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Over the last three days the ERA estimator SIERA has been introduced, complete with explanations of its origins and the derivation of its formula. Now comes one of the most important aspects of building a new metric: making sure it works and testing it against its peers. Any estimator should be both realistic in its modeling and accurate in its predictive ability. These are not mutually exclusive attributes, however, as you could have a situation where a regression on sock height and induced foul balls caught by middle-aged men holding babies somehow predicts park-adjusted ERA better than anything else. Sure, it tested well, but that type of modeling is absurd and illogical; those two variables should not be assumed to have any impact whatsoever on run prevention. This regression means nothing in spite of its hypothetical test results, but situations may also arise in which the most fundamentally modeled statistic tests poorly.

Realistic modeling is based on a combination of truths and assumptions, as we’ve discussed before; the former being that walk and strikeout rates are stable with the latter suggesting that HR/FB is comprised more of luck than skill. During the course of our post yesterday, it seems safe to say that our modeling is sound as the variables used make sense as far as perceived impact on what they seek to measure. The question then becomes one of how we can test the results to determine how it compares to other estimators currently on the market. For the purposes of this study, we used root mean square error testing, a simple but effective method that informs on the average error between an actual and a predicted array of data. In terms of calculations, RMSE is simple enough to do in Excel: take the difference between the actual and predicted term, square it, and take the square root of the average of those previously squared deltas. When compared to other predictors, the lower the RMSE the better.

With the why and how out of the way, who are we testing? To gauge how SIERA fares as an estimator we must compare it to its colleagues. In this forum, that group would consist of FIP, xFIP, QERA, tRA, raw ERA, and park-adjusted ERA itself. To further ensure data integrity, each of the above statistics was calculated from the same Retrosheet dataset. For FIP, the standard +3.2 constant was dependent upon the league and year in question and not a shell figure, with the same obviously true for the xFIP mark. Additionally, as it pertains to xFIP, we spoke with Dave Studeman of The Hardball Times in order to determine that the expected number of home runs to be substituted into the FIP formula is to be calculated through home runs per outfield flies, not the sum of those and popups. Coding for tRA was the final hurdle, but a bunch of help from the invaluable Colin Wyers helped in that regard. One other note on tRA: since it calculates RA/9 instead of ERA, an adjustment needed to be made that essentially resulted in the creation of tERA, which was the normal tRA value discounted by the difference-league and year dependent-between RA and ERA.

Before getting into the results it would be prudent to discuss some assumptions and goals here. To be blunt, our goal was to beat everyone at predicting park-adjusted ERA in the following season, regardless of HR/FB treatment, and beat everyone but FIP and tRA in terms of same-year predictive value. Though it may sound counterintuitive to openly seek a third-place finish in something like this, the rationale is that both FIP and tRA treat HR/FB as skill rather than luck, meaning that no adjustment at all is made to the home run variable; if we assume that a fluky high or low HR/FB is the true skill level of the pitcher and not in line for adjustment, then of course those metrics will be better at same-year predictivity (should totally be a word, damn that red squiggly line) than one that does apply an adjustment. As we discussed earlier in the week, the intra-class correlation for HR/FB is very low, both from an overall standpoint and one in which the individuals are isolated from their respective teams.

Both tRA and FIP use extra information to retroactively guess ERA, but if we wanted to do that we could use BABIP to more precisely derive an ERA prediction. Looking at tRA, we know that it uses the same information that FIP does, but also takes GB, LD, FB, and PU totals and estimates the expected number of runs and outs that each of these batted balls leads to on average.  The assumption is certainly one worth exploring-that pitchers control the rates of each of these batted balls, but that defense and luck determine whether they land in gloves or not.  The problem is that this is mostly going to hinge on the assumption that line-drive rates are persistent for pitchers, because line drives are outs far less frequently. Therefore, a pitcher’s line drive rate is going to affect his tRA significantly.  However, when we look at the ICC of the pitcher’s line-drive rate relative to the rest of his team, we only get .007. In this regard, tRA takes a luck-based stat used in FIP but adds another luck-laden metric in the rate of line drives and uses that as a main determinant of expected ERA.

The ideas are certainly sound, but assumptions must be tested, which is exactly what we did here with SIERA. If everything plays out the way we hoped, then tRA and FIP will best SIERA in post-dicting same year ERA but will lose at subsequent year predictive value. But the goal isn’t so much to lose to both of them in the same-year RMSEs but as much as it is to beat the other competitors that treat HR/FB similarly, which would be xFIP and QERA. With that series of disclaimers out of the way, the tables below show the same-year and subsequent-year RMSEs for the seven metrics in a variety of different categories and subsets. For starters, here is the table of overall results:

```
Stat    YR-Same YR-Next
SIERA    0.957   1.162
tRA      0.755   1.222
FIP      0.773   1.224
xFIP     1.168   1.319
QERA     1.070   1.248
ERA-Park  ----   1.430
ERA      0.094   1.434
```

