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\titel{Introduction.}
\par\noindent The procedures included perform variable selection and
maximum likelihood parameter estimation for the univariate and bivariate
probit model. Parameter estimation for the multivariate probit model is
performed by means of a sub-optimal likelihood based method. The procedures
are written in the mathematical programming language GAUSS.
The programming code is stored in the \tt BPM*.ARC \rm files. Examples can
be found in \tt BPM.E* \rm files.
Together with it goes a library
\tt BPM.L32\rm. It is intended to be used with GAUSS386, but upon
renaming this
library to \tt BPM.LIB \rm it can also be used with GAUSS.
The library can be activated by setting
$$\hbox{\tt library a:$\backslash$bpm}.$$
It is assumed that the diskette is inserted into drive \tt A:\rm. If this is
not the case, just replace all occurrences in \tt BPM.L32 \rm of \tt A: \rm
by the appropriate drive.


\titel{Parameter Estimation.}
\par\noindent
 The procedures performing parameter estimation are BIVPROB, BPROBIT,
BPROBRS, BPRS and MPROBIT. \medskip\volgende
BPROBIT and BPRS are the main bivariate estimation programs. Input can be
given either via global matrices or via a GAUSS dataset. BPROBIT assumes the
dependent variables are dichotomous, for BPRS they have to be
ordered categorical with  $r$ and $s$ categories respectively. Independent
variables may be continuous, discrete or both. All variables are given by
column vectors (grouped into matrices) or by names of dataset variables.
Dependent variables may be represented by means of  $2\times 2$
($r\times s$) contingency tables (e.g.~Ashford and Sowden's coal miners data),
with a particular vector of covariates
attached to it. If data are represented in this form, BIVPROB and BPROBRS
should be called instead. BPROBIT and BPRS transform their data into the latter
form and then call BIVPROB and BPROBRS. \medskip\volgende
For multivariate analysis, a set of $M$ dichotomous outcome variables may
be specified. A sub-optimal estimation methods consists of
determining maximum likelihood estimations for the
$M(M-1)/2$ pairwise correlation parameters. Though there exist implementations
to compute multivariate normal integrals, their low speed makes it not
justifiable to use a full maximum likelihood approach over
the sub-optimal method.
\medskip\volgende
Output of the estimation procedures is written to \tt BPROBIT.OUT \rm and
\tt MPROBIT.OUT \rm respectively.
In the bivariate probit estimation procedures, the correlation can be
a constant or a function of the covariates. In the $2\times 2$ table case,
the correlation may also be transformed into a real variable $\mu$,
avoiding problems with correlations outside $[-1,1]$. This not yet
implemented for $r\times s$ tables, but it can be done if there is any
interest.
\vfill\eject
\titel{Selection Procedures.}
\par\noindent
There are both univariate and bivariate selection procedures, using either
the likelihood ratio test or the score test.
UPROBSEL(UPROBS2) and UPROBSC(UPROBSC2) are
the univariate procedures,
BPROBSEL and BPROBSC(BPROBSC2)
are the bivariate ones.
Between parentheses are the names of special version
for the case of dichotomous outcome variables. The main procedures
assume the outcomes are ordered categorical.
\medskip\volgende
The input is given via a
GAUSS dataset. The output of the selection process is written to
\tt BPROBSEL.OUT \rm. If, after selecting variables, an analysis is
performed, the output is written to \tt BPROBIT.OUT \rm. It goes without
saying that the user should specify a significance level, together with
dependent variables and candidate covariates. The selection method
used is forward selection. Other selection methods can be
implemented if there is interest for it.

\titel{Profile Likelihood.}
\par\noindent
The profile likelihood procedures, designed for the bivariate case,
are PROFLIK and PRBLIK. A maximum likelihood curve is constructed for
a correlation interval. Apart from numerical output, there is also
a graphical representation. PROFLIK can be called directly or via
BPROBIT and BPRS, when \it\_rpar \rm is set equal to 2. In the first
case, data should be given by global matrices and the dependent variables
should be in contingecy table form, in the second case the inputs for
BPROBIT and BPRS are exactly as for parameter estimation. The program
will ask the user to specify the range for the correlation.
PRBLIK constructs the range for the correlation by itself by means of
a $\chi^2$ value for the likelihood ratio between maximum and bounds.


\titel{Case Influence.}
\par\noindent It is possible to obtain a first idea about case influence
by performing INFL. Inputs are given as for BPROBIT. Output
is only graphical.


\titel{Examples.}
\par\noindent
Run the files \tt BPM.E1 through BPM.E6. \rm
In the example files, it is assumed that the
library is \tt A:BPM.L32. \rm Adapt the name of the drive if needed. In
Appendix A we give the printout of the example files.

