ordinalbayes: Bayesian Ordinal Regression for High-Dimensional Data

Kellie J. Archer, Anna Eames Seffernick, Shuai Sun, Yiran Zhang

Introduction

The ordinalbayes R package was developed for fitting ordinal Bayesian models when there is a high-dimensional covariate space, such as when high-throughput genomic data are used in modeling the ordinal outcome. This package depends on the runjags R package and JAGS (version >=4.x.x) must be installed as well. See the JAGS and [runjags] (https://CRAN.R-project.org/package=runjags) for installation instructions. The package includes the function ordinalbayes which can be used to fit LASSO (model = "lasso"), normal spike-and-slab (model = "normalss"), double exponential spike-and-slab (model = "dess"), and regression-based variable inclusion indicator Bayesian models (model = "regressvi"). Variable selection can be performed using Bayes factor or using the posterior distributions of the variable inclusion indicators directly. This vignette describes the syntax required for each of our Bayesian models.

library("ordinalbayes")
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The package includes a two subsets of The Cancer Genome Atlas Cervical Squamous Cell Carcinoma and Endocervical Adenocarcinoma (TCGA-CESC) dataset: finalSet includes 2,009 transcripts while reducedSet includes 41 transcripts. Both datasets include the same set of subjects who belong to FIGO stages I (\(N=124\)), II (\(N=61\)), and III-IV (\(N=57\)). Additionally, the cesc data.frame is an object that combines the phenotypic and gene expression data into one object. To shorten run time, all illustrations will use cesc.

