This function performs Bayesian variable selection for high dimensional design matrix using iMOM prior for non-zero coefficients. It also performs adaptive hyperparameter selection for iMOM prior. Cleaning the data in a preprocessing step and before any data analysis is done by user preference. This function is for binary and survival time response datasets. In the former, MCMC is used to search in the model space while for the latter a stochastic search does that job. This function has the option to do all the mentioned tasks in a parallel fashion, exploiting hundreds of CPUs. It is highly recommended to use a cluster for this purpose. This function also supports fixing covariates in variable selection process, thus making them included in the final selected model with probability 1. Categorical variable are also supported by this function as input covariates to the selection process. They need to be well defined factor variables as part of the input data frame. For the output, this function reports necessary measurements that is common in Bayesian variable selection algorithms. They include Highest Posterior Probability model, median probability model and posterior inclusion probability for each of the covariates in the design matrix.
bvs(
X,
resp,
prep = TRUE,
logT = FALSE,
fixed_cols = NULL,
eff_size = 0.5,
family = c("logistic", "survival"),
hselect = TRUE,
nlptype = "piMOM",
r = 1,
tau = 0.25,
niter = 30,
mod_prior = c("unif", "beta"),
inseed = NULL,
cplng = FALSE,
ncpu = 4,
parallel.MPI = FALSE
)The n times p input data frame containing the
covariates in the design matrix. The columns should represent genes and rows
represent the observed samples. The column names are used as gene names so
they should not be left as NULL. Moreover, the minimum number of
columns allowed is 3. The input data frame can also contain categorical
covariates that are appropriately defined as factor variables in R.
For logistic regression models it is the binary response vector which could be either numeric or factor variable in R. For the Cox proportional hazard models this is a two column matrix where the first column contains survival time vector and the second column is the censoring status for each observation.
A boolean variable determining if the preprocessing step should
be performed on the design matrix or not. That step contains removing
columns that have NA's or all their elements are equal to 0, along
with standardizing non-binary columns. This step is recommended and thus the
default value is TRUE.
A boolean variable determining if log transform should be done on continuous columns before scaling them in the preprocessing step. Note that those columns should not contain any zeros or negative values.
A vector of indices showing the columns in the input
data frame that are not subject to the the selection procedure. These
columns are always in the final selected model. Note that if any of these
columns contain NA, they will be removed. Moreover, if a categorical
variable with k levels is chosen to be fixed, all k-1 dummy
variables associated with it will be selected in the final model.
This is the expected effect size in the model for a standardized design matrix, which is basically the coefficient value that is expected to occur the most based on some prior knowledge.
Determines the type of data analysis. logistic is for
binary outcome data where logistic regression modeling is used, whereas
survival is for survival outcome data using Cox proportional
hazard model.
A boolean variable indicating whether the automatic procedure
for hyperparameter selection should be run or not. The default value is
TRUE. Note that in this setting, r is always chosen to be 1.
Determines the type of nonlocal prior that is used in the analyses. It can be "piMOM" for product inverse moment prior, or "pMOM" for product moment prior. The default is set to piMOM prior.
The paramter r of the iMOM prior, when no automatic
procedure for hyperparameter selection is done. As a result, this is
relevant only when hselect = FALSE, otherwise it is ignored.
The paramter tau of the iMOM prior, when no automatic
procedure for hyperparameter selection is done. As a result, this is
relevant only when hselect = FALSE, otherwise it is ignored.
Number of iterations. For binary response data, this determines the number of MCMC iterations per CPU. For survival response data this is the number of iterations per temperature schedule in the stochastic search algorithm.
Type of prior used for the model space. unif is
for a uniform binomial and beta is for a beta binomial prior. In the
former case, both hyper parameters in the beta prior are equal to 1,
but in the latter case those two hyper parameters are chosen as explained in
the reference papers. The default choice for this variable is the uniform
prior.
The input seed for making the parallel processing
reproducible. This parameter is ignored in logistic regression models when
cplng = FALSE. The default value is NULL which means that each
time the search for model space is started from different starting points.
