# Significance of (GAM) regression coefficients when model likelihood is not significantly higher than null

I am running a GAM-based regression using the R package gamlss and assuming a zero-inflated beta distribution of the data. I have only a single explanatory variable in my model, so it's basically: mymodel = gamlss(response ~ input, family=BEZI).

The algorithm gives me the coefficient $k$ for the impact of the explanatory variable into the mean ($\mu$) and the associated p-value for $k(\text{input})=0$, something like:

Mu link function:  logit
Mu Coefficients:
Estimate  Std. Error  t value   Pr(>|t|)
(Intercept)  -2.58051     0.03766  -68.521  0.000e+00
input        -0.09134     0.01683   -5.428  6.118e-08


As you can see in the above example, the hypothesis of $k(\text{input})=0$ is rejected with high confidence.

I then run the null model: null = gamlss(response ~ 1, family=BEZI) and compare the likelihoods using a likelihood-ratio test:

p=1-pchisq(-2*(logLik(null)[1]-logLik(mymodel)[1]), df(mymodel)-df(null)).


In a number of cases, I get $p>0.05$ even when the coefficients at input are reported to be highly significant (as above). I find this quite unusual -- at least it never happened in my experience with linear or logistic regression (in fact, this also never happened when I was using zero-adjusted gamma with gamlss).

My question is: can I still trust the dependence between response and input when this is the case?

Many thanks!

-
Thanks, Christophe, for formatting my post! –  a11msp Apr 6 '11 at 11:46