**Persistence Equations**

Baseball analysts frequently like to distinguish between a player’s performance record and his “true talent”. The latter of these – a player’s “true talent” – is used to indicate how much of a player’s performance is due purely to the player’s skill. Not all events are equally indicative of a player’s skill. For example, clutch hitting is certainly “real” in the sense that some hits occur in more clutch situations than others (Bobby Thomson, 1951; Bucky Dent, 1978), but most analysts believe that very few players have any real “skill” at clutch hitting, beyond their general skill at hitting.Identifying “True Talent”

The most common technique in sabermetric analysis of evaluating what is a skill and what is not is to measure whether a particular skill persists from year to year. In terms of Player won-lost records, for example, one measure of the extent to which a particular factor is a skill could be the extent to which a player’s winning percentage persists from year to year.

In theory, the extent to which a factor persists should be a very good indicator of the extent to which that factor represents an actual skill. In practice, however, comparing one year to another can be problematic. A straight comparison of one year to another is only valid if the player’s underlying “true talent” skill level is the same both years. Otherwise, differences in “true talent” between the years will be incorrectly viewed as evidence of unexplained, or random, variance between the two years. In such a case, the extent to which something involves a real skill will likely be underestimated. But, of course, we know that players’ talent levels change from year to year. Young players improve as they get older, while old players lose their skills. Players may be injured one year but healthy the next. Or players’ actual value may be affected by a change in home ballpark, from, for example, a ballpark to which a player is particularly well-suited to one which adversely affects that player’s specific strengths.

Player won-lost records are not constructed from aggregated year-end data, however, but are, instead, constructed from play-by-play data. Rather than comparing results across years, therefore, it is possible for me to compare results across plays. As a general rule, players’ “true talent” should be much more stable from play to play than from year to year.My Persistence Equations

For the purpose of developing what I call “Persistence Equations”, I divide the plays that took place in a particular season into two pools: odd and even. That is, the first plate appearance of the season is identified as play number 1, the next plate appearance is play number 2, etc. Even-numbered plays (2, 4, …) go into one pool; odd-numbered plays go into the other one.

To evaluate the persistence of skills, I can then fit a simple equation which attempts to explain the relevant factor (e.g., winning percentage by component, inter-game win adjustment, etc.) on even plays as a function of the same factor for odd plays. That is,

(Factor A)_{Even} = a + b•(Factor A)_{Odd}

The constant term, a, can be thought of as a measure of the extent to which Factor A regresses toward the mean. That is, one could re-write the Persistence Equation as follows:

The coefficient b in the Persistence Equation measures the persistence of Factor A between the two samples (even plays v. odd plays). The value of Factor A in the odd and the even period here are both samples of Factor A’s true value. Sample statistics have a tendency to trend toward their long-run value as the sample size increases. Statisticians call this “regression to the mean”.Interpretation of Persistence Equation

The constant term, a, can be thought of as a measure of the extent to which Factor A regresses toward the mean. That is, one could re-write the Persistence Equation as follows:

(Factor A)_{Even} = b•(Factor A)_{Odd} + (1-b)•(Factor A)_{Baseline}

where (Factor A)_{Baseline} represents a baseline toward which Factor A regresses over time.

There are two relevant results in interpreting the extent to which Factor A persists. The one most commonly used by sabermetricians is the correlation coefficient, or r (or r^{2}). The value, R^{2}, measures the percentage of variation in the left-hand side variable – (Factor A)_{Even} – that can be explained by the right-hand side variable(s) – i.e., (Factor A)_{Odd}. This provides some indication of the magnitude of the persistence of Factor A.

