Gibbs Variable Selection
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Gibbs Variable Selection
Xiaomei Pan, Kellie Poulin, Jigang Yang, Jianjun Zhu
Topics
1. Overview of Variable Selection Procedures
2. Gibbs Variable Selection (GVS)
3. How to implement in WinBUGS
4. Recommendations
5. Appendices
Overview of Variable Selection Procedures
• Selecting the best model – The best likelihood– The link function– Priors– Variable Selection
Overview of Variable Selection Procedures
• The type of response variable typically narrows our choices for:
• The best likelihood• The link function
• Also, we often only have a relatively small number of priors to choose from
• If no prior information is known non-informative priors are chosen for the model parameters.
• If prior information is known, this will also narrow our choices for priors
Overview of Variable Selection Procedures
• However there may be many choices for what variables should be included in the model
• In many real world problems the number of candidate variables is in the tens to hundreds.
• For example 10 candidate variables leads to 1024 different possible linear models
• More models are possible if considering interactions and non-linear terms!
Overview of Variable Selection Procedures
• Things to keep in mind when doing variable selection– P-values for variables are no longer valid.
• Variable selection is a form of “data snooping” (recall “multiple comparison procedures”)
– Correlation between the predictors could lead to less than optimal results.
– It is best to use cross-validation techniques as a safeguard
– Best used in “prediction models” rather than “effect models”
Overview of Variable Selection Procedures
• Variable Selection Methods• Frequentists use methods such as
– Stepwise Regression – Mallow’s CP– Maximum R-squared
• What methods do Bayesians use?
Overview of Bayesian Variable Selection Procedures
Method Used For Ease of Use References*
Gibbs Variable Selection (GVS)
Variable Selection Moderately Easy 2, 3, 7.
SSVS (Stochastic Search Variable
Selection)
Variable Selection Somewhat difficult 2, 5.
Kuo and Mallick (Unconditional
Priors for Variable Selection )
Variable Selection Very Easy 2, 6.
Carlin and ChibGeneral Model Selection
Moderately Easy 1, 2.
Reversible JumpMostly Variable Selection
Moderately Difficult 4 and also see Matt Bognar’s thesis in 241 SH
* Refer to reverence slide
Overview of Bayesian Variable Selection Procedures
Method Advantages Disadvantages
Gibbs Variable Selection (GVS)
Pseudo-priors do not affect the posterior distribution. Easy to implement using WinBUGS.
Requires pseudo-priors on all model coefficients whose sole function is to increase the efficiency of the sampler.
SSVS (Stochastic Search Variable
Selection)
Can fit a wide variety of models. Allows the user to indicate which models they think are more likely.
Requires pseudo-priors on all model coefficients and candidate models.
Kuo and Mallick (Unconditional
Priors for Variable Selection )
Extremely straightforward, only required to specify the usual prior on full parameter vector( for the full model) and the conditional prior distributions replace the pseudo-priors.
No flexibility to alter the method to improve efficiency. If, for any parameter, the prior is diffuse compared with the posterior, the method becomes inefficient.
Carlin and ChibFlexible Gibbs sampling strategy for any situation involving model uncertainty.
Computationally demanding. Must specify efficient pseudo-priors becomes too time consuming if there are a large number of model under consideration
Reversible Jump*No need for pseudo-priors. Maybe faster than GVS.
Diffuse priors will often lead to the fewest parameter model being chosen. Cannot implement in WinBUGS.
* Thanks to Dr. Matt Bognar for insights into reversible jump. His thesis in 241 SH contains examples of using this method.
