# Test set

The following is based on a CCP4BB discussion around June 17, 2008 entitled: "How many reflections for R_{free}"

First of all, the test set is that set of reflections put aside for unbiased calculation of statistical quantities, in particular R_{free} and sigmaA.

The need to find a good compromise for the size of the test set has been discussed by Axel Brunger in a "Methods in Enzymology" (1997) paper. He writes:

In all test calculations to date, the free R value has been highly correlated with the phase accuracy of the atomic model. In practice, about 5-10% of the observed diffraction data (chosen at random from the unique reflections) become sequestered in the test set. The size of the test set is a compromise between the desire to minimize statistical fluctuations of the free R value and the need to avoid a deleterious effect on the atomic model by omission of too much experimental data.

## How precise is the estimate of R_{free} for a certain number of test set reflections?

The estimate for the relative error of R_{free} is 1/sqrt(n), where n is the size of the test set. So if n is 1000, and the R_{free} is 31%, you would expect its relative error to be 31%/sqrt(1000), which is about 1%.

I believe this is from a paper of Ian Tickle (FIXME: reference).

## How many reflections do you need to get a good estimate of the sigmaA values (as a function of resolution) needed to calibrate the likelihood target?

Randy Read's rule of thumb is this:

My impression is that you gain relatively little by adding more reflections, once you have a total of about 1000 or at most 2000 in the cross-validation set. However, giving up more than 10% of the data is probably a bad idea, even if the sigmaA estimates are somewhat less accurate. I've had reasonable results refining against data sets of 3000-5000 reflections, setting aside only 10% (i.e. 300-500 reflections) for cross-validation.

So here's the recipe I would use, for what it's worth: <10000 reflections: set aside 10% 10000-20000 reflections: set aside 1000 reflections 20000-40000 reflections: set aside 5% >40000 reflections: set aside 2000 reflections

I'm sure that with a bit of thought someone could come up with a smooth function that achieves something similar, but it seems adequate.

More quantitative estimates can be found in
K.Cowtan (2005) J. Appl. Cryst. 38, 193-198. Likelihood weighting of partial structure factors using spline coefficients
http://journals.iucr.org/j/issues/2005/01/00/zm5022/zm5022.pdf
Kevin Cowtan summarizes:

the result is that the number of reflections required varies with the level of error in the model (i.e. with sigmaa). For refinement close to convergence, one could use about 250 free reflections per sigmaA parameter (so 1500 would probably do). However when dealing with a very poor initial model, or, for example, when using sigmaA in a density modification calculation, then it may be necessary to use all the reflections.

This agrees with Randy's explanation:

In case anyone is interested, the reason this is a bit simplistic is that the number of reflections you need depends on how good your model is. If you look at the contribution to the likelihood function from one reflection, it is very broad for low sigmaA values and becomes sharper as the sigmaA values increase. This means that, if the true value of sigmaA is low, you need more reflections to get a precise estimate than if the true sigmaA value is high. This happens because, if sigmaA is low, any value of Fo could be expected for a particular Fc because the model predicts the data poorly, but if sigmaA is high, then there is a very restricted range of possible values for Fo given Fc. So to get stable refinement from a very poor model, you might need to set aside a larger number of reflections for cross-validated sigmaA estimation. Later on, when the model is better, you could afford to absorb some of those reflections into the working set.