[R] logistic regression: wls and unbalanced samples

[Prof Brian Ripley]

On Wed, 27 Apr 2011, peter dalgaard wrote:

> On Apr 27, 2011, at 00:22 , Andre Guimaraes wrote:
>
>> Greetings from Rio de Janeiro, Brazil.
>>
>> I am looking for advice / references on binary logistic regression
>> with weighted least squares (using lrm & weights), on the following
>> context:
>>
>> 1) unbalanced sample (n0=10000, n1=700);
>> 2) sampling weights used to rebalance the sample (w0=1, w1=14.29); e
>> 3) after modelling, adjust the intercept in order to reflect the
>> expected % of 1’s in the population (e.g., circa 7%, as opposed to
>> 50%).
>
> ??
>
> If the proportion of 1 in the population is about 7%, how exactly is 
> the sample "unbalanced". I don't see a reason to use weights at all 
> if the sample is representative of the population. The opposite 
> situation, where the sample is balanced (e.g. case-control), the 
> population not, and you are interested in the population values, 
> _that_ might require weighting, with some care because case 
> weighting and sample weighting are two different things so the s.e. 
> will be wrong. That sort of stuff handled by the survey package.
>
> However what you seem to be doing is to create results for an 
> artificial 50/50 population, then project back to the population you 
> were sampling from all along. I don't think this makes sense at all.

There are circumstances where it might.  It is quite common in pattern 
recognition for the proportions in the training set to not reflect the 
population.  And if the misclassification costs are asymmetric, you 
may want to weight the fit.

The case I encountered was SGA births.  By definition there are about 
10% 'successes', but false negatives are far more important than false 
positives (or one would simply predict all births as normal).  This 
means that you want accurate estimation of probabilities in the right 
tail of the population distribution, and plug-in estimation of 
logistic regression is biased.  One of many ways to reduce that bias 
is to re-weight the training set so the estimated probabilities of 
marginal cases are in the middle of the range.

Note that logistic regression is not normally fitted by 'weighted 
least squares' (not even by 'lrm' from some unstated package).

This is not a list for tutorials in advanced statistics, but one 
reference is my Pattern Recognition and Neural Networks book.

>
> -- 
> Peter Dalgaard
> Center for Statistics, Copenhagen Business School
> Solbjerg Plads 3, 2000 Frederiksberg, Denmark
> Phone: (+45)38153501
> Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com
>
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-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595