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23
Bayesian Tests And Model Diagnostics In Conditionally Independent Hierarchical Models
 Journal of the American Statistical Association
, 1994
"... Consider the conditionally independent hierarchical model (CIHM) where observations y i are independently distributed from f(y i j` i ), the parameters ` i are independently distributed from distributions g(`j), and the hyperparameters are distributed according to a distribution h(). The posterior ..."
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Consider the conditionally independent hierarchical model (CIHM) where observations y i are independently distributed from f(y i j` i ), the parameters ` i are independently distributed from distributions g(`j), and the hyperparameters are distributed according to a distribution h(). The posterior distribution of all parameters of the CIHM can be efficiently simulated by Monte Carlo Markov Chain (MCMC) algorithms. Although these simulation algorithms have facilitated the application of CIHM's, they generally have not addressed the problem of computing quantities useful in model selection. This paper explores how MCMC simulation algorithms and other related computational algorithms can be used to compute Bayes factors that are useful in criticizing a particular CIHM. In the case where the CIHM models a belief that the parameters are exchangeable or lie on a regression surface, the Bayes factor can measure the consistency of the data with the structural prior belief. Bayes factors can ...
Empirical Modeling of Genetic Algorithms
 EVOLUTIONARY COMPUTATION
, 2001
"... This paper addresses the problem of reliably setting genetic algorithm parameters for consistent labelling problems. Genetic algorithm parameters are notoriously difficult to determine. This paper proposes a robust empirical framework, based on the analysis of factorial experiments. The use of a gra ..."
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Cited by 6 (1 self)
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This paper addresses the problem of reliably setting genetic algorithm parameters for consistent labelling problems. Genetic algorithm parameters are notoriously difficult to determine. This paper proposes a robust empirical framework, based on the analysis of factorial experiments. The use of a graecolatin square permits an initial study of a wide range of parameter settings. This is followed by fully crossed factorial experiments with narrower ranges, which allow detailed analysis by logistic regression. The empirical models thus derived can be used first to determine optimal algorithm parameters, and second to shed light on interactions between the parameters and their relative importance. The initial models do not extrapolate well. However, an advantage of this approach is that the modelling process is under the control of the experimenter, and is hence very flexible. Refined models are produced, which are shown to be robust under extrapolation to up to triple the problem size.
The Cost of Adding Parameters to a Model
, 1996
"... For a general regression model with n independent observations we consider the variance of the estimate of a quantity of interest under two scenarios. One scenario is where all the parameters are estimated from the data, the other scenario is where a subset of the parameters are assumed known at the ..."
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Cited by 5 (1 self)
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For a general regression model with n independent observations we consider the variance of the estimate of a quantity of interest under two scenarios. One scenario is where all the parameters are estimated from the data, the other scenario is where a subset of the parameters are assumed known at their true values and the remaining parameters are estimated. We focus on quantities of interest which are defined on the scale of the response variable. We show that, under certain conditions, the ratio of a weighted sum across the design points of the variance of the quantity of interest is given by q=p, where q and p are the number of free parameters in the two scenarios. Thus, in this average sense, the inflation in variance associated with adding parameters, also interpreted as the cost of adding parameters to a model, is directly proportional to the number of parameters. We study models involving power transformations, nonlinear models and exponential family models. Key Words: BoxCox t...
On Selecting Parametric Link Transformation Families in Generalized Linear Models
 Journal of Statistical Planning and Inference
, 1995
"... The use of parametric link transformation families in generalized linear models (GLM) has been shown to improve substantially the fit of standard analyses using a fixed link in some data sets (see Czado [1993], for example). When link and regression parameters are globally orthogonal (Cox and Reid [ ..."
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Cited by 3 (3 self)
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The use of parametric link transformation families in generalized linear models (GLM) has been shown to improve substantially the fit of standard analyses using a fixed link in some data sets (see Czado [1993], for example). When link and regression parameters are globally orthogonal (Cox and Reid [1987]), then the variance inflation of the regression parameter estimates due to the additional estimation of the link is asymptotically zero. Parameter orthogonality also induces numerical stability which is seen in the reduction of computation time required for the calculation of parameter estimates. This stability remains a desirable property even for inferences which are conditional on a fixed link value. Czado and Santner [1992b] for binomial error and Czado [1992] for GLM's have shown that only local orthogonality can be achieved in general. This paper provides conditions on the link family to extend the notion of local orthogonality at a point to orthogonality in a neighborhood asympt...
Fractional regression models for second stage DEA efficiency analyses ∗
, 2010
"... Data envelopment analysis (DEA) is commonly used to measure the relative efficiency of decisionmaking units. Often, in a second stage, a regression model is estimated to relate DEA efficiency scores to exogenous factors. In this paper, we argue that the traditional linear or tobit approaches to sec ..."
