacting mass, and relate the elastic modulus of the mixture with the rheological degree of conversion. In this paper we use rheological data to fit Malkin and Kulichikin (1996) [1] based models to describe the crosslinking of chitosan. First, the self-crosslinking of chitosan is considered. Then, the agent-based crosslinking reaction promoted by genipin is addressed. We use dynamical rheological data to fit the reaction models. The model fitting problem generated using Maximum Likelihood principle with heteroscedastic prediction error variance is formulated as a Dynamic Optimization problem and subsequently solved with a sequential approach. Parametric confidence regions are computed using the linear approximation of the covariance matrix at the optimum. Further, the parameters correlation matrix is also determined and used to qualitatively infer about the practical identifiability. The reaction order obtained for self-crosslinking kinetics is 1.3375 ± (0.0151) - approximately of first order -, and is 2.2402 ± (0.0373) for hybrid crosslinking (approximately of second order). In both cases we prove the error variance model is heteroskedastic and the model is identifiable. The approach proposed herein can be extended to other polymer systems.

A Weighted-Least-Squares Estimation Approach to Comparing Trends in Age-Adjusted Cancer Rates Across Overlapping Regions

Li and Tiwari (2008) recently developed a corrected Z-test statistic for comparing the trends in cancer age-adjusted mortality and incidence rates across overlapping geographic regions, by properly adjusting for the correlation between the slopes of the fitted simple linear regression equations. One of their key assumptions is that the error variances have unknown but common variance. However, since the age-adjusted rates are linear combinations of mortality or incidence counts, arising naturally from an underlying Poisson process, this constant variance assumption may be violated. This paper develops a weighted-least-squares based test that incorporates heteroscedastic error variances, and thus significantly extends the work of Li and Tiwari. The proposed test generally outperforms the aforementioned test through simulations and through application to the age-adjusted mortality data from the Surveillance, Epidemiology, and End Results (SEER) Program of the National Cancer Institute.

A multiple comparison procedure for dose‐finding trials with subpopulations

Identifying subgroups of patients with an enhanced response to a new treatment has become an area of increased interest in the last few years. When there is knowledge about possible subpopulations with an enhanced treatment effect before the start of a trial it might be beneficial to set up a testing strategy, which tests for a significant treatment effect not only in the full population, but also in these prespecified subpopulations. In this paper, we present a parametric multiple testing approach for tests in multiple populations for dose‐finding trials. Our approach is based on the MCP‐Mod methodology, which uses multiple comparison procedures (MCPs) to test for a dose–response signal, while considering multiple possible candidate dose–response shapes. Our proposed methods allow for heteroscedastic error variances between populations and control the family‐wise error rate over tests in multiple populations and for multiple candidate models. We show in simulations that the proposed multipopulation testing approaches can increase the power to detect a significant dose–response signal over the standard single‐population MCP‐Mod, when the specified subpopulation has an enhanced treatment effect.

Profiling heteroscedasticity in linear regression models

AbstractDiagnostics for heteroscedasticity in linear regression models have been intensively investigated in the literature. However, limited attention has been paid on how to identify covariates associated with heteroscedastic error variances. This problem is critical in correctly modelling the variance structure in weighted least squares estimation, which leads to improved estimation efficiency. We propose covariate‐specific statistics based on information ratios formed as comparisons between the model‐based and sandwich variance estimators. A two‐step diagnostic procedure is established, first to detect heteroscedasticity in error variances, and then to identify covariates the error variance structure might depend on. This proposed method is generalized to accommodate practical complications, such as when covariates associated with the heteroscedastic variances might not be associated with the mean structure of the response variable, or when strong correlation is present amongst covariates. The performance of the proposed method is assessed via a simulation study and is illustrated through a data analysis in which we show the importance of correct identification of covariates associated with the variance structure in estimation and inference. The Canadian Journal of Statistics 43: 358–377; 2015 © 2015 Statistical Society of Canada

Bayesian Inference for the Finite Population Total from a Heteroscedastic Probability Proportional to Size Sample

