Homogeneous Error Variance Research Articles

Some inferential questions in regard to analysing two-way Layouts and associated linear model theory and practice

In analysing a well known data set from the literature which can be thought of as a two-way layout it transpires that a robust adaptive regression approach for identifying outliers fails to be sensitive enough to detect the possible interchange of two observations. On the other hand if one takes the classical approach of diagnostic checking one may also stop too early and be satisfied with a model that falls short of a more detailed analysis that takes account of heteroscedasticity in the data. An exact F-test for heteroscedasticity in the two way layout is compared with various more general tests proposed by Shukla. In conclusion it is noted that when modelling the particular form of heteroscedasticity countenanced here, the estimated column effects are unchanged from those estimated from the model assuming homogeneous error variance structure. It is only the estimated variances of these column effects that changes.

Genetic evaluation of growth traits in Iranian Kordi Sheep using random regression model with homogeneous and heterogeneous residual variances

The purpose of this study was estimation of genetic parameter using random regression model (RRM) with various error variance in Iranian Kordi sheep. The data (consisting of 7875 weight records from birth to 360 days of age) were collected during the period 2000 to 2013 from the rearing and breeding station of Kordi sheep in Shirvan, Iran. The independent variables were Legendre polynomials (LP) of age at weighing and orders of fit from 2 to 5 were considered. Analyses were carried out fitting sets of random regression coefficients due to direct additive genetic, direct and maternal permanent environmental effects, with heterogeneous and homogeneous error variances. To compare the model were used different criteria such as LogL, AIC, BIC and LRT. The best fitting RRM among homogeneous error variance was the model with a LP of fourth order for fixed effect, fourth order for direct additive genetic and fifth order for direct and maternal permanent environmental effects (model 4455). Among the models with heterogeneous error variances different, model 7 (Heterogeneous error variances of 72 various classes), was selected as the best model. The variances increased along the trajectory from 3.75 to 12.81, 4.43 to 30.28 and 1.49 to 8.49; 0.25 to 27.94, 0.03 to 12.32 and 0.15 to 22.66 for direct additive genetic, direct and maternal permanent environmental effect by homogeneous and heterogeneous error variances, respectively. The direct heritability ranged from 0.15 to 0.41 and 0.11 to 0.56 by homogeneous and heterogeneous error variances, respectively. Genetic correlation between adjacent test days was more than between distant test days. This research has demonstrated the possibility of application of RRM with heterogeneous error variance for genetic evaluation of Iranian Kordi Sheep.

Comment on Hoffman and Rovine (2007): SPSS MIXED can estimate models with heterogeneous variances.

Hoffman and Rovine (Behavior Research Methods, 39:101-117, 2007) have provided a very nice overview of how multilevel models can be useful to experimental psychologists. They included two illustrative examples and provided both SAS and SPSS commands for estimating the models they reported. However, upon examining the SPSS syntax for the models reported in their Table 3, we found no syntax for models 2B and 3B, both of which have heterogeneous error variances. Instead, there is syntax that estimates similar models with homogeneous error variances and a comment stating that SPSS does not allow heterogeneous errors. But that is not correct. We provide SPSS MIXED commands to estimate models 2B and 3B with heterogeneous error variances and obtain results nearly identical to those reported by Hoffman and Rovine in their Table 3. Therefore, contrary to the comment in Hoffman and Rovine's syntax file, SPSS MIXED can estimate models with heterogeneous error variances.

Adaptively weighted kernel regression

We develop a new kernel-based local polynomial methodology for nonparametric regression based on optimising a linear combination of several loss functions. Optimal weights for least squares and quantile loss functions can be chosen to provide maximum efficiency and these optimal weights can be estimated from data. The resulting estimators are at least as efficient as those provided by existing procedures, but can be much more efficient for many distributions. The data-based weights adapt to the tails of the error distribution resulting in a procedure which is both robust and resistant. Furthermore, the assumption of homogeneous error variance is not required. To illustrate its practical use, we apply the proposed method to model the motorcycle data.

