How important is batting order?

Everyone knows it’s good for a hitter to hit high in the lineup. But from a fantasy perspective, how much does where you hit affect production?

I did a little study of the American League in 2006, profiling all 14 teams according to production from each lineup spot (leadoff, #2, #3, etc) and what I found was pretty interesting.

• The AL average line was .275/.339/.437, and teams scored 804 runs on average.
• As is well known, team OPS is indeed the best predictor of the quality of an offense (runs scored), with an R-squared value of 0.846. However, a few teams defied logic, such as the Blue Jays, who had the second-highest OPS at .811 but scored only 809 runs, ranking 7th in the league. They hit .284/.348/.463, but undershot their OPS prediction by an incredible 65 runs.
• The best offense (New York at .285/.363/.461) scored 930 runs, while the worst offense (Tampa Bay at .255/.314/.420) scored only 689. New York outperformed its OPS prediction by 30 runs, while Tampa Bay’s offense underperformed by 30 runs.
• The Yankees as a team had 6372 plate appearances last season, while the Devil Rays only had 5962, a difference of 410 plate appearances. That’s a huge difference, about 2.5 extra batters per game.
• On average, the number 9 hitter gets 150 fewer plate appearances than the leadoff man. That’s a lot. You add 150 plate appearances to Robinson Cano last year, and we’re talking 50 doubles, 19 HR and 100 RBI. Of course, that assumes he could have kept up that level of production, which would be unlikely.
• For every spot you drop in the order, you would get about 18-20 fewer plate appearances if you played all 162 games. The smallest drop (12 PA) was moving from batting second to batting third, and the largest drop (26 PA) was from hitting eighth to hitting ninth. The leadoff man gets 21 more PA than the number two hitter.
• The leadoff hitter takes 12.2% of all PAs on the team, while #9 takes only 9.8%.

I’ve already tried simple fitting of BA/OBP/SLG and BA/OBP/ISO to total runs using a system of linear equations, but the constants don’t work out to be, well, constant. Apparently there are a lot of other factors which really affect how many runs you can put up.

I think there must be some way to extrapolate how a lineup would do, if you could supply the BA/OBP/SLG for each hitter in the projected lineup. It’d be interesting to see if you could optimize a batting order mathematically, or look at how adding (or subtracting) a given player would impact your offensive potential. Fun stuff. Anyone know how Baseball Prospectus and other stat agencies do it? Is it just simulation, or something else?