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








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