To assess the significance of the persistence, however, one must look at the significance of the persistence coefficient, b. The estimated value of b will have a standard error associated with it. If one divides b by this standard error, the resulting variable is called a t-statistic. The larger the t-statistic (in absolute value), the less likely that the true persistence coefficient is zero. As a (somewhat crude) rule of thumb, if the t-statistic is greater than 2, then we can be 95% certain that the true value of b is greater than zero (given that certain statistical assumptions about our equation are true).

where (Factor A)

There are two relevant results in interpreting the extent to which Factor A persists. The one most commonly used by sabermetricians is the correlation coefficient, or r (or r

To assess the significance of the persistence, however, one must look at the significance of the persistence coefficient, b. The estimated value of b will have a standard error associated with it. If one divides b by this standard error, the resulting variable is called a t-statistic. The larger the t-statistic (in absolute value), the less likely that the true persistence coefficient is zero. As a (somewhat crude) rule of thumb, if the t-statistic is greater than 2, then we can be 95% certain that the true value of b is greater than zero (given that certain statistical assumptions about our equation are true).

The basic Persistence Equation above:Complications Estimating Persistence Equations

(Factor A)_{Even} = a + b•(Factor A)_{Odd}

can be solved by Ordinary Least Squares (OLS), which is one of the most basic statistical regression procedures out there. There are, however, two additional complications associated with estimating Persistence Equations.

The first issue is that, in order to ensure that the estimated value b is not biased, the persistence equation should be fully specified. That is, if there are other variables that can be expected to affect (Factor A)_{Even}, these variables should be included on the right-hand side of the persistence equation along with (Factor A)_{Odd}. In general, this is not a big deal for most of the Persistence Equations that I estimate here, but it can be an issue in general regression analysis and is always worth keeping in mind.

The second issue is much more of an issue with the Persistence Equations that I estimate. The validity of OLS as an estimation technique is dependent on several assumptions about the distribution of the residual, or error, term in the persistence equation^{*}. One of these assumptions is that the variance of the error term is constant across all observations. That is, for example, OLS is only valid if the unexplained variation in player winning percentage is equal for all players. In this case, however, not only do we not want to assume this, but we actually know that it’s wrong. Unexplained variation declines as the number of player games increases. Fortunately, there is a very easy way to adjust for this. Instead of OLS, I use Weighted Least Squares (WLS). This weights each observation by the number of player games over which the Factor has been compiled^{**}, squared^{***}. In this way, the results for players with more games played are weighted more heavily than players with fewer games.

^{*} To be technically correct, the persistence equation should be written as follows:

can be solved by Ordinary Least Squares (OLS), which is one of the most basic statistical regression procedures out there. There are, however, two additional complications associated with estimating Persistence Equations.

The first issue is that, in order to ensure that the estimated value b is not biased, the persistence equation should be fully specified. That is, if there are other variables that can be expected to affect (Factor A)

The second issue is much more of an issue with the Persistence Equations that I estimate. The validity of OLS as an estimation technique is dependent on several assumptions about the distribution of the residual, or error, term in the persistence equation

(Factor A)_{Even} = a + b•(Factor A)_{Odd} + e

where e is the “error” or “residual” term that measures unexplained variation in (Factor A)_{Even}. The appropriateness of OLS is then dependent on a set of assumptions regarding the distribution of e.

^{**} The number of games is defined as the harmonic mean of the games over which (Factor A)_{Odd} and (Factor A)_{Even} are compiled.

^{***}The decision to square the number of games in the weighting matrix was determined by empirical experimentation, which considered several alternative weighting schemes, based on the number of games (total games, the log of games, games squared, et al.).

Persistence Equations form the basis for dividing shared Player Game Points between batters and baserunners as well as between pitchers and fielders for several components. I also calculate and discuss specific Persistence Equations for Inter-Game Win Adjustments as a measure of “clutch”.

*All articles are written so that they pull data directly from the most recent version of the Player won-lost database. Hence, any numbers cited within these articles should automatically incorporate the most recent update to Player won-lost records. In some cases, however, the accompanying text may have been written based on previous versions of Player won-lost records. I apologize if this results in non-sensical text in any cases.*

where e is the “error” or “residual” term that measures unexplained variation in (Factor A)

Persistence Equations form the basis for dividing shared Player Game Points between batters and baserunners as well as between pitchers and fielders for several components. I also calculate and discuss specific Persistence Equations for Inter-Game Win Adjustments as a measure of “clutch”.