AbstractSpurious correlations are a threat to the trustworthiness of natural language processing systems, motivating research into methods for identifying and eliminating them. However, addressing the problem of spurious correlations requires more clarity on what they are and how they arise in language data. Gardner et al (2021) argue that the compositional nature of language implies that all correlations between labels and individual “input features” are spurious. This paper analyzes this proposal in the context of a toy example, demonstrating three distinct conditions that can give rise to feature-label correlations in a simple PCFG. Linking the toy example to a structured causal model shows that (1) feature-label correlations can arise even when the label is invariant to interventions on the feature, and (2) feature-label correlations may be absent even when the label is sensitive to interventions on the feature. Because input features will be individually correlated with labels in all but very rare circumstances, domain knowledge must be applied to identify spurious correlations that pose genuine robustness threats.