This is the obvious followup to park factors.
Prerequisites for derivations: Data, park factors.
What are LEs?
Unlike with park factors, we don't really need to run league equivalencies to derive value. Instead, we adjust for league in order to figure out how a given performance translates to a different level of play. We normally apply the idea of a Major League Equivalency (MLE) to prospects to translate their statistics onto a major league scale, but the general idea will also apply to the Japan-MLB split, or even the AL-NL gap. The problem is that not only are the talent levels different in different leagues, the skills emphasised can also be dramatically dissimilar. In Japan, for example, the power tool is devalued relative to MLB, moreso that we might expect given the talent differential (this means that we should trust NPB batting averages and on-base-percentages more than we should trust slugging, or home runs). There are also some slightly strange skill interactions that go on in the minor leagues - high walk rates for batters in the lower levels will not always translate to good discipline in the majors unless they are accompanied by actual hitting ability and/or power.
How are they derived?
League equivalencies typically include a correction for park factors and one for the run environment. We've covered those concepts already (bear in mind, however, that park factors are a difficult thing to find for minor league stadiums), so let's skip those steps and focus on correcting for talent. How do we do this? Comparing talent in different leagues is obviously a difficult problem to deal with, but we'll handle it in much the same way as the positional adjustments by looking at players who are promoted or demoted. For the minor leagues we limit ourselves to mid-season transactions in order to avoid the significant changes in talent that can occur in young players. If we compare production between one league and another for all players transferring in both directions between leagues, we can determine the expected statistical line of a player moving up or down into a new league based on his current stats and our numbers. We allow for leagues emphasising different statistics by building our translations around the basic statistics rather than using a more general tool - this can then be used to reconstruct our more advances statistics. Anyway, once we have our league equivalency, we can do another one, and so on and so forth. Eventually we chain these all together and thus have a translatory path from rookie ball to the majors.
Of course, there are still unresolved issues. What of the skill interactions mentioned earlier? The Oakland Athletics appeared to fall afoul of this when translating prospect statistics into MLEs, as their all-discipline, no-power prospects fizzled out in the high minors. There's also the problem of dealing with players who are repeating levels, as this seems to give them an artificial performance boost compared to our estimates of their talent post-promotion. It's not clear how to deal with either concern without doing some major statistical lifting, so it's always a good idea to bear these extra factors in mind. Note also that translating a player's statistics to the majors is not the same thing as projecting future performance: it's simply looking at current performance in a new light and attempting to neutralise the environmental effects on a player.
Player projections (minor leaguers).