# Scale variable as count data - correct or not?

In this paper (freely available via PubMed central), the authors use negative binomial regression to model the score on a 10-item screening instrument scored 0-40. This procedure assumes count data, which clearly isn't the case here. I'd like your opinions on whether this approach is acceptable, because I sometimes use the same instrument or similar ones in my work. If not, I'd like to know if there are any acceptable alternatives. More details below:

The scale used is the Alcohol Use Disorders Identification Test (AUDIT), a 10-item questionnaire designed as a screening instrument for alcohol use disorder and hazardous/harmful drinking. The instrument is scored from 0 to 40, and the results are typically heavily left-skewed.

To my understanding, using count data assumes that all values that are "counted" are independent of each other - patients coming to an emergency ward each day, number of fatalities in a certain group, etc - they're all independent from each other, though dependent on underlying variables. Furthermore, I think there cannot be a maximum allowed count when using count data, though I think that this assumption can be relaxed when the theoretical maximum is very high when compared to the observed maximum in the data?

When using the AUDIT scale, we do not have a true count. We have 10 items with a maximum total score of 40, though that high scores are rarely seen in practice. The scores on the items are naturally correlated with each other.

The assumptions required to use count data are thus violated. But is this still an acceptable approach? How serious are the violations of the assumptions? Are there certain circumstances under which this approach can be considered more acceptable? Are there any alternatives to this approach that doesn't involve reducing the scale variable to categories?

The AUDIT instrument is essentially a Likert scale. A set of questions (Likert items), with answers often on a five-point scale, is designed to get at some underlying phenomenon. The sum of responses to the set of questions, the Likert scale, is then used as the measure of the underlying phenomenon. Although Likert items are often on a scale of "strongly disagree" to "strongly agree," the application to measure a tendency toward "Alcohol Use Disorders" in this "Identification Test" is straightforward.

As noted in the Likert scale Wikipedia page, "Whether individual Likert items can be considered as interval-level data, or whether they should be treated as ordered-categorical data is the subject of considerable disagreement in the literature, with strong convictions on what are the most applicable methods." This dispute probably dates back through most of the 80+ years since Likert first proposed the scale: is each step along the scale equivalent, both within and among the items that make up the scale? The issue has been addressed on Cross Validated, as in answers to this question, one of the earliest questions asked on this site.

If you accept the idea that the scale has steps that are uniform (or close enough to uniform for the application at hand, perhaps averaged out by adding 10 different items, as in AUDIT), then several approaches to analysis are possible. One is to consider the response on the scale as a series of steps chosen or not chosen to move up the scale, with the same probability of moving up each of the steps.

This allows one to think of "n-point Likert scale data as n trials from a binomial process," as in a 2010 question from @MikeLawrence. Although responses to that question were not terribly supportive of that idea, it was not hard to quickly find today a 2014 study that used and extended this approach successfully to distinguish sub-populations with different binomial probablilities. Although a binomial process is often used to model count data, it thus can be used to model the number, the count, of steps that an individual took along the scale of "Alcohol Use Disorders."

As @Scortchi noted in an answer to the question linked in the second paragraph, a limitation of the binomial model is that it imposes a particular relation between the mean and the variance of the response. The negative binomial removes that restriction, with loss of the easy interpretation provided by the simple binomial model. In the analysis, the extra parameter that needs to be fit uses up just one additional degree of freedom. In contrast, trying to specify different probabilities for each of the 40 Likert-item steps and their sum into the Likert scale would be daunting.

As @MatthewGraves noted in his answer to this question, whether the negative binomial model is appropriate is best answered by examining the residuals. In the original study that developed AUDIT, a value of 8 or more on the 40-point scale had quite reasonable specificity and sensitivity for distinguishing those diagnosed for "hazardous or harmful alcohol use," across 6 different countries. So perhaps a two-population binomial model based on high-risk and low-risk populations, similar to the 2014 study linked above, would be better.