head(cesc)
#>                              age_at_index cigarettes_per_day         race Stage
#> TCGA-VS-A950-01A-11R-A42T-07           42         0.00000000 not reported     3
#> TCGA-Q1-A73Q-01A-21R-A32P-07           46         0.18082192        white     1
#> TCGA-EK-A2H1-01A-11R-A180-07           20         0.00000000        white     1
#> TCGA-EA-A44S-01A-12R-A26T-07           31         0.00000000        white     3
#> TCGA-ZJ-AAX4-01A-11R-A42T-07           85         0.03123288        white     2
#> TCGA-EA-A97N-01A-11R-A38B-07           38         0.00000000        white     1
#>                              ENSG00000076344 ENSG00000077274 ENSG00000101888
#> TCGA-VS-A950-01A-11R-A42T-07        5.678609        6.720300       10.414790
#> TCGA-Q1-A73Q-01A-21R-A32P-07        2.624350        6.743087       10.320279
#> TCGA-EK-A2H1-01A-11R-A180-07        4.429953        6.716512        9.863641
#> TCGA-EA-A44S-01A-12R-A26T-07        7.021206        6.768850       10.330866
#> TCGA-ZJ-AAX4-01A-11R-A42T-07        4.278489        6.863422       10.332698
#> TCGA-EA-A97N-01A-11R-A38B-07        4.743899        6.748497       10.677080
#>                              ENSG00000115548 ENSG00000122884 ENSG00000125430
#> TCGA-VS-A950-01A-11R-A42T-07        12.34056        11.67348        6.970051
#> TCGA-Q1-A73Q-01A-21R-A32P-07        13.03217        11.69802        6.694007
#> TCGA-EK-A2H1-01A-11R-A180-07        11.88161        11.91802        9.736960
#> TCGA-EA-A44S-01A-12R-A26T-07        12.29227        11.47830        7.820151
#> TCGA-ZJ-AAX4-01A-11R-A42T-07        12.26797        10.52209        8.599117
#> TCGA-EA-A97N-01A-11R-A38B-07        12.69808        11.62760        7.292076
#>                              ENSG00000131370 ENSG00000135443 ENSG00000136457
#> TCGA-VS-A950-01A-11R-A42T-07        7.773317       0.2365704        4.474246
#> TCGA-Q1-A73Q-01A-21R-A32P-07        7.940475       2.8509120        4.224641
#> TCGA-EK-A2H1-01A-11R-A180-07        6.840253       0.1340269        4.000747
#> TCGA-EA-A44S-01A-12R-A26T-07        6.248571       1.9650794        2.658386
#> TCGA-ZJ-AAX4-01A-11R-A42T-07        7.377230       2.4958251        4.158171
#> TCGA-EA-A97N-01A-11R-A38B-07        8.462590       0.8706077        6.069735
#>                              ENSG00000138398 ENSG00000150636 ENSG00000161277
#> TCGA-VS-A950-01A-11R-A42T-07        11.96366        6.951354        8.397064
#> TCGA-Q1-A73Q-01A-21R-A32P-07        12.19146        7.057087        8.616144
#> TCGA-EK-A2H1-01A-11R-A180-07        12.03228        5.994565        7.664759
#> TCGA-EA-A44S-01A-12R-A26T-07        11.49463        6.076752        8.229867
#> TCGA-ZJ-AAX4-01A-11R-A42T-07        11.62506        6.153901        8.164465
#> TCGA-EA-A97N-01A-11R-A38B-07        11.91054        5.766807        7.599969
#>                              ENSG00000163510 ENSG00000164485 ENSG00000164651
#> TCGA-VS-A950-01A-11R-A42T-07        10.81206        5.523265        1.232149
#> TCGA-Q1-A73Q-01A-21R-A32P-07        10.87095        6.572502        7.002737
#> TCGA-EK-A2H1-01A-11R-A180-07        10.91258        5.190466        3.773971
#> TCGA-EA-A44S-01A-12R-A26T-07        10.14616        5.230594        4.125903
#> TCGA-ZJ-AAX4-01A-11R-A42T-07        10.51710        7.611485        4.626328
#> TCGA-EA-A97N-01A-11R-A38B-07        10.65186        6.434868        3.245165
#>                              ENSG00000166091 ENSG00000166342 ENSG00000171121
#> TCGA-VS-A950-01A-11R-A42T-07     -0.04265304       3.8745245        5.513192
#> TCGA-Q1-A73Q-01A-21R-A32P-07     -0.09322960       0.9864192        8.369090
#> TCGA-EK-A2H1-01A-11R-A180-07      0.46102302       3.5160175        6.690186
#> TCGA-EA-A44S-01A-12R-A26T-07      1.48172747       0.3112567        7.179220
#> TCGA-ZJ-AAX4-01A-11R-A42T-07     -0.02222228       3.6285145        6.507039
#> TCGA-EA-A97N-01A-11R-A38B-07      0.98318046       3.3422185        7.321927
#>                              ENSG00000177173 ENSG00000180229 ENSG00000188817
#> TCGA-VS-A950-01A-11R-A42T-07        4.446792        5.221334      -0.3054604
#> TCGA-Q1-A73Q-01A-21R-A32P-07        3.806744        5.043258       2.8640614
#> TCGA-EK-A2H1-01A-11R-A180-07        2.511395        3.628166      -0.4096528
#> TCGA-EA-A44S-01A-12R-A26T-07        3.072081        4.242614      -0.4280007
#> TCGA-ZJ-AAX4-01A-11R-A42T-07        3.435647        4.987061      -0.2773388
#> TCGA-EA-A97N-01A-11R-A38B-07        3.787959        5.867989       2.1377124
#>                              ENSG00000197360 ENSG00000203601 ENSG00000225449
#> TCGA-VS-A950-01A-11R-A42T-07      -0.9218773       0.6578841       0.8891913
#> TCGA-Q1-A73Q-01A-21R-A32P-07       0.1096139       1.3319538       2.0622014
#> TCGA-EK-A2H1-01A-11R-A180-07      -1.0322035      -0.2522735       0.1200411
#> TCGA-EA-A44S-01A-12R-A26T-07      -0.3961578       4.6452716       2.3121313
#> TCGA-ZJ-AAX4-01A-11R-A42T-07      -0.8921010      -0.1027297       1.6885515
#> TCGA-EA-A97N-01A-11R-A38B-07      -0.9470927       1.4453662       1.8141608
#>                              ENSG00000230201 ENSG00000233996 ENSG00000236138
#> TCGA-VS-A950-01A-11R-A42T-07      0.45515752      -0.8005787      2.39181764
#> TCGA-Q1-A73Q-01A-21R-A32P-07     -0.01038494      -0.8460940     -0.02090517
#> TCGA-EK-A2H1-01A-11R-A180-07     -0.04845732      -0.8570974      0.90786314
#> TCGA-EA-A44S-01A-12R-A26T-07     -0.56658738       0.2505256      0.87490415
#> TCGA-ZJ-AAX4-01A-11R-A42T-07     -0.50968290      -0.7739022      0.72861666
#> TCGA-EA-A97N-01A-11R-A38B-07     -0.53019894      -0.2257776      1.77938002
#>                              ENSG00000236819 ENSG00000250602 ENSG00000253923
#> TCGA-VS-A950-01A-11R-A42T-07       0.6838114        1.405047      0.47003057
#> TCGA-Q1-A73Q-01A-21R-A32P-07       6.4898812        3.254435      0.33945673
#> TCGA-EK-A2H1-01A-11R-A180-07      -0.6782753        3.336353      0.02637618
#> TCGA-EA-A44S-01A-12R-A26T-07      -0.6916469        2.931575      0.44931341
#> TCGA-ZJ-AAX4-01A-11R-A42T-07      -0.5467389        3.602136      0.23578770
#> TCGA-EA-A97N-01A-11R-A38B-07      -0.6117338        3.057354     -0.79423073
#>                              ENSG00000256980 ENSG00000259083 ENSG00000259134
#> TCGA-VS-A950-01A-11R-A42T-07        3.212629        2.073511       1.9863973
#> TCGA-Q1-A73Q-01A-21R-A32P-07        3.131681        2.553974       0.9570086
#> TCGA-EK-A2H1-01A-11R-A180-07        3.408530        2.962731       1.7540029
#> TCGA-EA-A44S-01A-12R-A26T-07        2.265136        2.087338       2.1799799
#> TCGA-ZJ-AAX4-01A-11R-A42T-07        1.691849        3.295767       1.5173814
#> TCGA-EA-A97N-01A-11R-A38B-07        1.590774        3.163264       2.2831056
#>                              ENSG00000260484 ENSG00000263612 ENSG00000264049
#> TCGA-VS-A950-01A-11R-A42T-07      -0.7171493     -0.06975943       1.0613205
#> TCGA-Q1-A73Q-01A-21R-A32P-07      -0.7653965      1.08025444       1.9822718
#> TCGA-EK-A2H1-01A-11R-A180-07      -0.1380986      6.20400949       1.1402154
#> TCGA-EA-A44S-01A-12R-A26T-07       2.4260667      0.49979689       0.8579997
#> TCGA-ZJ-AAX4-01A-11R-A42T-07      -0.6844628      0.79342833       0.1272949
#> TCGA-EA-A97N-01A-11R-A38B-07      -0.7394679      5.90066262       1.0206188
#>                              ENSG00000264954 ENSG00000265579 ENSG00000271711
#> TCGA-VS-A950-01A-11R-A42T-07     -0.58860794       2.8046547     0.157358427
#> TCGA-Q1-A73Q-01A-21R-A32P-07     -0.01342846      -0.1233080    -0.818584845
#> TCGA-EK-A2H1-01A-11R-A180-07      0.32064622       2.4109333    -0.001821686
#> TCGA-EA-A44S-01A-12R-A26T-07     -0.64840554       0.4594487     0.214261241
#> TCGA-ZJ-AAX4-01A-11R-A42T-07     -0.55822181       2.4464137    -0.747553857
#> TCGA-EA-A97N-01A-11R-A38B-07     -0.60127115       2.5372659    -0.242044960
#>                              ENSG00000272071 ENSG00000276517
#> TCGA-VS-A950-01A-11R-A42T-07        3.504165        4.384537
#> TCGA-Q1-A73Q-01A-21R-A32P-07        1.179076        5.068678
#> TCGA-EK-A2H1-01A-11R-A180-07        1.124490        5.115981
#> TCGA-EA-A44S-01A-12R-A26T-07        3.398700        5.139583
#> TCGA-ZJ-AAX4-01A-11R-A42T-07        1.348471        4.523211
#> TCGA-EA-A97N-01A-11R-A38B-07        1.259106        4.805752

The primary function for model fitting in the ordinalbayes package is ordinalbayes. The function arguments are

args(ordinalbayes)
#> function (formula, data, x = NULL, subset, center = TRUE, scale = TRUE, 
#>     a = 0.1, b = 0.1, model = "regressvi", gamma.ind = "fixed", 
#>     pi.fixed = 0.05, c.gamma = NULL, d.gamma = NULL, alpha.var = 10, 
#>     sigma2.0 = NULL, sigma2.1 = NULL, coerce.var = 10, lambda0 = NULL, 
#>     nChains = 3, adaptSteps = 5000, burnInSteps = 5000, numSavedSteps = 9999, 
#>     thinSteps = 3, parallel = TRUE, seed = NULL, quiet = FALSE) 
#> NULL

The ordinalbayes function accepts a model formula that specifies the ordinal outcome on the left-hand side of the equation and any unpenalized predictor variable(s) from the phenotypic dataset on the right-hand side of the \(\sim\) equation; if no unpenalized predictor variables are included, the model formula includes 1 (the intercept) on the right-hand side of the equation. Unpenalized predictors are those that we want to coerce into the model (e.g., age) so that no penalty is applied. When unpenalized predictors are included (or coerced) into the model, the user can specify the variance associated with those model parameters (default coerce.var=10).

