Comparison of designs for the three-fold nested random model

The quality of estimation of variance components depends on the design used as well as on the unknown values of the variance components. In this article, three designs are compared, namely, the balanced, staggered, and inverted nested designs for the three-fold nested random model. The comparison is based on the so-called quantile dispersion graphs using analysis of variance (ANOVA) and maximum likelihood (ML) estimates of the variance components. It is demonstrated that the staggered nested design gives more stable estimates of the variance component for the highest nesting factor than the balanced design. The reverse, however, is true in case of lower nested factors. A comparison between ANOVA and ML estimation of the variance components is also made using each of the aforementioned designs.     

Graphical technique for comparing designs for random models

Methods for comparing designs for a random (or mixed) linear model have focused primarily on criteria based on single-valued functions. In general, these functions are difficult to use, because of their complex forms, in addition to their dependence on the model's unknown variance components. In this paper, a graphical approach is presented for comparing designs for random models. The one-way model is used for illustration. The proposed approach is based on using quantiles of an estimator of a function of the variance components. The dependence of these quantiles on the true values of the variance components is depicted by plotting the so-called quantile dispersion graphs (QDGs), which provide a comprehensive picture of the quality of estimation obtained with a given design. The QDGs can therefore be used to compare several candidate designs. Two methods of estimation of variance components are considered, namely analysis of variance and maximum-likelihood estimation.     

Robust estimation of variance components

New robust estimates for variance components are introduced. Two simple models are considered: the balanced one-way classification model with a random factor and the balanced mixed model with one random factor and one fixed factor. However, the method of estimation proposed can be extended to more complex models. The new method of estimation we propose is based on the relationship between the variance components and the coefficients of the least-mean-squared-error predictor between two observations of the same group. This relationship enables us to transform the problem of estimating the variance components into the problem of estimating the coefficients of a simple linear regression model. The variance-component estimators derived from the least-squares regression estimates are shown to coincide with the maximum-likelihood estimates. Robust estimates of the variance components can be obtained by replacing the least-squares estimates by robust regression estimates. In particular, a Monte Carlo study shows that for outlier-contaminated normal samples, the estimates of variance components derived from GM regression estimates and the derived test outperform other robust procedures.     

Graphical evaluation of robust parameter designs based on extended scaled prediction variance and extended spherical average prediction variance

For any response surface design, there are locations in the design region where responses are estimated well and locations where estimation is relatively poor. Consequently, graphical evaluation—such as variance dispersion graphs and the fraction of design space—is used as an alternative to a single-valued criterion. Such plots are used to investigate and compare the prediction capabilities of certain response surface designs currently available to the researcher. In this article, we propose the extended scaled prediction variance and extended spherical average prediction variance as prediction methods. We also illustrate how graphical methods can be employed to evaluate robust parameter designs.     

Efficiencies in Variance Estimation for Complex Survey Data

Data from complex survey designs require special consideration, with regard to variance estimation and analysis, because of design components that include unequal selection probabilities, stratification, and clustering. Using data from the National Medical Care Expenditure Survey, which is characterized by a complex survey design, four variance estimation programs are compared: SESUDAAN/RATIOEST, SUPERCARP, PSALMS, and HESBRR. The comparisons concentrate on program capabilities, computational efficiency, and user facility.     

Some Recent Developments in the Analysis and Design of Quantitative Genetic Experiments

The design and analysis of experiments to estimate heritability when data are available on both parents and progeny and the offspring have a hierarchical structure is considered. The method of analysis is related to a multivariate analysis of variance and to weighted least squares. It is shown that genetical theory gives a simple interpretation of both maximum likelihood (ML) and Rao's minimum norm quadratic unbiased (MINQUE) methods of estimation of variance components in unbalanced designs.     

Estimation of variance components in staggered nested designs

SUMMARY Variance components are estimated by two different methods for a general p stage random-effects staggered nested design. In addition to estimation from an analysis of variance, a new approach is introduced. The main features of this new technique are its simplicity and its ability to yield non-negative estimates of the variance components. The performances of the two procedures are compared using simulation and the meansquared-error criterion.     

Mass imputation for two-phase sampling

Two-phase sampling is a cost-effective method of data collection using outcome-dependent sampling for the second-phase sample. In order to make efficient use of auxiliary information and to improve domain estimation, mass imputation can be used in two-phase sampling. Rao and Sitter (1995) introduce mass imputation for two-phase sampling and its variance estimation under simple random sampling in both phases. In this paper, we extend the Rao–Sitter method to general sampling design. The proposed method is further extended to mass imputation for categorical data. A limited simulation study is performed to examine the performance of the proposed methods.     

Generalized staggered nested designs for variance components estimation

Staggered nested experimental designs are the most popular class of unbalanced nested designs in practical fields. The most important features of the staggered nested design are that it has a very simple open-ended structure and each sum of squares in the analysis of variance has almost the same degrees of freedom. Based on the features, a class of unbalanced nested designs that is a generalization of the staggered nested design is proposed in this paper. Formulae for the estimation of variance components and their sums are provided. Comparing the variances of the estimators to the staggered nested designs, it is found that some of the generalized staggered nested designs are more efficient than the traditional staggered nested design in estimating some of the variance components and their sums. An example is provided for illustration.     

