[R] Validity of GLM using Gaussian family with sqrt link

Gerard M. Keogh GMKeogh at justice.ie
Thu Dec 11 17:41:09 CET 2008

```Hi all,

Just on this question :

can I assume any R internal defined function can be used to describe the
link (e.g. = "arctan") so long as its increasing and monotone?
How might abs work for example - (except at 0)?

And/or finally, can I define any old function in R called "myfun" and use
link="myfun" provided myfun is a sort of "nice" function?

Gerard

"Lam, Tzeng Yih"
<Tzengyih.Lam at ore
gonstate.edu>                                              To
Sent by:                  "Prof Brian Ripley"
r-help-bounces at r-         <ripley at stats.ox.ac.uk>
project.org                                                cc
r-help at r-project.org
Subject
11/12/2008 15:20          Re: [R] Validity of GLM using

Dear Prof. Ripley,

Thank you for your quick response.

(A)
> link-sqrt is a name and not accepted.  link="sqrt" is a literal character
string, and is.

I am not entirely sure whether I understand that statement but this is what
I found out. If I specify family=gaussian(link=sqrt), the glm() fails to
run because it is not a default link (so, I understand this part).
Following Venables and Ripley (2002):

>

Call:
glm(formula = cnt ~ herbc + herbht, family = gaussian(link = "sqrt"),
data = sotr, start = c(0.1, -0.004, 0.01))

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  0.462211   0.043475  10.632  < 2e-16 ***
herbc       -0.003315   0.001661  -1.996   0.0461 *
herbht       0.010241   0.001291   7.935 4.86e-15 ***

AIC: 3235.0

>

Call:
glm(formula = cnt ~ herbc + herbht, family = quasi(link = power(0.5)),
data = sotr, start = c(0.1, -0.004, 0.01))

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  0.462211   0.043475  10.632  < 2e-16 ***
herbc       -0.003315   0.001661  -1.996   0.0461 *
herbht       0.010241   0.001291   7.935 4.86e-15 ***

AIC: NA

Notice that the parameter estimates and corresponding standard errors are
identical. So, my interpretation is that family=gaussian(link="sqrt") is
identical as specify family=quasi(link=power(0.5)) in glm(). The exception
is that AIC (and thus maximized log-likelihood values) can be computed for

The questions are:
(A.1) Is this interpretation correct?
(A.2) If (A.1) is true, does family=gaussian(link="sqrt") implies that I am
doing a Generalized Linear Model with normal distribution and the link
function is: sqrt(mu) = b0+b1(herbc)+b2(herbht)?

(B)
> In less technical terms, in model 1 you compute the likelihood from
probabilities
> and in model 2 from probability densities, and the latter depend on the
> units of measurement.
Yes, you are correct and I understand it now. Although not as common these
days, some small mammal studies still use sqrt transformation of count as
response variable and carry out a linear model fitting with predictors (via
least squares). So, the exercise that I got into is to compare performances
of linear model with sqrt transformation of count and GLM with Poisson.
However, knowing that we can't compare logLik or AIC based on different
measures of responses. So, I thought that comparison under GLM framework
might be an approach closer to the intention.

Thank again for your quick respond and advices. I appreciate it very much.

Best regards,
TzengYih Lam

-----------------------------------
Ph.D. student
College of Forestry
Oregon State University

-----Original Message-----
From: Prof Brian Ripley [mailto:ripley at stats.ox.ac.uk]
Sent: Wed 12/10/2008 11:45 PM
To: Lam, Tzeng Yih
Cc: r-help at r-project.org
Subject: Re: [R] Validity of GLM using Gaussian family with sqrt link

link: a specification for the model link function.  This can be a
name/expression, a literal character string, a length-one
character vector or an object of class '"link-glm"' (such as
generated by 'make.link') provided it is not specified _via_
one of the standard names given next.

link-sqrt is a name and not accepted.  link="sqrt" is a literal character
string, and is.

b) Your first model is a model for integer observations, the second for
continuous observations.  As such, the log-likleihoods are computed with
respect to different reference measures and are not comparable.  In less
technical terms, in model 1 you compute the likelihood from probabilities
and in model 2 from probability densities, and the latter depend on the
units of measurement.

On Wed, 10 Dec 2008, Lam, Tzeng Yih wrote:

> Dear all,
>
> I have the following dataset: each row corresponds to count of forest
floor small mammal captured in a plot and vegetation characteristics
measured at that plot
>
>> sotr
>     plot cnt herbc herbht
> 1     1A1   0 37.08  53.54
> 2     1A3   1 36.27  26.67
> 3     1A5   0 32.50  30.62
> 4     1A7   0 56.54  45.63
> 5     1B2   0 41.66  38.13
> 6     1B4   0 32.08  37.79
> 7     1B6   0 33.71  30.62
> ...