For example, this call fits a regression-based variable inclusion indicator Bayesian model to predict the ordinal outcome Stage where cigarettes_per_day+age_at_index and age_at_index are included as unpenalized predictors (coerced into the model) and the expression of 41 genes are included as penalized predictors. The user should pass to x the genomic feature data (e.g., expression of genes from high-throughput assays) to be penalized in the fitted model, which are in columns 5-45 of the cesc data.frame. Here a fixed constant prior for \(\pi_j\) is set to 0.05. To shorten run time for demonstration purposes, we reduced the number of iterations for adaptation (adaptSteps), the number of iterations of the Markov chain to run (burnInSteps), and the number of saved steps per chain (numSavedSteps) for all examples.

fit<-ordinalbayes(Stage~cigarettes_per_day+age_at_index, data=cesc,x=cesc[, 5:45], model="regressvi", gamma.ind="fixed", pi.fixed=0.05, adaptSteps=500, burnInSteps=500,  numSavedSteps=999)
#> Welcome to JAGS 4.3.0 on Wed Apr  6 09:32:28 2022
#> JAGS is free software and comes with ABSOLUTELY NO WARRANTY
#> Loading module: basemod: ok
#> Loading module: bugs: ok
#> . . Reading data file data.txt
#> . Compiling model graph
#>    Resolving undeclared variables
#>    Allocating nodes
#> Graph information:
#>    Observed stochastic nodes: 242
#>    Unobserved stochastic nodes: 87
#>    Total graph size: 13583
#> . Reading parameter file inits1.txt
#> . Initializing model
#> . Adapting 500
#> -------------------------------------------------| 500
#> ++++++++++++++++++++++++++++++++++++++++++++++++++ 100%
#> Adaptation successful
#> . Updating 500
#> -------------------------------------------------| 500
#> ************************************************** 100%
#> . . . . . . Updating 2997
#> -------------------------------------------------| 2950
#> ************************************************** 99%
#> * 100%
#> . . . . Updating 0
#> . Deleting model
#> .

By default the genomic features are centered (center=TRUE) and scaled (scale=TRUE) and three chains are run (nChains). The user can subset the data set prior to model fitting, for example, subset=(race=="white").

LASSO Ordinal Model

The LASSO Bayesian ordinal model can be fit by specifying model="lasso" which assumes the penalized coefficients \(\beta_j\) for \(j=1,\ldots,P\) are from independent Laplace (or double exponential) distributions with parameter \(\lambda\) which is from a Gamma distribution with parameters a and b. The default parameters are a=0.01 and b=0.01.

fit.lasso<-ordinalbayes(Stage~cigarettes_per_day+age_at_index, data=cesc,x=cesc[, 5:45], model="lasso", adaptSteps=500, burnInSteps=500,  numSavedSteps=999)

Regression-Based Variable Inclusion Indicator Ordinal Model

Like the LASSO model, the regression-based variable inclusion indicator model assumes the penalized coefficients \(\beta_j\) for \(j=1,\ldots,P\) are from independent Laplace (or double exponential) distributions with parameter \(\lambda\) which is from a Gamma distribution with parameters a and b. Additionally, a variable inclusion indicator \(\gamma_j\) is assumed to follow a Bernoulli distribution with parameter \(\pi_j\). The user can use either a fixed (gamma.ind="fixed") or random (gamma.ind="random") prior for \(\pi_j\). When gamma.ind="fixed", the user can specify pi.fixed as the constant prior to be some value the (0, 1) interval (default is 0.05). Here there are no unpenalized covariates included in the model so the right-hand side of the model formula is 1.

fit.regressvi.fixed<-ordinalbayes(Stage~1, data=cesc,x=cesc[, 5:45], model="regressvi", gamma.ind="fixed", pi.fixed=0.05, adaptSteps=500, burnInSteps=500,  numSavedSteps=999)

When gamma.ind="random", the user must specify parameter values for the Beta distribution c.gamma (e.g., 0.01) and d.gamma (e.g. 0.19).

fit.regressvi.random<-ordinalbayes(Stage~1, data=cesc,x=cesc[, 5:45], model="regressvi", gamma.ind="random", c.gamma=0.01, d.gamma=0.19, adaptSteps=500, burnInSteps=500,  numSavedSteps=999)

Normal Spike-and-Slab Ordinal Model

The normal spike-and-slab Bayesian ordinal model can be fit by specifying model="normalss". When fitting this model the user is required to specify the variance for the spike by setting sigma2.0 to a small positive value (e.g., 0.01) and variance for the slab by setting sigma2.1 to a large positive value (e.g., 10). Additionally, a variable inclusion indicator \(\gamma_j\) is assumed to follow a Bernoulli distribution with parameter \(\pi_j\). The user can use either a fixed (gamma.ind="fixed") or random (gamma.ind="random") prior for \(\pi_j\). When gamma.ind="fixed", the user can specify pi.fixed as the constant prior to be some value the (0, 1) interval (default is 0.05). Here there are no unpenalized covariates included in the model so the right-hand side of the model formula is 1.

fit.normalss.fixed<-ordinalbayes(Stage~1, data=cesc,x=cesc[, 5:45], model="normalss", gamma.ind="fixed", pi.fixed = 0.05, sigma2.0=0.01, sigma2.1=10, adaptSteps=500, burnInSteps=500,  numSavedSteps=999)

When gamma.ind="random", the user must specify parameter values for the Beta distribution c.gamma (e.g., 0.01) and d.gamma (e.g. 0.19).

fitted.normalss.random<-ordinalbayes(Stage~1, data=cesc,x=cesc[, 5:45], model="normalss", gamma.ind="random", c.gamma = 0.01, d.gamma=0.19, sigma2.0=0.01, sigma2.1=10, adaptSteps=500, burnInSteps=500,  numSavedSteps=999)

Double Exponential Spike-and-Slab Ordinal Model

The double exponential spike-and-slab ordinal model can be fit by specifying model="dess". Like LASSO and , the slab is taken to be a double exponential distribution with parameter \(\lambda\) which follows a Gamma distribution with parameters a and b. When fitting this model the user is required to specify the parameter for the spike (\(\lambda_0\)) using lambda0 (e.g., 20). Additionally, a variable inclusion indicator \(\gamma_j\) is assumed to follow a Bernoulli distribution with parameter \(\pi_j\). The user can use either a fixed (gamma.ind="fixed") or random (gamma.ind="random") prior for \(\pi_j\). When gamma.ind="fixed", the user can specify pi.fixed as the constant prior to be some value the (0, 1) interval (default is 0.05). Here there are no unpenalized covariates included in the model so the right-hand side of the model formula is 1.

fit.dess.fixed<-ordinalbayes(Stage~1, data=cesc,x=cesc[, 5:45], model="dess", gamma.ind="fixed", pi.fixed = 0.05, lambda0=20, adaptSteps=500, burnInSteps=500,  numSavedSteps=999)

When gamma.ind="random", the user must specify parameter values for the Beta distribution c.gamma (e.g., 0.01) and d.gamma (e.g. 0.19).

