Quantile Dispersion Graphs Research Articles

Robust designs for multinomial models

Model misspecification in generalized linear models (GLMs) occurs usually when the linear predictor and/or the link function assumed are incorrect. This article discusses the effect of such misspecification on design selection for multinomial GLMs and proposes the use of quantile dispersion graphs to select robust designs. Due to misspecification in the model, parameter estimates are usually biased and the designs are compared on the basis of their mean squared error of prediction. Several numerical examples including a real data set are presented to illustrate the proposed methodology.

Comparison of three-level cluster randomized trials using quantile dispersion graphs

The purpose of this article is to evaluate and compare several three-level cluster randomized designs on the basis of their power functions. The power function of cluster designs depends on the intracluster correlations (ICCs), which are generally unknown at the planning stage. Thus, to compare these designs a prior knowledge of the ICCs is required. Three interval estimation methods are proposed for assigning joint confidence intervals to the two ICCs (corresponding to each cluster level). A detailed simulation study comparing the confidence intervals attained by the different techniques is given. The technique of quantile dispersion graphs is used for comparing the three-level cluster designs. For a given design, quantiles of the power function, are obtained for various effect sizes. These quantiles are functions of the unknown ICC coefficients. To address the dependence of the quantiles on the correlations, a confidence region is computed, and used as a parameter space. A three-level nested data set collected by the University of Michigan to study various school reforms on the achievements of students is used to illustrate the proposed methodology.

Robust Designs in Generalized Linear Models: A Quantile Dispersion Graphs Approach

This article studies design selection for generalized linear models (GLMs) using the quantile dispersion graphs (QDGs) approach in the presence of misspecification in the link and/or linear predictor. The uncertainty in the linear predictor is represented by a unknown function and estimated using kriging. For addressing misspecified link functions, a generalized family of link functions is used. Numerical examples are shown to illustrate the proposed methodology.

Comparison of designs for generalized linear models under model misspecification

The purpose of this article is to demonstrate the use of the quantile dispersion graphs (QDGs) approach for comparing candidate designs for generalized linear models in the presence of model misspecification in the linear predictor. The proposed design criterion is based on the mean-squared error of prediction which incorporates the prediction variance and the bias caused by fitting the wrong model. The method of kriging is used to estimate the unknown function assumed to be the cause of model misspecification. The QDGs approach is also useful in assessing the robustness of a given design to values of the unknown parameters in the linear predictor. Three numerical examples are presented to illustrate the application of the proposed methodology.

Quantile dispersion graphs to compare the efficiencies of cluster randomized designs

The purpose of this article is to compare efficiencies of several cluster randomized designs using the method of quantile dispersion graphs (QDGs). A cluster randomized design is considered whenever subjects are randomized at a group level but analyzed at the individual level. A prior knowledge of the correlation existing between subjects within the same cluster is necessary to design these cluster randomized trials. Using the QDG approach, we are able to compare several cluster randomized designs without requiring any information on the intracluster correlation. For a given design, several quantiles of the power function, which are directly related to the effect size, are obtained for several effect sizes. The quantiles depend on the intracluster correlation present in the model. The dispersion of these quantiles over the space of the unknown intracluster correlation is determined, and then depicted by the QDGs. Two applications of the proposed methodology are presented.

Comparison of Designs for Response Surface Models with Random Block Effects

The purpose of this article is to compare designs for response surface models with a random block effect. To assess the quality of prediction associated with a given design, the scaled prediction variance is considered as a design criterion. The proposed approach is based on using quantiles of this design criterion on concentric surfaces within the experimental region. The dependence of these quantiles on the unknown value of the ratio of two variance components, namely, the ones for the block effect and the experimental error, is depicted by plotting the so-called quantile dispersion graphs (QDGs). These plots provide a clear assessment of the quality of prediction associated with a given design. A numerical example is presented to illustrate the proposed methodology.

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.

Comparison of designs for multivariate generalized linear models

The purpose of this paper is to discuss response surface designs for multivariate generalized linear models (GLMs). Such models are considered whenever several response variables can be measured for each setting of a group of control variables, and the response variables are adequately represented by GLMs. The mean-squared error of prediction (MSEP) matrix is used to assess the quality of prediction associated with a given design. The MSEP incorporates both the prediction variance and the prediction bias, which results from using maximum likelihood estimates of the parameters of the fitted linear predictor. For a given design, quantiles of a scalar-valued function of the MSEP are obtained within a certain region of interest. The quantiles depend on the unknown parameters of the linear predictor. The dispersion of these quantiles over the space of the unknown parameters is determined and then depicted by the so-called quantile dispersion graphs. An application of the proposed methodology is presented using the special case of the bivariate binary distribution.

