AIC / BIC vs P-Values / MAM
On 10-08-04 10:55 AM, Chris Mcowen wrote:
Dear List, I was after some advice on model selection,
OK, you asked for it ...
I am using AIC model selection rather than P-value based stepwise regression as i feel it is more robust (Burnham& Anderson, 2002). However there seems to be a huge difference in my results.
In my opinion model selection via AIC shares most of the
disadvantages of p-value based model selection. "All model" selection is
slightly better than stepwise approaches because it is less susceptible
to getting stuck in some weird local branch, but whether you select
models via p-value or AIC *should* be based on whether you are trying to
test hypotheses or make predictions, and you should seriously question
whether you should be doing model selection in the first place. You
should *not* select a model and then make inferences about the
'significance' of what remains in the model ...
AIC is great but it's not a panacea.
Now -- on to "p vs AIC" question.
The factors with the highest p-values , and therefore retained in the MAM, when i did an explanatory stepwise regression, do not appear in the model with the lowest AIC value - do the two approaches generally not match? The factors retained by the MAM are theoretically what i would expect, so i am a bit surprised as to why the model with the lowest AIC doesn't contain them? I have ranked the AIC models with Akaike weights, but still the top ranked models don't incorporate the traits i would expect / retained in the MAM. LOWEST AIC MODEL model43<- lmer(threatornot~1+(1|order/family) + geophyte + seasonality + pollendispersal + woodyness, family=binomial)
model43
Generalized linear mixed model fit by the Laplace approximation
Formula: threatornot ~ 1 + (1 | order/family) + geophyte + seasonality + pollendispersal + woodyness
AIC BIC logLik deviance
1395 1430 -690.6 1381
Random effects:
Groups Name Variance Std.Dev.
family:order (Intercept) 0.37447 0.61194
order (Intercept) 0.00000 0.00000
Number of obs: 1116, groups: family:order, 43; order, 9
Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.40234 0.43237 0.931 0.35208
geophyte2 0.06453 0.19616 0.329 0.74218
seasonality2 -1.06900 0.34241 -3.122 0.00180 **
pollendispersal2 0.64474 0.31089 2.074 0.03809 *
woodyness2 0.47599 0.25646 1.856 0.06346 .
BEST STEPWISE MAM
Generalized linear mixed model fit by the Laplace approximation
Formula: threatornot ~ 1 + (1 | order/family) + breedingsystem * fruit + woodyness
AIC BIC logLik deviance
1409 1454 -695.3 1391
Random effects:
Groups Name Variance Std.Dev.
family:order (Intercept) 0.52475 0.7244
order (Intercept) 0.00000 0.0000
Number of obs: 1116, groups: family:order, 43; order, 9
Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.1290 0.4909 -2.300 0.0215 *
breedingsystem2 0.8123 0.4756 1.708 0.0876 .
breedingsystem3 0.9449 0.5246 1.801 0.0717 .
fruit2 1.3885 0.6221 2.232 0.0256 *
woodyness2 0.5484 0.2627 2.088 0.0368 *
breedingsystem2:fruit2 -1.6218 0.6577 -2.466 0.0137 *
breedingsystem3:fruit2 -1.6645 0.7449 -2.235 0.0255 *
The breedingsystem* fruit interaction, should, based on theory be important so why is it not in the model with the lowest AIC but is in the MAM?
I am not sure if it is because i did not set out my candidate models correctly, I did a different model for every combination of traits (2 to the power of 7) -1 as i was unsure of which models would be important. I was given the data, i didn't collect it, therefore i have to work with what i have.
My best guess as to what's going on here is that you have a good
deal of correlation among your factors (in this case, with
discrete factors, that means that some combinations of factors are
under/overrepresented in the data set), which means that quite
different combinations of factors can fit/explain the data approximately
equally well.
It's really hard to say without going through the data in detail.
My advice would be to (a) read [or skim] Frank Harrell's book on
Regression Modeling Strategies, particularly about the
dangers of model reduction; (b) if you're interested in **testing
hypotheses about which factors are important**, simply fit
the full model and base your inference on the estimates and confidence
intervals from the full model.
good luck,
Ben Bolker