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I am building a marketing model based on logistic regression. It's a customer attrition model. The event rate is very low i.e 0.1%. I have more than 1000 predictors. I know there is a rule - Minimum 10 events per predictor. I want to know - Does this rule exist before dimensionality reduction (feature extraction) with PCA and Information value? Should i consider this rule based on my original 1500 variables or does it exist for significant variables that came after applying variable selection techniques such as Stepwise Regression , PCA etc?

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A 20:1 rule is better, or use 15:1 as a compromise. This refers to the number of candidate variables, e.g., m/15 if m is the number of events. You are in trouble. Stepwise regression won't help. Your best bet is to use the first m/15 principal components and regress these against $Y$. When you can reduce dimensionality in a way that is masked to $Y$ you can count the number of candidate variables as equaling the dimensionality of the reduced space.

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There is no such rule as “10 events per predictor”. All you need is sufficient number of 1’s for the maximum likelihood to converge.

For example if you had a sample of 1 million observations with an event rate of 0.1% --you are probably good. In this case you have 1000 events. However, if you had a sample of 100,000 and you even rate is0.1%, then you may face convergence issues.

However, there are many ways get around this. One is over sampling. That is you take all the 1’s and take a fraction of the 0’s and use weighted maximum likelihood.

Example: Suppose, you took all the 1’s and half of the 0’s, you would create a weight variable that will have the value:

1 for observation where your dependent variable is 1

2 for observations where your dependent variable is 0

Use this weight variable in your logistics regression to weight the observations—this is called weighted MLE

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    $\begingroup$ Unfortunately none of that is correct. $\endgroup$ – Frank Harrell Aug 17 '15 at 21:42
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    $\begingroup$ This answer ignores the issue with the number of predictors, a big issue here for two reasons: a surprisingly small number of predictors can lead to convergence problems in logistic regression, and a model fit with too many predictors per event will be seriously overfit, unable to work in another sample from the same population even if it models the original data set very well. There are important reasons for this old rule of thumb, and for being even more conservative than 10:1. $\endgroup$ – EdM Aug 17 '15 at 22:03
  • $\begingroup$ @ Frank Harrell: Can you please show me a statistical proof or reference where it says " 10 events per predictor" in logistic regression? The only valid argument here is the degrees freedom will decrease as the number of regressors increases. $\endgroup$ – subra Aug 18 '15 at 16:06
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    $\begingroup$ Incorrect again. It can be detected with cross-validation but the disaster will not be fixed by cross-validation. My book Regression Modeling Strategies goes into detail about the 15:1 rule. See also citeulike.org/user/harrelfe/article/13467382 who advocate a 20:1 rule (200:1 if using machine learning instead). $\endgroup$ – Frank Harrell Aug 18 '15 at 16:30
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    $\begingroup$ Univariate analysis is inappropriate. Unsupervised learning does not count against the 20:1 rule, same with any method masked to $Y$. Don't know what information value is. The number of events, as stated previously, needs to be adequate for the reduced space, i.e., for the number of parameters to be estimated against $Y$. $\endgroup$ – Frank Harrell Aug 18 '15 at 17:53

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