I was thinking a lot about proportional-odds cumulative logit models last fall while designing a study to evaluate an intervention’s effect on meat consumption. After a fairly extensive pilot study, we had determined that participants can have quite a difficult time recalling precise quantities of meat consumption, so we were forced to move to a categorical response. (This was somewhat unfortunate, because we would not have continuous or even count outcomes, and as a result, might not be able to pick up small changes in behavior.) We opted for a question that was based on 30-day meat consumption: none, 1-3 times per month, 1 time per week, etc. - six groups in total. The question was how best to evaluate effectiveness of the intervention?
Since the outcome was categorical and ordinal - that is category 1 implied less meat consumption that category 2, category 2 implied less consumption that category 3, and so on - a model that estimates the cumulative probability of ordinal outcomes seemed like a possible way to proceed. Cumulative logit models estimate a number of parameters that represent the cumulative log-odds of an outcome; the parameters are the log-odds of categories 2 through 6 versus category 1, categories 3 through 6 versus 1 &amp; 2, etc. Maybe not the most intuitive way to interpret the data, but seems to plausibly fit the data generating process.
I was concerned about the proportionality assumption of the cumulative logit model, particularly when we started to consider adjusting for baseline characteristics (more on that in the next post). I looked more closely at the data generating assumptions of the cumulative logit model, which are quite frequently framed in the context of a continuous latent measure that follows a logistic distribution. I thought I’d describe that data generating process here to give an alternative view of discrete data models.
I know I have been describing a context that includes an outcome with multiple categories, but in this post I will focus on regular logistic regression with a binary outcome. This will hopefully allow me to establish the idea of a latent threshold. I think it will be useful to explain this simpler case first before moving on to the more involved case of an ordinal response variable, which I plan to tackle in the near future.
A latent continuous process underlies the observed binary process
For an event with a binary outcome (true or false, A or B, 0 or 1), the observed outcome may, at least in some cases, be conceived as the manifestation of an unseen, latent continuous outcome. In this conception, the observed (binary) outcome merely reflects whether or not the unseen continuous outcome has exceeded a specified threshold. Think of this threshold as a tipping point, above which the observable characteristic takes on one value (say false), below which it takes on a second value (say true).
The logistic distribution
Logistic regression models are used to estimate relationships of individual characteristics with categorical outcomes. The name of this regression model arises from the logistic distribution, which is a symmetrical continuous distribution. In a latent (or hidden) variable framework, the underlying, unobserved continuous measure is drawn from this logistic distribution. More specifically, the standard logistic distribution is typically assumed, with a location parameter of 0, and a scale parameter of 1. (The mean of this distribution is 0 and variance is approximately 3.29.)
Here is a plot of a logistic pdf, shown in relation to a standard normal pdf (with mean 0 and variance 1):
library(ggplot2)
library(data.table)
my_theme