Validity of GLM using Gaussian family with sqrt link

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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