Gibbs Variable Selection
• GVS Sampling Procedure
Likelihood:
Y[i] ~ dnorm(mu[i],tau)
mu[i]= β 0 + β 1* X1 *γ 1 + β 2* X2 *γ 2 + … + β p* Xp *γ p
Prior:
γ i ~ dbern(0.5) #γ =1 means this variable is selected
β i ~γ i*Real Prior+(1-γ i)*Pseudo Prior
tau ~ dgamma(1.0E-3,1.0E-3)
How to Implement in WinBUGS
• We adapted code from Ntzoufras, I. (2003)– This code and the paper is available on the
WinBUGS web site– The example showed variable selection for a
model with 3 candidate predictor variables– This code required the user to modify the
code extensively if they wanted to use it for their own data
How to Implement in WinBUGS
• Our WinBUGS code– Requires no changes in the model
specification– The user must only insert their data and
modify initial values.• p=number of x variables• N=Number of observations• Models=number of models (2^p)• Initial values for beta’s
How to Implement in WinBUGS
• Provided at the end of this presentation– Full WinBUGS code for variable selection– Code for fitting the full model in WinBUGs (for
development of Pseudopriors)• This code also only requires the user to change
the data and initial values– R code to assist in interpreting output from
WinBUGS– SAS Code used to develop sample data
How to Implement in WinBUGS
EXAMPLE• Data used for example
– Validated our code using published results for a model with 3 variables
– Created a simulated data set with 500 observations and 10 predictors
• 9 predictors were continuous• 1 predictor was binary• Created a version of the file with correlation between
predictors to test robustness of GVS to non-orthogonal data– Code used to create simulated data is in Appendix 1– Full correlation matrix (for correlated data) is in
Appendix 2
How to Implement in WinBUGS
Target model created in simulated data
Y= 2 * X1 + .7 * X6 + .22 * X10 -.7 * X5 + random error (normal 0,1)
How to implement in WinBUGS
Determine Likelihoods and
priors
Fit the full model to develop information
for pseudo-priors
Standardize X Matrix
Orthoganalize X Matrix**
Step 1 Step 2 Step 3
Prior to Running Our Code
** see recommendations
How to Implement GVS in WinBUGS
• Standardize X matrix– Centering the covariates will remove
correlation between the model coefficients– Dividing by the standard deviation allows for
comparison of coefficients on the same scale– May also make it easy to assign proper non-
informative priors– If the user wants to use informative priors this
may be a nuisance
Step 1
How to Implement GVS in WinBUGS
• Standardize X matrix– In SAS
proc standard data=input_file_name mean=0 std=1 out=output_file_name; var variable_names;run;
– In Winbugs
# This is at the top of the model code and is done automaticallyfor (j in 1:p) {
b[j] <- beta[j]/sd(x[,j]) for (i in 1:N) {
z[i,j] <- (x[i,j] - mean(x[,j]))/sd(x[,j]) }temp_mean[j]<-b[j]*mean(x[,j]) }
b0 <- beta0-sum(temp_mean[])
Step 1
How to Implement GVS in WinBUGS
• Orthoganalize X matrix**– This is recommended by those who designed GVS to
make variable selection more accurate– However, this makes interpretation of the coefficients
impossible– We do not include this step in the WinBUGS code but
suggest some alternative methods for handling correlation in our recommendations
Step 1
How to Implement GVS in WinBUGS
• Orthogonalize X matrix**– In our example we simulated data with correlations and GVS
seems to be able to handle some correlation– More analysis needs to be done to determine the sensitivity of
the procedure to correlations in the x matrix– We recommend variable clustering to remove highly correlated
variables– If you want to orthogonalize your X matrix, this is the appropriate
SAS code
proc princomp data=input_file_name out=output_file_name;var variable_names;run;
Step 1
How to Implement GVS in WinBUGS
• Fit full model to get Pseudo-priors– In SAS
– In Winbugs (full code is at end of document)
Step 2
proc reg data=input_file_name;model y=variable_list;run;