fit.dess.random<-ordinalbayes(Stage~1, data=cesc,x=cesc[, 5:45], model="dess", gamma.ind="random", c.gamma = 0.01, d.gamma=0.19, lambda0=20, adaptSteps=500, burnInSteps=500,  numSavedSteps=999)

Other Package Functions

Generic functions for the resulting ordinalbayes object are available for extracting meaningful results from the resulting MCMC chain. The print function returns several summaries from the MCMC output for each parameter monitored including: the 95th lower confidence limit for the highest posterior density (HPD) credible interval (Lower95), the median value (Median), the 95th upper confidence limit for the HPD credible interval (Upper95), the mean value (Mean), the sample standard deviation (SD), the mode of the variable (Mode), the Monte Carlo standard error (MCerr,) percent of SD due to MCMC (MC%ofSD), effective sample size (SSeff), autocorrelation at a lag of 30 (AC.30), and the potential scale reduction factor (psrf).

print(fit)
#>                            Lower95     Median     Upper95        Mean
#> alpha0[1]             -0.433475000  0.9298580 2.26676e+00  0.91275616
#> alpha0[2]              2.018540000  3.4247800 4.94128e+00  3.41229646
#> gamma.ENSG00000076344  0.000000000  0.0000000 1.00000e+00  0.32866200
#> gamma.ENSG00000077274  0.000000000  1.0000000 1.00000e+00  0.50984318
#> gamma.ENSG00000101888  0.000000000  1.0000000 1.00000e+00  0.70503837
#> gamma.ENSG00000115548  1.000000000  1.0000000 1.00000e+00  0.98298298
#> gamma.ENSG00000122884  0.000000000  0.0000000 1.00000e+00  0.05839173
#> gamma.ENSG00000125430  0.000000000  0.0000000 1.00000e+00  0.34367701
#> gamma.ENSG00000131370  0.000000000  0.0000000 1.00000e+00  0.38438438
#> gamma.ENSG00000135443  0.000000000  0.0000000 1.00000e+00  0.15382049
#> gamma.ENSG00000136457  0.000000000  0.0000000 1.00000e+00  0.18184852
#> gamma.ENSG00000138398  1.000000000  1.0000000 1.00000e+00  0.95261929
#> gamma.ENSG00000150636  0.000000000  0.0000000 1.00000e+00  0.48548549
#> gamma.ENSG00000161277  0.000000000  1.0000000 1.00000e+00  0.93693694
#> gamma.ENSG00000163510  0.000000000  0.0000000 1.00000e+00  0.05672339
#> gamma.ENSG00000164485  0.000000000  1.0000000 1.00000e+00  0.54721388
#> gamma.ENSG00000164651  0.000000000  1.0000000 1.00000e+00  0.54454454
#> gamma.ENSG00000166091  0.000000000  0.0000000 1.00000e+00  0.45812479
#> gamma.ENSG00000166342  0.000000000  0.0000000 1.00000e+00  0.14414414
#> gamma.ENSG00000171121  0.000000000  0.0000000 0.00000e+00  0.03970637
#> gamma.ENSG00000177173  1.000000000  1.0000000 1.00000e+00  0.95895896
#> gamma.ENSG00000180229  0.000000000  0.0000000 1.00000e+00  0.15682349
#> gamma.ENSG00000188817  0.000000000  0.0000000 1.00000e+00  0.40707374
#> gamma.ENSG00000197360  0.000000000  0.0000000 1.00000e+00  0.06039373
#> gamma.ENSG00000203601  0.000000000  0.0000000 1.00000e+00  0.09209209
#> gamma.ENSG00000225449  0.000000000  0.0000000 1.00000e+00  0.07874541
#> gamma.ENSG00000230201  0.000000000  0.0000000 1.00000e+00  0.36803470
#> gamma.ENSG00000233996  0.000000000  1.0000000 1.00000e+00  0.59793126
#> gamma.ENSG00000236138  0.000000000  0.0000000 0.00000e+00  0.04471138
#> gamma.ENSG00000236819  0.000000000  1.0000000 1.00000e+00  0.93393393
#> gamma.ENSG00000250602  0.000000000  0.0000000 1.00000e+00  0.32599266
#> gamma.ENSG00000253923  0.000000000  1.0000000 1.00000e+00  0.68935602
#> gamma.ENSG00000256980  0.000000000  0.0000000 1.00000e+00  0.05605606
#> gamma.ENSG00000259083  0.000000000  0.0000000 1.00000e+00  0.05038372
#> gamma.ENSG00000259134  0.000000000  1.0000000 1.00000e+00  0.69803136
#> gamma.ENSG00000260484  0.000000000  0.0000000 0.00000e+00  0.04304304
#> gamma.ENSG00000263612  0.000000000  0.0000000 1.00000e+00  0.18785452
#> gamma.ENSG00000264049  0.000000000  0.0000000 1.00000e+00  0.15215215
#> gamma.ENSG00000264954  0.000000000  0.0000000 1.00000e+00  0.21488155
#> gamma.ENSG00000265579  0.000000000  0.0000000 1.00000e+00  0.11177845
#> gamma.ENSG00000271711  0.000000000  0.0000000 1.00000e+00  0.08842176
#> gamma.ENSG00000272071  0.000000000  0.0000000 1.00000e+00  0.22288956
#> gamma.ENSG00000276517  0.000000000  1.0000000 1.00000e+00  0.94461128
#> beta.ENSG00000076344   0.000000000  0.0000000 7.43305e-01  0.17478649
#> beta.ENSG00000077274  -3.888170000 -0.1467700 0.00000e+00 -0.93716400
#> beta.ENSG00000101888  -0.987420000 -0.5514100 0.00000e+00 -0.47259610
#> beta.ENSG00000115548   0.456972000  0.9350130 1.49592e+00  0.93016342
#> beta.ENSG00000122884   0.000000000  0.0000000 1.51175e-01  0.02118685
#> beta.ENSG00000125430   0.000000000  0.0000000 7.73057e-01  0.18336678
#> beta.ENSG00000131370  -0.830980000  0.0000000 0.00000e+00 -0.21829392
#> beta.ENSG00000135443  -0.539741000  0.0000000 0.00000e+00 -0.06578075
#> beta.ENSG00000136457  -0.647580000  0.0000000 0.00000e+00 -0.09333314
#> beta.ENSG00000138398  -1.284970000 -0.8598300 0.00000e+00 -0.84204652
#> beta.ENSG00000150636   0.000000000  0.0000000 9.63402e-01  0.31474352
#> beta.ENSG00000161277   0.000000000  0.6928520 1.05848e+00  0.67259456
#> beta.ENSG00000163510  -0.095404800  0.0000000 1.89061e-02 -0.02167887
#> beta.ENSG00000164485  -0.872001000 -0.3073360 0.00000e+00 -0.32036501
#> beta.ENSG00000164651   0.000000000  0.3045340 8.91599e-01  0.32890516
#> beta.ENSG00000166091   0.000000000  0.0000000 8.86686e-01  0.26970016
#> beta.ENSG00000166342   0.000000000  0.0000000 4.72876e-01  0.05983235
#> beta.ENSG00000171121   0.000000000  0.0000000 0.00000e+00 -0.01181109
#> beta.ENSG00000177173   0.231326000  0.7670510 1.32043e+00  0.74704148
#> beta.ENSG00000180229   0.000000000  0.0000000 5.74341e-01  0.07259595
#> beta.ENSG00000188817  -0.918385000  0.0000000 6.01200e-04 -0.25678571
#> beta.ENSG00000197360  -0.177786000  0.0000000 5.28925e-02 -0.02169836
#> beta.ENSG00000203601  -0.012203000  0.0000000 3.93316e-01  0.03440799
