Correlations between chlorophyll a and total phosphorus in freshwater ecosystems were first documented in the 1960s and have been used since then to infer phosphorus limitation, build simple models, and develop management targets. Often these correlations are considered indicative of a cause-effect relationship. However, many scientists regard the use of these associations for modeling and inference to be misleading due to their potentially spurious nature. Using data from Saginaw Bay, Lake Huron, we examine the relationship among chlorophyll a, total phosphorus, and algal biomass measurements. We apply graphical models and recently developed "structure learning" principles that use conditional dependencies to help identify causal relationships among observational data. The spurious relationship suspected by some is not supported by our data, whereas a direct relationship between chlorophyll a and total phosphorus is always supported, and an additional indirect relationship with an algal biomass intermediary is plausible under some circumstances. Thus, we conclude that these correlations are useful for simple model building but encourage the use of modern statistical methods to avoid common model-assumption violations.