In my previous post, I described a continuous data generating process that can be used to generate discrete, categorical outcomes. In that post, I focused largely on binary outcomes and simple logistic regression just because things are always easier to follow when there are fewer moving parts. Here, I am going to focus on a situation where we have multiple outcomes, but with a slight twist - these groups of interest can be interpreted in an ordered way. This conceptual latent process can provide another perspective on the models that are typically applied to analyze these types of outcomes.
Categorical outcomes, generally
Certainly, group membership is not necessarily intrinsically ordered. In a general categorical or multinomial outcome, a group does not necessarily have any quantitative relationship vis a vis the other groups. For example, if we were interested in primary type of meat consumption, individuals might be grouped into those favoring (1) chicken, (2) beef, (3) pork, or (4) no meat. We might be interested in estimating the different distributions across the four groups for males and females. However, since there is no natural ranking or ordering of these meat groups (though maybe I am just not creative enough), we are limited to comparing the odds of being in one group relative to another for two exposure groups A and B, such as
\[\small{\frac{P(Beef|Group = A)}{P(Chicken|Group = A)} \ vs. \frac{P(Beef|Group = B)}{P(Chicken|Group = B)}}\].
Ordinal outcomes
Order becomes relevant when the categories take on meanings related strength of opinion or agreement (as in a Likert-type response) or frequency. In the motivating example I described in the initial post, the response of interest was the frequency meat consumption in a month, so the response categories could be (1) none, (2) 1-3 times per month, (3) once per week, (4) 2-6 times per week, (5) 1 or more times per day. Individuals in group 2 consume meat more frequently than group 1, individuals in group 3 consume meat more frequently than those both groups 1 &amp; 2, and so on. There is a natural quantitative relationship between the groups.
Once we have thrown ordering into the mix, we can expand our possible interpretations of the data. In particular it is quite common to summarize the data by looking at cumulative probabilities, odds, or log-odds. Comparisons of different exposures or individual characteristics typically look at how these cumulative measures vary across the different exposures or characteristics. So, if we were interested in cumulative odds, we would compare \[\small{\frac{P(Response = 1|Group = A)}{P(Response __ 1|Group = A)} \ \ vs. \ \frac{P(Response = 1|Group = B)}{P(Response __ 1|Group = B)}},\]
\[\small{\frac{P(Response \leq 2|Group = A)}{P(Response __ 2|Group = A)} \ \ vs. \ \frac{P(Response \leq 2|Group = B)}{P(Response __ 2|Group = B)}},\]
and continue until the last (in this case, fourth) comparison
\[\small{\frac{P(Response \leq 4|Group = A)}{P(Response = 5|Group = A)} \ \ vs. \ \frac{P(Response \leq 4|Group = B)}{P(Response = 5|Group = B)}}.\]
Multiple responses, multiple thresholds
The latent process that was described for the binary outcome is extended to the multinomial outcome by the addition of more thresholds. These thresholds define the portions of the density that define the probability of each possible response. If there are \(k\) possible responses (in the meat example, we have 5), then there will be \(k-1\) thresholds. The area under the logistic density curve of each of the regions defined by those thresholds (there will be \(k\) distinct regions) represents the probability of each possible response tied to that region. In the example here, we define five regions of a logistic density by setting the four thresholds. We can say that this underlying continuous distribution represents the probability distribution of categorical responses for a specific population, which we are calling Group A.
# preliminary libraries and plotting defaults
library(ggplot2)
library(data.table)
my_theme