Wednesday, November 30, 2005

The Fallacy of the Ecological Fallacy

From time to time, someone who has a background in formal statistics will claim that applying various measures tested at the team-level to individual players(usually a run estimator) is falling prey to the Ecological Fallacy and is thus invalid.

Not having a formal statistics background, it may be hazardous to talk about something that I don’t fully understand. But I can tell you that to the extent that I understand the ecological fallacy, the idea that it applies to individual runs created estimates is hokum.

According to this link, the ecological fallacy occurs when “making an unsupported generalization from group data to individual behavior”. They then use an example of voting. One community has 25% who make over \$100K a year, and 25% who vote Republican. Another has 75% who make over \$100K and 75% who vote Republican. To use this data to conclude that there is a perfect correlation between individuals voting Republican and making over \$100K would be the ecological fallacy. In fact, they show how the data could be distributed so that the correlation between individuals voting Republican and making over \$100K is actually negative.

People will then go on to claim that since Runs Created methods are tested on teams, it is wrong to apply them to individuals and assume accuracy. It is true that multiplicative methods like Runs Created and Base Runs make assumptions about how runs are created that are true when applied to teams but cannot be applied to individuals(the well-documented problem of driving yourself in; Barry Bonds’ high on base factor interacts with his high advancement factor in RC, but in reality interacts with the production of his teammates). It is also true that regression equations have many potential pitfalls when applied to teams, let alone taking team regressions and applying them to individuals. However, these limitations are well known by most sabermetricians(although some stubbornly continue to use James’ RC for individual hitters).

The ecological fallacy claim, though, is extended by some to every run estimator that is verified against team data. The claim is that there “need not be little to no connection between team-level functions and player-level functions”. I also saw a critic point out once that run estimators did not do a good job of predicting individual runs scored.

My retort was that the low temperature today in Mozambique did not do a good job of predicting individual runs scored either. To assume that the team runs scored function and the individual runs scored function are the same is to be ignorant of the facts of baseball. A walk and a single have an equal run-scoring value for an individual, and a home run will always have an individual run-scoring value of 1. This is not true for a team, because, except in the case of the home run, it takes another player to come along and drive his teammate in. In the team case, all of these individuals stats are aggregated. The home run by one batter not only scores him, it scores any teammates on base. And therefore the act of scoring runs, for a team, incorporates advancement value as well. A single will create more runs, in average circumstances, then will a walk.

Therefore, when we have a formula that estimates runs scored for a team, it does not estimate the same function as runs scored for a player. It instead approximates another function that we choose to call “runs created” or “runs produced” or what have you. Now it could be claimed, I suppose, that the runs created function cannot be applied to individuals? But why not? If a double creates .8 runs for a team, and a hitter hits a double, why can’t we credit him with creating .8 of the team’s runs? All we are doing is assigning what we know are properly generated coefficients for the team to the player who actually delivered them. Or you can look at it, in the case of theoretical team RC, that we are isolating the player’s contribution by comparing team runs scored with him to team runs scored without him.

Furthermore, the individual runs created function and the team runs scored function are the same function. They have to be. Who causes the team to score runs, the tooth fairy? In the case of the voting situation which was said to be the ecological fallacy, you are artificially forming groups of people that don’t actually interact with each other. I can vote Republican, and you can vote Republican, but we’re not working together in that. You can vote Democrat and I can still vote Republican; our choices are independent. Then you make this group that voted Republican, and look at the their income, and yes, you can reach misleading conclusions.

The point I’m trying to make is that voting is not a community-level function, and therefore it is wrong to attribute the community level data pattern to individuals. People vote as individuals, not as communities. But scoring runs is a team-level function. People create runs as teams, each contributing. If we use a different voting analogy, that of the electoral college, people cast electoral votes as states. And therefore we can break down how much of the electoral vote of Montana that each citizen was responsible for(one share of however many if they voted for the winning candidate, zero if they did not). And that’s what we are doing by looking at individual runs created.

I think the problem, and I don’t mean this to apply to all statisticians who dabble in sabermetrics, but to some, particularly those who don’t have a strong traditional sabermetric background to go along with their statistical knowledge, is that they tend to take all of the things they know can often happen in statistical practice and apply them to sabermetrics, without seeing whether the conditions are in place. In the same way, they will use statistical methods like regression when they are not necessary. If you are studying phenomenon that you don’t have a good theory on, then regression can be a great tool. But if you are studying a baseball offense, you’re better off constructing a logical expression of the run scoring process like Base Runs or using the base/out table to construct Linear Weights. You don’t need a regression to ascertain the run values of events--baseball offenses are complex, but they are not nearly as complex as many of the other phenomenons in the world.