Ordinal Forests

The prediction of the values of ordinal response variables using covariate data is a relatively infrequent task in many application areas. Accordingly, ordinal response variables have gained comparably little attention in the literature on statistical prediction modeling. The random forest method is one of the strongest prediction methods for binary response variables and continuous response variables. Its basic, tree-based concept has led to several extensions including prediction methods for other types of response variables. In this paper, the ordinal forest method is introduced, a random forest based prediction method for ordinal response variables. Ordinal forests allow prediction using both low-dimensional and high-dimensional covariate data and can additionally be used to rank covariates with respect to their importance for prediction. Using several real datasets and simulated data, the performance of ordinal forests with respect to prediction and covariate importance ranking is compared to competing approaches. First, these investigations reveal that ordinal forests tend to outperform competitors in terms of prediction performance. Second, it is seen that the covariate importance measure currently used by ordinal forest discriminates influential covariates from noise covariates at least similarly well as the measures used by competitors. In an additional investigation using simulated data, several further important properties of the OF algorithm are studied. The rationale underlying ordinal forests to use optimized score values in place of the class values of the ordinal response variable is in principle applicable to any regression method beyond random forests for continuous outcome that is considered in the ordinal forest method

A U-statistic estimator for the variance of resampling-based error estimators

We revisit resampling procedures for error estimation in binary classification in terms of U-statistics. In particular, we exploit the fact that the error rate estimator involving all learning-testing splits is a U-statistic. Therefore, several standard theorems on properties of U-statistics apply. In particular, it has minimal variance among all unbiased estimators and is asymptotically normally distributed. Moreover, there is an unbiased estimator for this minimal variance if the total sample size is at least the double learning set size plus two. In this case, we exhibit such an estimator which is another U-statistic. It enjoys, again, various optimality properties and yields an asymptotically exact hypothesis test of the equality of error rates when two learning algorithms are compared. Our statements apply to any deterministic learning algorithms under weak non-degeneracy assumptions. In an application to tuning parameter choice in lasso regression on a gene expression data set, the test does not reject the null hypothesis of equal rates between two different parameters

Block Forests:random forests for blocks of clinical and omics covariate data

Background In the last years more and more multi-omics data are becoming available, that is, data featuring measurements of several types of omics data for each patient. Using multi-omics data as covariate data in outcome prediction is both promising and challenging due to the complex structure of such data. Random forest is a prediction method known for its ability to render complex dependency patterns between the outcome and the covariates. Against this background we developed five candidate random forest variants tailored to multi-omics covariate data. These variants modify the split point selection of random forest to incorporate the block structure of multi-omics data and can be applied to any outcome type for which a random forest variant exists, such as categorical, continuous and survival outcomes. Using 20 publicly available multi-omics data sets with survival outcome we compared the prediction performances of the block forest variants with alternatives. We also considered the common special case of having clinical covariates and measurements of a single omics data type available. Results We identify one variant termed “block forest” that outperformed all other approaches in the comparison study. In particular, it performed significantly better than standard random survival forest (adjusted p-value: 0.027). The two best performing variants have in common that the block choice is randomized in the split point selection procedure. In the case of having clinical covariates and a single omics data type available, the improvements of the variants over random survival forest were larger than in the case of the multi-omics data. The degrees of improvements over random survival forest varied strongly across data sets. Moreover, considering all clinical covariates mandatorily improved the performance. This result should however be interpreted with caution, because the level of predictive information contained in clinical covariates depends on the specific application. Conclusions The new prediction method block forest for multi-omics data can significantly improve the prediction performance of random forest and outperformed alternatives in the comparison. Block forest is particularly effective for the special case of using clinical covariates in combination with measurements of a single omics data type

Interaction Forests: Identifying and exploiting interpretable quantitative and qualitative interaction effects

