Modern neural networks can assign high confidence to inputs drawn from outside the training distribution, posing threats to models in real-world deployments. While much research attention has been placed on designing new out-of-distribution (OOD) detection methods, the precise definition of OOD is often left in vagueness and falls short of the desired notion of OOD in reality. In this paper, we present a new formalization and model the data shifts by taking into account both the invariant and environmental (spurious) features. Under such formalization, we systematically investigate how spurious correlation in the training set impacts OOD detection. Our results suggest that the detection performance is severely worsened when the correlation between spurious features and labels is increased in the training set. We further show insights on detection methods that are more effective in reducing the impact of spurious correlation, and provide theoretical analysis on why reliance on environmental features leads to high OOD detection error. Our work aims to facilitate better understanding of OOD samples and their formalization, as well as the exploration of methods that enhance OOD detection. Code is available at https://github.com/deeplearning-wisc/Spurious_OOD.