# Multinomial logistic predictor matrix

I'm reading Tuerlinkx & Wang, "Models for Polytomous Data". They write: "Apart from the intercept, the predictors that are included in the predictor matrix X can be classified into seven groups: item predictors, person predictors, logit predictors, and the four combinations of them. ... A logit predictor has the same value in all rows of the predictor matrix referring to the same component of the vector-valued link function. Interactions between a logit predictor and any of the other predictors are also possible."

I don't understand what a logit predictor is. If it has the same value in all rows, then it sounds like it's just the intercept term. But from the first sentence it sounds like there's also another intercept term? And how can you have an interaction between an intercept and another predictor?

It'd be helpful to see a model matrix, but table 3.1 doesn't answer my questions.

In the section you've referenced, they're speaking of the class of mixed multinomial models. I don't think the authors' nomenclature is either intuitive or conventional.

What they're calling "person predictors" are random effects which are implicit model terms that handle unmeasured sources of variation between clusters, such as repeated measures within individuals, sites, or across time. I am not sure this is correct since these values are not explicitly part of the design matrix which we refer to as $\mathbb{X}$, however if the number of clusters are small we can adjust for them using fixed effects which may have been the authors' point. That is not a mixed model then.

I'm not sure, but the "logit predictors" are the fixed effects in the model, such as age, gender, or intervention status. The "item predictors" is a redundancy referring to breaking out a polytomous category like "race" into two or more columns of Y/N variables for specific race levels.

I would check out the text Longitudinal Data Analysis by Zeger, Heagerty, Liang, and Diggle for a good alternative slant on the topic.

• (+1) for neither intuitive nor conventional. This is almost certainly domain jargon. – tchakravarty Dec 9 '12 at 16:48
• (+1) Your comment is almost exactly what I was about to write! – Peter Flom Dec 9 '12 at 16:50
• In 2nd par of sec. 3.3, they say that $X$ is the fixed predictor matrix that can contain person predictors; random effects go in the matrix $Z$. And a "logit predictor" cannot be just a covariate such as age or gender, because it's value is the same for all rows of the predictor matrix. – Jack Tanner Dec 9 '12 at 17:28
• The specific language was "A logit predictor has the same value in all rows of the predictor matrix referring to the same component of the vector-valued link function " which I took to mean: if two individuals are both of the same age category (say 30-40), they'll have the same "logit predictor". Anyway, it's totally confusing language. The logit is a link function, and logit predictor sounds like the conventional "eta" (linear predictors) from the McCullogh and Nelder book. It's frustrating to read. – AdamO Dec 10 '12 at 0:07

The "vector-valued link function", see eqs. 3.3 and 3.5, is the McCullagh & Nelder $\eta$.