Questions on PROC BGLIMM procedure

[xliu1]

Hello, 

I am seeking experts' thoughts on this as I am new to this SAS procedure.

I am using proc bglimm procedure of Bayesian Hierarchical analysis to predict 6-year graduation count for each college. When the future denominator COHORT_6Y is known, for example, the numbers of cohort 2020 to cohort 2024 are already known, should  I use the actual cohort (i.e., COHORT_6YR) or should I estimate the cohort and use the estimates in the model? Does that make any difference for model results? For the covariates, FEMALE_PERC SCH100_PERC GPA2_PERC SENIOR_PERC, I chose to use the observed value of cohort 2019, which is the last historical cohort. 

%let seed=12345; 
%let nmc=50000; 
%let thin=10; 
%let nbi=10000; 

proc bglimm data=caa.combined_ftic_6yr_v1 
            seed=&seed nmc=&nmc thin=&thin nbi=&nbi
	        outpost=Post_Pred plots=all diagnostics=all dic;
class COLLEGE;
model GRAD_6YR / COHORT_6YR = time_c FEMALE_PERC SCH100_PERC GPA2_PERC SENIOR_PERC //* noint*/ dist=binomial link=logit;
random intercept time_c / subject=College type=un;
title "BGLIMM: Generate Forecasts for Future Cohorts";
run;
title;

[SAS_Rob]

In short, yes, it will make a difference if you use the observed vs estimated values for the number of trials.

Given the scenario you described you should use the known values rather than trying to estimate them.  Treating the denominator as fixed reduces the uncertainty and allows you to focus on estimating only the posterior of the probability for the binomial rather than having to also deal with the posterior of the denominator as well.

[xliu1]

Thanks for your prompt response. I want to follow up with an observation from my model analysis. The actual denominator COHORT_6YR has large standard deviation/swing year to year. However, the historical graduation rate is relatively stable. What we want to predict is the count on the numerator GRAD_6YR for the future cohort. In this situation, should I still use actual denominator or estimate it? What would be the reasonable approach for the model setup? 

[SAS_Rob]

If you want to model the count (numerator) then it would be appropriate to fit a Poisson model instead, using the log of the denominator as the offset. The approach would be similar to this example:
https://documentation.sas.com/doc/en/statug/v_035/statug_bglimm_examples03.htm

[xliu1]

Thank you. The example provided in the link is different from our case.  Graduation is a bounded success-out-of-cohort outcome, not an unrestricted event count. This is specified in the model statement. We are predicting GRAD_6YR out of COHORT_6YR. The numerator is bounded by the denominator. GRAD_6YR is a count of “successes” out of a known cohort size, COHORT_6YR, which is a binomial outcome, and will never exceeds one.  COHORT_6YR is a known denominator. For some colleges, it drastically change year to year.

[SAS_Rob]

Not sure that it matters whether the denominator bounds the numerator or not for the poisson model.  The key is in deciding whether you want to model it as a count (with an offset for the varying sample sizes) or as a probability and then fit the model accordingly.