Pitching runs I think came from Total Baseball. It was something love PR= pitchers innings/9* (lg ERA- indiv ERA). BBRef should be using adjustments for PPF and possibly other adjustments, but I couldn't find it.

If you used this PR= pitchers innings/9* (lg ERA- indiv ERA) * PPF, you will get close, but I can't get it exactly. I tried Grove for example in 1936 and got 65.9, but BBRef has 66.9. I think that was coincidence, though. I doubt they use that formula that way.

Moreover, I don't they use PR in the WAA calculation, since they have a separate Defense component. I would think that they use Runs scored, not earned runs.

Better methods would take actual innings in actual parks and allocate based on what each opponent scored (on average) adjusted by the park. That would give a weighted average. So, unlike the original TB formula, it would expect more scoring if you faced a high scoring team more than average and more scoring if you pitched in high scoring parks more than average. Implicitly, your own team would be removed from the data since as a pitcher you could not face them.

I guess I haven't helped!

You did help, actually, because I've been fooling around with a pitching runs metric. When I saw it on BBRef, I freaked out a little because I thought it was all in vain. Mine has to do with base runners, which translate into runs. Yet it is not a direct composition of runs. In fact, that's why I started the other thread about base runners and kept referencing 2007, as I've been looking at the 2007 closers.

When it comes to Pitching Runs, my favorite reference is Batters Faced Pitcher
as the denominator, with runs allowed [all runs, not just earned runs] as the
numerator.

It looks like a lazy man's shortcut at first glance; but that glance does reveal
quite a lot about the pitcher and the offensive climate when we are investigating
his work. I've selected 7 pitcher-seasons that would probably get nods for
exemplifying pitcher excellence. Great as they all are, we can determine a lot
from a sample performance line:

1. Pitcher relative work load, from the BFP denominator;
2. Offensive batting climate, from the LG average R/BFP percentage;
3. A direct and relevant comparative result by simply applying LG rate
to pitcher's BFP number;
4. Simple subtraction LG R - Pitcher R is simply Runs.

Here are the seven samples:

1. Grove, 1931: 84 [runs]/1160 [BFP] = .0724; LG = .1301 * Grove BFP
at 1160 = .1301*1160 = 151 Runs. 151-84 = Grove = 67 Runs Saved.

2. Gibson, 1968: 49/1161 = .0422; LG .0919 + 1161 = 107 Runs.
107-49 = Gibson + 58 Runs Saved.

3. W. Johnson, 1913: 56/1271 = .0441; LG = .1085* 1271 = 138.
138-56 = W. Johnson +82 Runs Saved.

4. R. Johnson, 2002: 78/1035 = .0754; LG = .1162*1035 = 120.
120-78 = +42 Runs Saved.

5. Guidry, 1978: 61/1057 = .0577; LG = .1107 * 1057 = 117 Runs.
117-61 = Guidry +56 Runs Saved.

6. P. Martinez, 2000: 44/817 = .0539; LG = .1344 * 817 = 110 Runs.
110-44 = Martinez +66 Runs Saved.

7. Verlander, 2011: 73/969 = .0753; LG = .1162 * 969 = 113 Runs.
113-73 = Verlander + 40 Runs Saved.

Added as a post script:

One wanting to refine the runs saved for each example above could relate
the numbers of runs saved, as presented, to each pitcher's individual IP
during the seasons being explored. Or, one could divide the IP by nine
and relate the runs saved to a "game" basis.

Example: Grove, 1931, at +67. Per IP = 67/288.2 = .232 RS/IP. Or,
288.2/9 = 32.02 "games;" so 67/32.02 = 2.09 RS/G. It's simple, direct
and versatile ... and specific to the pitcher and his LG, which could be
expanded to the season MLB averages rather than LG.