# How to analyze crossover trial with a scale as outcome variable?

I'm working on a pilot study for a new drug. We will use a crossover design in which half of the patients will be given the drug for the first 6 weeks, then, after a washout period of one week, they will get placebo for 6 weeks. The other half of the patients will get placebo for 7 weeks and then the drug for the remaining 6 weeks. The main outcome measure is the score on a certain instrument consisting of a number of items. This will be measured daily as this makes sense.

I'm thinking that I need to use a mixed model approach with random effects for each patient plus of course a variable on treatment and group (drug first or placebo first) and such. The problem I'm having is what kind of distribution to choose for the mixed model. I don't have the data yet, but I expect that most patients will score zero on instrument for the majority of days, and higher scores occasionally. A linear mixed model is thus not appropriate, and neither are binomial, poisson or negative binomial models since the last two concern counts. What about ordinal regression? Is there any way to use the proportion of points scored on the scale (30 points where 40 points is the maximum = 0.75)?

I use R for statistics.

• Some more information on the nature of the scale would help. Are there always 40 points on the scale each day, but in most cases the score is 0? If that's the case, how much do you care about differences among non-0 scores versus differences between 0-score and non-0-score cases? What type of distribution do you expect among the scores when they are non-zero?
– EdM
Aug 17, 2015 at 13:25
• These questions are difficult to answer, but I had a look at some other studies that use this particular scale to get an idea. I would expect that about 90% of scores are 0 and among the non-0-scores, one study reported a mean of about 9 and SD of about 5 (among non-zero scores), perhaps indicating a positively skewed normal distribution? Of course, using a binomial distribution and comparing any non-zero score to zero-scores (meaning an event has occurred vs no event occurred) would be the easiest way, but I do care about the score of each "event"...
– JonB
Aug 18, 2015 at 8:24

It's easy to get focused on the distribution of the outcome variable; I did in my comment. But that's not what's important.

What's important is the relation of the distribution of the outcome variable to the distribution of your independent variables. For example, there's no problem when 90% of cases have outcome scores of 0 if in the same 90% of cases the independent variables predict a 0 score. So in your study a linear model might be OK if the outcome score was closely related to some predictor variable with a typically 0 value (say, absence/presence of a fever on a particular day) and your drug altered the relation of that predictor to the score.

Thus much of the answer to this question depends on your understanding of the subject matter, the scoring system, and how the independent variables might be related to the score. What you seem to have in mind is a two-step model in which the independent variables first determine whether the score is non-0 and then determine the magnitude of the score if it is non-zero.

For count data this situation is handled by zero-inflated or hurdle models. You might consider whether your outcome score can be modeled adequately as a count variable even though you apparently think of this score as an integer-valued approximation to an underlying continuous variable. As you consider this, remember that you are modeling the relation of the score to your predictors; whether you really have count data might not matter if the relation is adequately modeled.

If you decide that you need to model the score as a continuous variable you can consider an approach where you first model the non-0 probability and then model the data for the non-0 cases, as in the example on this page for hurdle modeling of a continuously distributed variable. (It seems that there should be a way to include both analyses in a single mixture model rather than splitting, but I don't have any experience with that.) Another possibility would be to model the outcome as a continuous (not necessarily linear) function of your independent variables but with a threshold (included as a parameter to be determined from the data) below which the score is 0. My fear with the threshold approach is that the residuals in such a model might not be distributed very well.

• Thank you. I have heard about the zero-inflated and hurdle models before, but only when used with poisson or binomial distributions, if I recall correctly.
– JonB
Aug 18, 2015 at 16:08
• @JonasBerge If you want to consider your scale as an effectively continuous variable, I would suggest doing some more research (or asking your local competent statistician, or posing a more directed question on this site) about mixture models that would combine analyzing the probability of non-zero events together with the magnitude of the event (e.g., linear or other regression) in a single model. It seems that would be more efficient than doing 2 separate analyses as suggested in the paper that I linked, but this is getting beyond my expertise.
– EdM
Aug 18, 2015 at 20:38