Although interaction effects can be exploited to improve predictions and allow for valuable insights into covariate interplay, they are given little attention in analysis. We introduce interaction forests, which are a variant of random forests for categorical, continuous, and survival outcomes, explicitly considering quantitative and qualitative interaction effects in bivariable splits performed by the trees constituting the forests. The new effect importance measure (EIM) associated with interaction forests allows ranking of the covariate pairs with respect to their interaction effects' importance for prediction. Using EIM, separate importance value lists for univariable effects, quantitative interaction effects, and qualitative interaction effects are obtained. In the spirit of interpretable machine learning, the bivariable split types of interaction forests target well interpretable interaction effects that are easy to communicate. To learn about the nature of the interplay between identified interacting covariate pairs it is convenient to visualise their estimated bivariable influence. We provide functions that perform this task in the R package diversityForest that implements interaction forests. In a large-scale empirical study using 220 data sets, interaction forests tended to deliver better predictions than conventional random forests and competing random forest variants that use multivariable splitting. In a simulation study, EIM delivered considerably better rankings for the relevant quantitative and qualitative interaction effects than competing approaches. These results indicate that interaction forests are suitable tools for the challenging task of identifying and making use of well interpretable interaction effects in predictive modelling

Benchmark study of feature selection strategies for multi-omics data

BACKGROUND: In the last few years, multi-omics data, that is, datasets containing different types of high-dimensional molecular variables for the same samples, have become increasingly available. To date, several comparison studies focused on feature selection methods for omics data, but to our knowledge, none compared these methods for the special case of multi-omics data. Given that these data have specific structures that differentiate them from single-omics data, it is unclear whether different feature selection strategies may be optimal for such data. In this paper, using 15 cancer multi-omics datasets we compared four filter methods, two embedded methods, and two wrapper methods with respect to their performance in the prediction of a binary outcome in several situations that may affect the prediction results. As classifiers, we used support vector machines and random forests. The methods were compared using repeated fivefold cross-validation. The accuracy, the AUC, and the Brier score served as performance metrics. RESULTS: The results suggested that, first, the chosen number of selected features affects the predictive performance for many feature selection methods but not all. Second, whether the features were selected by data type or from all data types concurrently did not considerably affect the predictive performance, but for some methods, concurrent selection took more time. Third, regardless of which performance measure was considered, the feature selection methods mRMR, the permutation importance of random forests, and the Lasso tended to outperform the other considered methods. Here, mRMR and the permutation importance of random forests already delivered strong predictive performance when considering only a few selected features. Finally, the wrapper methods were computationally much more expensive than the filter and embedded methods. CONCLUSIONS: We recommend the permutation importance of random forests and the filter method mRMR for feature selection using multi-omics data, where, however, mRMR is considerably more computationally costly. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1186/s12859-022-04962-x

Synergistic Effects of Different Levels of Genomic Data for the Staging of Lung Adenocarcinoma: An Illustrative Study

Lung adenocarcinoma (LUAD) is a common and very lethal cancer. Accurate staging is a prerequisite for its effective diagnosis and treatment. Therefore, improving the accuracy of the stage prediction of LUAD patients is of great clinical relevance. Previous works have mainly focused on single genomic data information or a small number of different omics data types concurrently for generating predictive models. A few of them have considered multi-omics data from genome to proteome. We used a publicly available dataset to illustrate the potential of multi-omics data for stage prediction in LUAD. In particular, we investigated the roles of the specific omics data types in the prediction process. We used a self-developed method, Omics-MKL, for stage prediction that combines an existing feature ranking technique Minimum Redundancy and Maximum Relevance (mRMR), which avoids redundancy among the selected features, and multiple kernel learning (MKL), applying different kernels for different omics data types. Each of the considered omics data types individually provided useful prediction results. Moreover, using multi-omics data delivered notably better results than using single-omics data. Gene expression and methylation information seem to play vital roles in the staging of LUAD. The Omics-MKL method retained 70 features after the selection process. Of these, 21 (30%) were methylation features and 34 (48.57%) were gene expression features. Moreover, 18 (25.71%) of the selected features are known to be related to LUAD, and 29 (41.43%) to lung cancer in general. Using multi-omics data from genome to proteome for predicting the stage of LUAD seems promising because each omics data type may improve the accuracy of the predictions. Here, methylation and gene expression data may play particularly important roles