#> beta.ENSG00000225449  -0.320337000  0.0000000 0.00000e+00 -0.03102246
#> beta.ENSG00000230201   0.000000000  0.0000000 6.79358e-01  0.17816959
#> beta.ENSG00000233996   0.000000000  0.3746720 8.72514e-01  0.34556783
#> beta.ENSG00000236138   0.000000000  0.0000000 0.00000e+00  0.01276635
#> beta.ENSG00000236819  -1.350620000 -0.8553150 0.00000e+00 -0.82994589
#> beta.ENSG00000250602  -0.688813000  0.0000000 0.00000e+00 -0.16024025
#> beta.ENSG00000253923   0.000000000  0.4490010 8.10281e-01  0.38132075
#> beta.ENSG00000256980  -0.111667000  0.0000000 1.50226e-02 -0.01945728
#> beta.ENSG00000259083  -0.000288853  0.0000000 2.65257e-03 -0.01454637
#> beta.ENSG00000259134   0.000000000  0.5588050 1.04049e+00  0.48373660
#> beta.ENSG00000260484   0.000000000  0.0000000 0.00000e+00  0.01234166
#> beta.ENSG00000263612  -0.569360000  0.0000000 7.29816e-05 -0.08485093
#> beta.ENSG00000264049   0.000000000  0.0000000 4.87576e-01  0.05984188
#> beta.ENSG00000264954   0.000000000  0.0000000 5.71368e-01  0.09499381
#> beta.ENSG00000265579   0.000000000  0.0000000 4.26671e-01  0.04332112
#> beta.ENSG00000271711   0.000000000  0.0000000 3.21571e-01  0.03133108
#> beta.ENSG00000272071   0.000000000  0.0000000 6.21315e-01  0.10270013
#> beta.ENSG00000276517  -1.199630000 -0.7801000 0.00000e+00 -0.76318384
#> lambda                 0.651131000  1.4160400 2.35741e+00  1.46260996
#> cigarettes_per_day    -0.426335000  0.1025620 6.40773e-01  0.10095239
#> age_at_index          -0.011089100  0.0162206 4.23585e-02  0.01579986
#>                               SD Mode       MCerr MC%ofSD SSeff         AC.30
#> alpha0[1]             0.70262118   NA 0.061420785     8.7   131  4.040236e-01
#> alpha0[2]             0.75225406   NA 0.058550590     7.8   165  3.602857e-01
#> gamma.ENSG00000076344 0.46980521    0 0.020462713     4.4   527  1.045383e-01
#> gamma.ENSG00000077274 0.49998652    1 0.017777370     3.6   791  9.320344e-03
#> gamma.ENSG00000101888 0.45610161    1 0.024188016     5.3   356  1.358860e-01
#> gamma.ENSG00000115548 0.12935618    1 0.005109442     3.9   641 -4.052177e-03
#> gamma.ENSG00000122884 0.23452182    0 0.005461579     2.3  1844  2.032965e-02
#> gamma.ENSG00000125430 0.47501412    0 0.014968710     3.2  1007 -2.641471e-02
#> gamma.ENSG00000131370 0.48653059    0 0.020969202     4.3   538  5.610135e-02
#> gamma.ENSG00000135443 0.36083679    0 0.010944534     3.0  1087  1.974260e-02
#> gamma.ENSG00000136457 0.38578400    0 0.013444387     3.5   823  3.498624e-02
#> gamma.ENSG00000138398 0.21248729    1 0.012978779     6.1   268  2.716031e-01
#> gamma.ENSG00000150636 0.49987269    0 0.026435989     5.3   358  1.354807e-01
#> gamma.ENSG00000161277 0.24311692    1 0.009949310     4.1   597  5.165609e-02
#> gamma.ENSG00000163510 0.23135191    0 0.012065727     5.2   368  1.563362e-01
#> gamma.ENSG00000164485 0.49784892    1 0.020051600     4.0   616  8.233079e-02
#> gamma.ENSG00000164651 0.49809494    1 0.023571667     4.7   447  1.100634e-01
#> gamma.ENSG00000166091 0.49832653    0 0.022140968     4.4   507  1.240225e-01
#> gamma.ENSG00000166342 0.35129444    0 0.011021039     3.1  1016  1.471610e-02
#> gamma.ENSG00000171121 0.19530106    0 0.004733622     2.4  1702  4.494977e-03
#> gamma.ENSG00000177173 0.19841827    1 0.006197586     3.1  1025 -9.843752e-03
#> gamma.ENSG00000180229 0.36369495    0 0.011842343     3.3   943 -5.967982e-03
#> gamma.ENSG00000188817 0.49137081    0 0.020860697     4.2   555  9.661498e-02
#> gamma.ENSG00000197360 0.23825462    0 0.005182162     2.2  2114 -1.987087e-02
#> gamma.ENSG00000203601 0.28920416    0 0.008400792     2.9  1185 -1.286431e-02
#> gamma.ENSG00000225449 0.26938594    0 0.006999707     2.6  1481  4.508945e-03
#> gamma.ENSG00000230201 0.48235132    0 0.019704967     4.1   599  7.748766e-02
#> gamma.ENSG00000233996 0.49039750    1 0.022160180     4.5   490  9.492323e-02
#> gamma.ENSG00000236138 0.20670396    0 0.003993517     1.9  2679  4.189357e-03
#> gamma.ENSG00000236819 0.24843900    1 0.008728489     3.5   810 -1.260635e-02
#> gamma.ENSG00000250602 0.46882276    0 0.017193371     3.7   744  2.956125e-02
#> gamma.ENSG00000253923 0.46283450    1 0.019218596     4.2   580  6.061405e-02
#> gamma.ENSG00000256980 0.23006833    0 0.005440116     2.4  1789  1.103046e-02
#> gamma.ENSG00000259083 0.21877195    0 0.004419061     2.0  2451 -1.815343e-02
#> gamma.ENSG00000259134 0.45918834    1 0.024053126     5.2   364  1.764363e-01
#> gamma.ENSG00000260484 0.20298790    0 0.005152464     2.5  1552 -1.869395e-02
#> gamma.ENSG00000263612 0.39066114    0 0.010803767     2.8  1308 -4.700062e-03
#> gamma.ENSG00000264049 0.35922825    0 0.007486523     2.1  2302  3.402771e-02
#> gamma.ENSG00000264954 0.41080869    0 0.010826344     2.6  1440  5.426163e-02
#> gamma.ENSG00000265579 0.31514626    0 0.008325438     2.6  1433 -2.187072e-02
#> gamma.ENSG00000271711 0.28395467    0 0.006774902     2.4  1757 -2.008992e-03
#> gamma.ENSG00000272071 0.41625427    0 0.015910626     3.8   684  2.510316e-02
#> gamma.ENSG00000276517 0.22877560    1 0.008995935     3.9   647  7.038137e-02
#> beta.ENSG00000076344  0.27545182   NA 0.012747159     4.6   467  9.511276e-02
#> beta.ENSG00000077274  1.41669821   NA 0.062732731     4.4   510  8.087368e-02
#> beta.ENSG00000101888  0.35554978   NA 0.018770439     5.3   359  1.392913e-01
#> beta.ENSG00000115548  0.27431244   NA 0.009891473     3.6   769 -2.786978e-02
#> beta.ENSG00000122884  0.10297305   NA 0.002690444     2.6  1465  5.548370e-03
#> beta.ENSG00000125430  0.28356615   NA 0.010916030     3.8   675 -2.065471e-02
#> beta.ENSG00000131370  0.30968049   NA 0.014447889     4.7   459  4.965376e-02
#> beta.ENSG00000135443  0.17447576   NA 0.005856099     3.4   888  7.014074e-02
#> beta.ENSG00000136457  0.22033576   NA 0.007887998     3.6   780  7.115097e-02
#> beta.ENSG00000138398  0.30004735   NA 0.016354751     5.5   337  1.911974e-01