Inference for the population total from probability proportional to size (PPS) sampling provides a comparison of design-based and model-based approaches to survey inference, for an important practical design. The usual design-based approach weights sampled units by the inverse of their inclusion probabilities, using the Horvitz-Thompson or Hajek estimates. The model-based approach predicts the outcome for non-sampled units based on a regression of the outcome on the size variable. Zheng and Little (2003) showed that this regression approach, based on a flexible penalized spline regression model, can provide superior inferences to Horvitz-Thompson or generalized regression, in terms of both precision and confidence coverage. However, the sizes of non-sampled units are exploited in this approach, and this information is rarely included in public-use data files. Little and Zheng (2007) showed that when the sizes of non-sampled units are not available, the spline model, combined with a Bayesian bootstrap (BB) model for predicting the non-sampled sizes, can still provide superior inferences, though gains were reduced and less consistent. We further develop these methods by (a) including an unknown parameter to model heteroscedastic error variance in the spline model, an important modeling feature in the PPS setting; and (b) providing an improved Bayesian method for including summary information about the aggregate size of non-sampled units. Simulation studies suggest that the resulting Bayesian method, which includes information on the number and total size of the non-sampled units, recovers most of the information in the individual sizes of the non-sampled units, and provides significant gains over the traditional Horvitz-Thompson estimator. The method is applied to two public-use data sets from the U.S. Census Bureau as well as a data set from the U.S. Energy Information Administration.

Reverse attenuation in interaction terms due to covariate measurement error.

Covariate measurement error may cause biases in parameters of regression coefficients in generalized linear models. The influence of measurement error on interaction parameters has, however, only rarely been investigated in depth, and if so, attenuation effects were reported. In this paper, we show that also reverse attenuation of interaction effects may emerge, namely when heteroscedastic measurement error or sampling variances of a mismeasured covariate are present, which are not unrealistic scenarios in practice. Theoretical findings are illustrated with simulations. A Bayesian approach employing integrated nested Laplace approximations is suggested to model the heteroscedastic measurement error and covariate variances, and an application shows that the method is able to reveal approximately correct parameter estimates.

Stochastic approximation Monte Carlo Gibbs sampling for structural change inference in a Bayesian heteroscedastic time series model

We consider a Bayesian deterministically trending dynamic time series model with heteroscedastic error variance, in which there exist multiple structural changes in level, trend and error variance, but the number of change-points and the timings are unknown. For a Bayesian analysis, a truncated Poisson prior and conjugate priors are used for the number of change-points and the distributional parameters, respectively. To identify the best model and estimate the model parameters simultaneously, we propose a new method by sequentially making use of the Gibbs sampler in conjunction with stochastic approximation Monte Carlo simulations, as an adaptive Monte Carlo algorithm. The numerical results are in favor of our method in terms of the quality of estimates.

Reducing hydrologic model uncertainty in monthly streamflow predictions using multimodel combination

Model errors are inevitable in any prediction exercise. One approach that is currently gaining attention in reducing model errors is by combining multiple models to develop improved predictions. The rationale behind this approach primarily lies on the premise that optimal weights could be derived for each model so that the developed multimodel predictions will result in improved predictions. A new dynamic approach (MM‐1) to combine multiple hydrological models by evaluating their performance/skill contingent on the predictor state is proposed. We combine two hydrological models, “abcd” model and variable infiltration capacity (VIC) model, to develop multimodel streamflow predictions. To quantify precisely under what conditions the multimodel combination results in improved predictions, we compare multimodel scheme MM‐1 with optimal model combination scheme (MM‐O) by employing them in predicting the streamflow generated from a known hydrologic model (abcd model orVICmodel) with heteroscedastic error variance as well as from a hydrologic model that exhibits different structure than that of the candidate models (i.e., “abcd” model or VIC model). Results from the study show that streamflow estimated from single models performed better than multimodels under almost no measurement error. However, under increased measurement errors and model structural misspecification, both multimodel schemes (MM‐1 and MM‐O) consistently performed better than the single model prediction. Overall, MM‐1 performs better than MM‐O in predicting the monthly flow values as well as in predicting extreme monthly flows. Comparison of the weights obtained from each candidate model reveals that as measurement errors increase, MM‐1 assigns weights equally for all the models, whereas MM‐O assigns higher weights for always the best‐performing candidate model under the calibration period. Applying the multimodel algorithms for predicting streamflows over four different sites revealed that MM‐1 performs better than all single models and optimal model combination scheme, MM‐O, in predicting the monthly flows as well as the flows during wetter months.

The Heteroscedastic Graded Response Model with a Skewed Latent Trait: Testing Statistical and Substantive Hypotheses Related to Skewed Item Category Functions.

