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

[Prof Brian Ripley]

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