Chris Shepherd's most recent PSA on drafting running backs spurred a lot of discussion and nuanced arguments. There is not a purely analytical way to identify which position to draft when, as it varies by team need, relative draft strength of position, and the individual skill set of the player being considered.
However, using an analytical approach can help estimate a baseline value for drafting early/late for a specific position and thus provide a starting point to valuing a specific pick.
Using regular season data from 2000 - 2016, I plotted average rush yards per game for running backs by the round in which they were drafted.
As you can see, the yardage totals have a clear downward trend in rounds 1 - 4 and then level off in the later rounds (U = undrafted). So, at first blush it looks like drafting an RB earlier is better.
However, volume stats are always tricky and can easily be biased in many different ways. For example, it is common to draft backup positions in later rounds and so maybe all this chart is saying is that starters carry the ball more often than depth players.
Therefore, let's try to measure quality instead of quantity by using efficiency stats and look at 2 common metrics, yards per carry and explosive runs (eeew).
YPC (left hand axis) doesn't change much based on player draft round. It dips a bit for rounds 4, 6, and undrafted players, but it stays fairly level at about 4.2 for all other rounds.
Explosive run rate (defined as rushes >= 10 yards divided by total carries) is charted using the right hand axis. It also is fairly stable at around 10.2% with a notable spike in round 5 and drop in round 6.
These stats suggest there is not much value over replacement for RBs. However, YPC and explosive plays are pretty bad stats when it comes to measuring impact on wins. A much better stat is rushing success rate, which is basically the percent of carries that result in a TD, new set of downs or a good chunk of the yardage to go on 1st or 2nd downs.
Rushing success rate also supports a low value over replacement for running backs.
Now, this in no way some sort of proof that it doesn't matter when you pick an RB, but it does add credence to the idea that there is value in the later rounds (maybe a lot of value) and should provide food for thought when comparing to other positions that have much higher value over replacement.