Model {# Standardize code here#likelihood
for (i in 1:N){Y[i] ~ dnorm(mu[i],tau)for(j in 1:p){temp[i,j]<-beta[j]*z[i,j] }mu[i] <- beta0 + sum(temp[i,]) }
beta0 ~ dnorm(0,0.00001)for (j in 1:p) { beta[j] ~dnorm(0, 1.0E-6) }tau ~ dgamma(1.0E-3,1.0E-3)sigma <- sqrt(1/tau)}
How to Implement GVS in WinBUGS
• Fit full model to get Pseudo-priors– In SAS
Step 2
Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t|
Intercept 1 7.99442 0.04431 180.42 <.0001 x1 1 2.03194 0.05115 39.72 <.0001 x2 1 -0.01728 0.05096 -0.34 0.7347 x3 1 -0.09008 0.04458 -2.02 0.0439 x4 1 0.02241 0.04495 0.50 0.6183 x5 1 -0.38520 0.04590 -8.39 <.0001 x6 1 0.71948 0.04489 16.03 <.0001 x7 1 0.03432 0.04496 0.76 0.4456 x8 1 0.04153 0.04471 0.93 0.3534 x9 1 -0.01484 0.04682 -0.32 0.7515 x10 1 0.20915 0.04488 4.66 <.0001
How to Implement GVS in WinBUGS
• Fit full model to get Pseudo-priors– In WinBugs (note estimates are very similar and SAS runs in a fraction of the time)
Step 2
node mean sd MC error 2.50% median 97.50% start samplebeta[1] 2.031 0.0511 4.74E-04 1.931 2.032 2.132 500 9501beta[2] -0.0171 0.05108 5.09E-04 -0.1142 -0.01757 0.08328 500 9501beta[3] -0.09102 0.04418 4.50E-04 -0.1777 -0.09149 -0.00371 500 9501beta[4] 0.02227 0.04458 4.06E-04 -0.06568 0.02315 0.1091 500 9501beta[5] -0.3856 0.04629 4.51E-04 -0.4772 -0.386 -0.2954 500 9501beta[6] 0.7189 0.04544 4.43E-04 0.6307 0.7186 0.8083 500 9501beta[7] 0.03404 0.04576 5.22E-04 -0.05454 0.0342 0.1245 500 9501beta[8] 0.04132 0.04451 4.03E-04 -0.04581 0.04108 0.1306 500 9501beta[9] -0.01492 0.04676 4.58E-04 -0.1063 -0.01505 0.07692 500 9501beta[10] 0.2097 0.04496 4.44E-04 0.1207 0.2097 0.2975 500 9501
How to Implement GVS in WinBUGS
• Determine the Likelihood and Priors– Our GVS code is set up with the standard
likelihood and non-informative priors– Edit GVS code before running to put in data
for pseudo-priors
Step 3
How to implement in WinBUGS
model {# Standardize x's and coefficients
for (j in 1:p) {b[j] <- beta[j]/sd(x[,j]) ;for (i in 1:N) {
z[i,j] <- (x[i,j] - mean(x[,j]))/sd(x[,j]) ; }
temp_mean[j]<-b[j]*mean(x[,j])}
b0 <- beta0-sum(temp_mean[])
Running Our Adapted Code
How to implement in WinBUGS
#likelihood for (i in 1:N){
Y[i] ~ dnorm(mu[i],tau)for(j in 1:p){
temp[i,j]<-g[j]*beta[j]*z[i,j] }mu[i] <- beta0 + sum(temp[i,])
# residualsstres[i] <- (Y[i] - mu[i])/sigma
#if standardized residual is greater than 2.5, outlieroutlier[i] <- step(stres[i] - 2.5) + step(-(stres[i]+2.5) )
}
How to implement in WinBUGS
for (j in 1:p){ # Create indicators for the possible variables in the model
# for example x[3] show after intercept, x1,x2,x1+x2, which is pow(2,3-1)
TempIndicator[j]<-g[j]*pow(2, j-1) }
#Create a model number for each possible modelmdl<- 1+sum(TempIndicator[])
# calculate the percentage of time each model is selectedfor (j in 1 : models){
pmdl[j]<-equals(mdl, j) }
How to implement in WinBUGS
# diffuse normal prior on the intercept beta0 ~ dnorm(0,0.00001)
# if the parameter is not in the model an informative prior is used# this prior is calculated using a prior run of the full model
for (j in 1:p) {bprior[j]<-(1-g[j])*mean[j]tprior[j] <-g[j]*0.001+(1-g[j])/(se[j]*se[j])beta[j] ~ dnorm(bprior[j],tprior[j]) g[j]~ dbern(0.5) }
tau ~ dgamma(1.0E-3,1.0E-3)sigma <- sqrt(1/tau)
}
How to implement in WinBUGS
# Fit the full model and put the information for the mean and se of each # standardized beta here for use in the pseudo-priors
DATAmean[] se[]2.013 0.1120.149 0.112-0.259 0.094-0.161 0.096-0.293 0.1340.564 0.0970.094 0.098-0.023 0.097-0.018 0.1330.214 0.088
END
How to implement in WinBUGS
# Set the initial values for the beta's, the overall# precision and the variable selection indicators
Initial Valueslist(beta0 = 0, beta=c(0,0, 0,0,0,0,0,0,0,0), tau = .1, g=c(1,1,1,1,1,1,1,1,1,1))
# P=number of parameters# N=the number of obs# models= the number of possible models (2^p)
DATAlist(p = 10, N = 500, models = 1024Y =c(6.339,…..
