What is the relationship between quadraric and categorical logistic regression models? Consider two logistic regression models Y on x, one where x appears in the model as a categorical variable, and one where x appears in the predictor as both a linear term and as a quadratic term. 
What is the relationship of the log-likelihood for the quadratic model to the log-likelihood of the categorical model?
Assuming they are the same; see outout below
In short, what is the relationship between the quadratic and categorical model? How could you prove your assertion about this relationship?
EDIT: 
For reference, I have run both models in Stata and get the following outputs:
Model estimates for a model with respiratory category as a categorical variable are:

Model estimates for a model with respiratory category as a linear term, and a quadratic term are:

Using logistic explore the association between lung reactivity and risk of chronic respiratory disease.
The dataset contains information on a combined measure of lung function exposure respcat taking values 0, 1 or 2 representing the number of lung deficiencies: 0 = none, 1 = either airway closure (AC) or bronchial hyper- responsiveness (BHR) but not both and 2 = both AC and BHR. The dataset also contains a binary outcome variable copd indicating whether or not the individual has chronic obstructive pulmonary disease (COPD).
Stata code:
    logistic copd i.respcat
    gen respcat2 = respcat*respcat
    logistic copd respcat respcat2  

 A: You are in a setting where you have a predictor x, an outcome y and are trying to determine how - if at all - x affects y on the log odds scale.  
If you could argue that x is an ordinal predictor, so that there is a natural ordering among its categories, then the following considerations hold. 
Your first model, which treats x as categorical and dummy coded, essentially says: The effect of x on y on the log odds scale could look like anything (e.g., linear, nonlinear) - I am just going to recover it from the data and see what it actually looks like.
Your second model makes a very specific assumption about the nature of the effect of x on y on the log odds scale: The effect could be quadratic in nature. 
Your "predicament" is that the two models give you identical log-likelihoods so you don't know which one to choose as being "best" for your data if you were to compare them on the basis of AIC or BIC. 
The first model is referred in the literature as the "unconstrained" model (since it does not impose a specific shape on the effect of x on y on the log odds scale). The second model referred in the literature as the "constrained" model (since it imposes a specific shape on the effect of x on y on the log odds scale). 
First, I would suggest that you need to look at the model summary for your first and second model directly on the log odds scale, rather than the odds ratio scale, to better understand what is going on. 
Second, if the "constrained" model were better for your data, you would have expected a much lower AIC/BIC for it compared to the "unconstrained" model (e.g., at least 2-3 points lower). The fact that this is not the case suggests that the "constrained" model does not offer an improvement over the "unconstrained" one for these data. So you can stick with the "unconstrained" model. 
See Richard Williams's document on Ordinal Independent Variables for more details: https://www3.nd.edu/~rwilliam/stats3/OrdinalIndependent.pdf. 
