Bayesian Analysis
Bayesian methods treat parameters as random variables and define probability as "degrees of belief"
(that is, the probability of an event is the degree to which you believe the event is true). It follows
that probabilities are subjective and that you can make probability statements
about parameters. When performing a Bayesian analysis you begin with a prior belief regarding the probability
distribution of an unknown parameter. After learning information from observed data, you change or update
your belief about the unknown parameter and obtain a posterior distribution. In theory, Bayesian methods
offer simple alternatives to statistical inference—all inferences follow from the posterior distribution.
In practice, however, you can obtain the posterior distribution with straightforward analytical solutions only
in the most rudimentary problems. Most Bayesian analyses require sophisticated computations, including the use
of simulation methods. You generate samples from the posterior distribution and use these samples to estimate
the quantities of interest.
The SAS/STAT Bayesian analysis procedures include the following:
BCHOICE Procedure
The BCHOICE procedure fits Bayesian discrete choice models by using MCMC methods.
The procedure's capabilities include the following:
 fits the following types of models:
 multinomial logit
 multinomial probit
 nested logit
 multinomial logit with random effects
 multinomial probit with random effects
 samples directly from the full conditional distribution when possible
 supports the following sampling algorithms:
 MetropolisHastings approach of Gamerman
 random walk Metropolis
 latent variables via the data augmentation method
 provides a variety of Markov chain convergence diagnostics

 works with the postprocessing autocall macros that are designed for Bayesian posterior samples
 supports a CLASS statement for specifying classification variables
 supports a RESTRICT statement, enabling you to specify boundary requirements and order constraints
on fixed effects for logit models
 multithreaded
 creates an output data set that contains the posterior samples of all parameters
 creates an output data set that contains random samples from the posterior predictive distribution of the choice probabilities
 creates an output data set that corresponds to any output table
 supports BY group processing
 automatically produces graphs by using ODS Graphics

For further details, see
BCHOICE Procedure
FMM Procedure
The FMM procedure fits statistical models to data for which the distribution of the response
is a finite mixture of univariate distributions–that is, each response comes from one of
several random univariate distributions with unknown probabilities.
The following are highlights of the FMM procedure's features:
 model the component distributions in addition to the mixing probabilities
 fit finite mixture models by maximum likelihood or Bayesian methods
 fit finite mixtures of regression and generalized linear models
 define the model effects for the mixing probabilities and their link function
 model overdispersed data
 estimate multimodal or heavytailed densities
 fit zeroinflated or hurdle models to count data with excess zeros
 fit regression models with complex error distributions
 classify observations based on predicted component probabilities
 twenty five different response distributions
 linear equality and inequality constraints on model parameters

 specify the response variable by using either the response syntax or the events/trials syntax
 automated model selection for homogeneous mixtures
 weighted estimation
 control the performance characteristics of the procedure (for example, the number of CPUs, the number of threads for multithreading, and so on)
 obtain separate analyses on observations in groups
 create a data set that contains observationwise statistics that are computed after fitting the model
 create a SAS data set corresponding to any output table
 automatically create graphs by using ODS Graphics

For further details, see
FMM Procedure
GENMOD Procedure
The GENMOD procedure fits generalized linear models, as defined by Nelder and Wedderburn (1972). The class of generalized
linear models is an extension of traditional linear models that allows the mean of a population to depend on a linear predictor
through a nonlinear link function and allows the response probability distribution to be any member of an exponential family of
distributions. Many widely used statistical models are generalized linear models. These include classical linear models with normal
errors, logistic and probit models for binary data, and loglinear models for multinomial data. Many other useful statistical models
can be formulated as generalized linear models by the selection of an appropriate link function and response probability distribution.
The following are highlights of the GENMOD procedure's features:
 provides the following builtin distributions and associated variance functions:
 normal
 binomial
 Poisson
 gamma
 inverse Gaussian
 negative binomial
 geometric
 multinomial
 zeroinflated Poisson
 Tweedie
 provides the following builtin link functions:
 identity
 logit
 probit
 power
 log
 complementary loglog
 enables you to define your own link functions or distributions through DATA step
programming statements used within the procedure
 fits models to correlated responses by the GEE method

 perform Bayesian analysis for generalized linear models
 performs exact logistic regression
 performs exact Poisson regression
 enables you to fit a sequence of models and to perform Type I and Type III analyses
between each successive pair of models
 computes likelihood ratio statistics for userdefined contrasts
 computes estimated values, standard errors, and confidence limits for userdefined
contrasts and least squares means
 computes confidence intervals for model parameters based on either the profile
likelihood function or asymptotic normality
 produces an overdispersion diagnostic plot for zeroinflated models
 performs BY group processing, which enables you to obtain separate analyses on grouped observations
 creates SAS data sets that correspond to most output tables
 automatically generates graphs by using ODS Graphics

