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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