A test of significance for partial least squares regression

Partial least squares (PLS) regression is a commonly used statistical technique for performing multivariate calibration, especially in situations where there are more variables than samples. Choosing the number of factors to include in a model is a decision that all users of PLS must make, but is complicated by the large number of empirical tests available. In most instances predictive ability is the most desired property of a PLS model and so interest has centred on making this choice based on an internal validation process. A popular approach is the calculation of a cross‐validated r 2 to gauge how much variance in the dependent variable can be explained from leave‐one‐out predictions. Using Monte Carlo simulations for different sizes of data set, the influence of chance effects on the cross‐validation process is investigated. The results are presented as tables of critical values which are compared against the values of cross‐validated r 2 obtained from the user's own data set. This gives a formal test for predictive ability of a PLS model with a given number of dimensions.