Fig 1.
Alternative approaches to polygenic risk scoring vs hiPRS.
Strengths (first row) and Weaknesses (second row) of three main categories of PRS methods discussed in the Introduction. The green tick signals that the given point of strength applies to hiPRS as well. The blue arrow signals a point of weaknesses that hiPRS algorithm does not suffer, or some aspect that the algorithm was specifically designed to solve.
Fig 2.
(A) Input data is a list of genotype-level SNPs. (B) Focusing on the positive class only, the algorithm exploits FIM (apriori algorithm) to build a list of candidate interactions of any desired order, retaining those that have an empirical frequency above a given threshold δ. This leads to a filtered set of terms in the form of sequences of pairs of SNP and associated categorical level (i.e., allele frequency in this example). The sequences can include from a single SNP-allele pair up to a maximum number of pairs defined by the user (lmax). (C) The whole training data is then scanned, searching for these sequences and deriving a re-encoded dataset where interaction terms are binary features (i.e., 1 if sequence i is observed in j-th patient genotype, 0 otherwise). From this dataset we can compute the MI between each interaction and the outcome and (D) obtain a ranked list (Iδ) based on this metric. (E) Starting from the interaction at the top of Iδ, hiPRS constructs IK, selecting K (where K is user-specified) terms through the greedy optimization of the ratio between MI (relevance) and a suitable measure of similarity for interactions (redundancy) (cf. Algorithm 1, Materials and methods). This leads to a set of predictive, yet diverse, interactions that (F) we use to define the score weighting their contribution by fitting a LR model and retaining the corresponding β coefficients.
Fig 3.
Simulation data generating rules.
Graphical representation of the three rules that determine the positive class. Cases are obtained when A ∨ B ∨ C. More details on the generative model are provided in the Materials and Methods Section.
Fig 4.
hiPRS results on risk prediction against benchmark PRSs and ML approaches.
AUC (left) and AP (right) performance distributions of 30 independent trials. In grey the three traditional penalized PRSs approaches with additive effects only; in violet the two ML algorithms (SVM-Behravan and DNN-Badre); in pink glinternet algorithm for two model dimensions (3 interactions, i.e. 36 terms, and 8 interactions, i.e. 93 terms); in green hiPRS for K = 10 and K = 40.
Fig 5.
(A) Absolute frequency of the generative rules in the training data, limited to the positive class. (B) Interactions selected by hiPRS with K = 10 and corresponding β coefficients. (C) Coefficients of the glinternet model with 3 interaction terms: main effects are in gray, interactions in yellow. (D) Lists of SNPs selected by SVM-Behravan during its five internal cross validations, cf. Benchmark Algorithms in the Materials and Methods Section. Note: reported results are limited to one simulation among the 30 randomly generated datasets.
Fig 6.
Average performance of hiPRS in terms of AUC and AP for variable sample size (A.1 and A.2), class imbalance (B.1 and B.2) and missing heritability, i.e. noise (C.1 and C.2). Confidence bands are at the 95% level. The x-axis is in logarithmic scale for panels C.1 and C.2.
Fig 7.
Results and comparisons for a real biological setting.
AUC (left) and AP (right) performance distributions of 30 independent trials, sampled according to a real biological mechanism where SNPs regulate factors associated to atrial fibrillation. In grey, a traditional PRS with additive effects only; in violet the ML based algorithm, DNN-Badre; in pink glinternet (31 logistic terms); in green hiPRS (K = 13).
Table 1.
Fitting times in a real biological setting.
Fitting times for hiPRS and the benchmarks algorithms, averaged across 30 independent simulations.
Fig 8.
Left panel: AUCs obtained by hiPRS and a benchmark model during cross-validation (four folds). Average AUCs are 0.72 and 0.57 respectively for hiPRS and the benchmark model. Right panel: interactions selected by hiPRS, and corresponding effect-sizes, when fitting the model on the whole dataset. In grey are reported the pathways each SNP belongs to (cf. Materials and methods).
Fig 9.
Simulated atrial fibrillation data.
Schema of the distribution of cases (red section of the pie-plots) and controls (light blue section) in the case-control study described in [46], for each multilocus genotype combination of M235, T174M and ID. Dark gray cells are associated to higher risk of atrial fibrillation, while light gray cells are associated to protective combinations. The schema proposed here is based on the image reported in [45].
Table 2.
Patient characteristics of the subsample analysed in the DACHS case study.
Table 3.
SNPs considered in the DACHS case study.
Fig 10.
Fitting times of hiPRS for different values of δ and different numbers of SNPs (x-axis). For better readability, the y-axis is reported in logarithmic scale. Dashed-lines are obtained via least-squares.