How to implement in WinBUGS
• Output– You should monitor at least all of the Beta’s and
PMDL which is the percentage of the time each model was picked
– Models will be numbered 1 – 1024 in our example (the total number of models)
– To see which model number corresponds to which variables selected we created an R function which outputs the model names in order (appendix 5)
• To use this fuction type print.ind(number of x variables) Example: print.ind(10)
How to implement in WinBUGS
• Our Example– Ran 50,000
iterations– Some models
not visited– beta are the
standardized coefficients
– B are the raw coefficients
– Simulated • b1=2, • b5=-.7, • b6=.7, • b10=.22
Correlated Data node mean sd MC error 2.50% median 97.50%
b[1] 1.99 0.04395 2.14E-04 1.904 1.99 2.076
b[2] -0.01494 0.04601 2.16E-04 -0.1054 -0.0151 0.07503
b[3] -0.09533 0.04703 2.03E-04 -0.1878 -0.09507 -0.00378
b[4] 0.02211 0.04402 2.08E-04 -0.06457 0.02212 0.1083
b[5] -0.7717 0.08926 3.88E-04 -0.9483 -0.7714 -0.5961
b[6] 0.7387 0.04553 2.04E-04 0.65 0.7388 0.8278
b[7] 0.035 0.04602 2.23E-04 -0.05527 0.03524 0.1248
b[8] 0.04097 0.04408 1.92E-04 -0.04531 0.04119 0.128
b[9] -0.0145 0.04446 1.99E-04 -0.1007 -0.01441 0.07348
b[10] 0.2126 0.04531 1.94E-04 0.1234 0.2125 0.3015
beta[1] 2.016 0.04451 2.17E-04 1.929 2.016 2.102
beta[2] -0.01651 0.05087 2.39E-04 -0.1165 -0.01669 0.08294
beta[3] -0.09032 0.04456 1.92E-04 -0.1779 -0.09007 -0.00358
beta[4] 0.0225 0.0448 2.11E-04 -0.06572 0.02251 0.1102
beta[5] -0.3862 0.04467 1.94E-04 -0.4745 -0.386 -0.2983
beta[6] 0.7196 0.04435 1.99E-04 0.6332 0.7197 0.8064
beta[7] 0.03415 0.0449 2.17E-04 -0.05393 0.03439 0.1217
beta[8] 0.04161 0.04477 1.95E-04 -0.04602 0.04184 0.13
beta[9] -0.01521 0.04663 2.09E-04 -0.1056 -0.01511 0.07707
beta[10] 0.2096 0.04467 1.91E-04 0.1216 0.2094 0.2972
How to implement in WinBUGS
• Our Example– The model visited
97% of the time was the target model
– This is due to the strong correlations between the response and predictors.
– We also modeled data with weaker correlations.
