Date: Mon, 08 May 2000 19:13:43 +0200
From: Iñaki Trocóniz <itroconiz@unav.es>
Subject: Validation of models for categorical data

Dear Users,

To evaluate our logistic regression model we computed:

(i) for each dose the probability to get a certain score on the basis of the raw data

(ii) the typical probability to get a certain score for each dose, using the estimates of thetas (ETA=0), and we observed that the typical predicted profile described our raw data probabilities adequately.

(iii) the posterior expectation, and this expectation fitted our raw data probabilities much better than the typical profiles

(iv) the simulation expectation 100 times. The probabilities obtained were much closer to the ones we got from the posterior expectation than the ones we obtained using only the estimates of THETAs. Under the assumption that the ETAs used to compute the simulation expectation are normally distributed around 0 we would have expected that the results are closer to the typical population predicted probabilities.

Does anyone have an explanation for this result?

Inaki F. Troconiz Ph.D
Farmacia y Tecnologia Farmaceutica
Pamplona 31080, Spain
Tpn: +34 948 42 56 00 ext. 6507
Fax: +34 948 42 56 49
e-mail: itroconiz@unav.es

*****

From: LSheiner <lewis@c255.ucsf.edu>
Subject: Re: Validation of models for categorical data
Date: Mon, 08 May 2000 12:44:57 -0700

Inaki,

The logistic function is highly non-linear. What you are observing I think is that for random variable x, and arbtrary function f, where E(.) is the expectation operator, E(f(x)) != f(E(x)) When f is linear, then the above is an equality.

Here's an example:
p(y) = bernoulli(x)
logit(x) = theta + eta
theta = -3
omega = 3
p(E(x)) = .05
E(y) = .21

Lewis.

PS. Here's the S+ code I used to compute the above.
> exp(-3)/(1+exp(-3))
[1] 0.04742587
> nums _ rnorm(100,-3,3)
> mean(exp(nums)/(1+exp(nums)))
[1] 0.219993

--
_/ _/ _/_/ _/_/_/ _/_/_/ Lewis B Sheiner, MD (lewis@c255.ucsf.edu)
_/ _/ _/ _/_ _/_/ Professor: Lab. Med., Bioph. Sci., Med.
_/ _/ _/ _/ _/ Box 0626, UCSF, SF, CA, 94143-0626
_/_/ _/_/ _/_/_/ _/ 415-476-1965 (v), 415-476-2796 (fax)