Reliable rejection of local realism is highly desirable not only for understanding the foundation of quantum mechanics but also for facilitating quantum information tasks, such as quantum key distribution and randomness generation. The confidence for rejecting local realism is usually estimated by comparing Bell-inequality violation with experimental standard deviation. However, as we show, this method can overestimate the rejection confidence and so is not valid. We propose a prediction-based ratio (PBR) protocol to lower bound the rejection confidence directly from the full statistics of experimental results. The PBR protocol works even if the prepared quantum state, measurement settings, and local realistic models vary arbitrarily during an experiment. If the prepared state and measurement settings do not vary, the rejection-confidence bound is asymptotically tight. For comparison, confidence bounds derived from Bell-inequality violations using martingale theory [R. Gill, arXiv:quant-ph/0110137] are studied. It is found that bounds from martingale theory are not tight. We also apply the PBR protocol to the experimental loophole-free Bell test reported in [Shalm et al., PRL 115, 250402 (2015)]. Our result shows that one can reject the hypothesis of local realism at the significant level 1.5e-11.