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Cited by 3 (2 self)
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Data envelopment analysis (DEA) is commonly used to measure the relative efficiency of decisionmaking units. Often, in a second stage, a regression model is estimated to relate DEA efficiency scores to exogenous factors. In this paper, we argue that the traditional linear or tobit approaches to secondstage DEA analysis do not constitute a reasonable datagenerating process for DEA scores. Under the assumption that DEA scores can be treated as descriptive measures of the relative performance of units in the sample, we show that using fractional regression models are the most natural way of modeling bounded, proportional response variables such as DEA scores. We also propose generalizations of these models and, given that DEA scores take frequently the value of unity, examine the use of twopart models in this framework. Several tests suitable for assessing the specification of each alternative model are also discussed.
Bayesian Analysis Of A Random Link Function In Binary Response Regression
 Journal of the American Statistical Association Management
, 1994
"... Binary response regression is a useful technique for analyzing categorical data. Popular binary models use special link functions such as the logit or the probit link. We assume that the inverse link function H is a random member of the class of normal scale mixture cdfs. We propose three different ..."
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Binary response regression is a useful technique for analyzing categorical data. Popular binary models use special link functions such as the logit or the probit link. We assume that the inverse link function H is a random member of the class of normal scale mixture cdfs. We propose three different models for this random H : (i) H is a finite scale mixture with a Dirichlet distribution prior on the mixing distribution; (ii) H is a general scale mixture, the mixing distribution has a Dirichlet process prior; and (iii) H is a scale mixture of truncated normal distributions with the mixing distribution having a Dirichlet prior. We describe Bayesian analyses of these models using data augmentation and Gibbs sampling. Model diagnostics by cross validation of the conditional predictive distributions are proposed. These analyses are illustrated in two examples. Our proposed models match the performances of Bayesian probit and t link models in the first example whereas they outperform probit and t link models in the second example.
Noncanonical links in generalized linear models  when is the effort justified?
 Journal of Statistical Planning and Inference
, 2000
"... Generalized linear models (GLMs) allow for a wide range of statistical models for regression data. In particular, the logistic model is usually applied for binomial observations. Canonical links for GLM's such as the logit link in the binomial case, are often used because in this case minimal suffic ..."
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Cited by 2 (2 self)
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Generalized linear models (GLMs) allow for a wide range of statistical models for regression data. In particular, the logistic model is usually applied for binomial observations. Canonical links for GLM's such as the logit link in the binomial case, are often used because in this case minimal sufficient statistics for the regression parameter exist which allow for simple interpretation of the results. However, in some applications, the overall fit as measured by the pvalues of goodness of fit statistics (as the residual deviance) can be improved significantly by the use of a noncanonical link. In this case, the interpretation of the influence of the covariables is more complicated compared to GLM's with canonical link functions. It will be illustrated through simulation that the pvalue associated with the common goodness of link tests is not appropriate to quantify the changes to mean response estimates and other quantities of interest when switching to a noncanonical link. In particular, the rate of misspecifications becomes considerably large, when the inverse information value associated with the underlying parametric link model increases. This shows that the classical tests are often too sensitive, in particular, when the number of observations is large. The consideration of a generalized pvalue function is proposed instead, which allows the exact quantification of a suitable distance to the canonical model at a controlled error rate. Corresponding tests for validating or discriminating the canonical model can easily performed by means of this function. Finally, it is indicated how this method can be applied to the problem of overdispersion.
11 The Statistical Analysis of DiscreteResponse CV Data by
, 1998
"... rights reserved. Readers may make verbatim copies of this document for noncommercial purposes by any means, provided that this copyright notice appears on all such copies. ..."
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rights reserved. Readers may make verbatim copies of this document for noncommercial purposes by any means, provided that this copyright notice appears on all such copies.
Choosing the Link Function and Accounting for Link Uncertainty in Generalized Linear Models using Bayes Factors
, 2001
"... One important component of model selection... ..."
Notes And
"... ictor variable such as time or temperature. In its basic form, the classical linear model is unable to account for such effects. It is possible, however, to generalize the linear model, and allow it to overcome these limitations. The generalizations accept forms of nonnormal data, and also link the ..."
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ictor variable such as time or temperature. In its basic form, the classical linear model is unable to account for such effects. It is possible, however, to generalize the linear model, and allow it to overcome these limitations. The generalizations accept forms of nonnormal data, and also link the mean response to the linear predictor in a possibly nonlinear fashion. We call this the family of generalized linear models (GLiMs). The basic precept of a GLiM is to extend the linear model in two ways: (a) generalize to nonnormal parent distributions such as binomial, Poisson, or gamma; and/or (b) generalize to nonlinear functions that link the unknown means of the parent distribution with the predictor variables. In the former case, certain generalizations to nonnormal parent distributions can be grouped into a class of densities known as the exponential family. The density functions satisfy two basic requirements: (1) the random variable, Y, has a support