GLM Power regressions

[Tya Pal]

Hello! I ran the following GLM regression:

glm yvar xvar, f(gauss) link(power -4)

I would like to understand the underlying equation based on the results. I believe that the regression I ran follows the general equation of g(u)=u^-4. How would I create the specific equation for my regression using the stata output? I've included it below.



Many thanks,
Tya

[Weiwen Ng]

The graphic is completely illegible. Please read the FAQ, particularly the bit about not showing results in graphics files.

[Tya Pal]

Thank you, Weiwen, for your advice. I have pasted the results below.

glm yvar xvar, f(gauss) link(power -4)
  
Generalized linear models
No. of obs =
193

Optimization : ML
Residual df =
191


Scale parameter =
.0043681

Deviance = .8343152408
(1/df) Deviance =
.0043681

Pearson = .8343152408
(1/df) Pearson =
.0043681

Variance function: V(u) = 1
[Gaussian]

Link function : g(u) = u^(-4)
[Power]


AIC =
-2.585232

Log likelihood = 251.4748586
BIC =
-1004.34




OIM

yvar    Coef. Std. Err.
z
P>z [95% Conf.
Interval]




xvar 2536.339 269.7932
9.40
0.000 2007.554
3065.124

_cons 20.10067 1.883457
10.67
0.000 16.40916
23.79218

Example generated by -dataex-
input float(yvar xvar)
 .2672032  .5910986
.28835732 .59026927
.28401816 .58860195
.23398755  .5808591
 .3000235  .5805774
.20542994  .5799954
.25039405  .5738891
 .1682193  .5731346
 .1675029  .5663816
 .2082808  .5644914
.28327656  .5638836
 .1659431  .5589922
 .1637957  .5508529

[Weiwen Ng]

Anyway, back to your question. I'm not familiar with the power link, but you can't generate the specific equation used in your GLM with your Stata output. It is basic math, and I think the equation you wrote is roughly correct.

More specifically, let u = E(yvar | xvar). You've expressed the link function correctly.

Now, how do you understand the results, given that practically nobody knows what raising something to the power of -4 does? Use the margins command, which converts everything to the raw scale no matter what the link function is. So, margins gives you results in natural units of yvar. (Had you run any flavor of logistic regression, you would get results in probability terms.) Statalist's own Richard Williams has a nice explainer on margins here.