概化理论两侧面设计方差分量及其变异量估计方法比较

文章针对正态分布数据,对比Traditional方法、Bootstrap方法和MCMC方法在两侧面交叉设计(p×i×h)和两侧面嵌套设计(p×(i:h))下各个方差分量的估计精度,为实际应用提供参考。使用R软件模拟1000批数据,并在R软件上实现三种方法的方差分量及其变异量估计。结果表明:(1)相较于Traditional方法和MCMC方法,相同条件下,Bootstrap方法估计的方差分量及其变异量结果更为理想;(2)对于两侧面交叉设计和两侧面嵌套设计,在正态分布数据下,建议优先使用Bootstrap方法。     

Parametric bootstrap inferences for panel data models

This article presents parametric bootstrap (PB) approaches for hypothesis testing and interval estimation for the regression coefficients and the variance components of panel data regression models with complete panels. The PB pivot variables are proposed based on sufficient statistics of the parameters. On the other hand, we also derive generalized inferences and improved generalized inferences for variance components in this article. Some simulation results are presented to compare the performance of the PB approaches with the generalized inferences. Our studies show that the PB approaches perform satisfactorily for various sample sizes and parameter configurations, and the performance of PB approaches is mostly the same as that of generalized inferences with respect to the expected lengths and powers. The PB inferences have almost exact coverage probabilities and Type I error rates. Furthermore, the PB procedure can be simply carried out by a few simulation steps, and the derivation is easier to understand and to be extended to the incomplete panels. Finally, the proposed approaches are illustrated by using a real data example.     

The weighted jackknife for ratio estimation

Using the idea of impirical influence function, Hinkley (1977), the weighted jackknife technique is extended to ratio estimation. A weighted jackknife variance estimator for the ratio estimator is developed. Using the prediction theory approach, the properties of the weighted jackknifed variance estimator are examined. The implications of the failures of regression model on the behaviour of the weighted jackknifed variance estimator, for ratio estimation, are also studied.     

A Jackknife Method for Estimation of Variance Components

This paper concerns a method of estimation of variance components in a random effect linear model. It is mainly a resampling method and relies on the Jackknife principle. The derived estimators are presented as least squares estimators in an appropriate linear model, and one of them appears as a MINQUE (Minimum Norm Quadratic Unbiased Estimation) estimator. Our resampling method is illustrated by an example given by C. R. Rao [7] and some optimal properties of our estimator are derived for this example. In the last part, this method is used to derive an estimation of variance components in a random effect linear model when one of the components is assumed to be known.     

Linear Regression With Nested Errors Using Probability‐Linked Data

Probabilistic matching of records is widely used to create linked data sets for use in health science, epidemiological, economic, demographic and sociological research. Clearly, this type of matching can lead to linkage errors, which in turn can lead to bias and increased variability when standard statistical estimation techniques are used with the linked data. In this paper we develop unbiased regression parameter estimates to be used when fitting a linear model with nested errors to probabilistically linked data. Since estimation of variance components is typically an important objective when fitting such a model, we also develop appropriate modifications to standard methods of variance components estimation in order to account for linkage error. In particular, we focus on three widely used methods of variance components estimation: analysis of variance, maximum likelihood and restricted maximum likelihood. Simulation results show that our estimators perform reasonably well when compared to standard estimation methods that ignore linkage errors.     

Estimation using penalized quasilikelihood and quasi-pseudo-likelihood in Poisson mixed models

We consider two estimation schemes based on penalized quasilikelihood and quasi-pseudo-likelihood in Poisson mixed models. The asymptotic bias in regression coefficients and variance components estimated by penalized quasilikelihood (PQL) is studied for small values of the variance components. We show the PQL estimators of both regression coefficients and variance components in Poisson mixed models have a smaller order of bias compared to those for binomial data. Unbiased estimating equations based on quasi-pseudo-likelihood are proposed and are shown to yield consistent estimators under some regularity conditions. The finite sample performance of these two methods is compared through a simulation study.     

Note on umvsu-estimation under randomized response model

Considering a class ofs randomized response trials for eliciting sensitive information from a sample survey and a class of ordered sampling designs, a uniformly minimum variance unbiased estimator of population variance (of the sensitive character) has been obtained. This note indicates that a theorem (theorem 3.9) of Cassel, Sarndal and Wretman (1977) and the results in the present note can be extended to estimation of any symmetric function of population values in the field of direct response surveys and randomized response surveys respectively.     

分层线性模型的最大后验估计

最大后验估计（MAPE）和最大似然估计（MLE）都是重要的参数点估计方法。在介绍一般分层线性模型（HLM）MAPE方法的基础上,给出这种方法的期望最大化算法（EM）的具体步骤,运用对数似然函数的二阶导数推导了MAPE估计的方差估计量。同时运用数据模拟比较了EM算法下的MAPE和MLE。对于固定效应的估计,两种方法得到的估计量是一致的。当组数较少时,EM计算的MAPE的方差协方差成分比MLE的更靠近真实值,而且MAPE的迭代次数明显小于MLE。     

Quantile dispersion graphs for analysis of variance estimates of variance components

SUMMARY The exact distribution of an analysis of variance estimator of a variance component is obtained by determining its quantiles on the basis of R. B. Davies' algorithm. A plot of these quantiles provides useful information concerning the efficiency of the estimator, including the extent to which it can be negative. Furthermore, the variability in the values of each quantile is assessed by varying the values of the variance components for the model under consideration. The maximum and minimum of such quantile values can then be determined. A plot of the maxima and minima for various selected quantiles produces the so-called 'quantile dispersion graphs'. These graphs can be used to provide a comprehensive picture of the quality of estimation obtained with a particular design. They also provide an effective graphical tool for comparing designs on the basis of their estimation capabilities.     

Improved estimation of variance components in mixed models

Taking Albert's (1976) formulation of a mixed model ANOVA, we consider improved estimation of the variance components for balanced designs under squared error loss. Two approaches are presented. One extends the ideas of Stein (1964), The other is developed from the fact that variance components can be expressed as linear combinations of chi-square scale parameters. Encouraging simulation results are presented.