Btest {BayesVarSel}

Bayes factors and posterior probabilities for linear regression models

Description

It Computes the Bayes factors and posterior probabilities of a list of linear regression models proposed to explain a common response variable over the same dataset

Usage

Btest(
  models,
  data,
  prior.betas = "Robust",
  prior.models = "Constant",
  priorprobs = NULL,
  null.model = NULL
)

Arguments

models

A named list with the entertained models defined with their corresponding formulas. If the list is unnamed, default names are given by the routine. One model must be nested in all the others.

data

data frame containing the data.

prior.betas

Prior distribution for regression parameters within each model (to be literally specified). Possible choices include "Robust", "Robust.G", "Liangetal", "gZellner", "ZellnerSiow", "FLS", "intrinsic.MGC" and "IHG" (see details).

prior.models

Type of prior probabilities of the models (to be literally specified). Possible choices are "Constant" and "User" (see details).

priorprobs

A named vector ir list (same length and names as in argument models) with the prior probabilities of the models (used in combination of prior.models="User"). If the provided object is not named, then the order in the list of models is used to assign the prior probabilities

null.model

The name of the null model (eg. the one nested in all the others). By default, the names of covariates in the different models are used to identify the null model. An error is produced if such identification fails. This identification is not performed if the definition of the null model is provided, with this argument, by the user. Note that the (the null.model must coincide with that model with the largest sum of squared errors and should be smaller in dimension to any other model).

Details

The Bayes factors, BFi0, are expressed in relation with the simplest model (the one nested in all the others). Then, the posterior probabilities of the entertained models are obtained as

Pr(Mi | data)=Pr(Mi)*BFi0/C,

where Pr(Mi) is the prior probability of model Mi and C is the normalizing constant.

The Bayes factor B_i depends on the prior assigned for the parameters in the regression models Mi and Bvs implements a number of popular choices. The "Robust" prior by Bayarri, Berger, Forte and Garcia-Donato (2012) is the recommended (and default) choice. This prior prior can be implemented in a more stable way using the derivations in Greenaway (2019) and that are available in BayesVarSel since version 2.2.x setting the argument to "Robust.G".

Additional options are "gZellner" a prior which corresponds to the prior in Zellner (1986) with g=n. Also "Liangetal" prior is the hyper-g/n with a=3 (see the original paper Liang et al 2008, for details). "ZellnerSiow" is the multivariate Cauchy prior proposed by Zellner and Siow (1980, 1984), further studied by Bayarri and Garcia-Donato (2007). "FLS" is the (benchmark) prior recommended by Fernandez, Ley and Steel (2001) which is the prior in Zellner (1986) with g=max(n, p*p) p being the number of covariates to choose from (the most complex model has p+number of fixed covariates). "intrinsic.MGC" is the intrinsic prior derived by Moreno, Giron, Casella (2015) and "IHG" corresponds to the intrinsic hyper-g prior derived in Berger, Garcia-Donato, Moreno and Pericchi (2022).

With respect to the prior over the model space Pr(Mi) three possibilities are implemented: "Constant", under which every model has the same prior probability and "User". With this last option, the prior probabilities are defined through the named list priorprobs. These probabilities can be given unnormalized.

Limitations: the error "A Bayes factor is infinite.". Bayes factors can be extremely big numbers if i) the sample size is even moderately large or if ii) a model is much better (in terms of fit) than the model taken as the null model. We are currently working on more robust implementations of the functions to handle these problems. In the meanwhile you could try using the g-Zellner prior (which is the most simple one and results, in these cases, should not vary much with the prior) and/or using more accurate definitions of the simplest model.

Value

Btest returns an object of type Btest which is a list with the following elements:

BFio

A vector with the Bayes factor of each model to the simplest model.

PostProbi

A vector with the posterior probabilities of each model.

models

A list with the entertained models.

nullmodel

The position of the simplest model.

prior.betas

prior.betas

prior.models

prior.models

priorprobs

priorprobs

Author(s)

Gonzalo Garcia-Donato and Anabel Forte

Maintainer: <[e-mail]>

References

Bayarri, M.J., Berger, J.O., Forte, A. and Garcia-Donato, G. (2012)<DOI:10.1214/12-aos1013> Criteria for Bayesian Model choice with Application to Variable Selection. The Annals of Statistics. 40: 1550-1557.

Bayarri, M.J. and Garcia-Donato, G. (2007)<DOI:10.1093/biomet/asm014> Extending conventional priors for testing general hypotheses in linear models. Biometrika, 94:135-152.

Barbieri, M and Berger, J (2004)<DOI:10.1214/009053604000000238> Optimal Predictive Model Selection. The Annals of Statistics, 32, 870-897.

Berger, J., Garcıa-Donato, G., Moreno, E., and Pericchi, L. (2022). The intrinsic hyper-g prior for normal linear models. in preparation.

Fernandez, C., Ley, E. and Steel, M.F.J. (2001)<DOI:10.1016/s0304-4076(00)00076-2> Benchmark priors for Bayesian model averaging. Journal of Econometrics, 100, 381-427.

Greenaway, M. (2019) Numerically stable approximate Bayesian methods for generalized linear mixed models and linear model selection. Thesis (Department of Statistics, University of Sydney).

Liang, F., Paulo, R., Molina, G., Clyde, M. and Berger,J.O. (2008)<DOI:10.1198/016214507000001337> Mixtures of g-priors for Bayesian Variable Selection. Journal of the American Statistical Association. 103:410-423

Zellner, A. and Siow, A. (1980)<DOI:10.1007/bf02888369> Posterior Odds Ratio for Selected Regression Hypotheses. In Bayesian Statistics 1 (J.M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.) 585-603. Valencia: University Press.

Zellner, A. and Siow, A. (1984) Basic Issues in Econometrics. Chicago: University of Chicago Press.

Zellner, A. (1986)<DOI:10.2307/2233941> On Assessing Prior Distributions and Bayesian Regression Analysis with g-prior Distributions. In Bayesian Inference and Decision techniques: Essays in Honor of Bruno de Finetti (A. Zellner, ed.) 389-399. Edward Elgar Publishing Limited.

See Also

Bvs for variable selection within linear regression models

Examples

## Not run: 
#Analysis of Crime Data
#load data
data(UScrime)
#Model selection among the following models: (note model1 is nested in all the others)
model1<- y ~ 1 + Prob
model2<- y ~ 1 + Prob + Time
model3<- y ~ 1 + Prob + Po1 + Po2
model4<- y ~ 1 + Prob + So
model5<- y ~ .

#Equal prior probabilities for models:
crime.BF<- Btest(models=list(basemodel=model1,
	ProbTimemodel=model2, ProbPolmodel=model3,
	ProbSomodel=model4, fullmodel=model5), data=UScrime)

#Another configuration of prior probabilities of models:
crime.BF2<- Btest(models=list(basemodel=model1, ProbTimemodel=model2,
	ProbPolmodel=model3, ProbSomodel=model4, fullmodel=model5),
	data=UScrime, prior.models = "User", priorprobs=list(basemodel=1/8,
	ProbTimemodel=1/8, ProbPolmodel=1/2, ProbSomodel=1/8, fullmodel=1/8))
#same as:
#crime.BF2<- Btest(models=list(basemodel=model1, ProbTimemodel=model2,
	#ProbPolmodel=model3,ProbSomodel=model4, #fullmodel=model5), data=UScrime,
	#prior.models = "User", priorprobs=list(basemodel=1, ProbTimemodel=1,
	#ProbPolmodel=4, #ProbSomodel=1, fullmodel=1))

## End(Not run)