# High dimensional Bayesian variable selection using nonlocal priors 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. ```r 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 ) ``` ## Arguments - `X`: 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. - `resp`: 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. - `prep`: 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`. - `logT`: 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. - `fixed_cols`: 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. - `eff_size`: 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. - `family`: 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. - `hselect`: 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. - `nlptype`: 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. - `r`: 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. - `tau`: 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. - `niter`: 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. - `mod_prior`: 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. - `inseed`: 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. - `cplng`: This parameter is only used in logistic regression models, and indicating if coupling algorithm for MCMC output should be performed or not. - `ncpu`: 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. - `parallel.MPI`: 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. ## Returns 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` - **num_vis_models**: Number of unique models visited throughout the search of the model space. - **max_prob**: Maximum unnormalized probability among all visited models - **HPM**: 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. - **beta_hat**: The coefficient vector for the selected model. The first component is always for the intercept. - **MPM**: 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`. - **max_prob_vec**: 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. - **max_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`. The intercept column is not shown in this list as it is present in all of the models. - **inc_probs**: 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`. - **nlptype**: The type of nonlocal prior used in the analyses. - **des_mat**: 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. - **gene_names**: Names of the genes extracted from the design matrix. - **r**: The hyperparameter for iMOM prior density function, calculated using the proposed algorithm for the given dataset. - **tau**: The hyperparameter for iMOM prior density function, calculated using the proposed algorithm for the given dataset. 2) `family = logistic && cplng = TRUE` - **cpl_percent**: Shows what percentage of pairs of chains are coupled. - **margin_probs**: 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. - **chains**: 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`. - **nlptype**: The type of nonlocal prior used in the analyses. - **cpl_flags**: A `k` by `1` binary vector, showing which pairs are coupled, (=`1`) and which are not, (= `0`). - **beta_hat**: A `k` by `(p+1)` matrix where each row is the estimated coefficient for each modelin the rows of `Chains` variable. - **uniq_models**: A list showing unique models with the indices of the included covariates at each model. - **freq**: Frequency of each of the unique models. It is used to find the highest frquency model. - **probs**: Unnormalized probability of each of the unique models. - **des_mat**: 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. - **gene_names**: Names of the genes extracted from the design matrix. - **r**: The hyperparameter for iMOM prior density function, calculated using the proposed algorithm for the given dataset. - **tau**: The hyperparameter for iMOM prior density function, calculated using the proposed algorithm for the given dataset. 3) `family = survival` - **num_vis_models**: Number of visited models during the whole process. - **max_prob**: The unnormalized probability of the maximum model among all visited models. - **HPM**: 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`. - **MPM**: 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`. - **beta_hat**: The coefficient vector for the selected model reported in `HPM`. - **max_prob_vec**: 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` . - **max_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`. - **inc_probs**: 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`. - **nlptype**: The type of nonlocal prior used in the analyses. - **des_mat**: 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. - **start_models**: A `k` by `3` matrix showing the starting model for each worker CPU. Obviously `k` is equal to the number of CPUs. - **gene_names**: Names of the genes extracted from the design matrix. - **r**: The hyperparameter for iMOM prior density function, calculated using the proposed algorithm for the given dataset. - **tau**: The hyperparameter for iMOM prior density function, calculated using the proposed algorithm for the given dataset. ## Examples ```r ### 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 ``` ## References 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. ## See Also `ModProb`, `CoefEst` ## Author(s) Amir Nikooienejad

bvs function