Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas, Mar 1, 2009
We would like to thank the editor of RACSAM, Professor Manuel López Pellicer, for the opportunity... more We would like to thank the editor of RACSAM, Professor Manuel López Pellicer, for the opportunity he is offering to us of discussing this paper, and to congratulate Berger, Bernardo and Sun for an interesting and thought provoking paper. The paper is motivated by the observation that the uniform prior for R, say π(R|N) = 1/(N + 1), R = 0,. . ., N , gives poor results. It is shown that the posterior probability that all the N elements of the population are conforming, conditional on the event that all the observed n elements in the sample are conforming, is very small for N large, whatever moderate the sample size n should be. Then, a more reasonable prior π(R|M) is provided on the ground of being compatible with the Jeffreys prior for the parameter θ of the Binomial limiting distribution with parameters (n, θ), where θ = lim R→∞,N →∞ R/N. We enjoyed reading this clear argumentation. However, in the abstract it is recognized that "Bayesian solutions to this problem may be very sensitive to the choice of the prior, and there is no consensus as to the appropriate prior to use. " It seems to us that the natural consequence of this assertion-that we share-is to consider a class of priors and reporting their posterior answers, instead of considering the posterior answer for the single reference prior for R. In this discussion we try to add the robustness analysis that we feel is missing in the paper. For simplicity we will consider the limiting Binomial distribution Bi(r|n, θ), and the two problems addressed in the paper. Firstly, the testing problem
European Journal of Operational Research, Oct 1, 2016
The sampling information for the cost-effectiveness analysis typically comes from different healt... more The sampling information for the cost-effectiveness analysis typically comes from different health care centers, and, as far as we know, it is taken for granted that the distribution of the cost and the effectiveness does not vary across centers. We argue that this assumption is unrealistic, and prove that to not consider the sample heterogeneity will typically give misleading results. Consequently, a cost-effectiveness procedure for heterogeneous samples is here proposed. The proposed cost-effectiveness procedure consists of a Bayesian clustering to measure the sample heterogeneity, and a meta-analysis to account for the specific clustering structure of the data. Examples with real data illustrate this methodology for normal and lognormal models, and the results are compared with those we would obtain if homogeneity of the samples is assumed.
In this paper, we present two problems in meta-analysis. One is the model uncertainty generated b... more In this paper, we present two problems in meta-analysis. One is the model uncertainty generated by the available heterogenous sampling information. We claim that this model uncertainty has to be incorporated into the meta-inference, and propose a Bayesian clustering procedure for doing that. A second problem is that of choosing the linking distribution that relates the experimental sampling model and the meta-model. We claim that the join distribution for the experimental parameters and the meta-parameter has to be a copula in order to ensure that the Bayesian experimental model and meta-model are coherent. A general copula is proposed. Illustrative examples with real data set are given.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
The sampling information for the cost-effectiveness analysis typically comes from different healt... more The sampling information for the cost-effectiveness analysis typically comes from different health care centers, and, as far as we know, it is taken for granted that the distribution of the cost and the effectiveness does not vary across centers. We argue that this assumption is unrealistic, and prove that to not consider the sample heterogeneity will typically give misleading results. Consequently, a cost-effectiveness procedure for heterogeneous samples is here proposed. The proposed cost-effectiveness procedure consists of a Bayesian clustering to measure the sample heterogeneity, and a meta-analysis to account for the specific clustering structure of the data. Examples with real data illustrate this methodology for normal and lognormal models, and the results are compared with those we would obtain if homogeneity of the samples is assumed.
Abstract We put forward the idea that for model selection the intrinsic priors are becoming a cen... more Abstract We put forward the idea that for model selection the intrinsic priors are becoming a center of a cluster of a dominant group of methodologies for objective Bayesian Model Selection. The intrinsic method and its applications have been developed in the last two decades, and has stimulated closely related methods. The intrinsic methodology can be thought of as the long searched approach for objective Bayesian model selection and hypothesis testing. In this paper we review the foundations of the intrinsic priors, their general properties, and some of their applications.
International Journal of the Economics of Business, 2009
The economic literature on cost‐effectiveness analysis in the context of decisions by health tech... more The economic literature on cost‐effectiveness analysis in the context of decisions by health technology assessment agencies assumes as the quantity of interest a linear combination of the mean of the sampling distribution of the effectiveness and the cost. We argue that this is not always reasonable. Our reasons for this assertion are that (i) treatments are compared on the basis of mean values, and for some useful models the mean of the distribution of the cost, which is conditional on the available data, does not exist, and (ii) even for models for which the mean does exist, it might not constitute an accurate reflection of the distribution. This paper presents a general Bayesian cost‐effectiveness analysis of a single treatment, where the quantity of interest is the distribution, conditional on the data, of the net benefit. This approach permits a natural extension to several treatments, which enables us to make a statistical comparison. Illustrations with treatment comparisons for real and simulated data are given.
