# quantreg_or1: Bayesian quantile regression for ordinal quantile model with... In bqror: Bayesian Quantile Regression for Ordinal Models

## Description

This function estimates Bayesian quantile regression for ordinal quantile model with more than 3 outcomes and reports the posterior mean, posterior standard deviation, and 95 percent posterior credible intervals of (β, δ).

## Usage

 `1` ```quantreg_or1(y, x, b0, B0, d0, D0, mcmc, p, tune, display) ```

## Arguments

 `y` observed ordinal outcomes, column vector of dimension (n x 1). `x` covariate matrix of dimension (n x k) including a column of ones with or without column names. `b0` prior mean for normal distribution to sample β, default is 0. `B0` prior variance for normal distribution to sample β. `d0` prior mean of normal distribution to sample δ, default is 0. `D0` prior variance for normal distribution to sample δ. `mcmc` number of MCMC iterations, post burn-in. `p` quantile level or skewness parameter, p in (0,1). `tune` tuning parameter to adjust MH acceptance rate. `display` whether to print the final output or not, default is TRUE.

## Details

Function implements the Bayesian quantile regression for ordinal model with more than 3 outcomes using a combination of Gibbs sampling and Metropolis-Hastings algorithm.

Function initializes prior and then iteratively samples β, δ and latent variable z. Burn-in is taken as 0.25*mcmc and nsim = burn-in + mcmc.

## Value

Returns a list with components:

• `postMeanbeta`: vector with mean of sampled β for each covariate.

• `postMeandelta`: vector with mean of sampled δ for each cut point.

• `postStdbeta`: vector with standard deviation of sampled β for each covariate.

• `postStddelta`: vector with standard deviation of sampled δ for each cut point.

• `gamma`: vector of cut points including Inf and -Inf.

• `catt`

• `acceptancerate`: scalar to judge the acceptance rate of samples.

• `allQuantDIC`: results of the DIC criteria.

• `logMargLikelihood`: scalar value for log marginal likelihood.

• `beta`: matrix with all sampled values for β.

• `delta`: matrix with all sampled values for δ.

## References

Rahman, M. A. (2016). “Bayesian Quantile Regression for Ordinal Models.” Bayesian Analysis, 11(1): 1-24. DOI: 10.1214/15-BA939

Yu, K., and Moyeed, R. A. (2001). “Bayesian Quantile Regression.” Statistics and Probability Letters, 54(4): 437–447. DOI: 10.12691/ajams-6-6-4

Casella, G., and George, E. I. (1992). “Explaining the Gibbs Sampler.” The American Statistician, 46(3): 167-174. DOI: 10.1080/00031305.1992.10475878

Geman, S., and Geman, D. (1984). “Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images.” IEEE Transactions an Pattern Analysis and Machine Intelligence, 6(6): 721-741. DOI:10.1109/TPAMI.1984.4767596

Chib, S., and Greenberg, E. (1995). “Understanding the Metropolis-Hastings Algorithm.” The American Statistician, 49(4): 327-335. DOI: 10.2307/2684568

Hastings, W. K. (1970). “Monte Carlo Sampling Methods Using Markov Chains and Their Applications.” Biometrika, 57: 1317-1340. DOI: 10.2307/1390766

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26``` ``` set.seed(101) data("data25j4") x <- data25j4\$x y <- data25j4\$y k <- dim(x) J <- dim(as.array(unique(y))) D0 <- 0.25*diag(J - 2) output <- quantreg_or1(y = y,x = x, B0 = 10*diag(k), D0 = D0, mcmc = 40, p = 0.25, tune = 1) # Number of burn-in draws: 10 # Number of retained draws: 40 # Summary of MCMC draws: # Post Mean Post Std Upper Credible Lower Credible # beta_0 -2.6202 0.3588 -2.0560 -3.3243 # beta_1 3.1670 0.5894 4.1713 2.1423 # beta_2 4.2800 0.9141 5.7142 2.8625 # delta_1 0.2188 0.4043 0.6541 -0.4384 # delta_2 0.4567 0.3055 0.7518 -0.2234 # MH acceptance rate: 36 # Log of Marginal Likelihood: -554.61 # DIC: 1375.33 ```