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I have an outcome variable in my dataset that follows a Poisson distribution

Y   Frequency
0   52121
1   2831
2   34

0 - None, 1 - Moderate, 2- Severe : instances of infection.

my independent variables are: age, type of surgery, gender, location, BMI.

I am curious to know what would happen if I model this as an ordinal regression model 2>1>0 instead of Poisson model ? Any advise or suggestion is much appreciated.

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    $\begingroup$ What would your explanatory variables be? And on what basis do you suppose these data are Poisson? The frequency of $Y=2$ is less than half what it should be (which is significantly too low). Do you mean they might be conditionally Poisson based on controlling for other variables? $\endgroup$
    – whuber
    Sep 5 '19 at 20:31
  • $\begingroup$ @whuber good point, I have updated my question with a little bit of description about my y and independent variables. I am not sure how to test for conditional nature of Poisson distribution $\endgroup$ Sep 5 '19 at 21:27
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    $\begingroup$ You would conduct a Generalized Linear Model with a Poisson response and check the goodness of fit. $\endgroup$
    – whuber
    Sep 5 '19 at 21:41
  • $\begingroup$ What is your research question for these data? $\endgroup$ Sep 6 '19 at 14:31
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    $\begingroup$ @Isabella Based on the OP's reference to a Poisson distribution, I am interpreting $Y$ as a count. But see blog.stata.com/2011/08/22/…. $\endgroup$
    – whuber
    Sep 7 '19 at 13:41
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If the non-negative integers have true integer "meaning" (i.e. 3 represents a quantity three times what 1 represents), and particularly if they represent arrivals in a process, then Poisson regression is a good place to start. For example, if these are numbers of purchase orders from customers, and you want to predict the number of purchases a customer will make in a specific time frame from some other data you have on the customer, then Poisson regression could be a good fit.

Ordinal regression is for when you know that certain outcomes are "higher" or "lower" than others, but you're not sure by how much or if it even matters how much. For example, if the outcomes are ratings on a 1-5 scale, you know 4 is better than 2 but you can't necessarily say it's "twice as good" as 2. I don't know the scientific problem here (I always recommend explaining the scientific problem in the question) but I'd hazard a guess that ordinal regression isn't what you want here since there aren't many "0-1-2" type scales.

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According to the Frequentists' theory and MLE, the model and other following statistical tests only work correctly when it follows the real underlying data generating distribution.

If the data is sampled by Poisson distribution from your project context, Poison regression should be used rather then Ordinal regression.

Firstly, Poisson distribution is the aggregated Bernoulli event number during a long period. So the length of the time (so-called exposure) is an important factor for the total event number and Poisson regression is designed to re-scale such variable for each observation easily and correctly. While Ordinary regression considers the response as ordered factor, which is difficult to re-scale correctly through the link function.

Secondly, the $event Number = 2$ in Poisson regression means the exact observed number, while in the Ordinal regression means $event Number \geqslant 2$, which means the ordinal regression losing some information (such likelihood for 2 and very low likelihood for 3 or higher).

Generally speaking, Poisson regression uses all the information from $eventNumber=0, 1, 2, 3, 4 ...$ (in your case there is observed event number about 3, 4 ... are 0), all together to calibrate the underlying Bernoulli event with only one parameter $P(True|X)$. While the ordinal regression lose all these information. So the Poisson regression will be more robust with lower statistical error.

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