Most of the consistency analyses of Bayesian procedures for variable selection in regression refe... more Most of the consistency analyses of Bayesian procedures for variable selection in regression refer to pairwise consistency, that is, consistency of Bayes factors. However, variable selection in regression is carried out in a given class of regression models where a natural variable selector is the posterior probability of the models. In this paper we analyze the consistency of the posterior model probabilities when the number of potential regressors grows as the sample size grows. The novelty in the posterior model consistency is that it depends not only on the priors for the model parameters through the Bayes factor, but also on the model priors, so that it is a useful tool for choosing priors for both models and model parameters. We have found that some classes of priors typically used in variable selection yield posterior model inconsistency, while mixtures of these priors improve this undesirable behavior. For moderate sample sizes, we evaluate Bayesian pairwise variable selecti...
In one-sided testing, Bayesians and frequentists differ on whether or not there is discrepancy be... more In one-sided testing, Bayesians and frequentists differ on whether or not there is discrepancy between the inference based on the posterior model probability and that based on the p value. We add some arguments to this debate analyzing the discrepancy for moderate and large sample sizes. For small and moderate samples sizes, the discrepancy is measured by the probability of disagreement. Examples of the discrepancy on some basic sampling models indicate the somewhat unexpected result that the probability of disagreement is larger when sampling from models in the alternative hypothesis that are not located at the boundary of the hypotheses. For large sample sizes, we prove that the Bayesian one-sided testing is, under mild conditions, consistent, a property that is not shared by the frequentist procedure. Further, the rate of convergence is $$O(e^{nA})$$ , where A is a constant that depends on the model from which we are sampling. Consistency is also proved for an extension to multip...
Cost–effectiveness analysis of medical treatments is a statistical decision problem whose aim is ... more Cost–effectiveness analysis of medical treatments is a statistical decision problem whose aim is to choose an optimal treatment among a finite set of alternative treatments. It is assumed that the treatment selection is to be based on their cost and effectiveness.
Selecting a statistical model from a set of competing models is a central issue in the scientific... more Selecting a statistical model from a set of competing models is a central issue in the scientific task, and the Bayesian approach to model selection is based on the posterior model distribution, a quantification of the updated uncertainty on the entertained models. We present a Bayesian procedure for choosing a family between the Poisson and the geometric families and prove that the procedure is consistent with rate $$O(a^{n})$$ O ( a n ) , $$a>1$$ a > 1 , where a is a function of the parameter of the true model. An extension of this procedure to the multiple testing Poisson and negative binomial with r successes for $$r=1,\ldots ,L$$ r = 1 , … , L is also proved to be consistent with exponential rate. For small sample sizes, a simulation study indicates that the model selection between the above distributions is made with large uncertainty when sampling from a specific subset of distributions. This difficulty is however mitigated by the large consistency rate of the procedure.
Testing that some regression coefficients are equal to zero is an important problem in many appli... more Testing that some regression coefficients are equal to zero is an important problem in many applications. Homoscedasticity is not necessarily a realistic condition in this setting and, as a consequence, no frequentist test there exist. Approximate tests have been proposed. In this paper a Bayesian analysis of this problem is carried out, from a default Bayesian model choice perspective. Explicit expressions for intrinsic priors are provided, and it is shown that the corresponding Bayes factor is computed with the help of very simple numerical computations. Distribuciones a priori intrinsecas para modelos de regresion normales Resumen. En muchas aplicaciones es frecuente enfrentarse con el problema de contrastar si algunos coeficientes de regresion son nulos. Dicho problema se resuelve, bajo el punto de vista frecuentista, imponiendo la hipotesis de homocedasticidad. Sin embargo esta suposicion no es asumible en general, proporcionandose en tales casos tests aproximados. En este arti...
