The Hot Hand in Sports

A couple of months ago, Michigan State set an NCAA Division I-A football record for the greatest deficit overcome (35 points), in a game against Northwestern.

Tonight, Texas Tech reeled off an amazing comeback of its own, overcoming a 31-point deficit (38-7) to stun Minnesota 44-41 in overtime in the Tempe, Arizona-based Insight Bowl. The Red Raiders' rally set a bowl record for comebacks.

Interestingly, Michigan State and Texas Tech started their respective comebacks with similar amounts of time remaining. Northwestern scored on MSU to take a 38-3 lead with 9:54 left in the third quarter, whereas Minnesota went up 38-7 over TTU with 7:47 remaining in the third. It really looked like the Gophers had burrowed too deep a hole for the Red Raiders to climb out of.

One possible way to approach the Texas Tech comeback statistically is via the runs test. If we looked at the temporal sequence of the teams' scoring drives (whether for a touchdown or field goal) in regulation, it would look like this (M = Minnesota, T = Texas Tech):

M M M M T M M T T T T T

As can be seen from the color scheme, there were four "runs" in the sequence (a run being a stretch of one team scoring points without interruption by the other team). The fewer the runs, the more streakiness is present. I tested the Minnesota-Texas Tech sequence using an online runs-test calculator, typing in 1's instead of M's for Minnesota and 0's instead of T's for Texas Tech.

As explained in this document from North Carolina State University, the runs test determines how many runs would be expected by chance (which can then be compared to the actual number obtained), "given the proportion of the population in each of the two categories and given the sample size..."

In the Minnesota-Texas Tech analysis, there were significantly fewer runs than would be expected by chance (p < .05). One probably does not need a statistical test to be convinced that Texas Tech ended regulation play on a hot streak and Minnesota, on a cold one. Still, if you had never heard of the runs test before tonight, then the saying that, "You learn something every day," holds true, at least for tonight.

Every so often, one hears a reference to a coach or manager making a decision according to "the book," as though a definitive catalog of strategy for a given sport existed. Now, however, a trio of baseball authors has come along and written a volume entitled, appropriately enough, The Book, and they have a website to go with it.

The website has a blog component, whose topics include streakiness. It is through this blog that I learned about an online discussion on another board, where a contributor with the moniker "Dackle2" presented some statistics on what happens after baseball teams go through a particularly hot or cold 10-game stretch.

What looks like around 300,000 10-game sequences in Major League Baseball from 1871-2005 were extracted and classified according to teams' records during the stretch (i.e., from 0-10, 1-9, and 2-8 all the way through to 8-2, 9-1, and 10-0). Teams' winning percentages from the five games before and five games after the 10-game stretch were also noted.

If there were anything to the idea of momentum or carryover of streakiness, one would predict, for example, that after languishing through a 0-10 cold streak, teams would have an appreciably worse winning percentage in the five games after the 10-game losing streak than in the five games before. Five games, by themselves, do not constitute a great sample, but aggregating many five-game sequences over many teams and many years, the data would seem sufficient.

As seen in the linked document, however, teams did not play markedly worse immediately after their 0-10 stretches (.347) than they did immediately before them (.358). At the other extreme, teams that achieved 10-0 hot streaks did not play substantially better immediately afterwards (.620) than beforehand (.610). In fact, whichever 10-game breakdown you look at (e.g., 2-8, 5-5, 7-3), the average winning percentages for the five games before and five games after are virtually identical.

Like other studies going all the way back to the original "hot hand" research by Gilovich, Vallone, and Tversky in 1985, the present findings suggest that players and teams have characteristic baseline rates of success, and that short-term hot streaks do not lead to long-term success rates above baseline, nor do short-term cold streaks lead to long-term success rates below baseline.

One of the more interesting forms of streakiness, in my view, is seeing one team go on a run to build up a big lead, only to see the other team turn the tables and make its own spurt to come back and win the game, or at least make it close.

