PROC GLIMMIX: How to choose the distribution for response variable with limited range of value?

[Minhtrang]

Hi all,

My response variable is cognitive function (CASI) which is scored from 0 to 100. 

I want to model the relationship between CASI and some predictors.

One reccommendation is to transform CASI to CASI/100 and fit a logistic model.

So here CASI/100 ranges from 0 to 1. 

I plan to use PROC GLIMMIX in this case. But I'm not sure which distribution should I specify for this special response variable.

Is it beta distribution? I read somewhere that beta distribution doesn't accept value 0 or 1.

I would love to hear from your experience.

Thank you,

Trang

[Rick_SAS]

Yes, beta distribution. Unless responses are 0 or 100, you won't need to worry about whether the beta distribution "accepts values 0 or 1."

[Minhtrang]

Hi Rick,

If I choose beta distribution and I have some case that Casi/100 equals 0 or 1, although they are extremely rare, would SAS omit these cases from the analysis?
Thank you for your reply,
Trang

[Rick_SAS]

Yes, you are correct. The procedure will drop observations for which the response is not in (0,1) and will display the NOTE

NOTE: Some observations are not used in the analysis because of: not a
proportion, zero or negative response.

If you have 0 and 1 responses, perhaps beta is not the best model.

[Minhtrang]

Rick, so if I want to model this response variable in the most natural way, ie I can have some cases with Casi/100 equal to 0 or 1, what would be the most appropriate distribution?

[Ksharp]

I could suggest using GAMMA distribution for CASI variable.

and if you have many zero , try tweedie distribution.

OR

Try Poisson distribution  +  offset= option.  Make an offset variable which is 100 .

[Minhtrang]

Hi Ksharp,
Thank you for your suggestion.
However, CASI is skewed, so I'm not sure that gamma distribution works here.
maybe it's the reason that I need to transform CASI to CASI/100, which ranges from 0 to 1.
I'm thinking of beta distribution, but it doesn't allow 0 and 1.
I want to find the most appropriate distribution that can accept value of 0 and 1.

[sld · Accepted solution]

One approach is to gently rescale the data to lie strictly in (0, 1); see

https://www.ncbi.nlm.nih.gov/pubmed/16594767

A more elegant approach is the zero one inflated beta model; see

http://support.sas.com/resources/papers/proceedings12/325-2012.pdf

and an example

https://www.nefsc.noaa.gov/program_review/2015%20%20Review/BACKGROUND/B1A6SA%20Waring%20beta%20regre...

[Minhtrang]

Hi sld,

Thank you so much for your reference with one-zero inflated beta regression!

I know something about beta regression, but never thought of one- zero inflated one.

This is exactly I want.

The illustration of SAS used similar type of response variable (a score from 0 to 100) like mine.

They also divided it by 100 and gave detailed guidance of the Macro for this analysis.

Best,