• in these cases the target model was usually in the top 5 models and several models were visited with higher frequency
Correlated Data> print.ind(10) node mean sd
MC error
[1] "x1+ x5+ x6+ x10" pmdl[562] 0.9693 0.1726 7.61E-04
[1] "x1+ x5+ x6" pmdl[50] 0.01232 0.1103 4.93E-04
[1] "x1+ x3+ x5+ x6+ x10" pmdl[566] 0.009555 0.09728 4.04E-04
[1] "x1+ x5+ x6+ x7+ x10" pmdl[626] 0.00204 0.04512 1.89E-04
[1] "x1+ x5+ x6+ x8+ x10" pmdl[690] 0.00202 0.0449 2.05E-04
[1] "x1+ x4+ x5+ x6+ x10" pmdl[570] 0.001778 0.04213 1.82E-04
[1] "x1+ x5+ x6+ x9+ x10" pmdl[818] 0.001333 0.03649 1.70E-04
[1] "x1+ x2+ x5+ x6+ x10" pmdl[564] 0.001252 0.03537 1.52E-04
[1] "x1+ x3+ x5+ x6" pmdl[54] 1.01E-04 0.01005 4.47E-05
How to implement in WinBUGS
• Our Example– Frequentist
Stepwise method picks appropriate model but also includes x3 at the .05 level
– Note that the p-value of .0473 is not accurate (since we were data snooping)
– Parameter estimates are very similar
Correlated Data
Variable Estimate SE Type II SS F Value Pr > F
Intercept 0.19748 0.55106 0.12533 0.13 0.7202
x1 1.99288 0.04382 2018.143 2067.97 <.0001
x3 -0.09303 0.04678 3.85938 3.95 0.0473
x5 -0.77756 0.08863 75.11717 76.97 <.0001
x6 0.7403 0.04546 258.8251 265.22 <.0001
x10 0.20933 0.04509 21.03744 21.56 <.0001
StepVar.Entered
PartialR-Square
ModelR-Square C(p) F Value Pr > F
1 x1 0.7173 0.7173 354.643 1263.5 <.0001
2 x6 0.0874 0.8047 93.6354 222.44 <.0001
3 x5 0.0234 0.8281 25.2357 67.51 <.0001
4 x10 0.0074 0.8355 5.0224 22.21 <.0001
5 x3 0.0013 0.8368 3.091 3.95 0.0473
How to implement in WinBUGS
• Our Example– Ran 50,000
iterations– Some models
not visited– beta are the
standardized coefficients
– B are the raw coefficients
– Simulated • b1=2, • b5=-.7, • b6=.7, • b10=.22
Non-Correlated Data node mean sd
MC error 2.50% median 97.50% start
sample
b[1] 2.039 0.04615 2.20E-04 1.948 2.039 2.129 500 49501
b[2] 0.187 0.1436 6.75E-04 -0.09499 0.1865 0.4679 500 49501
b[3] -0.1483 0.1363 5.91E-04 -0.4163 -0.1475 0.1169 500 49501
b[4] -0.1588 0.1387 6.55E-04 -0.432 -0.1587 0.1124 500 49501
b[5] -0.5456 0.09244 4.02E-04 -0.729 -0.5455 -0.3641 500 49501
b[6] 0.7378 0.0473 2.11E-04 0.6455 0.7377 0.8308 500 49501
b[7] 0.00314 0.1303 6.29E-04 -0.2525 0.003829 0.2571 500 49501
b[8] 0.01064 0.1402 6.12E-04 -0.2638 0.01136 0.2876 500 49501
b[9] 0.1248 0.1325 5.93E-04 -0.1318 0.1249 0.3871 500 49501
b[10] 0.3615 0.048 2.02E-04 0.2668 0.3613 0.4557 500 49501
beta[1] 2.037 0.0461 2.20E-04 1.947 2.037 2.127 500 49501
beta[2] 0.1781 0.1368 6.43E-04 -0.09047 0.1776 0.4457 500 49501
beta[3] -0.1472 0.1354 5.87E-04 -0.4134 -0.1465 0.1161 500 49501
beta[4] -0.1543 0.1348 6.37E-04 -0.4199 -0.1542 0.1093 500 49501
beta[5] -0.2731 0.04627 2.01E-04 -0.3649 -0.273 -0.1822 500 49501
beta[6] 0.7182 0.04604 2.05E-04 0.6283 0.7181 0.8087 500 49501
beta[7] 0.00325 0.135 6.52E-04 -0.2617 0.00397 0.2665 500 49501
beta[8] 0.01025 0.1351 5.89E-04 -0.2542 0.01094 0.2771 500 49501