Last night alone in the NBA, there were at least four games that followed the above storyline:

Phoenix built a 47-22 lead over Golden State in the second quarter, only to see the Warriors rebound for an 80-72 lead (a 33-point turnaround from -25 to +8 on Golden State's part). Unfazed by blowing their huge lead, however, the Suns came back to win the game, their 13th straight victory.

Philly shaved a 17-point Dallas lead to just 3, although the Mavs pulled away again. It was the Sixers' 10th straight loss; Dallas had a 12-game winning streak earlier this season.

Sacramento erased a 16-point deficit to edge Utah.

Finally, in the late West Coast game, the Lakers came back from 21 behind to beat the Rockets.

A week ago, the Nets darted out to an 18-0 lead, but fell to Boston.

This recent lead-blowing even appears to transcend any particular sport. In NHL action last Monday, Washington squandered a 4-0 lead in falling to Pittsburgh 5-4.

The above scenarios were all that I planned to write about. But right now, upon checking the score of the Texas Tech-Arkansas men's basketball game, I see where the Red Raiders have jumped out to an 18-3 lead. Tech, to this point, has missed only one shot, whereas Arkansas has made only one. If the aforementioned games are any lesson, expect a Razorback run to get back into the game.

Update: Arkansas never got closer than eight, with Texas Tech then pulling away to win by 15.

Time to check in again on the three-point shooting of the Texas Tech men's basketball team. As noted in my November 30 posting (see below or in November 2006 archives), the Red Raiders were leading the nation in shooting percentage behind the arc, at 50.4. Citing the statistical concepts of extremity of outcomes in small sample sizes and regression to the mean, I predicted the team would drop off some.

It should also be noted that, over the past five seasons, none of the teams that led the nation in three-point percentage exceeded 44%.

2002 Oregon 42.4
2003 Illinois St. 44.0
2004 Birm. Southern 43.0
2005 Oklahoma St. 42.1
2006 Southern Utah 42.9

Since my last posting on this topic, Texas Tech has experienced a small drop in its accuracy from long distance, sitting currently in second place nationally at 47.8.

In the Red Raiders' three most recent games (season log), they had two poor outings from three-point land (4-14, .286 vs. Stanford, and 2-10, .200 vs. Louisiana Tech), followed by an 11-18 (.611) explosion vs. Centenary. I was at the Centenary game and noticed some fans marking each of Texas Tech's made treys by unveiling a succession of cloth signs with 3's on them. With the help of my faculty colleague Bo Cleveland (with later technical assistance by Rachna Mutreja), we were able to take the following photo of the display in full glory at game's end.





I'll continue to track the story. It's important to state that, even if Texas Tech's three-point percentage continues to drop -- as I predict it will -- the team could still lead the nation, as other teams will likely drop too. In other words, Texas Tech's anticipated drop would be in absolute terms, but not necessarily in relative terms.

In terms of individual players, BYU's Austin (Got the Range) Ainge is now down from his earlier 70.6 three-point percentage (which I cited in my previous write-up) to 54.2.

It was just a few days ago (see November 30 posting below) that I talked about how extreme patterns can occur when looking at a small number of observations (e.g., several batters hitting above .400 early in the baseball season). But, I warned, it's hard to maintain extremely high (or low) levels of performance over larger numbers of attempts.

Yesterday, however, a Division III men's basketball player did as much as can be done within a single game to contradict my assertions. What happened was that Lincoln University's Sami Wylie shot 51% on three-pointers.

If a player were to have shot (roughly) 50% on 10 three-point attempts, I would find that moderately interesting. In 20 attempts? More impressive. And so on as the number of shots from behind the arc increased.

Well, in Wylie's case, he shot 51% on 41 attempts from three-point land. Yes, he shot 41 times from downtown, making 21 treys! All told, he ended up with 69 points in Lincoln's 201-78 win over Ohio State-Marion.

ESPN.com's article likens the game to a scene from the movie Pleasantville, where "every shot from every conceivable angle goes in."

