First of all thank you both for the replies, my understanding is a lot clearer
after thinking about the example you showed.
One further question though, looking at the code for 'polr' in MASS suggests
that ordinal (and I would guess nominal too) data with levels >2 (ie not
binary) also has a saturated model log-likelihood of 0;
> res <- optim(s0, fmin, gmin, method = "BFGS", hessian = Hess, ...)
> .
> .
> .
> deviance <- 2 * res$value
So am I right in saying that Binary data isnt the only case where this is true?
It would make sense to me that for a multinomial model you could have a unique
factor for each data point and thus be able to create a likelihood of 1.
I have an example of this similar to the poisson one;
> library(nnet)
> y <- sample(1:3,replace=TRUE,size=10)
> factor <- as.factor(1:10)
> mod <- multinom(y~factor)
# weights: 33 (20 variable)
initial value 10.986123
iter 10 value 0.073035
final value 0.000086
converged
> mod
Call:
multinom(formula = y ~ factor)
Coefficients:
(Intercept) factor2 factor3 factor4 factor5 factor6 factor7 factor8
factor9 factor10
2 -14.33414 -21.50626 28.45961 28.45961 -9.18573 -9.18573 -9.18573 -9.18573
28.45961 -9.18573
3 -12.71240 -27.36982 -14.22512 -14.22512 23.75682 23.75682 23.75682 23.75682
-14.22512 23.75682
Residual Deviance: 0.0001713779
AIC: 40.00017
> logLik(mod)
'log Lik.' -8.568897e-05 (df=20)
> cbind(y,factor)
y factor
[1,] 1 1
[2,] 1 2
[3,] 2 3
[4,] 2 4
[5,] 3 5
[6,] 3 6
[7,] 3 7
[8,] 3 8
[9,] 2 9
[10,] 3 10
My understanding of this is that; if I observe a factor of '1' then the model
will say with probability 1 that the outcome will be 1 and so on for the other
rows in the dataset, and this shows in the estimated coefficients. I think the
reason the log-likelihood doesn’t return exactly 0 is that the fitting
algorithm used gets suitably close and then stops. Wouldn't make sense to
continue the algorithm until the coefficients where either 'Inf' or '-Inf'.
Please let me know your thoughts on this.
Thanks again,
Jeff
-----Original Message-----
From: peter dalgaard [mailto:pda...@gmail.com]
Sent: 21 April 2011 09:32
To: Juliet Hannah
Cc: Jeffrey Pollock; r-help@r-project.org
Subject: Re: [R] GLM output for deviance and loglikelihood
On Apr 21, 2011, at 05:14 , Juliet Hannah wrote:
> As you mentioned, the deviance does not always reduce to:
>
> D = -2(loglikelihood(model))
>
> It does for ungrouped data, such as for binary logistic regression.
To be precise, it only happens when the log likelihood of the saturated model
is 0, which for discrete models implies that the probability of the observed
data under the saturated model is 1. Binary data is pretty much the _only_
case where this is true (because individual fitted probabilities become either
zero or one).
--
Peter Dalgaard
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: pd....@cbs.dk Priv: pda...@gmail.com
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