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Magnetic Resonance Imaging (MRI) has become an important technique in the clinic for the visualization, detection, and diagnosis of various diseases. However, one bottleneck limitation of MRI is the relatively slow data acquisition process. Fast MRI based on k-space undersampling and high-quality image reconstruction has been widely utilized, and many deep learning-based methods have been developed in recent years. Although promising results have been achieved, most existing methods require fully-sampled reference data for training the deep learning models. Unfortunately, fully-sampled MRI data are difficult if not impossible to obtain in real-world applications. To address this issue, we propose a data refinement framework for self-supervised MR image reconstruction. Specifically, we first analyze the reason of the performance gap between self-supervised and supervised methods and identify that the bias in the training datasets between the two is one major factor. Then, we design an effective self-supervised training data refinement method to reduce this data bias. With the data refinement, an enhanced self-supervised MR image reconstruction framework is developed to prompt accurate MR imaging. We evaluate our method on an in-vivo MRI dataset. Experimental results show that without utilizing any fully sampled MRI data, our self-supervised framework possesses strong capabilities in capturing image details and structures at high acceleration factors.