For further details, see
GENMOD Procedure
LIFEREG Procedure
The LIFEREG procedure fits parametric models to failure time data that can be uncensored, right censored, left censored, or
interval censored. The models for the response variable consist of a linear effect composed of the covariates and a random
disturbance term. The distribution of the random disturbance can be taken from a class of distributions that includes the
extreme value, normal, logistic, and, by using a log transformation, the exponential, Weibull, lognormal, loglogistic, and
threeparameter gamma distributions. The following are highlights of the LIFEREG procedure's features:
 estimates the parameters by maximum likelihood with a NewtonRaphson
algorithm
 estimates the standard errors of the parameter estimates from the
inverse of the observed information matrix
 fits an accelerated failure time model that assumes that the effect
of independent variables on an event time distribution is multiplicative
on the event time
 computes least square means and least square mean differences for classification effects
 performs multiple comparison adjustments for the pvalues and confidence limits for the least
square mean differences
 estimates linear functions of the model parameters

 tests hypotheses for linear combinations of the model parameters
 performs samplingbased Bayesian analysis
 performs weighted estimation
 performs BY group processing, which enables you to obtain separate analyses on grouped observations
 creates a SAS data set that contains the parameter estimates, the maximized log likelihood,
and the estimated covariance matrix
 creates a SAS data set that corresponds to any output table
 automatically creates graphs by using ODS Graphics

For further details, see
LIFEREG Procedure
MCMC Procedure
The MCMC procedure is a general purpose Markov chain Monte Carlo (MCMC) simulation procedure that
is designed to fit a wide range of Bayesian models.
PROC MCMC procedure enables you to do the following:
 specify a likelihood function for the data, prior distributions for the parameters, and hyperprior distributions if you are fitting hierarchical models
 obtain samples from the corresponding posterior distributions, produces summary and diagnostic
statistics, and save the posterior samples in an output data set that can be used for further analysis
 analyze data that have any likelihood, prior, or hyperprior as long as these functions are programmable using the SAS data step functions
 enter parameters into a model linearly or in any nonlinear functional form
 fit dynamic linear models, state space models, autoregressive models, or other models that have a conditionally dependent structure
on either the randomeffects parameters or the response variable
 fit models that contain differential equations or models that require integration

 use an adaptive blocked randomwalk Metropolis algorithm that uses a normal or t proposal distribution by default
 use a Hamiltonian Monte Carlo algorithm with a fixed step size and predetermined number of steps
 use a NoUTurn sampler with the Hamiltonian algorithm
 create a user defined sampler as an alternative to the default algorithms
 create a data set that contains random samples from the posterior predictive distribution of the response variable
 perform BY group processing, which enables you to obtain separate analyses on grouped observations
 take advantage of multiple processors
 create a SAS data set that corresponds to any output table
 automatically create graphs by using ODS Graphics

For further details, see
MCMC Procedure
PHREG Procedure
The PHREG procedure performs regression analysis of survival data based on the Cox proportional hazards model.
Cox's semiparametric model is widely used in the analysis of survival data to explain the effect of explanatory variables on hazard rates.
The following are highlights of the PHREG procedure's features:
 fits a superset of the Cox model, known as the multiplicative hazards model or the AndersonGill model
 fits frailty models
 fits competing risk model of Fine and Gray
 performs stratified analysis
 includes four methods for handling ties in the failure times
 provides four methods of variable selection
 permits an offset in the model
 performs weighted estimation
 enables you to use SAS programming statements within the procedure to modify values of the explanatory variables or to create ne explanatory variables
 tests linear hypotheses about the regression parameters
 estimates customized hazard ratios
 performs graphical and numerical assessment of the adequacy of the Cox regression model

 creates a new SAS data set that contains the baseline function estimates at the event times of each stratum for every specified set of covariates
 outputs survivor function estimates, residuals, and regression diagnostics
 performs conditional logistic regression analysis for matched casecontrol studies
 fits multinomial logit choice models for discrete choice data
 performs samplingbased Bayesian analysis
 performs BY group processing, which enables you to obtain separate analyses on grouped observations
 creates an output data set that contains parameter and covariance estimates
 creates an output data set that contains userspecified statistics
 creates a SAS data set that corresponds to any output table
 automatically created graphs by using ODS Graphics

For further details, see
PHREG Procedure