Under which of the following conditions would you need to use the Spearman correlation coefficient and not the Pearson correlation coefficient?

The Spearman correlation coefficient is particularly useful when the relationship between two variables is not linear. It assesses how well the relationship between the two variables can be described using a monotonic function. This means that while the variables may not be related in a linear way, Spearman can still detect a relationship as long as it consistently increases or decreases.

In contrast, the Pearson correlation coefficient assumes that both variables are normally distributed and that there is a linear relationship between them. If such assumptions are violated—like in the case of non-linear relationships—using Pearson can lead to misleading results.

Therefore, when faced with non-linear relationships, opting for the Spearman correlation is more suitable as it ranks the data and focuses on the order of values rather than their specific magnitudes, enhancing the ability to detect relationships that are monotonic rather than strictly linear.