The MCMC Procedure

To specify a truncated distribution, you can use the LOWER= and/or UPPER= options. Almost all of the univariate standard distributions, including the GENERAL and DGENERALfunctions, take these optional truncation arguments. The binary, the uniform, and the tabled distributions do not support these truncation options. Multivariate distributions, such as the multivariate normal, do not support these options neither.

For example, you can specify the following:

prior alpha ~ normal(mean = 0, sd = 1, lower = 3, upper = 45);

or

parms beta;
a = 3; b = 7;
ll = (a + 1) * log(b / beta);
prior beta ~ general(ll, upper = b + 17);

The preceding statements state that if beta is less than b+17, the log of the prior density is ll, as calculated by the equation; otherwise, the log of the prior density is missing—the log of zero.

When the same distribution is applied to multiple parameters in a PRIOR statement, the LOWER= and UPPER= truncations apply to all parameters in that statement. For example, the following statements define a Poisson density for theta and gamma:

parms theta gamma;
lambda = 7;
l1 = theta * log(lambda) - lgamma(1 + theta);
l2 = gamma * log(lambda) - lgamma(1 + gamma);
ll = l1 + l2;
prior theta gamma ~ dgeneral(ll, lower = 1);

The LOWER=1 condition is applied to both theta and gamma, meaning that for the assignment to ll to be meaningful, both theta and gamma have to be greater than 1. If either of the parameters is less than 1, the log of the joint prior density becomes a missing value.

PROC MCMC calculates the normalizing constant in all truncated distributions (with exception to the GENERAL and the DGENERAL functions), and you can use parameters in the LOWER= or UPPER= option.

Note that if you use either the GENERAL or DGENERAL function, you must compute the normalizing constant in cases where it is required. A truncated distribution has the probability distribution

where is the density function and is the cumulative distribution function. In SAS functions, is the probability density function and is the cumulative distribution function. The following example shows how to construct a truncated gamma prior on theta, with SHAPE=3, SCALE=2, LOWER=A, and UPPER=B:

lp = logpdf('gamma', theta, 3, 2)
        - log(cdf('gamma', a, 3, 2) - cdf('gamma', b, 3, 2));
prior theta ~ general(lp);

This density specification is different from the following more naive definition, without taking into account the normalizing constant:

lp = logpdf('gamma', theta, 3, 2);
prior theta ~ general(lp, lower=a, upper=b);

If a or b is a parameter, you get very different results from the two formulations.

You can use either of two approaches to model censored data. One is to specify the marginal distribution, and the other is to treat the censored data as missing values.

Suppose you partition the data into four categories: uncensored (with observation x), left-censored (with observation xl), right-censored (with observation xr), and interval-censored (with observations xl and xr). The likelihood is the normal distribution with mean mu and standard deviation s. The following statements construct the corresponding log likelihood for the observed data:

if uncensored then
   ll = logpdf('normal', x, mu, s);
else if leftcensored then
   ll = logcdf('normal', xl, mu, s);
else if rightcensored then
   ll = logsdf('normal', xr, mu, s);
else /* this is the case of interval censored. */
   ll = log(cdf('normal', xr, mu, s) - cdf('normal', xl, mu, s));
model general(ll);

Alternatively, you can treat censored data as missing values and impute the values in the Markov chain. In the following statement, the CLOWER= and CUPPER= options are the censoring indicators:

model x ~ normal(mu, sd=1, clower=xl, cupper=xr);

Missing x values become parameters, and PROC MCMC samples according to the censoring information. Specify the MISSING=ACMODELY option in the PROC MCMC statement if the xl or xr variables contain missing values. This option enables PROC MCMC to draw missing response variables without discarding observations that have missing covariates. By default, PROC MCMC models missing values but throws away observations that have missing values in nonresponse variables.

See Normal Regression with Interval Censoring.