Then, knowing only the value of the measured variable, it is not possible to reconstruct both latent variables correctly. We therefore have to weight the measured variables and to select the optimal number of latent variables. We use the PLS-algorithm for this purpose. Low weights in important latent variables give a hint for measured variables that do not improve the prediction. Feature selection The PLS-approach is able to handle a high number of measured predictor variables relative to the number of samples. But certain response effects are very sensitive with regard to overfitting. Down-weighting the measured variables with a low contribution to the response, is not enough in this case. We therefore go one step further and suppress variables that contribute the least to the response of interest in the PLS-model. The contribution of a predictor variable to the response is measured by the VIP suggested previously [20,21], i.e. the variable’s importance in the projection of the data into such a space of latent variables, which corresponds best to the response. We rank the original predictor variables with respect to their VIP. Afterwards, low ranked variables are deleted from the model. In contrast to other methods [32], we do not use a fixed cut-off criterion. Instead, we select such a number of original variables, which yields the highest predictive power in a new refitted PLS- model. Thus, we run a nested optimization loop. The optimization on the higher nested scope is according to the number of original variables, whereas the optimization on the lower nested scope is subject to the number of latent variables. The computational costs of optimizing the number of latent variables increase with the number of original variables in the model. We therefore do not undertake those efforts for every possible number of original variables. The increment of our outer loop is ten, as a start, to get a general idea (cf. Figure S4 A). Afterwards, we reboot the outer loop with a refined increment of one, but break it after surpassing the two best arguments of the first boot by twenty steps (cf. Figure S4 B). This secures the identification of the optimal number of original variables, but it spares the refined search in unpromising regions with unjustified computational costs. Significance test To exclude the possibility that the predictive power is insignificant (which means that the in LOOV predicted response does not correlate significantly with the observed response), we perform a randomisation test. We build 1000 random permuta- tions of the true response. Then the Pearson correlation between the in LOOV predicted response and each permutation is calculated. The 1000 random correlations are compared to the predictive power, i.e. the correlation between the predicted response and the real response. Confidence intervals To judge the difference in predictive power between two distinct predictor sets, confidence intervals for the predictive power of optimized models are estimated using jackknife resampling. Given a predictor set and a response variable, we randomly select 358 out of our 359 samples and recompute the predictive power of the optimal variable selection on the resampled jackknife data. The resampling and the recomputation of predictive power are repeated 200 times. This results in a specific range of values for the predictive power of the regarded optimal variable selection for the particular response. We consider the central 99% of those 200 values as an estimation for the confidence interval. It stretches from the mean of the two smallest values to the mean of the two Heterosis Prediction Markers largest ones. If the estimated confidence intervals for different predictor sets or for different response variables are disjoint, we consider the discrepancy of the corresponding predictive powers significant. Occam’s razor In addition, the estimation of a confidence interval for the optimal predictive power enables us to follow Occam’s razor [36]. This widely approved principle denotes the simplest model, which contains the least variables, as the best among similarly descriptive models. The MDL [37] and the Bayesian Information Criterion [38] are just two examples that implement this principle. Since modern technologies, such as gene expression chips, confront us with a vast number of measured variables rising above the number of samples, it has become crucial, in recent years, to find a compromise between the acceptable loss of information and the desired number of variables [39]. In our approach, we use the VIP-ranking and the confidence interval to implement the principle. We look for the minimal number of highly ranked variables, whose predictive power is still located within the confidence interval estimated for the predictive power of the optimal variable selection (cf. Figure S4 C). We refer to the corresponding set of variables as the minimal variable selection. Response-specific predictors The predictor variables, which are always contained in the minimal variable selection, can be considered as biomarkers. Comparing the biomarkers for the two different testcross settings, we learn about common and specific effects, We refer to a certain metabolic marker as specific for heterosis prediction in one testcross population, if and only if it is contained in both, the minimal metabolic variable selection and the minimal variable selection of the combined genetic-metabolic approach for one testcross setting, but in none of the corresponding minimal variable selections for the other testcross setting. On the other hand, we consider a certain metabolic marker as highly predictive for heterosis in both testcross populations, if it is contained in the minimal metabolic and also in the minimal metabolic-genetic variable selection for both testcross settings (cf. Table 2). Analogous definitions can be applied to genetic markers. If we want to consider the order of importance of the biomarkers (cf. Tables S4 and S5), we consult their VIP in the corresponding minimal variable selections, which is in general slightly different from the VIP in the original PLS-model that included all measured predictor variables. Supporting Information Figure S1 Additional histograms of metabolic markers over the 359 RILs, in comparison to a random generation. The panels A-F show the six metabolic markers most important according to their VIP in the minimal metabolic and in the minimal combined genetic-metabolic models. Their deviation from a single normal distribution seems to abate with decreasing importance. Panels G- K show the 5 metabolic markers with the lowest mean VIP in the complete metabolic and the complete combined genetic-metabolic model for C24- and Col-heterosis, in comparison to a random generation of 359 numbers following a normal distribution with expectation zero and a standard deviation of one third (L). Found at: doi:10.1371/journal.pone.0005220.s001 (0.05 MB PDF) Figure S2 VIP of genetic markers in the complete (A) and in the minimal (B) genetic models. The figure shows the contribution of the 110 genetic markers, which have been arranged horizontally, PLoS ONE | www.plosone.org 10 April 2009 | Volume 4 | Issue 4 | e5220