Those interested in AUDIT specifically should examine that original study. For example, although the need for a morning drink might seem to measure something completely different from the frequency of drinking, as @SeanEaster surmised, morning drinking has a weighted mean correlation of 0.73 with a scale of measures of alcohol intake. (That result is not surprising to someone who has had friends with alcohol use disorders.) AUDIT seems to be a good example of the tradeoffs needed to develop an instrument that can be used reliably across multiple cultures.

• Thank you for a good answer. When looking at my own AUDIT data of more than 20000 individuals the shape looks close to a negative binomial distribution so it might be reasonable to use that distributional assumption, or perhaps a quasi-poisson model could be used? If we use a binomial distribution by consider the points as k successes out of 40 bernoulli trials, will we not have a serious problem with overdispersion? It looks that way in my data. Could quasi-binomial be an alternative?
– JonB
Oct 19, 2015 at 11:40
• Much depends on why you are modeling the 0-40 AUDIT scores and what heuristic interpretation you wish to place on the results. If all you want is a relation of AUDIT scores to other variables, with only limited interpretation of the distribution parameter values themselves, then use a distribution that provides well-behaved residuals; your suggestions are reasonable. Fitting a single binomial to the data is problematic, but a mixture of 2 binomials (high-risk and low-risk groups) with different p might be informative. Use your judgment based on your knowledge of the subject matter.
– EdM
Oct 19, 2015 at 12:31

The negative binomial distribution is preferred for "contagious" discrete events. A Poisson distribution is used when the discrete events are independent. These distributions are also fairly easy to truncate, by replacing the $x=40$ point with an $x\ge 40$ point, basically.

As a general comment, different flavors of regression have different priors for parameters (i.e. regularization) and different noise models. Standard least squares regression has a Gaussian noise model, negative binomial regression has a negative binomial noise model, and so on. The true test of whether or not a regression model is appropriate is whether or not the residual noise has the expected distribution.

So you can apply negative binomial regression to your data, calculate the residuals, and then plot them on a negative binomial probability plot, and get a sense of whether or not the model is appropriate. If the noise is structured in some other way, then we need to look for a noise model that fits that structure more closely.

Reasoning from the generative model to the noise structure is helpful--if we know the data is multiplicative instead of additive, for example, we reach for the lognormal instead of the normal--but if the expected generative model and the noise structure disagree, go with the data, not the expectation.

• Interesting, I didn't know that the events could be "contagious". What do you mean by replacing x=40 with x>=40, in practice? How do I do a negative binomial probability plot in R? I suppose you don't mean plot residuals against fitted values? Do you mean like a Q-Q plot?
– JonB
Oct 15, 2015 at 19:09
• @JonB Suppose you have a negative binomial with r=1 and success probability p=.9. The probability of surviving 40 trials exactly is 0.148%; the probability of surviving 40 or more trials is 1.48%. So one can define a well-formed probability on the domain [0,40] by using the negative binomial for [0,39] and then setting  so that it sums to one, which because the negative binomial distribution is well-formed is the probability that it's 40 or more. Oct 15, 2015 at 19:39
• @JonB Exactly, I mean like a Q-Q plot. I haven't done it in R before, but I hope this link will help. Oct 15, 2015 at 19:42
• I did an experiment on some data with AUDIT scores. When creating a q-q plot, I need to create a random vector of results from a negative binomial distribution. The mu/theta is given by my regression model, but how can I know what "size" to use? I'm sorry if this is an R-specific question.. Anyway, do you have a nice reference I which I can read more about applying negative binomial (and other distributions) to these kinds of scales constructed by summing several items that measure kind of the same process?
– JonB
Oct 15, 2015 at 20:20
• I did some additional experiments now. I simulated a dataset with two variables: x and y. 50% are x=0, 50% are x=1. Those who are x=0 have a 0.2 probability for y=1, and those who are x=1 have a 0.4 probability for y=1. I then ran a logistic regression and had a look at the residuals. The don't look binomially distributed at all. In fact, they (of course) take on 4 specific values. Are you certain that the residual pattern should always match the distributional assumption? Because in this instance, it's clearly wrong.
– JonB
Oct 16, 2015 at 6:22