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I'm having a hard time trying to understand what is the meaning of the coefficients of a production-constrained spatial interaction model re-specified as a Poisson linear regression model.

Following this guide to run a production-constrained spatial interaction model (section 2):

https://rpubs.com/adam_dennett/376877

Which is the practical part of this paper:

https://www.australianpopulationstudies.org/index.php/aps/article/view/38

The model is re-specified as a Poisson regression model.

Here's the output of the model in the guide.

enter image description here

I suppose that those coefficients can't be interpreted as in a linear regression, because the whole equation ins exponentiated.

Thanks in advance, any feedback will be highly appreciated.

guillermo_dangelo
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1 Answers1

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It can be interpreted the same way as linear regression except for the log (conditional) average number of counts (or what ever your outcome is). Alternatively, you can interpret the exponential of the coefficient as the multiplicate increase in the expected number of counts if you increase the covariate associated with the coefficient by a single unit. This is true because the Poisson model is of the form: $$E[Y|X_1, X_2, X_3] = \exp \left\{aX_1 + bX_2 + c X_3 \right\} $$ and thus: $$E[Y|X_1 + 1, X_2, X_3] = \exp \left\{a(X_1+1) + bX_2 + c X_3 \right\}$$ $$ = \exp \left\{a \right\} \exp \left\{ aX_1 + bX_2 + c X_3 \right\} $$ $$= \exp \left\{a \right\}E[Y|X_1, X_2, X_3].$$ So, $$\exp \left\{a \right\} = \frac{E[Y|X_1 + 1, X_2, X_3]}{E[Y|X_1 , X_2, X_3]}$$

You can also show $$\frac{d}{dx} \frac{E[Y|X_1 + x, X_2, X_3]}{E[Y|X_1 , X_2, X_3]} \big |_{x=0} = \exp \left\{a \right\},$$ or $$ \frac{E[Y|X_1 + dx, X_2, X_3]}{E[Y|X_1 , X_2, X_3]} \approx \exp \left\{a \right\} dx,$$ which can be interpreted as the incremental multiplicative change in the expected count.

Lars
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  • Thanks! But what about the coefficients of the origins ("Orig_code..." variables)? They are treated as dummy variables in the Poisson model, so, what does the coefficient mean? – guillermo_dangelo Sep 28 '21 at 18:05
  • @guillermo_dangelo Are only one of the Orig_code variables active at a time? By that I mean can only one of the Orig_code variables be 1 for each observation (with the remaining Orig_code variables being 0 for the observation) – Lars Sep 28 '21 at 22:41
  • Yes, that's the way it works if correctly understood the paper by A. Dennet. – guillermo_dangelo Oct 01 '21 at 18:49