“If Jose Valverde is not better in the ninth inning than Rick Porcello after eight innings, and Jose Valverde's a top closer, then we might as well not have a closer.”—Jim Leyland

 “Never is (Colon) your best chance, because on his best day, he’s never as good as Rivera.”—Mike Francesa

Here’s the situation: your starter has gone eight scoreless innings, but it’s a save situation. Do you trot him back out there for the ninth, or do you bring in your closer for the final three outs?

Presented with that very scenario last Wednesday and Sunday, respectively, both Jim Leyland and Joe Girardi removed their starters in favor of their closers. Mariano Rivera blew his save opportunity, while Jose Valverde converted his. Both managers ultimately presided over victories, but both drew criticism for not sticking with the starters who put their teams in a position to win.

So what's the correct decision? You go with your closer. It’s not really a question. If you, the reader, think otherwise, then you are going to be wrong more often than not. I am going to assume that you do think otherwise. That way, I can join the company of Francesa and Leyland, just as I’ve always dreamed of, and engage in a rant of my own directed against you.

Who is the better pitcher, in general, on a per-inning basis—the closer or the starter? Right, it’s the closer. Just look at their ERAs. What’s more, the starter is probably tired, and he’s probably run out of ways to get batters out. In the cases of Colon and Porcello, the starter wasn’t even that great to begin with.

Sure, you say, but the starter’s been throwing great. He’s maintained his velocity, he hasn’t made any mistakes, he’s getting stronger as the game goes on, he’s retired the guys due up each time he’s faced them. Meanwhile, the closer is an unknown commodity, and you can always bring him in if the starter gets into trouble.

And here’s what I say: How awesome would it be if I had that level of perception? If that were the case, and I could just tell from my living room that at any particular moment, a pitcher were about to pitch better than he usually does—or better than Mariano Rivera, for that matter—I would be a manager in the Major Leagues of Baseball, or at least a bookmaker or a doctor (the U.S. could always use more doctors). Because normally, pitchers who go eight scoreless are not that good in the ninth. It’s true, and I’ll get to that data in a minute.

I’m not saying that the starter hasn’t been excellent. The excellence is not what’s so hard to observe. (Mind you, I often have trouble observing it.) The starter has given up no runs in eight innings. Sure, we’ll give him that. We’ll allow that he’s been excellent. But you need a lot more than that to say that he will continue to be excellent.

Between 2000 and 2010, there were over 1700 instances in which a starting pitcher completed eight scoreless innings. In nearly two-thirds of those instances, the starter came back out for the ninth. The team allowed an average of 0.45 runs in those innings. Conversely, when the starter was pulled, his team allowed an average of 0.35 runs in the ninth.

There are sampling caveats. Teams that went to the bullpen might have been afforded better relievers. Managers like to allow starters their shot at a complete game, and therefore have motives other than run prevention.

Whatever the case, you would need your starter to be over half a run in ERA better than your best reliever to leave the starter in. Again, if you are watching a game where a pitcher has thrown eight scoreless, most of the time, bringing in a reliever is the better option in the ninth. Pinch-hitting for the pitcher is always the better option at that stage of the game, given that the objective is to win the game.

Why? Why is it that these pitchers who have pitched excellently do not outperform relievers?

Not all pitchers lose their stuff as the game wears on. The more universal phenomenon is that batters gain an advantage with each time they face the same pitcher by making “adjustments.” This is why relievers can survive with fewer pitchers than starters, because they have to face batters only once, whereas starters need to find a way to mix their pitches more as the game goes on.

This ground was covered in depth in The Book. By projecting the outcome of each plate appearance and comparing it to the actual outcome based on the time through the order, one can observe the magnitude of the times-through-the-order effect. I chose to come at this from a different angle, mainly because I didn’t want to take the time to project a million plate appearances.

The easiest mistake that people make in studying the times-through-the-order effect is looking at it on a league-wide basis and seeing that pitchers have better numbers the third and fourth time through the order than they do overall. This is obviously a sampling problem, as only the best pitchers make it through the third and fourth time through the order.

To control for this, I ran a fixed-effect regression to predict each pitcher’s linear weights based on his time facing a batter in a game. Essentially, the output showed that the average starting pitcher loses a half-a-run-per-nine in effectiveness each time through the order.

Some pitchers, like Zack Greinke, who expertly mixes his pitches, seemingly get more effective as the game goes on. Some, like CC Sabathia, who maintains his velocity throughout, show no observed split. The best—Pedro Martinez and Johan Santana and Roy Halladay and Felix Hernandez and Tim Lincecum—go through the order a third time, and even if they’ve lost something, they’re still as good as an average reliever. But those are the exceptions.

The rule is that Bartolo Colon and Rick Porcello are not the answers in the ninth inning. The rule is that we know how good Mariano Rivera and Jose Valverde are and that our powers of perception are not such that we can spot the rare starter who is better after eight scoreless than those two.

 I can’t put it better than these two did:

“You’re telling me that Joe Girardi should worry that Mariano can’t close the game…It’s the classic second-guess.”—Francesa

 “I’m glad I did it. And I’ll do it again tonight if we have the same (expletive) situation if I think it gives us the best (expletive) chance to win the (expletive) game.”—Leyland