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Parameterization of Gamma distribution used in PROC FMM

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Contributor
Posts: 25

Parameterization of Gamma distribution used in PROC FMM

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:

Intercept4.60500.00031614557.2<.000199.9813
Scale Parameter9.99310.01390

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

SAS Super FREQ
Posts: 3,546

Re: Parameterization of Gamma distribution used in PROC FMM

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 $\mu $ and dispersion parameter $\phi $ instead of the natural parameter $\theta $"

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 d...

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.

gammafit.png

Contributor
Posts: 25

Re: Parameterization of Gamma distribution used in PROC FMM

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

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