Most of the consistency analyses of Bayesian procedures for variable selection in regression refe... more Most of the consistency analyses of Bayesian procedures for variable selection in regression refer to pairwise consistency, that is, consistency of Bayes factors. However, variable selection in regression is carried out in a given class of regression models where a natural variable selector is the posterior probability of the models. In this paper we analyze the consistency of the posterior model probabilities when the number of potential regressors grows as the sample size grows. The novelty in the posterior model consistency is that it depends not only on the priors for the model parameters through the Bayes factor, but also on the model priors, so that it is a useful tool for choosing priors for both models and model parameters. We have found that some classes of priors typically used in variable selection yield posterior model inconsistency, while mixtures of these priors improve this undesirable behavior. For moderate sample sizes, we evaluate Bayesian pairwise variable selection procedures by comparing their frequentist Type I and II error probabilities. This provides valuable information to discriminate between the priors for the model parameters commonly used for variable selection.
An optimal Bayesian decision procedure for testing hypothesis in normal linear models based on in... more An optimal Bayesian decision procedure for testing hypothesis in normal linear models based on intrinsic model posterior probabilities is considered. It is proven that these posterior probabilities are simple functions of the classical F-statistic, thus the evaluation of the procedure can be carried out analytically through the frequentist analysis of the posterior probability of the null. An asymptotic analysis proves that, under mild conditions on the design matrix, the procedure is consistent. For any testing hypothesis it is also seen that there is a one-to-one mapping-which we call calibration curve-between the posterior probability of the null hypothesis and the classical p-value. This curve adds substantial knowledge about the possible discrepancies between the Bayesian and the p-value measures of evidence for testing hypothesis. It permits a better understanding of the serious difficulties that are encountered in linear models for interpreting the p-values. A specific illustration of the variable selection problem is given.
The random effect approach for meta-analysis was motivated by a lack of consistent assessment of ... more The random effect approach for meta-analysis was motivated by a lack of consistent assessment of homogeneity of treatment effect before pooling. The random effect model assumes that the distribution of the treatment effect is fully heterogenous across the experiments. However, other models arising by grouping some of the experiments are plausible. We illustrate on simulated binary experiments that the fully heterogenous model gives a poor meta-inference when fully heterogeneity is not the true model and that the knowledge of the true cluster model considerably improves the inference. We propose the use of a Bayesian model selection procedure for estimating the true cluster model, and Bayesian model averaging to incorporate into the meta-analysis the clustering estimation. A well-known meta-analysis for six major multicentre trials to assess the efficacy of a given dose of aspirin in post-myocardial infarction patients is reanalysed.
Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas, Mar 1, 2009
We would like to thank the editor of RACSAM, Professor Manuel López Pellicer, for the opportunity... more We would like to thank the editor of RACSAM, Professor Manuel López Pellicer, for the opportunity he is offering to us of discussing this paper, and to congratulate Berger, Bernardo and Sun for an interesting and thought provoking paper. The paper is motivated by the observation that the uniform prior for R, say π(R|N) = 1/(N + 1), R = 0,. . ., N , gives poor results. It is shown that the posterior probability that all the N elements of the population are conforming, conditional on the event that all the observed n elements in the sample are conforming, is very small for N large, whatever moderate the sample size n should be. Then, a more reasonable prior π(R|M) is provided on the ground of being compatible with the Jeffreys prior for the parameter θ of the Binomial limiting distribution with parameters (n, θ), where θ = lim R→∞,N →∞ R/N. We enjoyed reading this clear argumentation. However, in the abstract it is recognized that "Bayesian solutions to this problem may be very sensitive to the choice of the prior, and there is no consensus as to the appropriate prior to use. " It seems to us that the natural consequence of this assertion-that we share-is to consider a class of priors and reporting their posterior answers, instead of considering the posterior answer for the single reference prior for R. In this discussion we try to add the robustness analysis that we feel is missing in the paper. For simplicity we will consider the limiting Binomial distribution Bi(r|n, θ), and the two problems addressed in the paper. Firstly, the testing problem
European Journal of Operational Research, Oct 1, 2016
The sampling information for the cost-effectiveness analysis typically comes from different healt... more The sampling information for the cost-effectiveness analysis typically comes from different health care centers, and, as far as we know, it is taken for granted that the distribution of the cost and the effectiveness does not vary across centers. We argue that this assumption is unrealistic, and prove that to not consider the sample heterogeneity will typically give misleading results. Consequently, a cost-effectiveness procedure for heterogeneous samples is here proposed. The proposed cost-effectiveness procedure consists of a Bayesian clustering to measure the sample heterogeneity, and a meta-analysis to account for the specific clustering structure of the data. Examples with real data illustrate this methodology for normal and lognormal models, and the results are compared with those we would obtain if homogeneity of the samples is assumed.
