[R] logistic regression model with non-integer weights

[Ramón Casero Cañas]

Michael Dewey wrote:
> At 17:12 09/04/06, Ramón Casero Cañas wrote:
> 
> I am not sure what the problem you really want to solve is but it seems
> that
> a) abnormality is rare
> b) the logistic regression predicts it to be rare.
> If you want a prediction system why not try different cut-offs (other
> than 0.5 on the probability scale) and perhaps plot sensitivity and
> specificity to help to choose a cut-off?

Thanks for your suggestions, Michael. It took me some time to figure out
how to do this in R (as trivial as it may be for others). Some comments
about what I've done follow, in case anyone is interested.

The problem is a) abnormality is rare (Prevalence=14%) and b) there is
not much difference in the independent variable between abnormal and
normal. So the logistic regression model predicts that P(abnormal) <=
0.4. I got confused with this, as I expected a cut-off point of P=0.5 to
decide between normal/abnormal. But you are right, in that another
cut-off point can be chosen.

For a cut-off of e.g. P(abnormal)=0.15, Sensitivity=65% and
Specificity=52%. They are pretty bad, although for clinical purposes I
would say that Positive/Negative Predictive Values are more interesting.
But then PPV=19% and NPV=90%, which isn't great. As an overall test of
how good the model is for classification I have computed the area under
the ROC, from your suggestion of using Sensitivity and Specificity.

I couldn't find how to do this directly with R, so I implemented it
myself (it's not difficult but I'm new here). I tried with package ROCR,
but apparently it doesn't cover binary outcomes.

The area under the ROC is 0.64, so I would say that even though the
model seems to fit the data, it just doesn't allow acceptable
discrimination, not matter what the cut-off point.


I have also studied the effect of low prevalence. For this, I used
option ran.gen in the boot function (package boot) to define a function
that resamples the data so that it balances abnormal and normal cases.

A logistic regression model is fitted to each replicate, to a parametric
bootstrap, and thus compute the bias of the estimates of the model
coefficients, beta0 and beta1. This shows very small bias for beta1, but
a rather large bias for beta0.

So I would say that prevalence has an effect on beta0, but not beta1.
This is good, because a common measure like the odds ratio depends only
on beta1.

Cheers,

-- 
Ramón Casero Cañas

http://www.robots.ox.ac.uk/~rcasero/wiki
http://www.robots.ox.ac.uk/~rcasero/blog