Does Akaike play dice?
fairly common strategy in ecological research is to measure a
large number of covariates, put these together into a series of
models with all combinations of covariates, then look at
Akaike's Information Criterion (AIC) to see which is the best.
The idea is that the covariates that appear in the best model
(with lowest AIC) are really affecting the response variable.
This approach has the derogatory name of "dredging". But does it
In our introductory statistics "Boot Camps", we have an activity to generate data for logistic regression. Each participant tries to throw a sock into a box from different distances (2,3,4 and 5m) using each hand. After each attempt, the person rolls a 10-sided die and the score is recorded together with the distance and hand and whether the throw was successful. There are usually about 20 participants, so we have data for 160 attempts. (Activity based on trashball, see here.)
We then try a series of models with distance, hand and die score as predictors and success in getting the sock in the box as the response, and find which model has the lowest AIC. (We later add thrower ID as a blocking factor and gender to the range of covariates, but here we stick with the simple models.)
We have now done this on 26 occasions. The top model (lowest AIC) always included distance. On 10 occasions, hand was in the top model and on 9 occasions the die score. So the die score would seem to be just as good as hand as a predictor.
When the die score is not in the top model, a model with the die score is within 2 AIC units of the top, and thus would be judged to have substantial support from the data (Burnham & Anderson 2002, p.70).
Of course we know that the die score is just a random number and cannot influence the success of the throw, especially as it is rolled after the throw. So what is going on? How can the die score appear in the top model, and how can it have substantial support even when it's not in the top model? We'll deal with the second question first.
Models within 2 AIC units of the best
Adding an extra predictor to a linear model always results in a higher likelihood. This is true even if the predictor is just a vector of random numbers (eg, die rolls) as there is always some correlation between the predictor and the response, even if it is tiny. If we simply pick the model with the maximum likelihood, we'd always get the model with the most parameters. That's why we need to use AIC for model selection rather than likelihood.
Adding a completely unhelpful parameter to the model incurs a penalty of 2 units, so the difference in AIC will never be more than 2.
Models within 2 units of the best model should be checked to see if they differ from the best model by having one extra parameter. If that is the case, they are not really supported or competitive (Burnham & Anderson 2002, p.131). "Simply put, the uninformative parameter does not explain enough variation to justify its inclusion in the model and it should not be interpreted as having any ecological effect." (Arnold 2010, p.1175).
This applies if you have:
(1) response ~ distance, (2) response ~ distance + die, or (1) response ~ distance + hand, (2) response ~ distance + hand + die
since in these cases, die is doing nothing and you can safely remove the die model from the model set. But if you have:
(1) response ~ distance + hand, (2) response ~ distance + die, or (1) response ~ distance + die, (2) response ~ distance + hand
then the second model is not simply the first plus an extra parameter and you can't be certain which is best.
A spurious covariate in the top model
AIC attempts to find the best model for making predictions. "Prediction" doesn't necessarily involve forecasting the future, it also happens when you use a model to make inferences about parts of the population which were not included in your sample. The best way to assess the out-of-sample predictive accuracy of a model is to get a new, independent sample and see if the predictions match up to reality. But that's difficult, often impossible, so we have methods to do it based on our original sample. So we end up assessing out-of-sample accuracy using our original sample.
One way to do this is Leave One Out cross-validation (LOO). We leave out one observation, fit the model to all the rest, then predict the left-out observation. Do this for each of the observations one by one, so that we have a full set of predictions, and compare those with the actual data. Refitting the model hundreds of times is tedious and slow, so short-cuts have been developed; AIC gives very similar results to LOO without the extra computations.
An unavoidable risk with both AIC and LOO is that your single data set may have a correlation between a spurious predictor and the response. As a result, the spurious predictor really does help to predict the responses for this sample. For the 9 data sets where die was in the top model, the coefficient was >0.02 (4 cases) or <-0.02 (5 cases) while the variable ranges from 0 to 9.
My first thought was that this was due to sample size, and indeed pooling all 3696 observations avoided the spurious result. But when I split into 3 samples each with 1232 observations, one of those had in die in the top model. If 1200 is a small sample, we are all in trouble!
So, what can we do about this?
|Updated 25 December 2017 by Mike Meredith|