beta[9] 0.126 0.1339 5.99E-04 -0.1331 0.1262 0.391 500 49501
beta[10] 0.3483 0.04625 1.94E-04 0.2571 0.3481 0.439 500 49501
How to implement in WinBUGS
• Our Example– Oddly, the coefficient
for x10 and x5 are farther off than in the correlated data. This is by chance (found in simulated data review)
– The model visited most often is the target model
– This model is visited a slightly higher percentage of the time than in the correlated data
Non-Correlated Data> print.ind(10) node mean sd
[1] "x1+ x5+ x6+ x10" pmdl[562] 0.9904 0.09728
[1] "x1+ x4+ x5+ x6+ x10" pmdl[570] 0.002424 0.04918
[1] "x1+ x5+ x6+ x9+ x10" pmdl[818] 0.001616 0.04017
[1] "x1+ x5+ x6+ x7+ x10" pmdl[626] 0.001535 0.03915
[1] "x1+ x5+ x6+ x8+ x10" pmdl[690] 0.001455 0.03811
[1] "x1+ x2+ x5+ x6+ x10" pmdl[564] 0.001293 0.03593
[1] "x1+ x3+ x5+ x6+ x10" pmdl[566] 0.001111 0.03331
[1] "x1+ x6+ x10" pmdl[546] 6.06E-05 0.007785
[1] "x1+ x4+ x5+ x6+ x7+ x10" pmdl[634] 2.02E-05 0.004495
[1] "x1+ x4+ x5+ x6+ x8+ x10" pmdl[698] 2.02E-05 0.004495
[1] "x1+ x5+ x6+ x7+ x8+ x10" pmdl[754] 2.02E-05 0.004495
How to implement in WinBUGS
• Our Example– Frequentist Stepwise
method picks appropriate model
– Parameter estimates are very similar
Non-Correlated Data
Variable Estimate SE Type II SS F Value Pr > F
Intercept -1.68739 0.55768 9.68545 9.16 0.0026
x1 2.03861 0.04613 2065.806 1952.68 <.0001
x5 -0.54495 0.09201 37.11277 35.08 <.0001
x6 0.73788 0.04734 256.9933 242.92 <.0001
x10 0.36191 0.04785 60.5075 57.19 <.0001
StepVar.Entered
PartialR-Square
ModelR-Square C(p) F Value Pr > F
1 x1 0.695 0.695 335.019 1134.74 <.0001
2 x6 0.0914 0.7863 88.1258 212.5 <.0001
3 x10 0.0209 0.8072 33.2824 53.67 <.0001
4 x5 0.0128 0.82 0.5196 35.08 <.0001
Recommendations
• Orthogonalizing the X matrix makes interpretation of the coefficients impossible– If the correlations between the response and
explanatory variables are stronger then the correlations between the explanatory variables you may not need to orthogonalize the X matrix
– If correlations are high within the x matrix we recommend using a variable clustering procedure and then pick an explanatory variable from each cluster
• In SAS use Proc VARCLUS
Recommendations
• Remember the number of candidate models is 2 raised to the number of variables you have. – In our example 2^10-1024.– Sampler may have to many iterations
Conclusions
• Using the adapted WinBUGS code, it is now easy to implement GVS in the standard regression setting
• We are working on also adapting other code from the Ntzoufras paper (like Ridge Regression) to make it easy to implement
• In the case of standard linear regression with non-informative priors, the GVS method appears to give almost identical results to frequentist methods implemented in SAS.– SAS took seconds to fit these models while WinBUGS took
hours• The additional time to use GVS may only be warranted
when wanting to use informative priors.
