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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