You have indeed correctly described the way to work with crossvalidation. In fact, you are 'lucky' to have a reasonable validation set at the end, because often, crossvalidation is used to optimize a model, but no "real" validation is done.
As @Simon Stelling said in his comment, crossvalidation will lead to lower estimated errors (which makes sense because you are constantly reusing the data), but fortunately this is the case for all models, so, barring catastrophy (i.e.: errors are only reduced slightly for a "bad" model, and more for "the good" model), selecting the model that performs best on a crossvalidated criterion, will typically also be the best "for real".
A method that is sometimes used to correct somewhat for the lower errors, especially if you are looking for parsimoneous models, is to select the smallest model/simplest method for which the crossvalidated error is within one SD from the (crossvalidated) optimum. As crossvalidation itself, this is a heuristic, so it should be used with some care (if this is an option: make a plot of your errors against your tuning parameters: this will give you some idea whether you have acceptable results)
Given the downward bias of the errors, it is important to not publish the errors or other performance measure from the crossvalidation without mentioning that these come from crossvalidation (although, truth be told: I have seen too many publications that don't mention that the performance measure was obtained from checking the performance on the original dataset either --- so mentioning crossvalidation actually makes your results worth more). For you, this will not be an issue, since you have a validation set.
A final warning: if your model fitting results in some close competitors, it is a good idea to look at their performances on your validation set afterwards, but do not base your final model selection on that: you can at best use this to soothe your conscience, but your "final" model must have been picked before you ever look at the validation set.
Wrt your second question: I believe Simon has given your all the answers you need in his comment, but to complete the picture: as often, it is the bias-variance trade-off that comes into play. If you know that, on average, you will reach the correct result (unbiasedness), the price is typically that each of your individual calculations may lie pretty far from it (high variance). In the old days, unbiasedness was the nec plus ultra, in current days, one has accepted at times a (small) bias (so you don't even know that the average of your calculations will result in the correct result), if it results in lower variance. Experience has shown that the balance is acceptable with 10-fold crossvalidation. For you, the bias would only be an issue for your model optimization, since you can estimate the criterion afterwards (unbiasedly) on the validation set. As such, there is little reason not to use crossvalidation.
Setting NaNs to the mean of the training or test set is ok?
You need to define a procedure that you always follow. I see two valid options here:
- either use some value (e.g. mean) calculated for that subject (see also below)
- or some value calculated for the training set, basically a hyperparameter "value to be used for replacing NAs". This should not be calculated from the whole test set (independent testing also means that no parameters calculated from other test subjects should be used: the processing should not depend on the composition of the test set).
Edit:
Which method to follow (replace by value computed within subject or by value computed within training set) should IMHO be decided from the knowledge about the application and the data, we cannot tell you more than very general guidelines here.
Why not by value computed within test set?: That would mean that the value used to replace NAs in test subject A depends on whether subject A is tested together with subject B or subject C – which doesn't seem to be a desirable or sensible behaviour to me.
You may also want to look up "Imputation" which is the general term for techniques that fill in missing values.
Centering and scaling (standardization): if you have "external" (scientific) knowledge that suggests that a standardization within the subjects should take place, then go ahead with that. Whether this is sensible depends on your application and data, so we cannot answer this question.
For a more general discussion of centering and standardization, see e.g. Variables are often adjusted (e.g. standardised) before making a model - when is this a good idea, and when is it a bad one? and When conducting multiple regression, when should you center your predictor variables & when should you standardize them?
Now within each outer fold I plan to tune a classifier's parameter with help of another cross-validation.
With 4 subjects you probably won't be able to compare classifiers anyways: the uncertainty due to having only 4 test cases is far too high. Can't you fix this parameter by experience with similar data?
To illustrate the problem: assume you observe 4 correct classifications out of 4 test subjects. That gives you a point estimate of 100% correct predictions. If you look at confidence interval calculations for this, you get e.g. for the Agresti-Coull method a 95% ci of 45 - 105% (obviously not very precise with the small sample size), Bayes method with uniform prior makes it 55 - 100%. In any case it means that even if you observe perfect test results, it is not quite clear whether you can claim that the model is actually better than guessing. As long as you do not need to fear that fixing the parameter beforehand will produce a model that is clearly worse than guessing, you anyways cannot measure improvements in the practically important range.
The situation may be less drastic if you optimize e.g. Brier score but with 4 subjects I'd suspect that you still do not reach the precision you need for the expected improvement during the optimization.
Edit: Unfortunately, while 20 subjects are far more than 4, from a classifier validation statistics point of view, 20 is still very few.
We recently concluded that if you need to stick with the frequently used proportions for characterizing your classifier, at least in our field you cannot expect to have a useful precision in the test results with less than 75 - 100 test subjects (in the denominator of the proportion!). Again, you may be better off if you can switch to e.g. Brier's score, and with a paired design
for classifier comparison, but I'd call it lucky if that gains you a factor of 5 in sample size.
You can find our thoughts here: Beleites, C. and Neugebauer, U. and Bocklitz, T. and Krafft, C. and Popp, J.: Sample size planning for classification models. Anal Chim Acta, 2013, 760, 25-33.
DOI: 10.1016/j.aca.2012.11.007
accepted manuscript on arXiv: 1211.1323
AFAIK, dealing with the random uncertainty on test results during classifier optimization is an unsolved problem. (If not, I'd be extremely interested to papers about the solution!)
So my recommendation would be to do a preliminary experiment/analysis at the end of which you try to estimate the random uncertainty on the comparison results. If these do not allow to optimize (which I'd unfortunately expect to be the outcome), report this result and argue that in consequence you do not have any choice at the moment but fixing the hyper-parameters to some sensible (though not optimized) value.
Does the inner cross-validation necessarily need to be leave-one-subject-out as well?
If you do inner cross validation it would be better to do it subject-wise as well: without this, you'll get overly optimistic inner results. Which would not a problem iff the bias were constant. However, it usually isn't and you have the additional problem that due to the random uncertainty together with the optimistic bias you may observe many models that seem to be perfect. Among these you cannot distinguish (after all, they all seem to be perfect) which can completely mess up the optimization.
Again, with so few subjects I'd avoid this inner optimization and fix the parameter.
Best Answer
To answer this question, you should ask what you want your classifier to be able to do.
If I understand correctly, when you train your classifier on the 'measurement level' you would 'teach' the classifier to distinguish (classify) a set of features of a single subject. This is different from training it to classify any set of features independent of what subject it came from.
Assuming you want your classifier to be able to classify any set of features, independent of what subject those features came from, you should not do any cross-validation on the 'measurement level'.
In this same setting. I do not exactly understand why you would consider all the measurements of a single subject as 'one' (In the context of the leave-one-out cross-validation). Did you consider a single set of features (a single measurement), independent of what subject it came from, as being 'one' thing?
[EDIT]: I just discussed this problem with a colleague. Doing cross-validation on the entire dataset (independent of subject) will leak information.
If part of the measurements of a subject are included in the training set and if the other measurements of that same subject are in the testing set, this will overestimate the performance! Although, if you do not get any measurements from new subjects, this is not a problem.
In the case you do get measurements from new subjects (more likely, I think), then it is a good idea to include all the measurements of a subject in the testing set, if that subject was selected to be in that set.