[R] Validity of GLM using Gaussian family with sqrt link
Prof Brian Ripley
ripley at stats.ox.ac.uk
Thu Dec 11 18:04:27 CET 2008
Please do read the help page: fortune("WTFM") applies.
On Thu, 11 Dec 2008, Gerard M. Keogh wrote:
> 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)?
No.
> 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?
No.
>From the help page:
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.
You have to specify a *model link function*, as in
> gaussian(link="arctan")
Error in switch(link, logit = { : ‘arctan’ link not recognised
Only those known by name (to make.link) or an object of the specified
class can be used.
> 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
> Gaussian family with sqrt link
>
>
>
>
>
>
>
>
>
>
> 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):
>
>>
> summary(glm(cnt~herbc+herbht,data=sotr,family=gaussian(link="sqrt"),start=c(0.1,-0.004,0.01)))
>
> 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
>
>>
> summary(glm(cnt~herbc+herbht,data=sotr,family=quasi(link=power(0.5),variance=constant),start=c(0.1,-0.004,0.01)))
>
> 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
> family=gaussian(link="sqrt").
>
> 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
>
> a) There is a difference between link=sqrt and link="sqrt".
>
> 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
>>>
> gaus.sqrt<-glm(cnt~herbc+herbht,family=gaussian(link="sqrt"),data=sotr,start=c(0.1,-0.004,0.01))
>
>>> 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()
> accepting "sqrt" link. Although "sqrt" link is available for
> 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]]
>>
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>>
>
> --
> 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
>
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> ______________________________________________
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>
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--
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
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