Advanced Metrics: WAR, OPS+
Advanced Metrics: WAR, OPS+, and Plus-Minus
Modern sports analytics seeks a single number per player measuring total contribution above a replacement level — a freely available minor-league or bench substitute. Wins Above Replacement (WAR) combines offense, defense, baserunning, and positional adjustment into a common unit: wins a team would lose by replacing the player with a replacement-level peer for the same playing time.
The baseball construction is $\text{WAR} = (\text{Batting runs} + \text{Base running} + \text{Fielding} + \text{Positional adj} + \text{Replacement adj})/\text{Runs per win}$, with $\text{Runs per win} \approx 10$ in MLB. Basketball analogues include Box Plus-Minus (BPM), RAPM (regularized adjusted plus-minus), and VORP; football uses EPA (Expected Points Added) and DVOA.
OPS+ and ERA+ are simpler rate metrics: league- and park-adjusted indices centered at 100. An OPS+ of 150 means 50% better than league average; ERA+ of 120 means 20% better than average pitching. Always check the reliability of the metric before making decisions: WAR variants differ by 1–2 wins for the same player, and one-season defensive components are noisy.
$$\text{WAR} = \frac{\text{Batting} + \text{BsR} + \text{Fld} + \text{PosAdj} + \text{RepAdj}}{\text{Runs per Win}}$$
Runs per win scales as $\approx 9 + \sqrt{RS/G \cdot 9}$ (Pythagorean derivative); ≈10 in MLB.
$\mathrm{OPS} = \mathrm{OBP} + \mathrm{SLG}$.
$\mathrm{OPS+} = 100 \cdot (\mathrm{OBP}/\mathrm{lgOBP} + \mathrm{SLG}/\mathrm{lgSLG} - 1)$ adjusted for park factor. Scale: 100 = league average, 150 = 50% above.
Batting runs +35, BsR +3, Fld +5, PosAdj +2, RepAdj +20. Runs per win 10. WAR?
$\text{WAR} = (35 + 3 + 5 + 2 + 20)/10 = 65/10 = 6.5$.
A 6.5-WAR season is MVP-caliber.
Player OBP .400, SLG .550. League OBP .320, SLG .410. Park factor 1.0. Compute OPS+.
$\mathrm{OPS+} = 100(0.4/0.32 + 0.55/0.41 - 1) = 100(1.25 + 1.341 - 1) = 100(1.591) = 159.1$.
≈ 59% better than league average.
Player A's 5-year RAPM is +4.2 points per 100 possessions (±0.8 SE). Is he a star?
+4 per 100 is top-15 territory. 95% CI $[2.6, 5.8]$ clearly positive. Combined with minutes played (say 2000 per year), yields +84 net points per season — roughly 3 wins at NBA pace of 28 points per win.
Practice Problems
Show Answer Key
1. $(20+0-5-5+20)/10 = 3.0$ WAR — solid starter.
2. OPS+ 140 over 600 PA ≈ 30 batting runs ≈ 3 wins above average.
3. Play-by-play defensive sample sizes are small, and positioning effects are hard to credit individually; regressed UZR, DRS, OAA all have wide CIs.
4. $3.5 \times 9\text{M} = 31.5\text{M}$/year.
5. $\approx 9 + \sqrt{5\cdot 9} = 9 + 6.71 = 15.7$ — but empirical MLB is lower (~10) because of diminishing returns.
6. Different methodologies for defense and positional adjustments; averaging reduces systematic bias of any single source.