Power for homoscedastic normally distributed data.

Results based in 1000 replications for each value of β1 in the model given by Eqs (6) to (7) with sample size equal to 1000.

Library of Wroclaw University of Science and Technology scientific output (DONA database)

Data generating model given by Eqs (6) to (11) with N = 4.

Knowledge of contagion among economies is a relevant issue in economics. The canonical model of contagion is an alternative in this case. Given the existence of endogenous variables in the model, instrumental variables can be used to decrease the bias of the OLS estimator. In the presence of heteroskedastic disturbances this paper proposes the use of conditional volatilities as instruments. Simulation is used to show that the homoscedastic and heteroskedastic estimators which use them as instruments have small bias. These estimators are preferable in comparison with the OLS estimator and their asymptotic distribution can be used to construct confidence intervals.

Bias and RMSE (in parentheses) of various estimators in the homoscedastic case.

Data augmentation (DA) turns seemingly intractable computational problems into simple ones by augmenting latent missing data. In addition to computational simplicity, it is now well-established that DA equipped with a deterministic transformation can improve the convergence speed of iterative algorithms such as an EM algorithm or Gibbs sampler. In this article, we outline a framework for the transformation-based DA, which we call data transforming augmentation (DTA), allowing augmented data to be a deterministic function of latent and observed data, and unknown parameters. Under this framework, we investigate a novel DTA scheme that turns heteroscedastic models into homoscedastic ones to take advantage of simpler computations typically available in homoscedastic cases. Applying this DTA scheme to fitting linear mixed models, we demonstrate simpler computations and faster convergence rates of resulting iterative algorithms, compared with those under a non-transformation-based DA scheme. We also fit a Beta-Binomial model using the proposed DTA scheme, which enables sampling approximate marginal posterior distributions that are available only under homoscedasticity. Supplementary materials are available online.

Globally, more than 1.9 billion adults are overweight. Thus, obesity is a serious public health issue. Moreover, obesity is a major risk factor for diabetes mellitus, coronary heart disease, and cardiovascular disease. Recently, GWAS examining obesity and body mass index (BMI) have increasingly unveiled many aspects of the genetic architecture of obesity and BMI. Information on genome-wide genetic variants has been used to estimate the genome-wide polygenic score (GPS) for a personalized prediction of obesity. However, the prediction power of GPS is affected by various factors, including the unequal variance in the distribution of a phenotype, known as heteroscedasticity. Here, we calculated a GPS for BMI using LDpred2, which was based on the BMI GWAS summary statistics from a European meta-analysis. Then, we tested the GPS in 354,761 European samples from the UK Biobank and found an effective prediction power of the GPS on BMI. To study a change in the variance of BMI, we investigated the heteroscedasticity of BMI across the GPS via graphical and statistical methods. We also studied the homoscedastic samples for BMI compared to the heteroscedastic sample, randomly selecting samples with various standard deviations of BMI residuals. Further, we examined the effect of the genetic interaction of GPS with environment (GPS×E) on the heteroscedasticity of BMI. We observed the changing variance (i.e., heteroscedasticity) of BMI along the GPS. The heteroscedasticity of BMI was confirmed by both the Breusch-Pagan test and the Score test. Compared to the heteroscedastic sample, the homoscedastic samples from small standard deviation of BMI residuals showed a decreased heteroscedasticity and an improved prediction accuracy, suggesting a quantitatively negative correlation between the phenotypic heteroscedasticity and the prediction accuracy of GPS. To further test the effects of the GPS×E on heteroscedasticity, first we tested the genetic interactions of the GPS with 21 environments and found 8 significant GPS×E interactions on BMI. However, the heteroscedasticity of BMI was not ameliorated after adjusting for the GPS×E interactions. Taken together, our findings suggest that the heteroscedasticity of BMI exists along the GPS and is not affected by the GPS×E interaction.

