st: xtmelogit convergence issues and log transforming IVs

[William Hauser <whauseriii@gmail.com>]

Hi all,
I'm working with a dataset consisting of court cases nested within
judges nested within circuit.  The model is specified as 2 levels
(cases<judges) with circuit represented as 19 dummy variables (20
circuits, 1 omitted as reference).  The outcome is dichotomous so I'm
using the xtmelogit command.  Stata is version 12, intercooled.

The problem is that the model simply will not converge unless I
transform two of the predictor variables which are, in their
untransformed form, highly overdispersed.  These variables represent
the number of "points" the offender receives for their present offense
and for their prior record if they have one (more on that shortly).

Problem is, I'm not sure how to interpret the resulting odds ratios
for the log transformed predictor variables (crime seriousness and
prior record).

Using the natural log, calculated as ln(xvar), I think the coefficient
represents the change in odds for increasing x by a factor of ~2.7
(the value of e).  This would seem to be very unintuitive if correct.
Alternatively, I can use the log to base 1.10 of the x vars,
calculated as ln(xvar)/ln(1.10), which I think might be interpreted as
the odds ratio for a 10% change in x but I'm not at all sure.

So, what is the correct/best transformation for this application and
how do I interpret it?

There is also the vexing issue the log of 0.  For crime seriousness
there are no zeros since everyone committed a crime.  But for prior
record there are those with no prior record.  One solution that seems
to be roundly criticized is the addition of a constant such as .5 or 1
to all cases before logging.  Another solution is to keep the 0's as 0
and create a dummy coded as 1 for all cases with that 0 value (i.e.
those that would've been undefined or ".").  The syntax for the latter
solution looks like this,

gen log_priors=(ln(prior_record)/ln(1.10)
(a bunch of missing values result for all those cases where the
offender has no prior record)
replace log_priors=0 if prior_record==0
gen no_priors=0
replace no_priors=1 if prior_record==0

Anyone know if this is an acceptable solution or if perhaps another
transformation that is amenable to zeros is in order?

Any insight or guidance would be greatly appreciated.

Will Hauser

*
*   For searches and help try:
*   http://www.stata.com/help.cgi?search
*   http://www.stata.com/support/statalist/faq
*   http://www.ats.ucla.edu/stat/stata/