The Graded Response Model (GRM; Samejima, Estimation of ability using a response pattern of graded scores, Psychometric Monograph No.17, Richmond, VA: The Psychometric Society, 1969) can be derived by assuming a linear regression of a continuous variable, Z, on the trait, θ, to underlie the ordinal item scores (Takane & de Leeuw in Psychometrika, 52:393-408, 1987). Traditionally, a normal distribution is specified for Z implying homoscedastic error variances and a normally distributed θ. In this paper, we present the Heteroscedastic GRM with Skewed Latent Trait, which extends the traditional GRM by incorporation of heteroscedastic error variances and a skew-normal latent trait. An appealing property of the extended GRM is that it includes the traditional GRM as a special case. This enables specific tests on the normality assumption of Z. We show how violations of normality in Z can lead to asymmetrical category response functions. The ability to test this normality assumption is beneficial from both a statistical and substantive perspective. In a simulation study, we show the viability of the model and investigate the specificity of the effects. We apply the model to a dataset on affect and a dataset on alexithymia.

Estimating Riparian Understory Vegetation Cover with Beta Regression and Copula Models

Abstract Understory vegetation communities are critical components of forest ecosystems. As a result, the importance of modeling understory vegetation characteristics in forested landscapes has become more apparent. Abundance measures such as shrub cover are bounded between 0 and 1, exhibit heteroscedastic error variance, and are often subject to spatial dependence. These distributional features tend to be ignored when shrub cover data are analyzed. The beta distribution has been used successfully to describe the frequency distribution of vegetation cover. Beta regression models ignoring spatial dependence (BR) and accounting for spatial dependence (BRdep) were used to estimate percent shrub cover as a function of topographic conditions and overstory vegetation structure in riparian zones in western Oregon. The BR models showed poor explanatory power (pseudo-R2 ≤ 0.34) but outperformed ordinary least-squares (OLS) and generalized least-squares (GLS) regression models with logit-transformed response in terms of mean square prediction error and absolute bias. We introduce a copula (COP) model that is based on the beta distribution and accounts for spatial dependence. A simulation study was designed to illustrate the effects of incorrectly assuming normality, equal variance, and spatial independence. It showed that BR, BRdep, and COP models provide unbiased parameter estimates, whereas OLS and GLS models result in slightly biased estimates for two of the three parameters. On the basis of the simulation study, 93–97% of the GLS, BRdep, and COP confidence intervals covered the true parameters, whereas OLS and BR only resulted in 84–88% coverage, which demonstrated the superiority of GLS, BRdep, and COP over OLS and BR models in providing standard errors for the parameter estimates in the presence of spatial dependence.

Regression Calibration with Heteroscedastic Error Variance

The problem of covariate measurement error with heteroscedastic measurement error variance is considered. Standard regression calibration assumes that the measurement error has a homoscedastic measurement error variance. An estimator is proposed to correct regression coefficients for covariate measurement error with heteroscedastic variance. Point and interval estimates are derived. Validation data containing the gold standard must be available. This estimator is a closed-form correction of the uncorrected primary regression coefficients, which may be of logistic or Cox proportional hazards model form, and is closely related to the version of regression calibration developed by Rosner et al. (1990). The primary regression model can include multiple covariates measured without error. The use of these estimators is illustrated in two data sets, one taken from occupational epidemiology (the ACE study) and one taken from nutritional epidemiology (the Nurses' Health Study). In both cases, although there was evidence of moderate heteroscedasticity, there was little difference in estimation or inference using this new procedure compared to standard regression calibration. It is shown theoretically that unless the relative risk is large or measurement error severe, standard regression calibration approximations will typically be adequate, even with moderate heteroscedasticity in the measurement error model variance. In a detailed simulation study, standard regression calibration performed either as well as or better than the new estimator. When the disease is rare and the errors normally distributed, or when measurement error is moderate, standard regression calibration remains the method of choice.

Testing hypotheses on coefficients of variation from a series of two-armed experiments

We consider the problem of testing hypotheses on the difference of the coefficients of variation from several two-armed experiments with normally distributed outcomes. In particular, we deal with testing the homogeneity of the difference of the coefficients of variation and testing the equality of the difference of the coefficients of variation to a specified value. The test statistics proposed are derived in a limiting one-way classification with fixed effects and heteroscedastic error variances, using results from analysis of variance. By way of simulation, the performance of these test statistics is compared for both testing problems considered.