Globally, more than 1.9 billion adults are overweight. Thus, obesity is a serious public health issue. Moreover, obesity is a major risk factor for diabetes mellitus, coronary heart disease, and cardiovascular disease. Recently, GWAS examining obesity and body mass index (BMI) have increasingly unveiled many aspects of the genetic architecture of obesity and BMI. Information on genome-wide genetic variants has been used to estimate the genome-wide polygenic score (GPS) for a personalized prediction of obesity. However, the prediction power of GPS is affected by various factors, including the unequal variance in the distribution of a phenotype, known as heteroscedasticity. Here, we calculated a GPS for BMI using LDpred2, which was based on the BMI GWAS summary statistics from a European meta-analysis. Then, we tested the GPS in 354,761 European samples from the UK Biobank and found an effective prediction power of the GPS on BMI. To study a change in the variance of BMI, we investigated the heteroscedasticity of BMI across the GPS via graphical and statistical methods. We also studied the homoscedastic samples for BMI compared to the heteroscedastic sample, randomly selecting samples with various standard deviations of BMI residuals. Further, we examined the effect of the genetic interaction of GPS with environment (GPS×E) on the heteroscedasticity of BMI. We observed the changing variance (i.e., heteroscedasticity) of BMI along the GPS. The heteroscedasticity of BMI was confirmed by both the Breusch-Pagan test and the Score test. Compared to the heteroscedastic sample, the homoscedastic samples from small standard deviation of BMI residuals showed a decreased heteroscedasticity and an improved prediction accuracy, suggesting a quantitatively negative correlation between the phenotypic heteroscedasticity and the prediction accuracy of GPS. To further test the effects of the GPS×E on heteroscedasticity, first we tested the genetic interactions of the GPS with 21 environments and found 8 significant GPS×E interactions on BMI. However, the heteroscedasticity of BMI was not ameliorated after adjusting for the GPS×E interactions. Taken together, our findings suggest that the heteroscedasticity of BMI exists along the GPS and is not affected by the GPS×E interaction.

Four scenarios of the data generating model given by Eqs (6) to (11) with N = 3.

We consider the problem of estimating the density of a random variable when precise measurements on the variable are not available, but replicated proxies contaminated with measurement error are available for sufficiently many subjects. Under the assumption of additive measurement errors this reduces to a problem of deconvolution of densities. Deconvolution methods often make restrictive and unrealistic assumptions about the density of interest and the distribution of measurement errors, for example, normality and homoscedasticity and thus independence from the variable of interest. This article relaxes these assumptions and introduces novel Bayesian semiparametric methodology based on Dirichlet process mixture models for robust deconvolution of densities in the presence of conditionally heteroscedastic measurement errors. In particular, the models can adapt to asymmetry, heavy tails, and multimodality. In simulation experiments, we show that our methods vastly outperform a recent Bayesian approach based on estimating the densities via mixtures of splines. We apply our methods to data from nutritional epidemiology. Even in the special case when the measurement errors are homoscedastic, our methodology is novel and dominates other methods that have been proposed previously. Additional simulation results, instructions on getting access to the dataset and R programs implementing our methods are included as part of online supplementary materials.

We used several tests to test the hypotheses that: 1) the data are linear; 2) the variance is constant (i.e., homoscedastic); and 3) the frequency distribution of residuals is normal. For clarity, only p-values are reported. Most assumptions are met for regressions of log-transformed mass on log-transformed length, but regressions of mass on length cubed exhibited neither homoscedasticity nor normality. Abbreviations for each of the BCIs are provided in Table 1.

Moderation analysis has many applications in social sciences. Most widely used estimation methods for moderation analysis assume that errors are normally distributed and homoscedastic. When these assumptions are not met, the results from a classical moderation analysis can be misleading. For more reliable moderation analysis, this article proposes two robust methods with a two-level regression model when the predictors do not contain measurement error. One method is based on maximum likelihood with Student's t distribution and the other is based on M-estimators with Huber-type weights. An algorithm for obtaining the robust estimators is developed. Consistent estimates of standard errors of the robust estimators are provided. The robust approaches are compared against normal-distribution-based maximum likelihood (NML) with respect to power and accuracy of parameter estimates through a simulation study. Results show that the robust approaches outperform NML under various distributional conditions. Application of the robust methods is illustrated through a real data example. An R program is developed and documented to facilitate the application of the robust methods.

P-values for Figs 2B and 3C were calculated using a two-tailed homoscedastic student’s t-test. P-values for Figs 5B and 6A were calculated using edgeR compared to unwounded controls. (XLSX)

We tested three assumptions, whether 1) data are linear, 2) variance is constant (i.e., homoscedastic), and 3) frequency distribution of residuals are normal. Only p-values are reported.