I'm wanted to look at pitch sequencing this week and see how pitchers pitch in certain situations. What happens after a certain pitcher starts a hitter off with a fastball? What pitch do they throw for the second pitch? What if they start him off with a curve? Whats the most common first pitch to a batter? Do certain pitchers follow predictable patterns of pitches? Josh Beckett has dominated the ALCS so far, so I thought he was a good choice to start with.

Of the 1016 pitches that PITCH f/x has recorded for Beckett, he has thrown 67% fastballs, 27% curves and 6% changeups. He throws his fastball more than an average pitcher does, partly because he only has three pitches and partly because his fastball is such a good pitch. On the first pitch to a batter, Beckett pretty much throws his pitches at their normal frequencies (69% FB/23% CB/8% CH). It gets a little more interesting after he has thrown one pitch though. If Beckett starts the hitter off with a fastball (and the batter doesn't put it into play), the second pitch that Beckett throws is slightly more likely to be another fastball. Of the 155 pitches he has thrown after a first pitch fastball, 73% of them have also been fastballs. When Beckett throws a curve on the first pitch (and it isn't put in play, which happened on 61 pitches), his second pitch is a fastball only 53% of the time.

This is where I start to get a little hazy with the math, but if the decision to throw a fastball or not or every pitch were independent and Beckett has a 67% chance of throwing a fastball on any pitch, then given 155 pitches, you would be 95% confident that the range of fastball frequencies would be between 61-73%, which is what happened for the pitches after a first pitch fastball. However, when looking at the same 95% confidence interval for the pitch after a first pitch curveball (61 pitches), you get a range of 57-77% fastballs, but he actually only threw his fastball 54% of the time in those situations. Beckett significantly deviates from his "normal" pattern of pitching and throws fewer fastballs after he starts a hitter with a curveball.

This is easier to understand in a table, so here's a table with all the information from the previous paragraph. The numbers quoted above were frequencies that he threw different pitches. The way to read the table is that after a first pitch fastball that wasn't put in play, Beckett threw 155 pitches, 73% fastballs, 21% curveballs and 6% changeups.

        Overall   First    After First    After First    After First
Pitch   Freq.     Pitch    Pitch FB       Pitch CB       Pitch CH
FB      .67       .69      .73            .54*           .44*
CB      .28       .23      .21            .44            .17
CH      .05       .08      .06            .02            .39
==============================================================
N        1014      265      155            61             18
*-significant at 5% level, given number of pitches thrown in that situation and overall average.

There are plenty of obscure relationships between Beckett's pitches, such as what happens when he starts a batter off with two fastballs or curveballs, but before looking at those relationships, I need to make sure that my assumption of independence between pitches isn't going to be a problem. There are plenty of reasons why Beckett would throw more curves and change ups on the second pitch to a batter that he started off with a curveball. If a batter had a tough time hitting off speed pitches, it would make sense that Beckett would give him several in a row. In fact, if he starts a hitter off with two curveballs in a row, the chance that the third pitch is a fastball is 58%.

The assumption that his decision to throw a each pitch is independent isn't totally realistic, because the situation and type of hitter will impact his decision about which pitches to throw, but it doesn't really impact my results. The distributions will be different depending on the situation (I'd be more surprised if they weren't), but I'm more interested in how he changes his pitching patterns in certain situations, rather than if he changes or not. Is he throwing more fastballs on the first pitch than is expected? Does he follow up fastballs with curveballs? What does he throw after a fastball is fouled off? The assumption that he has a static 67% chance to throw a fastball on any pitch might end up being more of a problem, but I think that can be fixed with some regression toward the average values in each situation.

C.C. Sabathia is a power pitcher who throws his fastball more than average. Overall, for all pitchers, fastballs are thrown 55% of the time and 57% of first pitches are fastballs. Sabathia throws his fastball 61% of the time, but on his first pitches, he leans even more on his heater, throwing it 78% of the time. This is a significant difference given his overall average, but whether he starts the hitter off with a fastball or curveball, by the second pitch Sabathia is back to throwing pitches at their normal frequencies. The one oddity on the second pitch of an at-bat occurs if he starts the hitter off with a changeup. After a first pitch changeup, Sabathia throws a fastball only 44% of the time and throws more changeups instead. Sabathia's chart is below.

        Overall   First    After First    After First    After First
Pitch   Freq.     Pitch    Pitch FB       Pitch CB       Pitch CH
FB      .61       .78*     .59            .59            .44*
CB      .21       .08      .22            .32            .08
CH      .17       .14      .19            .09            .47
==============================================================
N        1101      298      199            22             36
*-significant at 5% level, given number of pitches thrown in that situation and overall average.

Greg Maddux is another roughly three pitch pitcher, but he has a slightly different pitches than either Sabathia or Beckett, as well as a pretty different overall style of pitching. Instead of just listing more frequencies for Maddux, his frequency table is below. The interesting things to notice here are how much he throws his fastball as the first pitch of an at-bat, and that if he starts an at-bat with a cutter there’s a good chance he’ll come back with a cutter as the second pitch of the at-bat.

        Overall   First    After First    After First    After First
Pitch   Freq.     Pitch    Pitch FB       Pitch CH       Pitch CT
FB      .66       .75*     .65            .68            .33*
CT      .15       .12      .16            .12            .59
CH      .19       .12      .19            .21            .07
==============================================================
N        1112      345      216            22             36
*-significant at 5% level, given number of pitches thrown in that situation and overall average.

It seems to me that pitchers would be most effective if they didn’t fall into tendencies regarding pitch sequencing. Beckett, Sabathia and Maddux are all essentially three pitch pitchers who throw fastballs more than average. They all throw slightly different amounts of fastballs, but on the first pitch of an at-bat, Sabathia and Maddux throw proportionally more fastballs than they do overall. Hitters are already probably looking for a fastball from these pitchers, but they can afford to look even more on the first pitch. On the first pitch of an at-bat, Sabathia and Maddux don’t exactly become 1-dimensional pitchers, but they do remove some of the uncertainty regarding pitch selection from a hitter’s mind, although they could be varying the location enough on the first pitch to make up for it. Beckett is much more in line with his overall pitch frequencies on the first pitch. He does throw 67% fastballs, so hitters should still be looking fastball on the first pitch, but no more than at any other time they face him.

The next step in this vein of research is to expand from looking at just three pitchers to all pitchers. Ideally, I would know what the average fastball (and other pitches) frequency is in the different sequencing situations I looked at, maybe split by hand orby type of pitcher. In addition to seeing if the pitch frequencies differed from a binomial distribution, I could also see how much they differed from the average frequencies in those situations. Using a static value for the frequency a pitcher throws a pitch is also not totally accurate and with average values for each situation, I could regress each pitcher’s situational frequency and get a better approximation of his true frequencies.