In case it is set to a number, it initializes the RNG for the first task and
subsequent tasks to get separate substreams.
This parameter is only used in logistic regression models, and indicating if coupling algorithm for MCMC output should be performed or not.
This is the number of cpus used in parallel processing. For logistic regression models this is the number of parallel coupled chains run at the same time. For survival outcome data this is the number of cpus doing stochastic search at the same time to increase th enumber of visited models.
A boolean variable determining if MPI is used for
parallel processing or not. Note that in order to use this feature, your
system should support MPI and Rmpi and doMPI packages should
already be installed. The default is set to FALSE but in case your
system have the requirements, it is recommended to set this parameter to
TRUE as it is more efficient and results in faster run-time.
It returns a list containing different objects that depend on the
family of the model and the coupling flag for logistic regression models.
The following describes the objects in the output list based on different
combinations of those two input arguments.
1) family = logistic && cplng = FALSE
Number of unique models visited throughout the search of the model space.
Maximum unnormalized probability among all visited models
The indices of the model with highest posterior
probability among all visited models, with respect to the columns in
the output des_mat. This is not necessarily the same as the input
design matrix due to some changes to categorical variables. The names of
the selected columns can be checked using gene_names.
The corresponding design matrix is also one of the outputs that can be
checked in des_mat. If the output is character[0] it means
none of the variables of the design matrix is selected in the HPM and
HPM contains only the intercept.
The coefficient vector for the selected model. The first component is always for the intercept.
The indices of median probability model. According to the paper
Barbieri et. al., this is defined to be the model consisting of those
variables whose posterior inclusion probability is at least 1/2. The order
of columns is similar to that is explained for HPM.
A 1000 by 1 vector of unnormalized
probabilities of the first 1000 models with highest posterior probability
among all visited models. If the total number of visited models is less than
1000, then the length of this vector would be equal to num_vis_models
. Note that the intercept is always used in calculating the probabilities
in this vector.
A list containing models corresponding to
max_prob_vec vector. Each entry of this list contains the indices of
covariates for the model with posterior probability reported in the
corresponding entry in max_prob_vec. The intercept column is not
shown in this list as it is present in all of the models.
A vector of length p+1 containing the posterior
inclusion probability for each covariate in the design matrix. The order of
columns is with respect to processed design matrix, des_mat.
The type of nonlocal prior used in the analyses.
The design matrix used in the analysis where fixed columns
are moved to the beginning of the matrix and if prep=TRUE, the
columns containing NA are all removed. The reported indices in
selected models are all with respect to the columns of this matrix.
Names of the genes extracted from the design matrix.
The hyperparameter for iMOM prior density function, calculated using the proposed algorithm for the given dataset.
The hyperparameter for iMOM prior density function, calculated using the proposed algorithm for the given dataset.
Shows what percentage of pairs of chains are coupled.
A k by 1 vector of marginal probabilities
where element i shows the maximum marginal probability of the
data under the maximum model for the \(i^{th}\) pair of chains. k
is the number of paired chains which is the same as number of CPUs.
A k by p binary matrix, where each row is the
model for the \(i^{th}\) pair of chains. Note that the index of nonzero
elements are not necessarily in the same order as the input design matrix,
X, depending on existence of fixed columns in selection procedure.
As a result, always match the indices to the columns of the design matrix
that is reported as an output in des_mat.
The type of nonlocal prior used in the analyses.
A k by 1 binary vector, showing which pairs
are coupled, (=1) and which are not, (= 0).
A k by (p+1) matrix where each row is the
estimated coefficient for each modelin the rows of Chains variable.
A list showing unique models with the indices of the included covariates at each model.
Frequency of each of the unique models. It is used to find the highest frquency model.
Unnormalized probability of each of the unique models.
The design matrix used in the analysis where fixed columns
are moved to the beginning of the matrix and if prep=TRUE, the
columns containing NA are all removed. The reported indices in
selected models are all with respect to the columns of this matrix.
Names of the genes extracted from the design matrix.
