Investigating Ranks, Monotonicity, and Spearman's Rho with R

The Problem I have a bunch of data that can be categorized into many small groups. Each small group has a set of values for an ordered set of intervals. Having observed that the values for most groups seem to increase with the order of the interval, I hypothesize that their is a statistically-significant, monotonically increasing trend. An Analogy To make this abstract problem more relatable, imagine the following scenario.