Iterative Weighted Semiparametric Least Squares Estimation in Repeated Measurement Partially Linear Regression Models

Consider a repeated measurement partially linear regression model with an unknown vector parameter β 1, an unknown function g(·), and unknown heteroscedastic error variances. In order to improve the semiparametric generalized least squares estimator (SGLSE) of β, we propose an iterative weighted semiparametric least squares estimator (IWSLSE) and show that it improves upon the SGLSE in terms of asymptotic covariance matrix. An adaptive procedure is given to determine the number of iterations. We also show that when the number of replicates is less than or equal to two, the IWSLSE can not improve upon the SGLSE. These results are generalizations of those in [2] to the case of semiparametric regressions.

An asymptotic theory for semiparametric generalized least squares estimation in partially linear regression models

Consider a partially linear regression model with an unknown vector parameter β, an unknown functiong(·), and unknown heteroscedastic error variances. In this paper we develop an asymptotic semiparametric generalized least squares estimation theory under some weak moment conditions. These moment conditions are satisfied by many of the error distributions encountered in practice, and our theory does not require the number of replications to go to infinity.

Additive biomass equations for native eucalypt forest trees of temperate Australia

Biomass additivity is a desirable characteristic of a system of equations for predicting component as well as total tree biomass since it eliminates the inconsistency between the sum of predicted values for components such as stem, bark, branch and leaf and the prediction for the total tree. Besides logical consistency, a system of additive biomass equations when estimated by taking into account the inherent correlation among the biomass components has greater statistical efficiency than separately estimated equations for individual components. Using mostly small sample data from both published and unpublished sources, a system of non-linear additive biomass equations was developed for 15 native eucalypt forest tree species of temperate Australia. Diameter at breast height was used as the independent variable for all 15 species, while the combined variable of diameter and tree height was used for 14 species with height data. The system of additive equations provided more accurate biomass estimates than the common approach of separately fitting total tree and component biomass equations using log transformed data through least squares regression. Residual error variances were collectively estimated for each species by pooling small sample data across species and using indicator variables to represent the scale factor for each species in a residual variance function. This method overcame a common problem in estimating heteroscedastic error variance in non-linear biomass equations with additive error terms for small samples. From the estimated residual variance functions, approximate confidence bands containing about 95% of the observed data about the mean curve of predicted biomass were derived for all biomass components of each species. This system of additive biomass equations will prove to be useful for biomass estimation of native eucalypt forests of temperate Australia.

Delete-group Jackknife Estimate in Partially Linear Regression Models with Heteroscedasticity

Consider a partially linear regression model with an unknown vector parameter β, an unknown function g(·), and unknown heteroscedastic error variances. Chen, You[23] proposed a semiparametric generalized least squares estimator (SGLSE) for β, which takes the heteroscedasticity into account to increase efficiency. For inference based on this SGLSE, it is necessary to construct a consistent estimator for its asymptotic covariance matrix. However, when there exists within-group correlation, the traditional delta method and the delete-1 jackknife estimation fail to offer such a consistent estimator. In this paper, by deleting grouped partial residuals a delete-group jackknife method is examined. It is shown that the delete-group jackknife method indeed can provide a consistent estimator for the asymptotic covariance matrix in the presence of within-group correlations. This result is an extension of that in [21].

Testing for homogeneity in a general one-way classification with fixed effects: power simulations and comparative study

In an unbalanced one-way layout with fixed effects and heteroscedastic error variances, several tests are available for testing the homogeneity hypothesis of the group means. While it is rather straightforward to study the performance of these tests under the null hypothesis by way of simulation, it is difficult to assess the power of the tests under the alternative hypothesis due to the many possibilities of choosing a particular alternative hypothesis, especially with increasing number of groups. An alternative way of simulating the power of the homogeneity tests is proposed. Only the distance of the alternative hypothesis to the null hypothesis has to be specified.

A note on weak form market efficiency in security prices: evidence from the Hong Kong stock exchange

This paper employs variance ratio tests with both homoscedastic and heteroscedastic error variances to examine the random walk hypothesis for the Hang Seng Index on the Hong Kong Stock Exchange. The empirical investigation leads us to suggest that the Hang Seng follows a random walk model and consequently that the index is weak form efficient. This conclusion offers both confirmatory and conflicting support for the conclusions of previous research, which has investigated the presence of random walks in the indices of both developed and emerging markets.