In this paper, we present two problems in meta-analysis. One is the model uncertainty generated b... more In this paper, we present two problems in meta-analysis. One is the model uncertainty generated by the available heterogenous sampling information. We claim that this model uncertainty has to be incorporated into the meta-inference, and propose a Bayesian clustering procedure for doing that. A second problem is that of choosing the linking distribution that relates the experimental sampling model and the meta-model. We claim that the join distribution for the experimental parameters and the meta-parameter has to be a copula in order to ensure that the Bayesian experimental model and meta-model are coherent. A general copula is proposed. Illustrative examples with real data set are given.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
The sampling information for the cost-effectiveness analysis typically comes from different healt... more The sampling information for the cost-effectiveness analysis typically comes from different health care centers, and, as far as we know, it is taken for granted that the distribution of the cost and the effectiveness does not vary across centers. We argue that this assumption is unrealistic, and prove that to not consider the sample heterogeneity will typically give misleading results. Consequently, a cost-effectiveness procedure for heterogeneous samples is here proposed. The proposed cost-effectiveness procedure consists of a Bayesian clustering to measure the sample heterogeneity, and a meta-analysis to account for the specific clustering structure of the data. Examples with real data illustrate this methodology for normal and lognormal models, and the results are compared with those we would obtain if homogeneity of the samples is assumed.
Abstract We put forward the idea that for model selection the intrinsic priors are becoming a cen... more Abstract We put forward the idea that for model selection the intrinsic priors are becoming a center of a cluster of a dominant group of methodologies for objective Bayesian Model Selection. The intrinsic method and its applications have been developed in the last two decades, and has stimulated closely related methods. The intrinsic methodology can be thought of as the long searched approach for objective Bayesian model selection and hypothesis testing. In this paper we review the foundations of the intrinsic priors, their general properties, and some of their applications.
International Journal of the Economics of Business, 2009
The economic literature on cost‐effectiveness analysis in the context of decisions by health tech... more The economic literature on cost‐effectiveness analysis in the context of decisions by health technology assessment agencies assumes as the quantity of interest a linear combination of the mean of the sampling distribution of the effectiveness and the cost. We argue that this is not always reasonable. Our reasons for this assertion are that (i) treatments are compared on the basis of mean values, and for some useful models the mean of the distribution of the cost, which is conditional on the available data, does not exist, and (ii) even for models for which the mean does exist, it might not constitute an accurate reflection of the distribution. This paper presents a general Bayesian cost‐effectiveness analysis of a single treatment, where the quantity of interest is the distribution, conditional on the data, of the net benefit. This approach permits a natural extension to several treatments, which enables us to make a statistical comparison. Illustrations with treatment comparisons for real and simulated data are given.
Most of the consistency analyses of Bayesian procedures for variable selection in regression refe... more Most of the consistency analyses of Bayesian procedures for variable selection in regression refer to pairwise consistency, that is, consistency of Bayes factors. However, variable selection in regression is carried out in a given class of regression models where a natural variable selector is the posterior probability of the models. In this paper we analyze the consistency of the posterior model probabilities when the number of potential regressors grows as the sample size grows. The novelty in the posterior model consistency is that it depends not only on the priors for the model parameters through the Bayes factor, but also on the model priors, so that it is a useful tool for choosing priors for both models and model parameters. We have found that some classes of priors typically used in variable selection yield posterior model inconsistency, while mixtures of these priors improve this undesirable behavior. For moderate sample sizes, we evaluate Bayesian pairwise variable selecti...
In one-sided testing, Bayesians and frequentists differ on whether or not there is discrepancy be... more In one-sided testing, Bayesians and frequentists differ on whether or not there is discrepancy between the inference based on the posterior model probability and that based on the p value. We add some arguments to this debate analyzing the discrepancy for moderate and large sample sizes. For small and moderate samples sizes, the discrepancy is measured by the probability of disagreement. Examples of the discrepancy on some basic sampling models indicate the somewhat unexpected result that the probability of disagreement is larger when sampling from models in the alternative hypothesis that are not located at the boundary of the hypotheses. For large sample sizes, we prove that the Bayesian one-sided testing is, under mild conditions, consistent, a property that is not shared by the frequentist procedure. Further, the rate of convergence is $$O(e^{nA})$$ , where A is a constant that depends on the model from which we are sampling. Consistency is also proved for an extension to multip...
Cost–effectiveness analysis of medical treatments is a statistical decision problem whose aim is ... more Cost–effectiveness analysis of medical treatments is a statistical decision problem whose aim is to choose an optimal treatment among a finite set of alternative treatments. It is assumed that the treatment selection is to be based on their cost and effectiveness.
