Real amply soft sets and parameter-dependent mean operators with an application to prostate cancer risk assessment

This study develops a real amply soft set framework for prostate cancer risk assessment with explicit emphasis on parameter-dependent modelling. Real amply soft sets are formulated as two-layer full-domain mappings: a value layer assigns real subsets or intervals to active parameters, while an endpoint-preference layer assigns criterion-specific positive real intervals that make the proposed parameter-dependent average (mean) operator sensitive to the lower and upper endpoints of clinical intervals. This construction is designed for nonconstant parameter profiles and allows parameter-specific modelling beyond fixed-valued soft set representation. The clinical validation is rebuilt using the open PI-CAI clinical marksheet. After complete-case filtering, 1040 patients with age, PSA, PSA density, prostate volume, and clinically significant prostate cancer labels are analyzed. Variables are mapped within training folds by clinical screening bounds, prostate volume is converted to an inverse risk-oriented coordinate, thresholds are selected by ROC analysis, and performance is reported using accuracy, sensitivity, specificity, precision, F1-score, and ROC-AUC. The real amply soft set parameter-dependent mean (RASS-PDM) model is compared with PSA-only scoring, PSA-density-only scoring, logistic regression, a linear SVM, and a decision tree. In five-fold stratified cross-validation, the clinically weighted RASS-PDM score obtains ROC-AUC 0.745 , while the AUC-calibrated RASS-PDM score obtains ROC-AUC 0.766 . Logistic regression and linear SVM obtain slightly higher AUCs ( 0.784 and 0.787 ), indicating that the proposed method is best interpreted as a transparent interval-based decision-support score rather than as a replacement for optimized statistical classifiers. The framework also includes scale invariance, boundedness, corrected monotonicity, and Lipschitz stability results for the parameter-dependent mean.