# Latent Variables

Suppose $Y$ is an ordinal variable such that $Y = 1,2,3,4$ corresponds to levels of impairment. So $Y=1$ is the last impaired and $Y = 4$ is the most impaired. What is the purpose of latent variables? That is what is the purpose of the following $$\alpha_{j-1} \leq Z \leq \alpha_{j} \Longleftrightarrow Y = j$$

Does this juts mean that if we added $1$ to the $Y$ so that $Y = 2,3,4,5$, then $Y=2$ corresponds to the least impaired and $Y = 5$ corresponds to the most impaired?

The idea is that the levels $Y = 1,2,3,4$ of impairment are really just an ordinal approximation to some true (but unmeasured, and hence latent) continuous measure of impairment, $Z$. We assume that if $Z$ is in the interval $[\alpha_{j-1}, \alpha_j]$ then you will observe $Y = j$. This can allow your model to account for the fact that someone whose impairment is rated as $Y_i = 2$, but is "barely two" (i.e., $Z_i$ is close to the $\alpha_{1}$ cutoff) might be quite different from another person whose impairment is also rated as $Y_j = 2$, but is impaired at "almost a three" level ($Z_j$ is close to $\alpha_2$).
The purpose is to be able to use the underlying $Z$ in more complicated analyses, such as ordinal regression modeling (an approximation of regression of $Z$ on covariates) or polychoric correlations (correlation between two underlying latent $Z$-like variables).