On Mon, 14 Jun 2010, Joris Meys wrote:
Hi,
Marcs explanation is valid to a certain extent, but I don't agree with
his conclusion. I'd like to point out "the curse of
dimensionality"(Hughes effect) which starts to play rather quickly.
Ahem!
... minimal, self-contained, reproducible code ...
The OPs situation may not be an impossible one:
set.seed(54321)
dat <- as.data.frame(matrix(rnorm(1800*50),nc=50))
dat$y <- rbinom(1800,1,plogis(rowSums(dat)/7))
fit <- glm(y~., dat, family=binomial)
1/7 # the true coef
[1] 0.1428571
sd(coef(fit)) # roughly, the common standard error
[1] 0.05605597
colMeans(coef(summary(fit))) # what glm() got
Estimate Std. Error z value Pr(>|z|)
0.14590836 0.05380666 2.71067328 0.06354820
# a trickier case:
set.seed(54321)
dat <- as.data.frame(matrix(rnorm(1800*50),nc=50))
dat$y <- rbinom(1800,1,plogis(rowSums(dat))) # try again with coef==1
fit <- glm(y~., dat, family=binomial)
colMeans(coef(summary(fit)))
Estimate Std. Error z value Pr(>|z|)
0.982944012 0.119063631 8.222138491 0.008458002
Finer examination of the latter fit will show some values that differ too
far from 1.0 to agree with the asymptotic std err.
sd(coef(fit)) # rather bigger than 0.119
[1] 0.1827462
range(coef(fit))
[1] -0.08128586 1.25797057
And near separability may be playing here:
cbind(
+ table(
+ cut(
+ plogis(abs(predict(fit))),
+ c( 0, 0.9, 0.99, 0.999, 0.9999, 0.99999, 1 ) ) ) )
[,1]
(0,0.9] 453
(0.9,0.99] 427
(0.99,0.999] 313
(0.999,0.9999] 251
(0.9999,0.99999] 173
(0.99999,1] 183
Recall that the observed information contains a factor of
plogis( predict(fit) )* plogis( -predict(fit))
hist(plogis( predict(fit) )* plogis( -predict(fit)))
So the effective sample size here was much reduced.
But to the OP's question, whether what you get is reasonable depends on
what the setup is.
I wouldn't call the first of the above cases 'highly unstable'.
Which is not to say that one cannot generate difficult cases (esp. with
correlated covariates and/or one or more highly influential covariates)
and that the OPs case is not one of them.
HTH,
Chuck
The curse of dimensionality is easily demonstrated looking at the
proximity between your datapoints. Say we scale the interval in one
dimension to be 1 unit. If you have 20 evenly-spaced observations, the
distance between the observations is 0.05 units. To have a proximity
like that in a 2-dimensional space, you need 20^2=400 observations. in
a 10 dimensional space this becomes 20^10 ~ 10^13 datapoints. The
distance between your observations is important, as a sparse dataset
will definitely make your model misbehave.
Even with about 35 samples per variable, using 50 independent
variables will render a highly unstable model, as your dataspace is
about as sparse as it can get. On top of that, interpreting a model
with 50 variables is close to impossible, and then I didn't even start
on interactions. No point in trying I'd say. If you really need all
that information, you might want to take a look at some dimension
reduction methods first.
Cheers
Joris
On Mon, Jun 14, 2010 at 2:55 PM, Marc Schwartz <marc_schwa...@me.com> wrote:
On Jun 13, 2010, at 10:20 PM, array chip wrote:
Hi, this is not R technical question per se. I know there are many excellent
statisticians in this list, so here my questions: I have dataset with ~1800
observations and 50 independent variables, so there are about 35 samples per
variable. Is it wise to build a stable multiple logistic model with 50
independent variables? Any problem with this approach? Thanks
John
The general rule of thumb is to have 10-20 'events' per covariate degree of
freedom. Frank has suggested that in some cases that number should be as high
as 25.
The number of events is the smaller of the two possible outcomes for your
binary dependent variable.
Covariate degrees of freedom refers to the number of columns in the model
matrix. Continuous variables are 1, binary factors are 1, K-level factors are K
- 1.
So if out of your 1800 records, you have at least 500 to 1000 events, depending
upon how many of your 50 variables are K-level factors and whether or not you
need to consider interactions, you may be OK. Better if towards the high end of
that range, especially if the model is for prediction versus explanation.
Two excellent references would be Frank's book:
?http://www.amazon.com/Regression-Modeling-Strategies-Frank-Harrell/dp/0387952322/
and Steyerberg's book:
?http://www.amazon.com/Clinical-Prediction-Models-Development-Validation/dp/038777243X/
to assist in providing guidance for model building/validation techniques.
HTH,
Marc Schwartz
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--
Joris Meys
Statistical consultant
Ghent University
Faculty of Bioscience Engineering
Department of Applied mathematics, biometrics and process control
tel : +32 9 264 59 87
joris.m...@ugent.be
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Charles C. Berry (858) 534-2098
Dept of Family/Preventive Medicine
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