#> beta.ENSG00000150636  0.36437043   NA 0.020641486     5.7   312  1.700882e-01
#> beta.ENSG00000161277  0.26488946   NA 0.009287531     3.5   813  3.373874e-02
#> beta.ENSG00000163510  0.11887246   NA 0.007776245     6.5   234  2.653924e-01
#> beta.ENSG00000164485  0.33036993   NA 0.013893643     4.2   565  1.014205e-01
#> beta.ENSG00000164651  0.34032661   NA 0.016851767     5.0   408  1.309196e-01
#> beta.ENSG00000166091  0.33254468   NA 0.017099486     5.1   378  1.361102e-01
#> beta.ENSG00000166342  0.16326897   NA 0.005313827     3.3   944 -7.578358e-05
#> beta.ENSG00000171121  0.07220400   NA 0.001938587     2.7  1387  2.977390e-02
#> beta.ENSG00000177173  0.26274186   NA 0.008128530     3.1  1045  1.509765e-02
#> beta.ENSG00000180229  0.19029833   NA 0.006804792     3.6   782  4.409283e-03
#> beta.ENSG00000188817  0.34617478   NA 0.015648911     4.5   489  9.317959e-02
#> beta.ENSG00000197360  0.10346679   NA 0.002497440     2.4  1716 -1.300475e-02
#> beta.ENSG00000203601  0.12756907   NA 0.004482872     3.5   810  3.506626e-03
#> beta.ENSG00000225449  0.12468281   NA 0.003594500     2.9  1203  4.730983e-03
#> beta.ENSG00000230201  0.25853009   NA 0.010907662     4.2   562  1.095835e-01
#> beta.ENSG00000233996  0.32464712   NA 0.016420794     5.1   391  1.241967e-01
#> beta.ENSG00000236138  0.07335294   NA 0.001712755     2.3  1834  4.658866e-03
#> beta.ENSG00000236819  0.34651100   NA 0.013774235     4.0   633  4.639616e-02
#> beta.ENSG00000250602  0.25416918   NA 0.009952624     3.9   652  4.187546e-02
#> beta.ENSG00000253923  0.29531999   NA 0.011780088     4.0   628  6.499906e-02
#> beta.ENSG00000256980  0.09790173   NA 0.002288114     2.3  1831 -6.266798e-04
#> beta.ENSG00000259083  0.07436149   NA 0.001745832     2.3  1814 -7.809889e-03
#> beta.ENSG00000259134  0.37337270   NA 0.020512857     5.5   331  1.792189e-01
#> beta.ENSG00000260484  0.06829827   NA 0.001804714     2.6  1432 -3.442721e-02
#> beta.ENSG00000263612  0.19762962   NA 0.005877525     3.0  1131 -6.527597e-03
#> beta.ENSG00000264049  0.15801594   NA 0.003378146     2.1  2188  3.442352e-02
#> beta.ENSG00000264954  0.20159813   NA 0.005570215     2.8  1310  5.387916e-02
#> beta.ENSG00000265579  0.13953408   NA 0.004193964     3.0  1107 -1.658003e-02
#> beta.ENSG00000271711  0.11335734   NA 0.002821477     2.5  1614 -1.293232e-03
#> beta.ENSG00000272071  0.21480477   NA 0.009355345     4.4   527  4.264340e-02
#> beta.ENSG00000276517  0.29458176   NA 0.011540368     3.9   652  5.260914e-02
#> lambda                0.44499403   NA 0.017943188     4.0   615  1.038871e-01
#> cigarettes_per_day    0.27636237   NA 0.006482322     2.3  1818 -1.333869e-02
#> age_at_index          0.01402138   NA 0.001193205     8.5   138  4.131138e-01
#>                            psrf
#> alpha0[1]             1.0113915
#> alpha0[2]             1.0154169
#> gamma.ENSG00000076344 1.0013954
#> gamma.ENSG00000077274 1.0005987
#> gamma.ENSG00000101888 0.9997783
#> gamma.ENSG00000115548 1.0561920
#> gamma.ENSG00000122884 1.0012792
#> gamma.ENSG00000125430 1.0068505
#> gamma.ENSG00000131370 1.0011319
#> gamma.ENSG00000135443 1.0101614
#> gamma.ENSG00000136457 1.0029154
#> gamma.ENSG00000138398 1.0029560
#> gamma.ENSG00000150636 1.0034633
#> gamma.ENSG00000161277 1.0046685
#> gamma.ENSG00000163510 1.0019082
#> gamma.ENSG00000164485 1.0038864
#> gamma.ENSG00000164651 0.9998362
#> gamma.ENSG00000166091 1.0010285
#> gamma.ENSG00000166342 1.0006914
#> gamma.ENSG00000171121 1.0134976
#> gamma.ENSG00000177173 1.0152291
#> gamma.ENSG00000180229 1.0061620
#> gamma.ENSG00000188817 1.0100037
#> gamma.ENSG00000197360 1.0037385
#> gamma.ENSG00000203601 1.0019510
#> gamma.ENSG00000225449 1.0015737
#> gamma.ENSG00000230201 1.0032263
#> gamma.ENSG00000233996 1.0011983
#> gamma.ENSG00000236138 1.0048129
#> gamma.ENSG00000236819 1.0183495
#> gamma.ENSG00000250602 1.0019062
#> gamma.ENSG00000253923 1.0012277
#> gamma.ENSG00000256980 1.0061124
#> gamma.ENSG00000259083 1.0017067
#> gamma.ENSG00000259134 1.0027180
#> gamma.ENSG00000260484 1.0003839
#> gamma.ENSG00000263612 1.0019489
#> gamma.ENSG00000264049 1.0007694
#> gamma.ENSG00000264954 0.9998891
#> gamma.ENSG00000265579 1.0030010
#> gamma.ENSG00000271711 1.0042908
#> gamma.ENSG00000272071 1.0109562
#> gamma.ENSG00000276517 1.0144901
#> beta.ENSG00000076344  1.0010221
#> beta.ENSG00000077274  1.0019150
#> beta.ENSG00000101888  0.9998463
#> beta.ENSG00000115548  1.0046313
#> beta.ENSG00000122884  1.0085651
#> beta.ENSG00000125430  1.0083874
#> beta.ENSG00000131370  1.0011497
#> beta.ENSG00000135443  1.0152429
#> beta.ENSG00000136457  1.0066863
#> beta.ENSG00000138398  1.0000278
#> beta.ENSG00000150636  1.0024666
#> beta.ENSG00000161277  1.0013281
#> beta.ENSG00000163510  1.0040441
#> beta.ENSG00000164485  1.0051341
#> beta.ENSG00000164651  1.0005512
#> beta.ENSG00000166091  1.0010422
#> beta.ENSG00000166342  1.0012695
#> beta.ENSG00000171121  1.0249745
#> beta.ENSG00000177173  1.0020057
#> beta.ENSG00000180229  1.0067688
#> beta.ENSG00000188817  1.0145703
#> beta.ENSG00000197360  1.0123973
#> beta.ENSG00000203601  1.0017999
#> beta.ENSG00000225449  1.0002578
#> beta.ENSG00000230201  1.0029182
#> beta.ENSG00000233996  1.0024614
#> beta.ENSG00000236138  1.0158148
#> beta.ENSG00000236819  1.0044316
#> beta.ENSG00000250602  1.0019545
#> beta.ENSG00000253923  1.0006083
#> beta.ENSG00000256980  1.0346782
#> beta.ENSG00000259083  1.0007234
#> beta.ENSG00000259134  1.0056051
#> beta.ENSG00000260484  1.0010880
#> beta.ENSG00000263612  1.0027956
#> beta.ENSG00000264049  0.9999775
#> beta.ENSG00000264954  1.0001266
#> beta.ENSG00000265579  1.0033739
#> beta.ENSG00000271711  1.0059616
#> beta.ENSG00000272071  1.0161004
#> beta.ENSG00000276517  1.0079891
#> lambda                1.0065106
#> cigarettes_per_day    0.9998849
#> age_at_index          1.0120707

