# Model selection and model performance in logistic regression

I have a question about model selection and model performance in logistic regression. I have three models that are based on three different hypotheses. The first two models (lets name them z and x) only have one explanatory variable in each model, and the third (lets name it w) is more complicated. I’m using AIC for variable selection for the w model and then AIC for comparing which of the three models that explain the dependent variable best. I’ve found that the w model has the lowest AIC and now want to do some performance statistics on that model to get some idea about the predictive power of the model. Since all I know is that this model is better than the other two but not how good it is.

Since I’ve used all data to learn the model (to be able to compare all three models) how do I go about with model performance? From what I’ve gathered I can’t just do a k-fold cross validation on the final model I got from model selection using AIC but need to start from the beginning with all explanatory variables included, is this correct? I’d think that it is the final model I’ve chosen with AIC that I want to know how well it performs, but do realize that I’ve trained on all data so the model might be biased. So if I should start from the beginning with all explanatory variables in all folds I will get different final models for some folds, can I just choose the model from the fold which gave the best predictive power and apply that to the full data set to compare AIC with the two other models (z and x)? Or how does it work?

Second part of my question is a basic question about over-parameterization. I have 156 data points, 52 is 1’s the rest are 0’s. I have 14 explanatory variables to choose from for the w model, I realize that I can’t include all due to over-parameterization, I’ve read that you should only use 10% of the group of the dependent variable with fewest observations which only would be 5 for me. I’m trying to answer a question in ecology, is it ok to select the starting variables which I think explains the dependent best simply based on ecology? Or how do I choose the starting explanatory variables? Doesn’t feel right to completely exclude some variables.

So I really have three questions:

• Could it be ok to test performance on a model trained on the full data set with cross-validation?
• If not, how do I choose the final model when doing cross-validation?
• How do I choose the starting variables so I want over-parameterize?

Sorry for my messy questions and my ignorance. I know that similar questions have been asked but still feel a little confused. Appreciate any thoughts and suggestions.

It's true that it is better to use a test set of data to validate your model. However, you can still say how well your model performed on your data, as long as you are honest about what you did. What you cannot really do is say that it will do this well on other data: It likely won't. Unfortunately, a lot of published articles at least hint at this incorrect notion.

is it ok to select the starting variables which I think explains the dependent best simply based on ecology?

Not only is it OK, it is better than any automated scheme. Indeed, these could also be the final variables. It depends, somewhat, on the extent of knowledge in the field. If not much is known about what you are researching, then a more exploratory approach may be necessary. But if you have good reason to think that certain variables should be in the model, then by all means, put them in. And I would argue for leaving them there, even if not significant.

If you are going to do model selection then I think you're better doing an exhaustive search and weighting each model rather than cherry picking. You only have 14 variables, which is certainly feasible - 16384 different models is not prohibitively large, especially because the sample size is small. I would also look at normalised weights, defined by:

$$w_m=\left[\sum_{l}\exp\left(-\frac{1}{2}[AIC_l - AIC_m]\right)\right]^{-1}$$

These weights assume AIC is negative twice the log likelihood plus two times the number of betas. If the best model has a weight close to $1$ then just use that. otherwise you should average your results aross models with total weight close to $1$. What usually happens is that a "core" group of variables should be always included, with uncertainty over a "non-core" set, and a third set of unimportant variables which never appear in models with high weight.

You could also replace AIC with BIC or some other penalty based IC to see how much the weights depend on the specific complexity penalty used.

• Using AIC on all possible models is a process with extreme multiplicity for which I wonder about the performance. Speaking in broad generalities it is not always logical to think about this as a variable selection problem but rather as a penalization (shrinkage) problem. – Frank Harrell Aug 10 '12 at 12:53
• Is there a process which doesn't have extreme multiplicity in model selection? You are dealing with a massive discrete space - this invariably leads to a large number of comparisons. I think the question is more whether or not the implicit prior over the models is a reasonable one. – probabilityislogic Aug 11 '12 at 2:39
• Well put. But I think of most model selection exercises as unnecessary (i.e., parsimony is not your friend) and the result of having no priors at all. – Frank Harrell Aug 11 '12 at 13:37
• I also agree, I think Bayes Factors are best used for issues of model structure, such as whether to use normal or t distribution for example. They are not useless for covariate selection, but inefficient compared to shrinkage. – probabilityislogic Aug 11 '12 at 23:28
• Sorry for my late comment, but do you know of any easy way to calculate this in R? I have the AIC:s in a list or matrix. I'm fairly new to R so any complicated function building is hard. Thanks! – mael Aug 29 '12 at 12:44

To answer "Could it be ok to test performance on a model trained on the full data set with cross-validation?" NO, I don't think this is OK. You should fit all the 3 models to the same subset of your dataset. Then do the cross-validation to see which one is better.

• So if I understand you right I should only use one training and one test set for all models? Can I still use my 5 variables for the last model then or is it a risk for over-parameterize? And isn't it a risk with just one training and test set - since it will be very much depending on where the split will be with the relative little data I have - or is that nothing to worry about? Otherwise it does feel like this would be the most correct way of doing it. – mael Aug 10 '12 at 7:48

Could it be ok to test performance on a model trained on the full data set with cross-validation?

I think not. Perhaps a better method would be to evaluate each of the three models using repeated cross validation. Given that you have chosen your features based on prior knowledge you do not need to worry about feature selection. This method allows you to evaluate the performance of the model.

If not, how do I choose the final model when doing cross-validation?

Once you have evaluated the performance of your model using repeated cross-validation you can then train the final model using all available data.

How do I choose the starting variables so I want over-parameterize?

If I understand correctly: As suggested by a contributor above you can either add in your features based on prior knowlegde of the area or else you need to perform feature selection within the cross-validation to avoid overfitting. This same feature selection procedure would then be applied to all data when training the final model. You cannot use this model to report the generalized performance of the model, this must come from the cross-validation estimate.