RE: st: Omit Constant from Count Models

["Habiger, Matt" <MHabiger@ndbh.com>]

Thanks for the response Maarten. In this instance, all rhs variables are nonnegative so my statement is true in this instance but certainly in not generally true. Hence, when running the predict command no values are below the exponentiated constant. With that said, it is the case for this data set that dropping the constant leads to better predictions (see distributions below). Thanks for the article reference and comment on a priori evidence! I'll have to consider if there is any reasonable a priori argument for dropping the constant. 

Days	Actual	     No constant		Constant
1	8%		2%		0%
2	13%		19%		0%
3	15%		21%		0%
4	16%		18%		2%
5	11%		13%		66%
6	8%		9%		25%
7	5%		6%		4%
8	3%		3%		2%
9	3%		2%		0%
10	3%		1%		0%

-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Maarten Buis
Sent: Friday, September 14, 2012 8:32 AM
To: statalist@hsphsun2.harvard.edu
Subject: Re: st: Omit Constant from Count Models

On Fri, Sep 14, 2012 at 2:58 PM, Habiger, Matt  wrote:
> I'm hoping somebody can inform me of what impact(s) omitting a constant term from count models, such as poisson or negative binomial, have? Does it impact t-statistics or the validity of coefficient estimates?
>
> I'm modeling the number of days a patient spends in a hospital for a given year and the constant is causing the predicted visits distribution to start at ~4 days (exp(1.46)). In the actual data, roughly 25% of days are below 4 (only those with visits are being modeled). When I drop the constant my estimates are much closer to resembling the actual distribution. Below are the outputs from two models for reference.
>
>
> Truncated negative binomial regression            Number of obs   =       1334
> Truncation point: 0                               LR chi2(5)      =      99.50
> Dispersion     = mean                             Prob > chi2     =     0.0000
> Log likelihood = -3639.7431                       Pseudo R2       =     0.0135
>
> ------------------------------------------------------------------------------
>     inpunits |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
> -------------+--------------------------------------------------------
> -------------+--------
>   claims2009 |   .0047266   .0020829     2.27   0.023     .0006442    .0088091
> previnpunits |   .0290032    .007004     4.14   0.000     .0152755    .0427308
>        age09 |   .0041179   .0016468     2.50   0.012     .0008903    .0073455
>   inplow_ind |   .1627803    .080974     2.01   0.044     .0040741    .3214865
> inphigh_ind |   .2904038    .078893     3.68   0.000     .1357764    .4450312
>        _cons |   1.469507   .0654664    22.45   0.000     1.341195    1.597818
> -------------+--------------------------------------------------------
> -------------+--------
>     /lnalpha |   -.418825   .0693098                     -.5546696   -.2829804
> -------------+--------------------------------------------------------
> -------------+--------
>        alpha |   .6578193   .0455933                       .574262    .7535346
> ----------------------------------------------------------------------
> -------- Likelihood-ratio test of alpha=0:  chibar2(01) = 2745.11 
> Prob>=chibar2 = 0.000

The statement that "the constant is causing the predicted visits distribution to start at ~4 days (exp(1.46))" is not quite true. Your results say that for a (hypothetical) observation with the value 0 on the variables claims2009, previnpunits, age09, inplow_ind, inphigh_ind you would predict that such an person would stay about 4 days in hospital. Depending on these variables, this can be a gross extrapolation.

In general you do not want to leave the constant out. The idea that leaving the constant out will lead to better predictions is certainly wrong. But don't take my word for it, try it out: estimate the model with and without the constant, use -predict- to predict the expected days in hospital for each of these models and plot both against the observed days in hospital.

As always there are exceptions. I think the most common valid reason for leaving out the constant is when you put it back in through the backdoor by the way you enter categorical variables, e.g.: M.L. Buis
(2012) "Stata tip 106: With or without reference", The Stata Journal, 12(1), pp. 162-164. You may also be modeling a physical process, where there is very strong a priori evidence that there can be no constant.
But any process involving humans is just too random to fall in that class of models.

Hope this helps,
Maarten

---------------------------------
Maarten L. Buis
WZB
Reichpietschufer 50
10785 Berlin
Germany

http://www.maartenbuis.nl
---------------------------------

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