Appendix 1 Simulated Data Creation Code
*Correlated Data;
data hw.simulate (drop= i j);
array x{10};do i= 1 to 500;
* Make 10 normal(J,1) variables;
do j= 1 to 10;x{j}=rannor(12458)+j;
end;
* Turn one into a binary variable;if x5 le 5 then x5=1;else x5=0;
*create correlation between two predictors;x2=.5*x1+rannor(5);if x5=1 then x9=rannor(5)+.5;else x9=rannor(5);
*create y from x1, x5, x6, and x10;
y=2*x1+.7*x6+.22*x10-.7*x5+rannor(1);
output ;end;run;
*Non-correlated Data;
data hw.simulate_nocorr (drop= i j);
array x{10};do i= 1 to 500;
* Make 10 normal(J,1) variables;
do j= 1 to 10;x{j}=rannor(12458)+j;
end;
* Turn one into a binary variable;if x5 le 5 then x5=1;else x5=0;
*create y from x1, x5, x6, and x10;
y=2*x1+.7*x6+.22*x10-.7*x5+rannor(1);
output ;end;run;
Appendix 2 Simulated Data Correlation Matrix
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 yx1 0.479 0.032 0.025 0.004 0.047 -0.055 -0.039 0.153 0.063 0.847x2 0.479 -0.027 -0.012 -0.028 0.109 -0.002 -0.021 0.093 0.046 0.433x3 0.032 -0.027 -0.031 -0.038 0.019 0.023 0.034 0.013 -0.041 -0.001x4 0.025 -0.012 -0.031 0.054 0.015 -0.003 0.079 0.113 0.056 0.032x5 0.004 -0.028 -0.038 0.054 -0.007 -0.044 -0.022 0.235 0.066 -0.151x6 0.047 0.109 0.019 0.015 -0.007 0.079 -0.021 -0.056 -0.009 0.335x7 -0.055 -0.002 0.023 -0.003 -0.044 0.079 0.057 -0.096 -0.077 -0.007x8 -0.039 -0.021 0.034 0.079 -0.022 -0.021 0.057 0.009 -0.050 -0.022x9 0.153 0.093 0.013 0.113 0.235 -0.056 -0.096 0.009 -0.019 0.065x10 0.063 0.046 -0.041 0.056 0.066 -0.009 -0.077 -0.050 -0.019 0.125y 0.847 0.433 -0.001 0.032 -0.151 0.335 -0.007 -0.022 0.065 0.125
Appendix 3 Code to Fit full model in WinBUGS
#This code can be run with modification# only to the data and initial values
Model {# Standardize x's and coefficients
for (j in 1:p) {b[j] <- beta[j]/sd(x[,j]) for (i in 1:N) {
z[i,j] <- (x[i,j] - mean(x[,j]))/sd(x[,j]) }temp_mean[j]<-b[j]*mean(x[,j]) }
b0 <- beta0-sum(temp_mean[]) #likelihood
for (i in 1:N){Y[i] ~ dnorm(mu[i],tau)
for(j in 1:p){temp[i,j]<-beta[j]*z[i,j] }mu[i] <- beta0 + sum(temp[i,]) }
#priorsbeta0 ~ dnorm(0,0.00001)
for (j in 1:p) {beta[j] ~dnorm(0, 1.0E-6)
}tau ~ dgamma(1.0E-3,1.0E-3)sigma <- sqrt(1/tau) }
DATA#p= the number of explanatory variables#N=the number of observations#Put your response variables and explanatory variables here
list(p = 10, N = 500, Y =c(6.339,…, 8.4532),x = structure(.Data =c(0.3534,…, 10.6885),.Dim = c(500,10)))
Initial Values#enter initial values for the betas.