Heterogeneity of error variance and its influence on genotype comparison in multi-location trials

Multi-location trials play an important role when evaluating genotypes at many stages of plant-breeding programs, as well as when recommending varietal plant production. The most common procedure for analyzing these trials is based on the assumption that the residual error variance is homogenous across all considered locations. However, this may often be unrealistic, and therefore limit the accuracy of genotype evaluations or the reliability of varietal recommendations. The objectives of this study were to investigate how frequently and how seriously the problem of heterogeneous variances appears across locations in multi-location trials, and to evaluate the impact of the analytical procedure with different considerations about error variations when comparing genotype effects. A series of 16 multi-location trials from a corn-breeding program in the north of China were simultaneously analyzed from 2005 to 2008 using a randomized complete block design at each location; the analysis used models with homogeneous residual variances, as well as models with heterogeneous residual variances to take into account that different locations may have different levels of precision. The results showed that the residual error variances strongly varied across locations in all of the considered trials. The model with heterogeneous residual error variances was significantly (α<0.001) more appropriate than the one with homogeneous error variances in all of the trials for both fixed and random locations, according to model fitting statistics. Ignoring error variance differences across locations in the analysis procedure of multi-location trials could result in a large Type I error rate for comparisons (F-test) of the main effect of genotypes, especially in case of random location main effects and genotype–location interactions, locations, and genotype–location interactions, as well as an inflation or deflation of statistical Type I error rates for comparisons (t-test) of simple genotype effects depending on specific locations.

On the size of the F-test for the one-way random model with heterogeneous error variances

Traditional analysis of variance tests are based on the assumption of homogeneous error variances, which often fails in real experimental situations. Violation of this assumption affects not only the power of the standard F-test, but also its size. When a design is unbalanced, the effect of unequal error variances is even more complex. In this paper, we study the effect of heterogeneous error variances on the size of the F-test concerning the among-group variance component in an unbalanced random one-way model. We also provide a method for computing the true critical value of the F-test for a given level of significance.

A simulation study for a linear measurement error model when error variances vary between measurements

When predicting scores in the Draize eye irritation test based on measurements of in vitro alternative tests, we are often faced with estimating parameters in a linear measurement error model with heterogeneous error variances. This article proposes a new statistical method for parameter estimation to address this issue. The proposed method is an extension of an earlier proposal that applied a linear measurement error model with homogeneous error variances, to cases with heterogeneous error variances. A simulation study to examine the performance of the proposed method was conducted in a framework that was adaptable to the data, which was obtained in a validation study of alternative methods to animal experiments conducted in Japan. The proposed method reduced the biases of estimates in comparison with an ordinary regression analysis method and three other methods under the assumption of homogeneous error variances. Although the proposed method did not fit the real data well, the resulting prediction formula was far better than those obtained by other methods.

Use of covariance structures for temporal errors in the analysis of a three-course wheat rotation and tillage trial

Crop rotation serves as a mechanism for developing sustainable crop production systems. Crop-rotation trials are used to identify agronomic input factors suitable for use in a cropping system. In crop-rotation trials, experimental errors within the same plot over time are correlated. The form of the covariance structure of the plot errors may be specific to the data from a rotation trial, but is unknown and is generally assumed. Statistical analyses are usually based on the assumption that plot errors are independent, or have constant covariance. An experiment was conducted using wheat-based, three-course rotations containing tillage treatment subplots over 12 years at ICARDA's experimental station at Tel Hadya, a moderately dry area in northern Syria. This study examined several covariance structures for temporal errors arising over the rotation plots and tillage subplots, in order to model wheat yield data. Eighteen covariance structures were examined, and the best pair was selected using the Akaike Information Criterion. The best pair comprised first-order autocorrelation and homogeneous variance for temporal errors in rotation plots, and uniform correlation with heterogeneous variances for temporal errors in tillage subplots. Using the 12 years of data obtained for wheat yield and the best pair of covariance structures, the tillage and rotation effects were found to be statistically significant and to have significant interactions with the cycle of rotation. The precision of the means calculated differed from those calculated using a control structure based on homogeneous error variances and constant correlation. The cumulative yield build-up over time differed significantly over the rotations and the tillage methods. An increasing yield trend was observed for the bread wheat rotation, while a yield decline was observed in durum wheat when the rotation was repeated. When evaluating the effects of input factors in crop rotations, we therefore recommend that the covariance structures be examined and that a suitably chosen structure be used.

Estimation of genetic parameters for ultrasound-predicted percentage of intramuscular fat in Angus cattle using random regression models.