\titel{Description of Procedures.}
\par\noindent
In Appendix B we describe the main procedures. In Appendix C the auxiliary
procedures are described. 


\vfill\eject
\titel{Description of Globals.}
\par\noindent
It is possible to control initial
values, output, significance tests and the modelling of the correlation
by setting global variables, all beginning with an underscore.
The relevant globals are listed in the
description of the procedures. You may note that there are other globals
used by the program, but unlisted here. They are for internal use
only.\medskip\noindent
It is possible to call BPMSET to reset all global values to default. This
affects not only the ones described below, but also the internal globals.
It is a good practice to call BPMSET whenever a procedure is stopped in
an abnormal way (e.g.~by CTRL-BREAK).

\bigskip

\glob{\_draw}{0}{The profile likelihood procedures PROFLIK and PRBLIK
 give no graphical output.}
\glob{}{1}{The profile likelihood procedures do give graphical output
(default).}
\bigskip

\glob{\_eps}{}{Tolerance for gradients. If the absolute value of all elements
of a gradient vector are below \it\_eps\rm, Newton-Raphson
iteration stops.}
\bigskip

\glob{\_file}{0}{The
 auxiliary output device for the bivariate probit estimation
procedures is the file \tt BPROBIT.OUT \rm (default).}
\glob{}{1}{The auxiliary output device is the printer via \tt LPT1\rm.}
\glob{}{2}{No auxiliary output.}
\bigskip

\glob{\_final}{0}{At the end of the bivariate
 selection procedures, no analysis is
performed.}
\glob{}{1}{At the end of the bivariate
selection procedures, an analysis is performed
with the selected variables (default).}
\bigskip

\glob{\_imat}{0}{Tsiatis' test uses program defined partitioning matrix
(default).}
\glob{}{1}{Tsiatis' test uses user defined partitioning matrix. If this is
the case, the user will be asked to give the name of a partitioning matrix
while running BIVPROB and BPROBIT.}
\bigskip

\glob{\_initval}{0}{The user is not asked for an opinion about the initial
estimates, proposed by the bivariate probit estimation procedures.}
\glob{}{1}{The user is asked for an opinion about the initial estimates,
proposed by the bivariate probit estimation procedures. It is possible to
alter or to replace proposed initial estimates (default).}
\bigskip

\glob{\_itfile}{0}{No report of the current iteration step to the output file
(default).}
\glob{}{1}{A short report of the current iteration step is sent to the
output file \tt BPROBIT.OUT\rm.}
\bigskip

\glob{\_itscrn}{0}{No report of the current iteration step on screen.}
\glob{}{1}{A short report of the current iteration step is sent to the
screen (default)}
\bigskip

\glob{\_maxit}{}{Upper bound for the number of Newton-Raphson iterations.}
\bigskip

\glob{\_restric}{}{Matrix that has 2 columns if \it\_rpar \rm is 0 and 3
columns otherwise, indicating which covariates should be taken into account
for respectively first and second dependent variable and for the correlation.
The matrix is considered to be boolean. By default \it\_restric \rm is a
scalar 1, which means that all covariates are to be considered. If the
matrix has the wrong shape, it is set to default.}
\bigskip

\glob{\_rhonorm}{0}{The correlation $\rho$ is transformed into a real variable
$\mu$.}
\glob{}{1}{The correlation $\rho$ is not transformed (default).}
\glob{}{}{This global has effect only on the 2 by 2 table case.}
\bigskip
\glob{\_rpar}{0}{The correlation is constant over the covariate values
 (default).}
\glob{}{1}{The correlation depends on the covariates. In the case of 2 by 2
tables, the way in which it
depends on the covariates can be set by \it\_rhonorm \rm.}
\glob{}{2}{This option is only meaningful when calling BPROBIT or BPROBRS.
It indicates that profile likelihood should be performed instead of
ordinary parameter estimation.}
\bigskip

\glob{\_sfile}{0}{The output of the selection procedures goes to the file
\tt BPROBSEL.OUT \rm (default).}
\glob{}{1}{The output of the selection procedures goes to the printer via
\tt LPT1\rm.}
\bigskip

\glob{\_signif}{0}{Significance tests are not performed (default).}
\glob{}{1}{Significance tests are performed by BIVPROB, BPROBIT, BPROBRS and
BPRS.}
\bigskip

\glob{\_tsiatis}{0}{Tsiatis' Goodness-of-fit test is not performed (default).}
\glob{}{1}{Tsiatis' Goodness-of-fit test is performed by BIVPROB and
BPROBIT.}
\bigskip
















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