[R] Validity of GLM using Gaussian family with sqrt link
Prof Brian Ripley
ripley at stats.ox.ac.uk
Thu Dec 11 08:45:02 CET 2008
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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