Our goals came to fruition, as SIERA beat xFIP and QERA in the same-year RMSE test while besting everyone else in terms of predicting park-adjusted ERA in the following year. The latter is very important as a big purpose of these estimators is to base ERA around repeatable skills that would conceivably lead to better results the next time out. Next, we will break the RMSE test results down into a number of subsets to add a level of granularity to the discussion. These subsets were not chosen at random, either, with each being tested for a specific purpose. Most of these purposes involve specific interactions of skills, thus the name Skill-Interactive Earned Run Average. For starters, here are the pitchers with above average strikeout rates:

>= AVG SO/PA

```
Stat    YR-Same YR-Next
SIERA    0.929   1.135
tRA      0.704   1.191
FIP      0.748   1.191
xFIP     1.191   1.275
QERA     1.032   1.191
ERA-Park  ----   1.401
ERA      0.084   1.404
```

When looking at the crop of pitchers with an above average SO/PA, the standing of SIERA relative to the overall group remains unchanged. Next up, the group with an SO/PA greater than or equal to one standard deviation from the mean, classified as really high strikeout guys:

>= AVG SO_PA + 1 SD

```
Stat    YR-Same YR-Next
SIERA    0.866   1.218
tRA      0.689   1.229
FIP      0.722   1.216
xFIP     1.214   1.289
QERA     0.972   1.222
ERA-Park  ----   1.430
ERA      0.071   1.432
```

Here, FIP pulls ever so slightly ahead, but remains very close to SIERA in predicting park-adjusted ERA the following year.  SIERA uses a quadratic term on strikeouts, which makes it particularly good at estimating ERA for medium-high levels of strikeouts but does not add anything particularly helpful for very-high levels of strikeouts. Shifting the focus to walks, how do things shake out when looking at below average walk rates (i.e. pitchers with good control)?

<= AVG BB_PA

```
Stat    YR-Same YR-Next
SIERA    0.871   1.071
tRA      0.725   1.133
FIP      0.719   1.125
xFIP     1.105   1.168
QERA     0.915   1.073
ERA-Park  ----   1.329
ERA      0.085   1.336
```

Interesting results surface here, as SIERA and QERA are very similar as it pertains to walk rates below the league average. Looking at the pitchers with very low walk rates, xFIP and QERA actually predict next-year ERA better than SIERA, while the latter continues to best both of them at same-year predictions. Moving onto ground ball rates, both above average and above one standard deviation from the mean:

> = AVG GB_PA

```
Stat    YR-Same YR-Next
SIERA    1.079   1.153
tRA      0.761   1.205
FIP      0.773   1.202
xFIP     1.071   1.216
QERA     1.088   1.234
ERA-Park  ----   1.419
ERA      0.099   1.422
```

>= AVG GB_PA + 1 SD

```
Stat    YR-Same YR-Next
SIERA    1.003   1.193
tRA      0.844   1.203
FIP      0.823   1.226
xFIP     1.063   1.217
QERA     1.173   1.272
ERA-Park  ----   1.456
ERA      0.091   1.453
```

Same story, different metrics. Next up is a table looking at interactions between skills. It looks for low strikeout but high grounder and high walk pitchers, the kinds of hurlers we would expect to allow plenty of baserunners and rely on fielded grounders to wipe the slate clean:

>= AVG GB_PA, BB_PA & <= AVG SO_PA

```
Stat    YR-Same YR-Next
SIERA    1.123   1.299
tRA      0.883   1.318
FIP      0.875   1.305
xFIP     1.178   1.408
QERA     1.294   1.477
ERA-Park  ----   1.551
ERA      0.121   1.553
```

>= AVG GB_PA, BB_PA

```
Stat    YR-Same YR-Next
SIERA    0.876   1.064
tRA      0.698   1.141
FIP      0.708   1.127
xFIP     0.982   1.090
QERA     0.912   1.065
ERA-Park  ----   1.323
ERA      0.086   1.329
```

In both of these tables, we see the results we would expect. SIERA, being a Skill-Interactive Earned Run Average, does exactly what it should here: beat other estimators at measuring the skill components of pitcher performance that interact with each other. Moving on to elite pitchers:

<= 3.50 ERA-Park

```
Stat    YR-Same YR-Next
SIERA    1.221   1.142
tRA      0.833   1.208
FIP      0.873   1.203
xFIP     1.601   1.235
QERA     1.439   1.180
ERA-Park  ----   1.536
ERA      0.063   1.535
```

With its ability to properly estimate the effects of very strong skill levels, SIERA again beats other estimators that treat HR/FB as luck neutral in predicting same-year park-adjusted ERA and leads all other estimators in predicting next-year park-adjusted ERA.

Looking through all of these different tests, it is apparent not only that SIERA is the best ERA estimator currently available, but specifically that it is exceptionally strong at measuring the skill level of specialized kinds of pitchers. To make this less abstract, tomorrow’s fifth and final article in our introductory series will discuss three specific examples of pitchers who are unique in their skill sets, and are particularly troublesome for other estimators. SIERA will perform excellently with all three, which should leave you with a solid understanding of what SIERA does and why it is so important.

10/17
10/17
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