I would like to model a depression scale score, which is rather heavily right skewed, from a larger number of predictor variables (~15), including one three-way interaction (and the corresponding subordinate two-way interaction terms).
Because of the outcome distribution, I do not think a standard multiple regression is appropriate. Transforming the variable is also not a sensible option since there is only a limited number of outcome values (between 0 and 15). I tried median regression, but it seems like my sample size (about 700) is not large enough to give stable estimates with the three-way interaction included (I get rather strange results, whereas it seems to work without this three-way interaction).
My question then is, is there any modelling technique which you could recommend in this situation? I am currently thinking about robust regression techniques (e.g. Huber M-estimation), but maybe there are other, better options?
Try to find a conditional probability mass function for your outcome and model the relationships, then you can estimate your model using MCMC approach.
I think the interpretation of the scores as interval-scaled data is problematic, and I especially doubt the interpretability of interactions! This seems to make no sense and can severely misguide you!
Use the data as what it is: an ordinally scaled variable. The appropriate analysis technique is an ordered logistic regression. Here (I think) you can well include interactions.
Thanks for your feedback. MCMC is a technique which I am totally unfamiliar with and therefore would kind of hesitate to use it. I'm familiar with the "ordinal-interval-scale" debate concerning psychometric scales and tend to take the position that you can consider psychometric scales as measuring a latent continuous trait or state. But given that all items are binary, maybe a binomial-logistic regression could also be a possibility... for proportional odds / ordinal logistic regression, I would be kind of worried about the number of parameters you need to introduce, as there would be one extra parameter for each level of the dependent variable if I'm not mistaken?
Yes, I just saw it: 16 different levels in the response (am I correct?) is in fact a bit much. The results will be hart to interpret.
What way to go is reasonable will depend on the definition of the scale and on the (assumed) distribution of the values. Not sure if you can re-cose the scores in ]0;1[ and do a beta-regression or a simple logistic regression.
You need to provide more details about what your variables are.
Say, you have a model Y=f(X1,X2,X3)
What is the domain of Y, X1, X3, X3? What values can those variables take?
You could try bootstrapping a linear regression to see whar comes out of it. It's not a perfect technique, but it's highly interpretable and easy to understand for most reader. With your large sample seize you should also get stable coefficient.
Dear Hans,
Bottai's work, "Logistic quantile regression for bounded response" would be a good approach for your problem, because when you use a transformation (some link function) for mean regression, the problem is that the mean of the transformation is not the transformation of the mean.
For quantile regression this is not true (and that is good) because quantiles are invariant to transformation (continuous monotone functions), so the quantile from the transformation is the transformation of the quantile, letting you to computed easily the predicted values for your original response. The interpretation are similar than for logistic regression for binary response.
I have implemented this model in R, available in the "lqr" package, specifically in the function "Log.best.lqr" and "Log.lqr" too. It fits a parametric quantile regression using a family of sweked distribution, including skewed versions of normal, t, laplace, slash and contaminated normal distribution.
Please, see the attach. There's a draft I wrote in portuguese :(.
Hope it would be helpful.
Best, Chris.
https://cran.r-project.org/web/packages/lqr/lqr.pdf
Article Logistic Quantile Regression for Bounded Outcomes
Thanks Christian, this appears to be a very interesting approach to me, I'll definitely take a look at your papers!
Dear Hans,
Perhaps, an interesting alternative is to assume that you are dealing with a bounded response variable in your case (0-15) and rescale it to the unit interval by using, for example, (x - 0)/(15-0), where x is your observation. After this transformation, you get a bounded response variable that can be modelled by using the beta distribution. An additional problem for your case is that you will have values at the border, i.e exact 0's or exact 1's. Thus, you can use the quasibinomial model as a approximation. You can do that using the glm function with the argument family = quasibiomial. An advantage of the quasi-likelihood approach used to fit the model is that the standard error for the regression parameters are in general robust to model misspecification. If you want an even more robust approach you can construct confidence interval for the regression parameters obtained by the quasibinomial model using non-parametric bootstrap.
If the measure of depression is the outcome, then modeling as a count variable using Poisson or negative binomial regressions could be a solution. The difference between both is that negative binomial works bets when there is overdispersion.
Another choice could be modelling in structural equations modeling usin WLS-family estimators, however it depends much on the sample size.
you may use a new approach
Article TO DETERMINE SKEWNESS, MEAN AND DEVIATION WITH A NEW APPROAC...
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