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Objectives: To investigate machine-learning classifiers and interpretable models using chest CT for detection of COVID-19 and differentiation from other pneumonias, ILD and normal CTs. Methods: Our retrospective multi-institutional study obtained 2096 chest CTs from 16 institutions (including 1077 COVID-19 patients). Training/testing cohorts included 927/100 COVID-19, 388/33 ILD, 189/33 other pneumonias, and 559/34 normal (no pathologies) CTs. A metric-based approach for classification of COVID-19 used interpretable features, relying on logistic regression and random forests. A deep learning-based classifier differentiated COVID-19 via 3D features extracted directly from CT attenuation and probability distribution of airspace opacities. Results: Most discriminative features of COVID-19 are percentage of airspace opacity and peripheral and basal predominant opacities, concordant with the typical characterization of COVID-19 in the literature. Unsupervised hierarchical clustering compares feature distribution across COVID-19 and control cohorts. The metrics-based classifier achieved AUC=0.83, sensitivity=0.74, and specificity=0.79 of versus respectively 0.93, 0.90, and 0.83 for the DL-based classifier. Most of ambiguity comes from non-COVID-19 pneumonia with manifestations that overlap with COVID-19, as well as mild COVID-19 cases. Non-COVID-19 classification performance is 91% for ILD, 64% for other pneumonias and 94% for no pathologies, which demonstrates the robustness of our method against different compositions of control groups. Conclusions: Our new method accurately discriminates COVID-19 from other types of pneumonia, ILD, and no pathologies CTs, using quantitative imaging features derived from chest CT, while balancing interpretability of results and classification performance, and therefore may be useful to facilitate diagnosis of COVID-19.