The MCMC Procedure

Some Useful SAS Functions

Table 59.42: Some Useful SAS Functions

SAS Function	Definition
`abs` (x) ;	$\|x\|$
`airy` (x) ;	Returns the value of the AIRY function.
`beta` (x1, x2) ;	$\int _{0}^{1} z^{x1 - 1} (1-z)^{x2 - 1} dz$
`call logistic` (x) ;	$\frac{\exp (x)}{1+\exp (x)}$
`call softmax` (x1,...,xn) ;	Each element is replaced by ${\exp (x_ j)}/{\sum \exp (x_ j)}$
`call stdize` (x1,...,xn) ;	Standardize values
`cdf` () ;	Cumulative distribution function
`cdf` (’normal’, x, 0, 1) ;	Standard normal cumulative distribution function
`comb` (x1, x2) ;	$\frac{x1!}{x2! (x1-x2)!}$
`constant` (’.’) ;	Calculate commonly used constants
`cos` (x) ;	cosine(x)
`css` (x1, ..., xn) ;	$\sum _ i (x_ i - \bar{x})^2$
`cv` (x1, ..., xn) ;	std(x) / mean(x) * 100
`dairy` (x) ;	Derivative of the AIRY function
`dimN` (m) ;	Returns the numbers of elements in the Nth dim of array m
x1 eq x2	Returns 1 if x1 = x2; 0 otherwise
x1**x2	$x1^{x2}$
`geomean` (x1, ..., xn) ;	$\exp \left(\frac{ \log (x_1) + \cdots + \log (x_ n)}{n} \right)$
`difN` (x) ;	Returns differences between the argument and its Nth lag
`digamma` (x1) ;	$\frac{\Gamma (x1)}{\Gamma (x1)}$
`erf` (x) ;	$\frac{2}{\sqrt {\pi }} \int _0^ x \exp (-z^2) dz$
`erfc` (x) ;	1 - erf(x)
`fact` (x) ;	$x!$
`floor` (x) ;	Greatest integer $\leq x$
`gamma` (x) ;	$\int _0^{\infty } z^{x-1}\exp (-1) dz$
`harmean` (x1, ..., xn) ;	$\frac{n}{1/x_1 + \cdots 1/x_ n}$
`ibessel` (nu, x, kode) ;	Modified Bessel function of order nu evaluated at x
`jbessel` (nu, x) ;	Bessel function of order nu evaluated at x
`lagN` (x) ;	Returns values from a queue
`largest` (k, x1, ..., xn) ;	Returns the $k^{th}$ largest element
`lgamma` (x) ;	$\ln (\Gamma (x))$
`lgamma` (x+1) ;	$\ln (x!)$
`log` (x, logN(x)) ;	$\ln (x)$
`logbeta` (x1, x2) ;	lgamma( $x_1$ ) + lgamma( $x_2$ ) - lgamma( $x_1+x_2$ )
`logcdf` () ;	Log of a left cumulative distribution function
`logpdf` () ;	Log of a probability density (mass) function
`logsdf` () ;	Log of a survival function
`max` (x1, x2) ;	Returns $x_1$ if $x_1 > x_2$ ; $x_2$ otherwise
`mean` (of x1-xn) ;	$\sum _ i x_ i / n$
`median` (of x1-xn) ;	Returns the median of nonmissing values
`min` (x1, x2) ;	Returns $x_1$ if $x_1 < x_2$ ; $x_2$ otherwise
`missing` (x) ;	Returns 1 if x is missing; 0 otherwise
`mod` (x1, x2) ;	Returns the remainder from $x_1/x_2$
`n` (x1, ..., xn) ;	Returns number of nonmissing values
`nmiss` (of y1-yn) ;	Number of missing values
`quantile` () ;	Computes the quantile from a specific distribution
`pdf` () ;	Probability density (mass) functions
`perm` (n, r) ;	$\frac{n!}{(n-r)!}$
`put` () ;	Returns a value that uses a specified format
`round` (x) ;	Rounds x
`rms` (of x1-xn) ;	$\sqrt {\frac{x_1^2 + \cdots x_ n^2 }{n}}$
`sdf` () ;	Survival function
`sign` (x) ;	Returns –1 if $x < 0$ ; 0 if $x = 0$ ; 1 if $x > 0$
`sin` (x) ;	sine(x)
`smallest` (s, x1, ..., en ) ;	Returns the $s^{th}$ smallest component of $x_1, \cdots , x_ n$
`sortn` (of x1-xn) ;	Sorts the values of the variables
`sqrt` (x) ;	$\sqrt {x}$
`std` (x1, ..., xn) ) ;	Standard deviation of $x_1, \cdots , x_ n$ (n-1 in denominator)
`sum` (of x:) ;	$\sum _ i x_ i$
`trigamma` (x) ;	Derivative of the DIGAMMA(x) function
`uss` (of x1-xn) ;	Uncorrected sum of squares

Here are examples of some commonly used transformations:

logit

mu = beta0 + beta1 * z1;
call logistic(mu);

log
```
w = beta0 + beta1 * z1;
mu = exp(w);
```

probit

w = beta0 + beta1 * z1;
mu = cdf(`normal', w, 0, 1);

cloglog

w = beta0 + beta1 * z1;
mu = 1 - exp(-exp(w));