Split-Sample Model Validation

Methods used to obtain unbiased estimates of future performance of statistical prediction models and classifiers include data splitting and resampling. The two most commonly used resampling methods are cross-validation and bootstrapping. To be as good as the bootstrap, about 100 repeats of 10-fold cross-validation are required.

As discussed in more detail in Section 5.3 of Regression Modeling Strategies Course Notes and the same section of the RMS book, data splitting is an unstable method for validating models or classifiers, especially when the number of subjects is less than about 20,000 (fewer if signal:noise ratio is high). This is because were you to split the data again, develop a new model on the training sample, and test it on the holdout sample, the results are likely to vary significantly. Data splitting requires a significantly larger sample size than resampling to work acceptably well. See also Section 10.11 of BBR.

There are also very subtle problems:

  1. When feature selection is done, data splitting validates just one of a myriad of potential models. In effect it validates an example model. Resampling (repeated cross-validation or the bootstrap) validate the process that was used to develop the model. Resampling is honest in reporting the results because it depicts the uncertainty in feature selection, e.g., the disagreements in which variables are selected from one resample to the next.
  2. It is not uncommon for researchers to be disappointed in the test sample validation and to ask for a “re-do” whereby another split is made or the modeling starts over, or both. When reporting the final result they sometimes neglect to mention that the result was the third attempt at validation.
  3. Users of split-sample validation are wise to recombine the two samples to get a better model once the first model is validated. But then they have no validation of the new combined data model.

There is a less subtle problem but one that is ordinarily not addressed by investigators: unless both the training and test samples are huge, split-sample validation is not nearly as accurate as the bootstrap. See for example the section Studies of Methods Used in the Text here. As shown in a simulation appearing there, bootstrapping is typically more accurate than data splitting and cross-validation that does not use a large number of repeats. This is shown by estimating the “true” performance, e.g., the R2 or c-index on an infinitely large dataset (infinite here means 50,000 subjects for practical purposes). The performance of an accuracy estimate is taken as the mean squared error of the estimate against the model’s performance in the 50,000 subjects.

Data are too precious to not be used in model development/parameter estimation. Resampling methods allow the data to be used for both development and validation, and they do a good job in estimating the likely future performance of a model. Data splitting only has an advantage when the test sample is held by another researcher to ensure that the validation is unbiased.

Many investigators have been told that they must do an “external” validation, and they split the data by time or geographical location. They are sometimes surprised that the model developed in one country or time does not validate in another. They should not be; this is an indirect way of saying there are time or country effects. Far better would be to learn about and estimate time and location effects by including them in a unified model. Then rigorous internal validation using the bootstrap, accounting for time and location all along the way. The end result is a model that is useful for prediction at times and locations that were at least somewhat represented in the original dataset, but without assuming that time and location effects are nil.

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