The present study included 3,358 observations of 675 bulls and heifers from the Iowa State University beef cattle breeding project. Data were collected over a 3-yr period between 1998 and 2000. Each year, cattle were scanned four to six times for ultrasound-predicted percentage of intramuscular fat (UPFAT) and other ultrasound traits, starting at a minimum age of 28 wk. The objective of the current study was to estimate variance components, heritability, and repeatability of UPFAT in young bulls and heifers. Data were subjected to random-regression animal models that included fixed effects of contemporary group, fixed Legendre polynomial of age at measurement, and random regression coefficients on Legendre polynomial of age at measurement for animals' direct genetic and direct permanent environmental effects. Phenotypic and genetic models involving different levels of polynomial fit for the animal component were considered. A model fitting a linear effect of Legendre polynomial of age at a measurement for animal direct genetic and direct permanent environmental effects and a homogeneous error variance described the present data adequately. Heritability of UPFAT ranged from 0.32 at 28 wk of age to a maximum of 0.53 at 63 wk. Repeatability of UPFAT increased from a minimum of 0.60 at ages of 28 to 39 wk to a maximum of 0.80 at ages 61 to 63 wk. Heritability and repeatability of yearling UPFAT were 0.50 and 0.71, respectively. With the exception of minor differences at earlier ages, fitting heterogeneous error variances did not have an effect on genetic parameter estimates for most ages of measurement. The present results showed an optimal heritability and repeatability of UPFAT measures around 52 wk and through at least 63 wk of age. This suggested that differences in UPFAT measures during this period also are good measures of differences in marbling genetic potential of Angus cattle.

Robust Regression Approach to Estimating Fish Mortality Rates with a Cohort-Based Model

Natural mortality rate (M) and fishing catchability coefficient (q) were estimated by using the Paloheimo pairwise linear regression model with simulated and real catch–effort data. The four estimation methods used were least squares (LS), least absolute value (LAV), least median of squares (LMS), and LMS-based reweighted least squares (RLS). When catch or effort data or both contained homogeneous error variances, mean squared errors (MSE) of LS and RLS were significantly smaller than those of LAV and LMS. However, there was no significant difference in MSE between LS and RLS. When catch–effort data contained variances inconsistent among years, MSEs of the robust methods were much smaller than those of LS. Reweighted least squares substantially reduced the variances in the LS-estimated q and M, and it had the smallest MSE among the four estimation methods. We suggest using the following procedures in estimating q and M based on the Paloheimo pairwise regression model: (1) applying LMS to the model to identify outliers in the data; and (2) weighting the defined outliers by a factor of 0 and normal data points by a factor of 1 and applying LS to the model to estimate q and M and their associated variances.

Weighted Versus Unweighted Estimates Using Scheffé's Mixture Model for Symmetrical Error Variances Patterns.

When fitting the first-degree Scheffe polynomial and the assumption of homogeneous error variance is suspect, one suggestion made in this paper is to place the design points in a symmetrical arrangement and along the component axes. For if the magnitude of the error variance (or observation variance) follows one of several conceivable symmetrical patterns, the use of the symmetrical axial designs will result in the existence of certain relationships between the parameter estimates obtained using an unweighted analysis versus using a weighted analysis. Some results which carry over to the parameter estimates in a second-degree polynomial for q = 3, 4 and 5 are presented along with two real examples.

A Test for Homogeneity of Regression Without Homogeneity of Variance

A likelihood ratio test procedure is described for testing for homogeneity of regression coefficients and population means across treatment groups without assuming homogeneous error variances. Maximum likelihood estimation equations are obtained assuming the criterion scores to be normally distributed, and a method for their solution is described. Examples of situation for which the procedure is appropriate are given, and extensions of the procedure are discussed. The procedure has been used on hypothetical and real data.

Testing the Homogeneity of Variances in a Two-Way Classification

In a two-way classification with one observation per cell, heterogeneity of variances may occur as considered by Box [1954], Graybill [1954], Hartley [1950a], and Russell and Bradley [1958]. We are interested in testing whether the column variances are all equal while the row variances are assumed to be the same. Russell and Bradley [1958] have considered this problem and derived the likelihood ratio test when there are three columns. The test was used in checking the assumption of homogeneous error variances in randomized block designs under certain conditions and in comparing the precisions of analytical methods in quantitative experimentation as well as the consistencies of judges in subjective experimentation. Hartley [1950a] derived a test based on the range and gave an approximate distribution of the test statistic. We consider the following model:

Analysis of covariance structures.

A general method is presented for estimating variance components when the experimental design has one random way of classification and a possibly unbalanced fixed classification. The procedure operates on a sample covariance matrix in which the fixed classes play the role of variables and the random classes correspond to observations. Cases are considered which assume (i) homogeneous and (ii) nonhomogeneous error variance, and (iii) arbitrary scale factors in the measurements and homogeneous error variance. The results include maximum-likelihood estimations of the variance components and scale factors, likelihood-ratio tests of the goodness-of-fit of the model assumed for the design, and large-sample variances and covariances of the estimates. Applications to mental test data are presented. In these applications the subjects constitute the random dimension of the design, and a classification of the mental tests according to objective features of format or content constitute the fixed dimensions.