Two-part model ordinal data

Question

A response scale has the categories (strongly agree, midly agree, midly disagree, strongly disagree, do not know). A two-part model uses a logistic regression model for the probability of a DK (don't know) response and a separate ordinal model per the ordered categories conditional on response in one of those categoeries.

I'd like to fit these two part simultaneously but I'm stucked.

I think of it as an Hurdle model for the Poisson situation, where the 0 process is different than the counting process and there's no mixture (so no Zero inflation).

I wonder if it exists a way to deal with situation in R.

Someone would told me to treat DK response either as missing values or neutral values (so put them in the central order). But I'd like to use a model like the Hurdle one for counting data. So a logit model (DK versus notDK) and a Proportional Odds Model per the ordinal part.

I've looked for some literature on the argument but I have found nothing in particular. Any suggestion? Does it make sense, anyway, as approach?

Answer 1

I am finding in my reading about hurdle models is that they may not be as simultaneous as they seem to me, despite being done in one R command. (See my question recently posted here on CV.)

I have dealt with this before, and this was my reasoning:

The only way that I would put "don't know" in the middle is that if the measurement tool actually positioned it in the middle of the scale. I don't agree with this measurement style, but I think that participants might interpret "don't know" to be "neutral" if the scale looks like, on the survey that the participant sees:

strongly disagree - mildly disagree - don't know - mildly agree - strongly agree

If this is the case, then I think it would be fair to just do one ordinal model, although I think the problem is with the measurement being confusing to the participant—not necessarily a statistical issue.

However, if the measurement tool placed "don't know" on the outside like I've seen others do, e.g.:

strongly disagree - mildly disagree - mildly agree - strongly agree || don't know

Then I think your two-step approach is totally valid. It is similar to and in the spirit of hurdle models (as far as I know about them—see the above link for evidence that I am still somewhat confused). The only thing I might change is:

1. Do a multinomial logistic regression, using "don't know" as the reference category and collapsing the two agree values together and the two disagree values together. That way you can predict the odds of people moving from don't know to agree and don't know to disagree. This can be done by recoding strongly/mildly agree into "agree" and recode strongly/mildly disagree into "disagree," then making "don't know" the first level in a factor variable that represents this three-category dependent variable. I've only fit these models in Stan, but it seems pretty straightforward using the multinom function from the nnet package, although I would suggest reading up on it to see if everything matches what you think it is doing.

2. Do an ordered logistic regression, excluding the cases that are "don't know" from the sample. This can be done by assigning your dependent variable as factor with levels ordered correctly, and then using the clm function from the ordinal package:

   model2 <- ordinal::clm(dv ~ iv, data = dat)

This two-step approach is similar to the hurdle model in that you are explicitly modeling two different kinds of processes: First, having no opinion vs. agreeing or disagreeing; Second, level of agreement.

Lastly, if you are interested in trying to predict whether or not people have an opinion—regardless of direction (disagree or agree)—then you should do a logistic regression predicting don't know versus any other response. It depends on what you are interested in at that point.

Answer 2

It is very difficult to know how to position or analyze the "don't know" answer. It is often best to not change the definition of the variable by creating a new category for it, but rather to use multiple imputation to multiply impute values after setting "don't know"s to missing. The multiple imputation model would be a richer model that would include variables you don't even want in the outcome model. This would preserve the meaning of the data, while properly penalizing you for not knowing some of the values.

Answer 3

Let $\pi_1, \cdots, \pi_J$ denote the probabilities of the ordinal categories [$\pi_1 = P(y=1) = P(y \le 1)$ and $\pi_2 = P(y=2) = P(y \le 2) - P (y \le 1)$] and $\pi_{DK}$ denote the probability of don't know. We would like to model $\pi_{DK}$ as a logistic regression model and $\pi_1, \cdots, \pi_J$ as a cumulative logistic model subject to the constrain that $\pi_1 + \cdots + \pi_J + \pi_{DK} = 1$.

Usually cumulative logistic models only differ in intercept parameters, but we can allow for a different slope matrix and intercept parameter for the don't know category. Specifically define \begin{eqnarray} \pi_{DK} &=& \frac{\exp\left(\alpha_0 + \beta_{\ast}^{\prime}\boldsymbol{x}\right)}{1+\exp\left(\alpha_0 + \beta_{\ast}^{\prime}\boldsymbol{x}\right)} \\ \pi_1 &=& \frac{\exp\left(\alpha_1 + \beta^{\prime}\boldsymbol{x}\right)}{1+\exp\left(\alpha_1 + \beta^{\prime}\boldsymbol{x}\right)} \\ \pi_j &=& \frac{\exp\left(\alpha_j + \beta^{\prime}\boldsymbol{x}\right)}{1+\exp\left(\alpha_j + \beta^{\prime}\boldsymbol{x}\right)} - \frac{\exp\left(\alpha_{j-1} + \beta^{\prime}\boldsymbol{x}\right)}{1+\exp\left(\alpha_{j-1} + \beta^{\prime}\boldsymbol{x}\right)} \quad \mbox{for} \quad j=2,...,J-1 \\ \pi_J &=& 1 - \pi_{DK} - \sum_{j=1}^{J-1}\pi_{j} \end{eqnarray} Next define a multinomial loglikelihood as usual, but with these probabilities.