Please help to understand Agresti's book

Question

I need some help to understand some topics in Agresti's Categorical Data Analysis. In section 6.3.1 (p 231), he provided a model like:
$$ \text{logit}(\pi_{ik})=\alpha+\beta x_i+\beta_k^Z $$ where $i$=1,2, $k$ = 1, $\ldots$, $K$

Basically we have 3 categorical variables $X$, $Y$, & $Z$ ($Z$ is a confounding variable), and we are trying to test the conditional independence of $X$ & $Y$ for each level of $Z$.

He says that "this model assumes that [the] $XY$ conditional odds ratio [is the] same at each category of $Z$ namely exp($\beta$)"

I cannot understand this statement. How is the odds ratio exp($\beta$)? What about $\beta_k^Z$? isn't this also varying for each level of $Z$? Then the odds ratio must be different $Z$-wise?

What I am missing here?

Thanks for your time

Answer 1

The key to understanding lies in the word "conditional" in the phrase "conditional odds ratio". Basically, the statement means that the model assumes that $\beta$ doesn't vary across different levels of $Z$.

If conditional independence of $Y$ and $X$ given $Z$ holds, then $\beta = 0$, so testing for $\beta = 0$ tests one of the implications of conditional independence. However, one could construct a situation where $\beta$ varies with $Z$ in such a way that, on average, $\beta = 0$, but for any given $Z$, $\beta \ne 0$. In that case, the assumption of conditional independence would be false, yet the test would have a hard time rejecting it, because of the model misspecification due to assuming that $\beta$ doesn't vary with $Z$. Hence the importance of pointing out the assumption.