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st: RE: RE: gamma family - alpha and theta parameters - xtgee
From
"Guillermo A. Sandoval (UofT)" <[email protected]>
To
"[email protected]" <[email protected]>
Subject
st: RE: RE: gamma family - alpha and theta parameters - xtgee
Date
Wed, 29 May 2013 15:07:09 +0000
Many thanks Alan. This is very useful. I did use gamma and log as the link function. I suppose what you found also applies. see my output below. It looks to me that a is 1 in my case.
GEE population-averaged model Number of obs = 475
Group and time vars: FacNo Year Number of groups = 123
Link: log Obs per group: min = 2
Family: gamma avg = 3.9
Correlation: AR(1) max = 4
Wald chi2(40) = 612.84
Scale parameter: 1.002232 Prob > chi2 = 0.0000
G
----
Guillermo A. Sandoval
PhD Candidate in Health Services Research
Institute of Health Policy, Management and Evaluation
University of Toronto
________________________________________
From: [email protected] [[email protected]] on behalf of Feiveson, Alan H. (JSC-SK311) [[email protected]]
Sent: May 29, 2013 10:24 AM
To: [email protected]
Subject: st: RE: gamma family - alpha and theta parameters - xtgee
In partial response to Guillermo's question, I simulated some gamma data and tried - glm- to see what the "scale" parameter in Stata is an estimate of:
rgamma(a, b) Description: returns gamma(a,b) random variates, where a is the gamma shape parameter and b is the scale parameter.
. gen y = rgamma(2,.01)
. summ y
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
y | 10000 .0199965 .014057 .0003844 .1184808
. glm y,fam(gamma) link(iden) nolog
Generalized linear models No. of obs = 10000
Optimization : ML Residual df = 9999
Scale parameter = .4941738
Deviance = 5407.647596 (1/df) Deviance = .5408188
Pearson = 4941.243902 (1/df) Pearson = .4941738
Variance function: V(u) = u^2 [Gamma]
Link function : g(u) = u [Identity]
AIC = -5.824201
Log likelihood = 29122.00461 BIC = -86686.55
------------------------------------------------------------------------------
| OIM
y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_cons | .0199965 .0001406 142.25 0.000 .0197209 .020272
------------------------------------------------------------------------------
Note that the "scale" parameter value of .4941738 is not 0.01 (the value of "b") but appears to be an estimate of 1/a = 0.5.
Note that this does not depend on the value of "b":
. replace y = rgamma(2,34567)
(10000 real changes made)
. glm y,fam(gamma) link(iden) nolog
Generalized linear models No. of obs = 10000
Optimization : ML Residual df = 9999
Scale parameter = .4963884
Deviance = 5469.254185 (1/df) Deviance = .5469801
Pearson = 4963.387852 (1/df) Pearson = .4963884
Variance function: V(u) = u^2 [Gamma]
Link function : g(u) = u [Identity]
AIC = 24.32117
Log likelihood = -121604.8625 BIC = -86624.94
------------------------------------------------------------------------------
| OIM
y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_cons | 70297.13 495.2803 141.93 0.000 69326.4 71267.86
------------------------------------------------------------------------------
However if I change the value of "a" (for any "b"):
. replace y = rgamma(10,1232)
(10000 real changes made)
. glm y,fam(gamma) link(iden) nolog
Generalized linear models No. of obs = 10000
Optimization : ML Residual df = 9999
Scale parameter = .1023587
Deviance = 1038.561081 (1/df) Deviance = .1038665
Pearson = 1023.484523 (1/df) Pearson = .1023587
Variance function: V(u) = u^2 [Gamma]
Link function : g(u) = u [Identity]
AIC = 20.85373
Log likelihood = -104267.6349 BIC = -91055.63
------------------------------------------------------------------------------
| OIM
y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_cons | 12416.28 39.72747 312.54 0.000 12338.41 12494.14
------------------------------------------------------------------------------
The "scale" parameter estimate is now close to 0.1, which is 1/"a".
One more test:
. replace y = rgamma(2.5,.765)
(10000 real changes made)
. glm y,fam(gamma) link(iden) nolog
Generalized linear models No. of obs = 10000
Optimization : ML Residual df = 9999
Scale parameter = .4056914
Deviance = 4317.23078 (1/df) Deviance = .4317663
Pearson = 4056.508667 (1/df) Pearson = .4056914
Variance function: V(u) = u^2 [Gamma]
Link function : g(u) = u [Identity]
AIC = 3.291668
Log likelihood = -16457.33854 BIC = -87776.96
------------------------------------------------------------------------------
| OIM
y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_cons | 1.907386 .012149 157.00 0.000 1.883575 1.931198
------------------------------------------------------------------------------
Again, the "scale" parameter estimate is close to 1/"a" = 0.4.
Bottom line: With a gamma family, the "scale" parameter in -glm- (and also in -xtgee-) is an estimate of 1/"a", and does not appear to depend on the value of the "real" scale parameter "b".
Al Feiveson
-----Original Message-----
From: [email protected] [mailto:[email protected]] On Behalf Of Guillermo A. Sandoval (UofT)
Sent: Tuesday, May 28, 2013 6:37 PM
To: [email protected]
Subject: st: gamma family - alpha and theta parameters - xtgee
Dear
I am using xtgee with gamma distribution. I understand that for gamma distribution, one needs to define the parameters alpha and theta, which define the shape of the distribution function. Please see http://en.wikipedia.org/wiki/Gamma_distribution. My variable is continuous (mortality rates in hospitals) and distributes pretty much like the one in wekipedia that says k=1 and theta=2.
In xtgee, it looks like one cannot define these parameters. I guess my question is, is this necessary (if yes, how)? or xtgee actually estimates them and I don't really need to worry about that?
Thanks in advance
Guillermo
----
Guillermo A. Sandoval
PhD Candidate in Health Services Research
Institute of Health Policy, Management and Evaluation University of Toronto
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