Geometric least squares?

[Gianfranco Di Gennaro]

Dear all,
I have a question.
I've got a geometrically distributed outcome (it's an antibody titer) and I want to test whether the geometric mean changes in three different groups of patients.

I wonder whether a linear regression is appropriate (exponentiating the coefficients with eform) or if an estimation method based on "geometric least squares" is available.

Thank you.
Gianfranco

[Nick Cox]

I'd use a generalised linear model here with log link here.

[Jeff Wooldridge]

To add to Nick’s suggestion: You can use Poisson regression with a log link (exponential mean) and it’s fully robust to misspecification of the distribution. Use robust standard errors. That’s an easy solution. More complicated but not too hard is to use an exponential mean and the geometric distribution, which, like the Poisson, is in the linear exponential family. So it’s also completely robust to distributional misspecification. Actually, since the geometric is a special case of neg binomial, you might be able to fix the appropriate parameter at unity and use glm.

[Jeff Wooldridge]

I think this does it:

glm y x1 ... xK, fam(nb) link(log) vce(robust)

The default is to set the dispersion parameter to unity, which gives the geometric.

[Gianfranco Di Gennaro]

Thank you so much to Nick and Jeff
My data are distriubuted as follows:






I have the following three geometric means:

Group 0: 2583.70
Group 1: 858.12
Group 2: 1676.84


I tried an unadjusted model. How do you interpret it? I think I should consider to report results as irr.

Thank you again.
I wonder whether a quantile model could be an option, to account for skeweness of data.

glm IgGT3 i.group, fam(nb) link(log) vce(robust)

------------------------------------------------------------------------------
             |                 OIM
       IgGT3 | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
       group |
          1  |  -.7624329     .32741    -2.33   0.020    -1.404145   -.1207211
          2  |  -.4044871   .2782252    -1.45   0.146    -.9497985    .1408242
             |
       _cons |    8.15122   .1890095    43.13   0.000     7.780768    8.521672
------------------------------------------------------------------------------

[Gianfranco Di Gennaro]

Dear Jeff Wooldridge , since my right skewed outcome is not discrete isn't it better to use a gamma distribution?

glm y x1 ... xK, fam(gamma) link(log) vce(robust)
Thank you so much!

[Nick Cox]

The attachment in #5 is an empty file.

Do you have any zeros in the data? Otherwise why not show us the data on log scale, as in these examples using qplot from the Stata Journal?

webuse grunfeld, clear

set scheme s1color

local opts ysc(log) yla(1000 300 100 30) aspect(1) xla(0 0.25 "0.25" 0.5 "0.5" 0.75 "0.75" 1)

qplot invest if company <= 3 , over(company) `opts' legend(row(1)) name(QP1, replace) 

qplot invest if company <= 3 , by(company, row(1) note("")) `opts' name(QP2, replace)

(The code is a tidied up script after a few iterations around graph choices.)

Clearly any other graph will be fine that shows each distribution in detail in a way that allows easy and effective comparison with the others. (Here, and elsewhere, box plots are typically not as helpful as more detailed views.)

Here the quantile plots are propaganda for a view that log scale is natural for the variable in question, as leading to

* approximately symmetric distributions

* approximately equal variability

* additive shifts between distributions


We don't have to fuss about the relative importance of those, as we get all at once. But I think the last is most important given the kind of testing that is likely.

[Gianfranco Di Gennaro]

Thank you Nick Cox.
I will follow your advice.
Here's below my original distribution



I hope you can see it. I often have serious problems in attaching files in Statalist

[Nick Cox]

All those files are empty. What does work best are .png attachments. Not .gph. The forum software was not written by StataCorp and does not understand Stata file formats.

Can you post the data?

[Gianfranco Di Gennaro]

Thank you Nick Cox
This time it should work.
It's a continuous variable ranging from 0.02 to 2.63

I wonder whether to use a gamma distribution with link "power -1", or gamma with "log" link. In this latter case it's the same of a Poisson distribution, isn'it?
Addictional trouble, interpreting gamma regression coefficients is not easy in this case.
Thank so much for your time!
Gianfranco