topic Re: Parameterization of Gamma distribution used in PROC FMM in Statistical Procedures

Parameterization of Gamma distribution used in PROC FMM

BjoernHolzhauer — Tue, 11 Aug 2015 16:18:58 GMT

I am trying to figure out the parameterization of Gamma distribution used in PROC FMM. When I think of the typical Gamma(shape=alpha , rate=beta) parameterization, I at first assumed alpha = (FMM's scale) and beta = (FMM's scale) / (FMM's intercept) based on the likelihood shown in the documentation. But that does not seem true and the documentation says that by default a log-link function is used. However, it is not clear at all from the documentation whether this means I need to use exp(FMM's intercept) and perhaps some other transformation on FMM's scale?

Below is an example of what I might be doing:

data test;

do i=1 to 1000000;

y = 10*rangam(123,10);

output;

end;

run;

proc fmm;

model y = / dist=gamma k=1;

run;

This gave me:

Intercept	4.6050	0.000316	14557.2	<.0001	99.9813
Scale Parameter	9.9931	0.01390

If I now wanted to use the PDF function to simply re-plot the density, what are the shape and scale???

data replot;

do x=0 to 100 by 1;

density = PDF("GAMMA",x,shape,scale);

output;

end;

run;

proc sgplot data=replot;

series x=x y=density;

run;

I would be grateful, if anyone could tell me exactly how PROC FMM really parameterizes the gamma distribution.

Many thanks,

Björn

Re: Parameterization of Gamma distribution used in PROC FMM

Rick_SAS — Tue, 11 Aug 2015 19:09:33 GMT

The parameterization is in terms of the mean and dispersion, not the standard shape and scale parameters for the gamma distribution.

This is the same as GENMOD does, and the doc says

"Probability distributions of the response Y in generalized linear models are usually parameterized in terms of the mean and dispersion parameter instead of the natural parameter "

But that's okay, because you can convert between the two parameters. If alpha, beta are the shape and scale parameters, respectively, for the gamma distribution, then

intercept = mean = alpha*beta

dispersion = alpha;

There is some interesting geometry in the conversion. If you like math, you might enjoy reading an article I wrote about converting between mean/std and the mu/sigma parameters for the lognormal distribution.

The other important aspect of this problem is that the DIST= option activates the LINK= option. In your code, you were implicitly using LINK=LOG, which is the default link function for the gamma distribution. The following code should show you how to convert to the standard gamma parameters:

%let shape = 11; /* gamma shape */
%let scale = 4; /* gamma scale */

data Sim;
call streaminit(12345);
do i = 1 to 10000;
y = &scale * rand("Gamma", &shape);
output;
end;
run;

ods graphics off;
proc fmm data=Sim ITDETAILS;
model y = / dist=gamma k=1 link=identity; /* note LINK= ! */
run;

/* convert intercept/dispersion estimates to shape/scale estimates */
data Params;
fmm_int = 44.1356; /* COPY/PASTE from the PROC FMM output */
fmm_scale = 10.9620;
call symputx("gam_scale", fmm_int / fmm_scale);
call symputx("gam_shape", fmm_scale);
run;
%put gam_scale= &gam_scale;
%put gam_shape= &gam_shape;

If you read the article "How to overlay a custom density curve on a histogram in SAS" you can overlay the estimated gamma density on the simulated data.

Re: Parameterization of Gamma distribution used in PROC FMM

BjoernHolzhauer — Wed, 12 Aug 2015 06:52:16 GMT

Thanks! That gets me around having to understand exactly how the model is parameterized in case of the log-link function.

Re: Parameterization of Gamma distribution used in PROC FMM

acordes — Thu, 25 Oct 2018 22:34:29 GMT

I'm quite lost with the maths behind this conversion.
For the weibull estimates from proc fmm the mapping goes as well like this?
intercept = mean = alpha*beta

dispersion = alpha;

I'd like to understand it but I don't know where to start.

Re: Parameterization of Gamma distribution used in PROC FMM

Rick_SAS — Fri, 26 Oct 2018 13:22:06 GMT

See my response in the comments of this blog post.

Re: Parameterization of Gamma distribution used in PROC FMM

StatDave — Fri, 26 Oct 2018 14:59:25 GMT

As discussed in this note, the parameterization of the gamma distribution that you are used to is directly used by PROC NLMIXED, so this is much easier in that procedure as below:

proc nlmixed; 
model y~gamma(a,lambda);
run;
data replot;
     do x=0 to 250 by 1;
          density = PDF("GAMMA",x,9.9931,10.0051);
     output;
     end;
run;
proc sgplot data=replot;
  series x=x y=density;
run;

See also this note on the gamma in GENMOD and the functions like PDF.