The summary function provides the following output: * alphamatrix, the MCMC output for the threshold parameters; * betamatrix, the MCMC output for the penalized parameters; * zetamatrix, The MCMC output for the unpenalized parameters (if included); * gammamatrix, the MCMC output for the variable inclusion parameters (not available when model = "lasso");
* gammamean, the posterior mean of the variable inclusion indicators (not available when model = "lasso"); * gamma.BayesFactor, Bayes factor for the variable inclusion indicators (not available when model = "lasso"); * Beta.BayesFactor, Bayes factor for the penalized parameters; and * lambdamatrix, the MCMC output for the penalty parameter (not available when model="normalss").

summary.fit<-summary(fit)
names(summary.fit)
#> [1] "alphamatrix"       "betamatrix"        "zetamatrix"       
#> [4] "gammamatrix"       "gammamean"         "gamma.BayesFactor"
#> [7] "Beta.BayesFactor"  "lambdamatrix"
head(summary.fit$gammamatrix)
#>      gamma.ENSG00000076344 gamma.ENSG00000077274 gamma.ENSG00000101888
#> 1001                     0                     0                     1
#> 1004                     0                     0                     1
#> 1007                     0                     0                     1
#> 1010                     0                     0                     1
#> 1013                     0                     0                     1
#> 1016                     0                     0                     1
#>      gamma.ENSG00000115548 gamma.ENSG00000122884 gamma.ENSG00000125430
#> 1001                     1                     0                     0
#> 1004                     1                     0                     0
#> 1007                     1                     0                     0
#> 1010                     1                     0                     0
#> 1013                     1                     0                     1
#> 1016                     1                     0                     0
#>      gamma.ENSG00000131370 gamma.ENSG00000135443 gamma.ENSG00000136457
#> 1001                     1                     0                     0
#> 1004                     1                     0                     0
#> 1007                     1                     0                     1
#> 1010                     0                     0                     1
#> 1013                     0                     0                     1
#> 1016                     0                     0                     0
#>      gamma.ENSG00000138398 gamma.ENSG00000150636 gamma.ENSG00000161277
#> 1001                     1                     1                     1
#> 1004                     1                     1                     1
#> 1007                     1                     1                     1
#> 1010                     1                     1                     1
#> 1013                     1                     1                     1
#> 1016                     1                     1                     1
#>      gamma.ENSG00000163510 gamma.ENSG00000164485 gamma.ENSG00000164651
#> 1001                     0                     0                     1
#> 1004                     0                     0                     1
#> 1007                     0                     0                     1
#> 1010                     0                     0                     1
#> 1013                     0                     0                     0
#> 1016                     0                     0                     0
#>      gamma.ENSG00000166091 gamma.ENSG00000166342 gamma.ENSG00000171121
#> 1001                     0                     0                     0
#> 1004                     0                     0                     0
#> 1007                     1                     0                     0
#> 1010                     1                     0                     0
#> 1013                     0                     0                     0
#> 1016                     1                     0                     0
#>      gamma.ENSG00000177173 gamma.ENSG00000180229 gamma.ENSG00000188817
#> 1001                     1                     0                     0
#> 1004                     1                     0                     1
#> 1007                     1                     1                     0
#> 1010                     1                     0                     0
#> 1013                     1                     0                     0
#> 1016                     1                     0                     1
#>      gamma.ENSG00000197360 gamma.ENSG00000203601 gamma.ENSG00000225449
#> 1001                     0                     0                     0
#> 1004                     0                     0                     0
#> 1007                     0                     0                     0
#> 1010                     0                     0                     0
#> 1013                     0                     0                     0
#> 1016                     0                     0                     0
#>      gamma.ENSG00000230201 gamma.ENSG00000233996 gamma.ENSG00000236138
#> 1001                     0                     0                     0
#> 1004                     0                     1                     0
#> 1007                     0                     1                     1
#> 1010                     0                     1                     0
#> 1013                     0                     1                     0
#> 1016                     0                     0                     0
#>      gamma.ENSG00000236819 gamma.ENSG00000250602 gamma.ENSG00000253923
#> 1001                     1                     1                     1
#> 1004                     1                     1                     1
#> 1007                     1                     0                     1
#> 1010                     1                     0                     0
#> 1013                     1                     0                     1
#> 1016                     1                     1                     1
#>      gamma.ENSG00000256980 gamma.ENSG00000259083 gamma.ENSG00000259134
#> 1001                     0                     0                     1
#> 1004                     0                     0                     1
#> 1007                     0                     0                     1
#> 1010                     0                     0                     0
#> 1013                     0                     0                     0
#> 1016                     0                     0                     0
#>      gamma.ENSG00000260484 gamma.ENSG00000263612 gamma.ENSG00000264049
#> 1001                     0                     0                     0
#> 1004                     0                     0                     0
#> 1007                     0                     0                     1
#> 1010                     0                     0                     0
#> 1013                     0                     0                     0
#> 1016                     0                     0                     0
#>      gamma.ENSG00000264954 gamma.ENSG00000265579 gamma.ENSG00000271711
#> 1001                     1                     0                     0
#> 1004                     0                     1                     1
#> 1007                     0                     0                     0
#> 1010                     0                     0                     0
#> 1013                     0                     0                     0
#> 1016                     1                     0                     0
#>      gamma.ENSG00000272071 gamma.ENSG00000276517
#> 1001                     0                     1
#> 1004                     0                     1
#> 1007                     0                     1
#> 1010                     1                     1
#> 1013                     0                     1
#> 1016                     0                     1

