st: Re: Test for statistically significant difference in Poisson coefficients or predicted number of events

[Tony Love <tonyplove@gmail.com>]

> Hi, I am working on a project for my advisor, so please don't crucify me for missing some elementary concept.   I understand most of the elementary concepts regarding Poisson in Stata, but I have what I think is a question of higher difficulty.
>
> I have been working with the use of factor variables, for example, estimating a Poisson model using a Stata command like 'poisson num_awards i.cond math'.  This would return the expected difference in log count between each condition and the reference group (cond=1).
> The output might look something like this if there were 3 conditions:
>
> poisson num_awards i.cond math, vce(robust)
>
> Iteration 0:   log pseudolikelihood = -182.75759
> Iteration 1:   log pseudolikelihood = -182.75225
> Iteration 2:   log pseudolikelihood = -182.75225
>
> Poisson regression                                Number of obs   =        200
>                                                   Wald chi2(3)    =      80.15
>                                                   Prob > chi2     =     0.0000
> Log pseudolikelihood = -182.75225                 Pseudo R2       =     0.2118
>
> ------------------------------------------------------------------------------
>              |               Robust
>   num_awards |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
> -------------+----------------------------------------------------------------
>         cond |
>           2  |   1.083859   .3218538     3.37   0.001     .4530373    1.714681
>           3  |   .3698092   .4014221     0.92   0.357    -.4169637    1.156582
>              |
>         math |   .0701524   .0104614     6.71   0.000     .0496485    .0906563
>        _cons |  -5.247124   .6476195    -8.10   0.000    -6.516435   -3.977814
> ------------------------------------------------------------------------------
>
> likewise, using the 'margins' command and its option 'at means', will produce predicted number of events of interest in each condition if holding math at its mean.
>
> margins cond, atmeans
>
> Adjusted predictions                              Number of obs   =        200
> Model VCE    : Robust
>
> Expression   : Predicted number of events, predict()
> at           : 1.cond          =        .225 (mean)
>                2.cond          =        .525 (mean)
>                3.cond          =         .25 (mean)
>                math            =      52.645 (mean)
>
> ------------------------------------------------------------------------------
>              |            Delta-method
>              |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
> -------------+----------------------------------------------------------------
>         cond |
>           1  |    .211411   .0627844     3.37   0.001     .0883558    .3344661
>           2  |   .6249446   .0887008     7.05   0.000     .4510943    .7987949
>           3  |   .3060086   .0828648     3.69   0.000     .1435966    .4684205
> ------------------------------------------------------------------------------
>
>
> My question is this:  Are you aware of a test to determine whether the differences in either the coefficients produced in the first example or the predicted number of events in the second example are statistically significantly different from one another?  That is, if I compare the coef for cond1 to cond2 and cond1 to cond3 and cond2 to cond3 OR predicted number of events for cond1 to cond2 and cond1 to cond3 and cond2 to cond3, are those differences statistically significant from one another, and if not, which ones are not?
>
> Thanks in advance for any reply.

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