불균형 일원랜덤효과모형에서 EQDGs를 이용한 MIVQUE 추정량 비교

본 논문에서는 불균형 일원랜덤효과모형에서 분산성분에 대한 MIVQUE 추정량들의 효율에 대하여 연구하였다. MIVQUE 추정량에 대한 초기 추정치로 3가지 추정치가 사용되었으며 이들을 이용하여 얻어지는 MIVQUE 추정량의 효율성을 척도화된 추정량의 경험적 분위수를 이용하는 EQDGs 플롯을 이용하여 비교하였다. 모의실험 결과 집단간 분산의 초기치를 0으로 하고 집단내 분산의 초기치를 1로 사용한 MIVQUE 추정량의 효율이 추정량의 안정성 관점에서 다른 초기치를 사용한 MIVQUE 추정량에 비하여 약간 효율적인 것으로 나타났다. In this study, the MIVQUE estimators of variance components for the one-way random model with unbalanced data are investigated. In order to compare the efficiency of MIVQUE estimators obtained by using three priori estimates, the Empirical Quantile Dispersion Graphs (EQDGs) are used. From the results of Monte-Carlo study, the MIVQUE estimator using <TEX>${\sigma}^2_{\alpha}\;=\;0\;and\;{\sigma}^2_{varraho}=1$</TEX> as the priori estimate performs well relative to other estimators.

Use of the tail moment of the lymphocytes to evaluate DNA damage in human biomonitoring studies.

The Comet assay has gained increasing popularity for use in human biomonitoring or epidemiologic studies; however, one of the shortcomings of the Comet assay is a lack of agreement on a single appropriate Comet parameter that is capable of adequately describing observed DNA damages. Among the tail parameters of Comet features, the most frequently used are the tail moments (both the Olive tail moment and the extent tail moment), the tail DNA, and the tail length. Some studies comparing Comet parameters have been found in cell toxicity research, but there are few comparative studies that use human biomonitoring or epidemiologic data. In this study, we evaluate those four tail parameters in both high and low DNA damaged cells with the use of epidemiologic data. To do this, a new graphical approach, the so-called quantile dispersion graphs (QDGs) are used. In a comparison of an exposed group and a control group, either the tail moment or tail DNA is preferable to the tail length. With respect to providing smaller variability in quantiles for the amount of DNA damage, however, the tail moment is the preferred parameter for both groups. Moreover, the tail moment provides the most stable estimates for DNA damage because it has a larger degree of uniformity in quantile dispersions. To study high degrees of damage from toxic exposure using B cells or G cells, however, the tail DNA showed more significant discrepancies than the other parameters, in terms of both the mean differences and the graphical differences between the two groups. In view of this result, it is suggested that both the tail moment and the tail DNA be presented as tail parameters in human biomonitoring studies.

Quantile dispersion graphs for evaluating and comparing designs for logistic regression models

Designs for fitting a generalized linear model depend on the unknown parameters of the model. The use of any design optimality criterion would therefore require some prior knowledge of the parameters. In this article, a graphical technique is proposed for comparing and evaluating designs for a logistic regression model. Quantiles of the scaled mean-squared error of prediction are obtained on concentric surfaces inside a region of interest, R. For a given design, these quantiles depend on the model's parameters. Plots of the maxima and minima of the quantiles, over a subset of the parameter space, produce the so-called quantile dispersion graphs (QDGs). The plots provide a comprehensive assessment of the overall prediction capability of the design within the region R. They also depict the dependence of the design on the model's parameters. The QDGs can therefore be conveniently used to compare several candidate designs. Two examples are presented to illustrate the proposed methodology.

Quantile dispersion graphs for the comparison of designs for a random two-way model

The choice of optimal designs for the estimation of variance components depends on the method of estimation, the model used, and the values of the variance components themselves. Traditional comparisons of such designs cannot therefore be made without some prior knowledge of the variance components. This problem was circumvented by the introduction of the so-called quantile dispersion graphs (QDGs) in Khuri (1997) and Lee and Khuri (1999). The present article provides an extension of the use of the QDGs to the comparison of designs for an unbalanced random two-way model without interaction. Two methods of estimation of the variance components are considered, namely, the analysis of variance (ANOVA) and maximum likelihood (ML).

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.

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.