To identify which penalized features using Bayes factor at a given threshold (e.g., 5):

names(which(summary.fit$Beta.BayesFactor>5))
#>  [1] "ENSG00000076344" "ENSG00000077274" "ENSG00000101888" "ENSG00000115548"
#>  [5] "ENSG00000125430" "ENSG00000131370" "ENSG00000138398" "ENSG00000150636"
#>  [9] "ENSG00000161277" "ENSG00000164485" "ENSG00000164651" "ENSG00000166091"
#> [13] "ENSG00000177173" "ENSG00000188817" "ENSG00000230201" "ENSG00000233996"
#> [17] "ENSG00000236819" "ENSG00000250602" "ENSG00000253923" "ENSG00000259134"
#> [21] "ENSG00000264954" "ENSG00000272071" "ENSG00000276517"

or

names(which(summary.fit$gamma.BayesFactor>5))
#>  [1] "ENSG00000076344" "ENSG00000077274" "ENSG00000101888" "ENSG00000115548"
#>  [5] "ENSG00000125430" "ENSG00000131370" "ENSG00000138398" "ENSG00000150636"
#>  [9] "ENSG00000161277" "ENSG00000164485" "ENSG00000164651" "ENSG00000166091"
#> [13] "ENSG00000177173" "ENSG00000188817" "ENSG00000230201" "ENSG00000233996"
#> [17] "ENSG00000236819" "ENSG00000250602" "ENSG00000253923" "ENSG00000259134"
#> [21] "ENSG00000264954" "ENSG00000272071" "ENSG00000276517"

Alternatively, a threshold for \(\bar{\gamma}_j\) could be used for variable selection.

names(which(summary.fit$gammamean>0.5))
#>  [1] "ENSG00000077274" "ENSG00000101888" "ENSG00000115548" "ENSG00000138398"
#>  [5] "ENSG00000161277" "ENSG00000164485" "ENSG00000164651" "ENSG00000177173"
#>  [9] "ENSG00000233996" "ENSG00000236819" "ENSG00000253923" "ENSG00000259134"
#> [13] "ENSG00000276517"
coefficients<-coef(fit)
coefficients$gamma[which(summary.fit$gamma.BayesFactor>5)]
#> ENSG00000076344 ENSG00000077274 ENSG00000101888 ENSG00000115548 ENSG00000125430 
#>       0.3286620       0.5098432       0.7050384       0.9829830       0.3436770 
#> ENSG00000131370 ENSG00000138398 ENSG00000150636 ENSG00000161277 ENSG00000164485 
#>       0.3843844       0.9526193       0.4854855       0.9369369       0.5472139 
#> ENSG00000164651 ENSG00000166091 ENSG00000177173 ENSG00000188817 ENSG00000230201 
#>       0.5445445       0.4581248       0.9589590       0.4070737       0.3680347 
#> ENSG00000233996 ENSG00000236819 ENSG00000250602 ENSG00000253923 ENSG00000259134 
#>       0.5979313       0.9339339       0.3259927       0.6893560       0.6980314 
#> ENSG00000264954 ENSG00000272071 ENSG00000276517 
#>       0.2148815       0.2228896       0.9446113
coefficients$gamma[which(summary.fit$Beta.BayesFactor>5)]
#> ENSG00000076344 ENSG00000077274 ENSG00000101888 ENSG00000115548 ENSG00000125430 
#>       0.3286620       0.5098432       0.7050384       0.9829830       0.3436770 
#> ENSG00000131370 ENSG00000138398 ENSG00000150636 ENSG00000161277 ENSG00000164485 
#>       0.3843844       0.9526193       0.4854855       0.9369369       0.5472139 
#> ENSG00000164651 ENSG00000166091 ENSG00000177173 ENSG00000188817 ENSG00000230201 
#>       0.5445445       0.4581248       0.9589590       0.4070737       0.3680347 
#> ENSG00000233996 ENSG00000236819 ENSG00000250602 ENSG00000253923 ENSG00000259134 
#>       0.5979313       0.9339339       0.3259927       0.6893560       0.6980314 
#> ENSG00000264954 ENSG00000272071 ENSG00000276517 
#>       0.2148815       0.2228896       0.9446113

To obtain model predictions,

phat<-predict(fit)
table(phat$class, cesc$Stage)
#>    
#>       1   2   3
#>   1 121  16   2
#>   2   3  31  14
#>   3   0  14  41

The plot function provides a trace of the sampled output and optionally the density estimate for each variable in the chain. This function additionally adds the appropriate beta and gamma labels for each penalized variable name.

plot(fit)

References

  1. Zhang, Y.; Archer, K.J. Bayesian variable selection for high-dimensional data with an ordinal response: identifying genes associated with prognostic risk group in acute myeloid leukemia. BMC Bioinformatics 2021, 22, 539.
  2. Zhang, Y.; Archer, K.J. Bayesian penalized cumulative logit model for high-dimensional data with an ordinal response. Statistics in Medicine 2021, 40, 1453–1481.