st: Proper standard error for a difference in rates for a Poisson distributedcount?

[David Harless <dwharles@vcu.edu>]

Dear Statalisters:
I am writing to ask for advice on how to calculate (whether there exists?) a 
proper standard error for a mean *difference* in rates from a Poisson 
distributed count.  Suggestions or citations would be greatly appreciated.

Here are the details:
I have counts of the number of failures by groups over time.  In each period, 
there is also a score (from 3 to 5) which is claimed as an (ordinal) rating of 
safety of the group in the time period. Example:

. list group period score expos failures in 17/23, sepby(group) noobs
  +----------------------------------------------+
  | group   period   score      expos   failures |
  |----------------------------------------------|
  |     7        1       4    10.0092         17 |
  |     7        2       5    15.3863         11 |
  |----------------------------------------------|
  |     8        1       5   143.2803         93 |
  |     8        2       4    32.9646         24 |
  |     8        3       3    31.2676         21 |
  |     8        4       4    89.2825         63 |
  |     8        5       3    26.6402          6 |
  +----------------------------------------------+

My main analysis is poisson regression with fixed effects for the groups 
(xtpoisson, fe) that would include dummy variables for the scores.  But I would 
also like to provide a descriptive table that illustrated the change in failure 
rate as the score changed -- and have that table include a correct standard error.

My problem would be straightforward if I simply wanted to present mean failure 
rates by score:
. gen rate=failures/expos
. table score [aweight=expos] , c(m rate) format(%9.3f)
----------------------
 mean(rate)
----------+-----------
        3 |      0.964
        4 |      0.847
        5 |      0.875
----------------------

And the proper standard errors may be obtained using -ci- :

. sort score
. by score: ci failures, exposure(expos) poisson
-> score = 3
                                                         -- Poisson  Exact --
    Variable |   Exposure        Mean    Std. Err.       [95% Conf. Interval]
-------------+---------------------------------------------------------------
    failures |       4684    .9637415    .0143443          .93583    .9922742

And so on.

But I want my table to show the means of the *differences* in failure rates (by 
group) for different scores.  Along the lines of:

. tsset group period
       panel variable:  group, 1 to 153
        time variable:  period, 1 to 7

. gen d_rate=d.rate
(153 missing values generated)

. gen l_score=l.score
(153 missing values generated)

. gen sum_expos=expos+l.expos
(153 missing values generated)

. table l_score score [aweight=sum_expos], c(m d_rate ) f(%9.3f)
----------------------------------
  l_score |      3       4       5
----------+-----------------------
        3 | -0.029  -0.069  -0.074
        4 | -0.026  -0.009  -0.038
        5 |          0.135  -0.025
----------------------------------

I've already made an assumption in weighting by the sum of exposures over the 
two periods in which the difference in rates are calculated.  Perhaps there is a 
better way to incorporate this information, but the mean of differences in rates 
must be calculated in a way that reflects different exposures.

So my question is:  Can one calculate a proper standard error for a difference 
in rates for a Poisson distributed count, as in the above example?

Thanks,
Dave Harless
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