Bayesian inference for ordinal (or binary) regression models under a
proportional odds assumption.
stan_polr( formula, data, weights, ..., subset, na.action = getOption("na.action", "na.omit"), contrasts = NULL, model = TRUE, method = c("logistic", "probit", "loglog", "cloglog", "cauchit"), prior = R2(stop("'location' must be specified")), prior_counts = dirichlet(1), shape = NULL, rate = NULL, prior_PD = FALSE, algorithm = c("sampling", "meanfield", "fullrank"), adapt_delta = NULL, do_residuals = NULL ) stan_polr.fit( x, y, wt = NULL, offset = NULL, method = c("logistic", "probit", "loglog", "cloglog", "cauchit"), ..., prior = R2(stop("'location' must be specified")), prior_counts = dirichlet(1), shape = NULL, rate = NULL, prior_PD = FALSE, algorithm = c("sampling", "meanfield", "fullrank"), adapt_delta = NULL, do_residuals = algorithm == "sampling" )
Arguments
| formula, data, subset | Same as |
|---|---|
| weights, na.action, contrasts, model | Same as |
| ... | Further arguments passed to the function in the rstan
package ( |
| method | One of 'logistic', 'probit', 'loglog', 'cloglog' or 'cauchit',
but can be abbreviated. See |
| prior | Prior for coefficients. Should be a call to |
| prior_counts | A call to |
| shape | Either |
| rate | Either |
| prior_PD | A logical scalar (defaulting to |
| algorithm | A string (possibly abbreviated) indicating the
estimation approach to use. Can be |
| adapt_delta | Only relevant if |
| do_residuals | A logical scalar indicating whether or not to
automatically calculate fit residuals after sampling completes. Defaults to
|
| x | A design matrix. |
| y | A response variable, which must be a (preferably ordered) factor. |
| wt | A numeric vector (possibly |
| offset | A numeric vector (possibly |
Value
A stanreg object is returned
for stan_polr.
A stanfit object (or a slightly modified
stanfit object) is returned if stan_polr.fit is called directly.
Details
The stan_polr function is similar in syntax to
polr but rather than performing maximum likelihood
estimation of a proportional odds model, Bayesian estimation is performed
(if algorithm = "sampling") via MCMC. The stan_polr
function calls the workhorse stan_polr.fit function, but it is
possible to call the latter directly.
As for stan_lm, it is necessary to specify the prior
location of \(R^2\). In this case, the \(R^2\) pertains to the
proportion of variance in the latent variable (which is discretized
by the cutpoints) attributable to the predictors in the model.
Prior beliefs about the cutpoints are governed by prior beliefs about the
outcome when the predictors are at their sample means. Both of these
are explained in the help page on priors and in the
rstanarm vignettes.
Unlike polr, stan_polr also allows the "ordinal"
outcome to contain only two levels, in which case the likelihood is the
same by default as for stan_glm with family = binomial
but the prior on the coefficients is different. However, stan_polr
allows the user to specify the shape and rate hyperparameters,
in which case the probability of success is defined as the logistic CDF of
the linear predictor, raised to the power of alpha where alpha
has a gamma prior with the specified shape and rate. This
likelihood is called “scobit” by Nagler (1994) because if alpha
is not equal to \(1\), then the relationship between the linear predictor
and the probability of success is skewed. If shape or rate is
NULL, then alpha is assumed to be fixed to \(1\).
Otherwise, it is usually advisible to set shape and rate to
the same number so that the expected value of alpha is \(1\) while
leaving open the possibility that alpha may depart from \(1\) a
little bit. It is often necessary to have a lot of data in order to estimate
alpha with much precision and always necessary to inspect the
Pareto shape parameters calculated by loo to see if the
results are particularly sensitive to individual observations.
Users should think carefully about how the outcome is coded when using
a scobit-type model. When alpha is not \(1\), the asymmetry
implies that the probability of success is most sensitive to the predictors
when the probability of success is less than \(0.63\). Reversing the
coding of the successes and failures allows the predictors to have the
greatest impact when the probability of failure is less than \(0.63\).
Also, the gamma prior on alpha is positively skewed, but you
can reverse the coding of the successes and failures to circumvent this
property.
References
Nagler, J., (1994). Scobit: An Alternative Estimator to Logit and Probit. American Journal of Political Science. 230 -- 255.
See also
stanreg-methods and
polr.
The vignette for stan_polr.
http://mc-stan.org/rstanarm/articles/
Examples
if (!grepl("^sparc", R.version$platform)) { fit <- stan_polr(tobgp ~ agegp, data = esoph, method = "probit", prior = R2(0.2, "mean"), init_r = 0.1, seed = 12345, algorithm = "fullrank") # for speed only print(fit) plot(fit) }#> Chain 1: ------------------------------------------------------------ #> Chain 1: EXPERIMENTAL ALGORITHM: #> Chain 1: This procedure has not been thoroughly tested and may be unstable #> Chain 1: or buggy. The interface is subject to change. #> Chain 1: ------------------------------------------------------------ #> Chain 1: #> Chain 1: #> Chain 1: #> Chain 1: Gradient evaluation took 4.8e-05 seconds #> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.48 seconds. #> Chain 1: Adjust your expectations accordingly! #> Chain 1: #> Chain 1: #> Chain 1: Begin eta adaptation. #> Chain 1: Iteration: 1 / 250 [ 0%] (Adaptation) #> Chain 1: Iteration: 50 / 250 [ 20%] (Adaptation) #> Chain 1: Iteration: 100 / 250 [ 40%] (Adaptation) #> Chain 1: Iteration: 150 / 250 [ 60%] (Adaptation) #> Chain 1: Iteration: 200 / 250 [ 80%] (Adaptation) #> Chain 1: Success! Found best value [eta = 1] earlier than expected. #> Chain 1: #> Chain 1: Begin stochastic gradient ascent. #> Chain 1: iter ELBO delta_ELBO_mean delta_ELBO_med notes #> Chain 1: 100 -131.324 1.000 1.000 #> Chain 1: 200 -128.970 0.509 1.000 #> Chain 1: 300 -128.146 0.342 0.018 #> Chain 1: 400 -127.573 0.257 0.018 #> Chain 1: 500 -127.682 0.206 0.006 MEDIAN ELBO CONVERGED #> Chain 1: #> Chain 1: Drawing a sample of size 1000 from the approximate posterior... #> Chain 1: COMPLETED. #> stan_polr #> family: ordered [probit] #> formula: tobgp ~ agegp #> observations: 88 #> ------ #> Median MAD_SD #> agegp.L -0.2 0.2 #> agegp.Q -0.1 0.3 #> agegp.C -0.1 0.3 #> agegp^4 0.0 0.2 #> agegp^5 0.0 0.2 #> #> Cutpoints: #> Median MAD_SD #> 0-9g/day|10-19 -0.5 0.1 #> 10-19|20-29 0.2 0.2 #> 20-29|30+ 0.8 0.2 #> #> ------ #> * For help interpreting the printed output see ?print.stanreg #> * For info on the priors used see ?prior_summary.stanreg