>
> I am interested in comparing fit of different specification of
> Generalized Linear Models (although there are some issues with using AIC
> or BIC for comparison, but this is the question that I like to post
> here). Here are two of the several models that I am interested in:
>
> (1) Poission log-linear model
>> pois<-glm(cnt~herbc+herbht,family=poisson,data=sotr)
>> summary(pois)
> Call:
> glm(formula = cnt ~ herbc + herbht, family = poisson, data = sotr)
>
> Coefficients:
>             Estimate Std. Error z value Pr(>|z|)
> (Intercept) -1.341254   0.089969 -14.908   <2e-16 ***
> herbc       -0.007303   0.003469  -2.105   0.0353 *
> herbht       0.024064   0.002659   9.051   <2e-16 ***
> ---
>    Null deviance: 1699.0  on 1180  degrees of freedom
> Residual deviance: 1569.8  on 1178  degrees of freedom
> AIC: 2311.4
>
>
> (2) Gaussian with sqrt link model
>>

>> summary(gaus.sqrt)
> Call:
> glm(formula = cnt ~ herbc + herbht, family = gaussian(link = "sqrt"),
>    data = sotr, start = c(0.1, -0.004, 0.01))
>
> Coefficients:
>             Estimate Std. Error t value Pr(>|t|)
> (Intercept)  0.462211   0.043475  10.632  < 2e-16 ***
> herbc       -0.003315   0.001661  -1.996   0.0461 *
> herbht       0.010241   0.001291   7.935 4.86e-15 ***
> ---
>    Null deviance: 1144.6  on 1180  degrees of freedom
> Residual deviance: 1062.9  on 1178  degrees of freedom
> AIC: 3235.0
>
>> logLik(gaus.sqrt)
> 'log Lik.' -1613.524 (df=4)
>
>> From the glm() help file that I read, family=gaussian() accepts the
links "identity", "log" and "inverse". There is no mentioning of gaussian()
family=poisson()
>
> A. Therefore, is the code in (2) actually computing Maximum Likelihood
> Estimates (MLE) of the coefficients using Gaussian family with "sqrt"
> link or is it computing MLE of something else?
>
> B. If the code in (2) is computing the MLE with gaussian(link="sqrt"),
> then will the maximized value of log-likelihood function using logLik()
> be valid (other than the issue that the dispersion parameter is counted
> as a parameter in aic() within glm())?
>
> Thank you in advance and I appreciate it very much for any advices that
are offered.
>
> Best regards,
> TzengYih Lam
>
>
> TzengYih Lam, PhD Student
> College of Forestry
> Oregon State University
>
>
>
>
>
>
>
>
>            [[alternative HTML version deleted]]
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>

--
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

[[alternative HTML version deleted]]

______________________________________________
R-help at r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.

**********************************************************************************
The information transmitted is intended only for the person or entity to which it is addressed and may contain confidential and/or privileged material. Any review, retransmission, dissemination or other use of, or taking of any action in reliance upon, this information by persons or entities other than the intended recipient is prohibited. If you received this in error, please contact the sender and delete the material from any computer.  It is the policy of the Department of Justice, Equality and Law Reform and the Agencies and Offices using its IT services to disallow the sending of offensive material.
Should you consider that the material contained in this message is offensive you should contact the sender immediately and also mailminder[at]justice.ie.

Is le haghaidh an duine nó an eintitis ar a bhfuil sí dírithe, agus le haghaidh an duine nó an eintitis sin amháin, a bheartaítear an fhaisnéis a tarchuireadh agus féadfaidh sé go bhfuil ábhar faoi rún agus/nó faoi phribhléid inti. Toirmisctear aon athbhreithniú, atarchur nó leathadh a dhéanamh ar an bhfaisnéis seo, aon úsáid eile a bhaint aisti nó aon ghníomh a dhéanamh ar a hiontaoibh, ag daoine nó ag eintitis seachas an faighteoir beartaithe. Má fuair tú é seo trí dhearmad, téigh i dteagmháil leis an seoltóir, le do thoil, agus scrios an t-ábhar as aon ríomhaire. Is é beartas na Roinne Dlí agus Cirt, Comhionannais agus Athchóirithe Dlí, agus na nOifígí agus na nGníomhaireachtaí a úsáideann seirbhísí TF na Roinne, seoladh ábhair cholúil a dhícheadú.
Más rud é go measann tú gur ábhar colúil atá san ábhar atá sa teachtaireacht seo is ceart duit dul i dteagmháil leis an seoltóir láithreach agus le mailminder[ag]justice.ie chomh maith.
***********************************************************************************

```