list(beta0 = 0, beta=c(0,0, 0,0,0,0,0,0,0,0), tau = .1)
Appendix 4 GVS variable selection code
************************************************************************** # Code from Gibbs Variable Selection Using BUGS# code of Ioannis Ntzoufras for variable selection with 3 # predictor variables was modified to allow for # variable selection for any number of variables# the user only need to modify the inits and data
model {# Standardize x's and coefficients
for (j in 1:p) {b[j] <- beta[j]/sd(x[,j]) ;for (i in 1:N) {z[i,j] <- (x[i,j] - mean(x[,j]))/sd(x[,j]) ; }temp_mean[j]<-b[j]*mean(x[,j])}
b0 <- beta0-sum(temp_mean[]) #likelihood
for (i in 1:N){Y[i] ~ dnorm(mu[i],tau)
for(j in 1:p){temp[i,j]<-g[j]*beta[j]*z[i,j] }mu[i] <- beta0 + sum(temp[i,])
# residualsstres[i] <- (Y[i] - mu[i])/sigma #if standardized residual is greater than 2.5, outlieroutlier[i] <- step(stres[i] - 2.5) + step(-(stres[i]+2.5) ) }
for (j in 1:p){ # Create indicators for the possible variables in the model
TempIndicator[j]<-g[j]*pow(2, j-1) }
#Create a model number for each possible modelmdl<- 1+sum(TempIndicator[])
# calculate the percentage of time each model is selectedfor (j in 1 : models){
pmdl[j]<-equals(mdl, j) }
# Priors# diffuse normal prior on the intercept beta0 ~ dnorm(0,0.00001)
# if the parameter is not in the model an informative prior is used# this prior is calculated using a prior run of the full model
for (j in 1:p) {bprior[j]<-(1-g[j])*mean[j]tprior[j] <-g[j]*0.001+(1-g[j])/(se[j]*se[j])beta[j] ~ dnorm(bprior[j],tprior[j]) g[j]~ dbern(0.5)
}tau ~ dgamma(1.0E-3,1.0E-3)sigma <- sqrt(1/tau)
}
Appendix 4 GVS variable selection code
# Fit the full model and put the information for# the mean and se of each beta here for use in # the pseudo priors
mean[] se[]2.013 0.1120.149 0.112-0.259 0.094-0.161 0.096-0.293 0.1340.564 0.0970.094 0.098-0.023 0.097-0.018 0.1330.214 0.088
END
# Set the initial values for the beta's, the overall# precision and the variable selection indicators
Initial Valueslist(beta0 = 0, beta=c(0,0, 0,0,0,0,0,0,0,0), tau = .1, g=c(1,1,1,1,1,1,1,1,1,1))
# P=number of parameters# N=the number of obs# models= the number of possible models (2^p)
list(p = 10, N = 500, models=1024,Y =c(6.339, 4.4347, …),x = structure(.Data =c(0.3534,…10.6885),.Dim = c(500,10)))
Appendix 5 R code to name models
ind<-function(p){ if(p == 0) { return(t <- 0) } else if(p == 1) {return(t <- rbind(0, 1)) } else if(p == 2) { return(t <- rbind(c(0, 0), c(1, 0), c(0, 1), c(1, 1))) } else { t <- rbind(cbind(ind(p - 1), rep(0, 2^(p - 1))), cbind( ind(p - 1), rep(1, 2^(p - 1)))) return(t) }}print.ind<-function(p){ t <- ind(p) print("intercept") for(i in 2:nrow(t)) { e <- NULL L <- T for(j in 1:ncol(t)) { if(t[i, j] == 1 & L == T) { e <- paste(e, "x", j, sep = "") L <- F } else if(t[i, j] == 1 & L == F) { e <- paste(e, "+ x", j, sep = "") }} print(e)}}
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3. Dellaportas, P., Forster, J., & Ntzoufras, I. (2002), “On Bayesian Model and Variable Selection using MCMC”, Statistics and Computing 12:27-36.
4. Green, P. (1995) “Reversible Jump Markov Chain Monte Carlo Computation and Bayesian Model Determination”, Biometrika, Vol. 82 No. 4 711-732.
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