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When We estimate logistic regression using MLE we try to minimize $-2ln(likelihood)$, which is equivalent to minimization of sum of squared deviance residual.

Is it possible to fit logistic regression in such a way that number of positive targets in first decile is maximized? Decile is calculated using predicted score. The order in other nine deciles should not be relevant at all.

Update 2013-03-28 15:35
Maybe I will describe this problem from a bit more business perspective. Lets say we have a huge database of potencial customers. Some of them we have already called and we know whether they bought our product or not (this is out target). There is limit on number of phone calls that we can make and we want to maximize the sales. The limit is equal to 10% of customer number.

We can build logistic regression model and call 10% of people with highest scores, but intuitively I feel that something better can be done. MLE will also maximize the likelihood of those who probably won't buy. If we focus more on those with highest score then we might obtain better result. From top of my head I can think of following method:
1) Build logistic regression model on whole population.
2) Build logistic regression model using only 20% of people with highest scores in first model.

I'm not sure whether this solution makes sense, but I just wanted to show my way of thinking.

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    $\begingroup$ We maximize the log likelihood because statistical theory has proven this to be an optimum objective function in several senses, when there is no prior information about model parameters. To maximize something else implies that you have a special loss/cost/utility function. What is your utility function? And what is magical about a decile vs. an octile vs. a quartile for your purpose? This seems arbitrary. $\endgroup$
    – Frank Harrell
    Commented Mar 28, 2013 at 13:46
  • $\begingroup$ @FrankHarrell: There is nothing magical about the decile, it is more like an example of what I want to achieve. I've added my motivation, maybe this will be helpful. $\endgroup$
    – Tomek Tarczynski
    Commented Mar 28, 2013 at 14:46
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    $\begingroup$ What you need is a standard lift curve. Sort customers by descending predicted probability and call as many as you can afford. Deciles won't help. $\endgroup$
    – Frank Harrell
    Commented Mar 28, 2013 at 20:34

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Partially answered in comments:

What you need is a standard lift curve. Sort customers by descending predicted probability and call as many as you can afford. Deciles won't help. – Frank Harrell

We maximize the log likelihood because statistical theory has proven this to be an optimum objective function in several senses, when there is no prior information about model parameters. To maximize something else implies that you have a special loss/cost/utility function. What is your utility function? And what is magical about a decile vs. an octile vs. a quartile for your purpose? This seems arbitrary. – Frank Harrell

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