Background: The high dimensionality of radiomic feature sets, the variability in radiomic feature types and potentially high computational requirements all underscore the need for an effective method to identify the smallest set of predictive features for a given clinical problem. Purpose: Develop a methodology and tools to identify and explain the smallest set of predictive radiomic features. Materials and Methods: 89,714 radiomic features were extracted from five cancer datasets: low-grade glioma, meningioma, non-small cell lung cancer (NSCLC), and two renal cell carcinoma cohorts (n=2104). Features were categorized by computational complexity into morphological, intensity, texture, linear filters, and nonlinear filters. Models were trained and evaluated on each complexity level using the area under the curve (AUC). The most informative features were identified, and their importance was explained. The optimal complexity level and associated most informative features were identified using systematic statistical significance analyses and a false discovery avoidance procedure, respectively. Their predictive importance was explained using a novel tree-based method. Results: MEDimage, a new open-source tool, was developed to facilitate radiomic studies. Morphological features were optimal for MRI-based meningioma (AUC: 0.65) and low-grade glioma (AUC: 0.68). Intensity features were optimal for CECT-based renal cell carcinoma (AUC: 0.82) and CT-based NSCLC (AUC: 0.76). Texture features were optimal for MRI-based renal cell carcinoma (AUC: 0.72). Tuning the Hounsfield unit range improved results for CECT-based renal cell carcinoma (AUC: 0.86). Conclusion: Our proposed methodology and software can estimate the optimal radiomics complexity level for specific medical outcomes, potentially simplifying the use of radiomics in predictive modeling across various contexts.