# Appropriate way to treat [0,1]-distributed variables in HLM

Brief intro: I'm not really sure how to appropriately treat the dependent variables in a set of hierarchical linear models that I'm trying to run. In my models, Level 1 units are children and Level 2 units are schools (we don't have classroom info so there is no nesting level for this).

Background: There are five sets of models, one for each dependent variable. The DVs are all behavioral or cognitive outcome measures that are measured at Level 1. I'm using the HLM 7 software, but I could just as easily use Mplus or R. Each of the five DVs were assessed in the same way. Let's take depression as an example of one: this construct had about 6-8 items (presented to the children as scenarios), and for each item there were 2 response options that could be classified as "the depression options," and 2 response options that could be classified as "not the depression options." The 2 items in each set are treated as equivalent in terms of measurement -- that is, no ordinal ranking from least-to-most depressed. The final form of the depression outcome (as with the other 4 outcomes) is a proportion, which can be interpreted as the proportion of items/scenarios to which the child responded in a way indicative of being depressed. Personally, I hate that the assessment was done in this way, but there's nothing I can do about it now.

Problem: However, I may be able to get permission to score the assessments in a different way, provided I can get access to the item-level data. Right now I only have access to the final proportions. Obviously the default in HLM and in other software is to treat the outcomes as normally distributed. The only distribution appropriate for the [0,1] interval that I know of is the beta, and I do not know of any R packages to implement HLM for beta-distributed variables, nor do I think Mplus or HLM can handle this. I tried using the logit transformation to put the proportions back onto a scale that I could theoretically treat as normal, but with only 6-8 items per DV I still get something that is rather discrete. Not to mention that the 0 and 1 responses, even when converted first to .0001 and .9999 before logit transforming, give me these extreme values that are quite far away from the other transformed values. I've attached histograms to illustrate what I mean.

Question: What are some appropriate ways to treat an outcome variable on the [0,1] interval that takes on only a small handful of observed values when I want to run a set of hierarchical linear models? I'd REALLY like to use the HLM 7 software for this (the P.I. prefers it), but if the best way to treat these types of variables can only be implemented in R then I'm comfortable with that option as well.

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Neither normal nor beta ;-)

Let's say that there are 6 items (presented to the children as scenarios) and that each response may be 1 (may be depressed) or 0. Thus you have seven possible outcomes: 0 (0/6), 0.167 (1/6), 0.333 (2/6), 0.5 (3/6), 0.667 (4/6), 0.833 (5/6), and 1 (6/6).

You can think of these outcomoes as count data (how many ones?), thus you could try a Poisson regression to be able to predict a count, which becomes the final form of the depression outcome when divided by 6.

But there may be 6 to 8 items. No problem, if you know how many scenarios were presented to each child. For example, if your dataset is:

• id
• final score
• nitem
• etc.

you can define a new variable ones as final_score x nitem, then fit a Poisson model with exposure. A simple (not multilevel) model would be, in R:

fit <- glm(ones ~ <predictors>, data=dataset, family=poisson, offset=log(nitem))


Indeed, in most applications of Poisson regression the counts can be interpreted relative to some baseline. For example, the number of traffic accidents at street intersections must be interpreted relative to the number of vehicles that travel through each intersection (see Gelman & Hill, chap. 6).

At least, this is what I'd try. Other ideas are welcome.

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Very clever use of the Poisson. I'd add that you should always check whether $\mu =\sigma$ in your data; a lot of counts in practice tend to be overdispersed ($\mu>\sigma$) and would therefore be more appropriately modeled with the negative binomial. –  ssdecontrol Jul 4 at 14:33
Thank you for this idea! I will give this a shot within the next couple of days. And thanks @ssdecontrol for the note on overdispersion. –  psychometriko Jul 4 at 14:50
Well, my answer was just a hint. I hope that there are very few depressed children in your sample, and that a zero-inflated Poisson/negative-binomial would be much better ;-) –  Sergio Jul 4 at 15:11

If you end up with the [0,1] averaged distribution, you could use a zero-one-inflated beta model as well. Flexible, easy to implement.

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