Selecting a statistical model from a set of competing models is a central issue in the scientific... more Selecting a statistical model from a set of competing models is a central issue in the scientific task, and the Bayesian approach to model selection is based on the posterior model distribution, a quantification of the updated uncertainty on the entertained models. We present a Bayesian procedure for choosing a family between the Poisson and the geometric families and prove that the procedure is consistent with rate $$O(a^{n})$$ O ( a n ) , $$a>1$$ a > 1 , where a is a function of the parameter of the true model. An extension of this procedure to the multiple testing Poisson and negative binomial with r successes for $$r=1,\ldots ,L$$ r = 1 , … , L is also proved to be consistent with exponential rate. For small sample sizes, a simulation study indicates that the model selection between the above distributions is made with large uncertainty when sampling from a specific subset of distributions. This difficulty is however mitigated by the large consistency rate of the procedure.
Testing that some regression coefficients are equal to zero is an important problem in many appli... more Testing that some regression coefficients are equal to zero is an important problem in many applications. Homoscedasticity is not necessarily a realistic condition in this setting and, as a consequence, no frequentist test there exist. Approximate tests have been proposed. In this paper a Bayesian analysis of this problem is carried out, from a default Bayesian model choice perspective. Explicit expressions for intrinsic priors are provided, and it is shown that the corresponding Bayes factor is computed with the help of very simple numerical computations. Distribuciones a priori intrinsecas para modelos de regresion normales Resumen. En muchas aplicaciones es frecuente enfrentarse con el problema de contrastar si algunos coeficientes de regresion son nulos. Dicho problema se resuelve, bajo el punto de vista frecuentista, imponiendo la hipotesis de homocedasticidad. Sin embargo esta suposicion no es asumible en general, proporcionandose en tales casos tests aproximados. En este arti...
Most of the consistency analyses of Bayesian procedures for variable selection in regression refe... more Most of the consistency analyses of Bayesian procedures for variable selection in regression refer to pairwise consistency, that is, consistency of Bayes factors. However, variable selection in regression is carried out in a given class of regression models where a natural variable selector is the posterior probability of the models. In this paper we analyze the consistency of the posterior model probabilities when the number of potential regressors grows as the sample size grows. The novelty in the posterior model consistency is that it depends not only on the priors for the model parameters through the Bayes factor, but also on the model priors, so that it is a useful tool for choosing priors for both models and model parameters. We have found that some classes of priors typically used in variable selection yield posterior model inconsistency, while mixtures of these priors improve this undesirable behavior. For moderate sample sizes, we evaluate Bayesian pairwise variable selection procedures by comparing their frequentist Type I and II error probabilities. This provides valuable information to discriminate between the priors for the model parameters commonly used for variable selection.
An optimal Bayesian decision procedure for testing hypothesis in normal linear models based on in... more An optimal Bayesian decision procedure for testing hypothesis in normal linear models based on intrinsic model posterior probabilities is considered. It is proven that these posterior probabilities are simple functions of the classical F-statistic, thus the evaluation of the procedure can be carried out analytically through the frequentist analysis of the posterior probability of the null. An asymptotic analysis proves that, under mild conditions on the design matrix, the procedure is consistent. For any testing hypothesis it is also seen that there is a one-to-one mapping-which we call calibration curve-between the posterior probability of the null hypothesis and the classical p-value. This curve adds substantial knowledge about the possible discrepancies between the Bayesian and the p-value measures of evidence for testing hypothesis. It permits a better understanding of the serious difficulties that are encountered in linear models for interpreting the p-values. A specific illustration of the variable selection problem is given.
The random effect approach for meta-analysis was motivated by a lack of consistent assessment of ... more The random effect approach for meta-analysis was motivated by a lack of consistent assessment of homogeneity of treatment effect before pooling. The random effect model assumes that the distribution of the treatment effect is fully heterogenous across the experiments. However, other models arising by grouping some of the experiments are plausible. We illustrate on simulated binary experiments that the fully heterogenous model gives a poor meta-inference when fully heterogeneity is not the true model and that the knowledge of the true cluster model considerably improves the inference. We propose the use of a Bayesian model selection procedure for estimating the true cluster model, and Bayesian model averaging to incorporate into the meta-analysis the clustering estimation. A well-known meta-analysis for six major multicentre trials to assess the efficacy of a given dose of aspirin in post-myocardial infarction patients is reanalysed.
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