# Finding an optimal distribution to fit to a highly skewed data vector (DV with missing values) in R

I am analyzing data in R (and am a novice).

I have a highly skewed data vector in a dataframe with missing values that I hope to set as the dependent variable in a regression. The values in this vector are skewed because it has a polarizing response tendency (its range is from 1 to 7, but mostly receives very low response values). To proceed, I intend to fit an appropriate distribution to this data and then account for missing data (perhaps via MI or FIML).

Here is the data:

And here is what a Cullen and Frey graph suggests (via the descdist function):

For this data, the Tukey Ladder of Powers does not produce a normal distribution. I was considering fitting a negative binomial or Poisson distribution to the data (which barely gives normally-distributed data), but I think this violates the assumptions of these distributions because my data is not count data. Although a beta distribution is suggested by the graph, I think that my data does not fit the assumptions of this distribution either (its range extends beyond 0 and 1). I was considering that a correctly attuned (correctly-set α and β values) inverse gamma distribution might be the way to go (as can be observed in this website: https://distribution-explorer.github.io/continuous/inverse_gamma.html).

Of course, these considerations are speculative—I don't know how to correctly proceed. Any suggestions would be greatly appreciated.

• "Intend to fit" is a procedure for tackling a problem. Please tell us what that problem might be. Often, this kind of distribution fitting is a poor substitute for addressing the problem directly.
– whuber
Apr 19, 2022 at 16:55
• I think that I will need to impute in instances of missing data, which will require knowledge of a distribution that will fit the data. Next, I will run a moderated regression to test my hypothesis, wherein the data vector presented in this question will be the dependent variable. I don't know if this answers your question about specifics. Apr 19, 2022 at 23:53
• It suggests considering a different approach, because the univariate distribution of the dependent variable is scarcely relevant: you want to model the conditional distribution on the explanatory variables.
– whuber
Apr 20, 2022 at 1:51
• Ah, OK. Thanks. Sorry for the weak stats knowledge. I'm in a program where they do not teach details (I think because so many of the graduate students complain about how hard stats is—it's a psychology program.) So, to be clear, are you suggesting that I should model each IV's predictive value of this DV, which could then be used to fill in missing data either via MI or FIML (or something of the like)? If so, would you have a link or term suggestion to look up so that I could learn more specifics about predicting missing data in regression models? Apr 20, 2022 at 12:30
• "Multiple imputation" is the term you're looking for. (Maybe that's what you mean by "MI.") "MICE" is a particular form of it. The many low (zero?) values suggest possibly looking into "zero-inflated" forms of regression. Regardless, you should start with a thorough exploratory data analysis (EDA) to study how your response appears to be related to the potential explanatory variables.
– whuber
Apr 20, 2022 at 12:57