The hyperparameter for iMOM prior density function, calculated using the proposed algorithm for the given dataset.
The hyperparameter for iMOM prior density function, calculated using the proposed algorithm for the given dataset.
Number of visited models during the whole process.
The unnormalized probability of the maximum model among all visited models.
The indices of the model with highest posterior
probability among all visited models, with respect to the columns in
des_mat. As a result, always look at the names of the selected
columns using gene_names. The corresponding design matrix is one of
the outputs that can be checked in des_mat.
The indices of median probability model. According to the paper
Barbieri et. al., this is defined to be the model consisting of those
variables whose posterior inclusion probability is at least 1/2. The order
of columns is similar to that is explained for HPM.
The coefficient vector for the selected model reported in
HPM.
A 1000 by 1 vector of unnormalized
probabilities of the first 1000 models with highest posterior probability
among all visited models. If the total number of visited models is less than
1000, then the length of this vector would be equal to num_vis_models
.
A list containing models corresponding to
max_prob_vec vector. Each entry of this list contains the indices of
covariates for the model with posterior probability reported in the
corresponding entry in max_prob_vec.
A p by 1 vector containing the posterior
inclusion probability for each covariate in the design matrix. The order of
columns is with respect to processed design matrix, des_mat.
The type of nonlocal prior used in the analyses.
The design matrix used in the analysis where fixed columns
are moved to the beginning of the matrix and if prep=TRUE, the
columns containing NA are all removed. The reported indices in
selected models are all with respect to the columns of this matrix.
A k by 3 matrix showing the starting model
for each worker CPU. Obviously k is equal to the number of CPUs.
Names of the genes extracted from the design matrix.
The hyperparameter for iMOM prior density function, calculated using the proposed algorithm for the given dataset.
The hyperparameter for iMOM prior density function, calculated using the proposed algorithm for the given dataset.
Nikooienejad, A., Wang, W., and Johnson, V. E. (2016). Bayesian variable selection for binary outcomes in high dimensional genomic studies using nonlocal priors. Bioinformatics, 32(9), 1338-1345. Nikooienejad, A., Wang, W., & Johnson, V. E. (2020). Bayesian variable selection for survival data using inverse moment priors. Annals of Applied Statistics, 14(2), 809-828. Johnson, V. E. (1998). A coupling-regeneration scheme for diagnosing convergence in Markov chain Monte Carlo algorithms. Journal of the American Statistical Association, 93(441), 238-248. Shin, M., Bhattacharya, A., and Johnson, V. E. (2017). Scalable Bayesian variable selection using nonlocal prior densities in ultrahigh dimensional settings. Statistica Sinica. Johnson, V. E., and Rossell, D. (2010). On the use of non-local prior densities in Bayesian hypothesis tests. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72(2), 143-170. Barbieri, M. M., and Berger, J. O. (2004). Optimal predictive model selection. The annals of statistics, 32(3), 870-897.
# NOT RUN {
### Simulating Logistic Regression Data
n <- 200
p <- 40
set.seed(123)
Sigma <- diag(p)
full <- matrix(c(rep(0.5, p*p)), ncol=p)
Sigma <- full + 0.5*Sigma
cholS <- chol(Sigma)
Beta <- c(-1.9,1.3,2.2)
X <- matrix(rnorm(n*p), ncol=p)
X <- X%*%cholS
beta <- numeric(p)
beta[c(1:length(Beta))] <- Beta
XB <- X%*%beta
probs <- as.vector(exp(XB)/(1+exp(XB)))
y <- rbinom(n,1,probs)
colnames(X) <- paste("gene_",c(1:p),sep="")
X <- as.data.frame(X)
### Running 'bvs' function without coupling and with hyperparamter selection
### procedure
bout <- bvs(X, y, family = "logistic", nlptype = "piMOM",
mod_prior = "beta", niter = 50)
### Highest Posterior Model
bout$HPM
### Estimated Coefficients:
bout$beta_hat
### Number of Visited Models:
bout$num_vis_models
# }
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