Orthogonal projections to latent structures discriminant analysis (OPLS-DA)

OPLS-DA is based on the principle of partial least squares (PLS) regression; PLS itself is an extension of the multiple linear regression model (Wold, 1966). The goal of PLS regression is to predict response variable(s) Y from a (large) set of predictor variables X. PLS regression reduces the set of predictor variables to a smaller set of uncorrelated components and then performs least squares regression on these components. In the process, both variables X and Y are projected to a new space. This process is called projection to latent (hidden) structures (Abdi, Williams, 2010). Compared to multivariate regression, PLS has fewer assumptions (it can use predictor variables that are collinear and not independent) (Tobias, 1995). An additional modification to the PLS regression was developed by Trygg and Wold (Trygg, Wold, 2002), who introduced a way to remove systematic variation from the predictor variable dataset X that is not correlated to the response variable dataset Y, i.e., to remove variation within X that is orthogonal to Y. The advantage of such orthogonal projections to latent structures (OPLS) method is that a single latent variable (designated “T”) is used as a predictor of the Y. All variability in X is separated into predictive (T) and uncorrelated information (T_orthogonal), and the two components can be analyzed separately. In OPLS discriminant analysis, the Y is a binary class-designating variable, and the analysis aims to find the best separation between classes of objects along the T axis, while all variation unrelated to class separation is distributed along the T_orthogonal axes (Westerhuis et al., 2010). Similar to DFA, each variable’s loading indicates its contribution to the OPLS model. The generated model can then be applied to a new object to predict its class given the values of its X variables. The OPLS-DA output provides measures of the model fit (R²), model predictive power (Q²), and model accuracy based on a cross-validation procedure. Examples of OPLS-DA use in multivariate ecological analyses can be found in (Ramadan et al., 2014) and (Shankar et al., 2013).