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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 de ciencies: 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).

What is the relationship of the log-likelihood for the quadratic model to the log likelihood of the categorical model?

I note these are the same value (-155.4) but am unsure how to explain this

What is the relationship between a quadratic model and categorical model? How could you prove the assertion about this relationship?

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:

enter image description here

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

enter image description here I am assuming Stata will not allow me to comapre the quadratic and categorical models using the likelihood ratio test procedure, as for the likelihood ratio statistic to be valid, the simpler model must be a special case of the first. The likelihood ratio test would only work if covariates in the quadratic model were the same as in the categorical model ?

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Consider that you have three groups (0,1,2), and with both a quadratic model and a "categorical" model, you have 3 d.f. -- either way fitting the average outcome in each group exactly.

They're different parameterizations of the same fit. For ordered models there can sometimes be advantages in parameterizing it using orthogonal polynomials (in particular, if the levels progress smoothly you might find that a lower order fit is a good description).

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