First, a disclosure: I received my undergraduate degree at UCLA, in 1984.

Going into this year's annual USC-UCLA football game, played earlier today, the rivalry over the past 15 years had been as streaky as is possible for a situation where both teams had enjoyed stretches of dominance.

1991 UCLA
1992 UCLA
1993 UCLA
1994 UCLA
1995 UCLA
1996 UCLA
1997 UCLA
1998 UCLA
1999 USC
2000 USC
2001 USC
2002 USC
2003 USC
2004 USC
2005 USC

With UCLA's eight straight wins, followed by USC's seven, we don't need a statistical test to tell us that the number of observed runs (uninterrupted streaks by one team) is the minimum possible -- two -- given that each team has won at least once.

USC was widely expected to beat UCLA today en route to the (mythical) national championship game, in the process tying UCLA's earlier eight-game winning streak in the Battle of Los Angeles.

But it wasn't to be. UCLA 13, USC 9.

Being a faculty member at Texas Tech University, I periodically check out the Internet discussion boards related to the school's sports teams. It was there that I learned a few hours ago that the Red Raider men's basketball team is, at the moment, leading the NCAA in three-point shooting percentage.

Texas Tech has made 58 of 115 attempts from behind the arc (50.4%). While I was looking at the team statistics, I decided to peruse the individual three-point shooting statistics, as well.

Excluding three players who are each 2-for-2 (100%) on three-pointers due to insufficient attempts, the current national leader among individuals is BYU's Austin Ainge, who's hit 12-17 (70.6%). (For those who are wondering, Austin is indeed the son of former NBA guard Danny.) I guess you can say the young Ainge has the range!

Neither Texas Tech's 50% success rate as a team, nor Ainge's 70% rate, is likely to hold up for the season. Last year's three-point percentage leaders at the end of the season were Southern Utah (team) at 42.9% and Northern Arizona's Stephen Sir (individual) at 48.9%.

The current season is about one-fourth of the way through. What we're likely seeing, therefore, is the extremity of results associated with small numbers of observations. This concept was first brought to my attention by Geoff Fong in the spring of 1984, when he was on the faculty at Northwestern and I was visiting during my tour of prospective graduate schools (I ultimately chose Michigan).

Geoff was telling me about his research on statistical reasoning, and he pointed out how, early in every Major League Baseball season, the list of batting leaders will tend to have several players hitting above .400, yet there would be virtually no chance of any player ending the season at that level (the last player to hit .400 or better for a season was, of course, Ted Williams in 1941).

This statistical document describes the small-numbers phenomenon a bit more technically:

...all other things being equal, variation is more pronounced with small samples than with large ones. The larger your sample, the more stable your results will be. They will be less subject to the possibility that another study would produce greatly different results. A corollary is that large samples are less likely to produce extreme results. For example, assuming that you have a fair coin, it's much more difficult to get all heads when you toss a coin 50 times than when you toss it only two or three times.

Let's use last year's Texas Tech three-point success rate of .390 as a baseline for this year's squad (though there has been some change in personnel, most of the Red Raiders' outside shooters are still on the team, including offensive stalwart Jarrius [Jay] Jackson).

Using an online calculator for what is known as a binomial probability, we can ask how likely it is that a .390 three-point shooting team (which is what this year's Red Raiders are assumed to be, based on last year) could make 58 (or more) treys in 115 attempts. The answer is .008, a little less than 1-in-100, so what the Red Raiders are doing is already very rare statistically. Eventually, we may have to reject our "null hypothesis" that Texas Tech really has an underlying .390 probability on making threes.

As noted above, however, the larger the sample, the less susceptibility to unusually high or low success rates. To approximate a full season's worth of shots (i.e., a larger sample) instead of just a quarter season, I multiplied by four, Texas Tech's current number of made threes (58 X 4 = 232) and number of attempts (115 X 4 = 460). The ratio of 232/460 is the same as the Raiders' current three-point percentage of 50.4, but would be a much longer-term accomplishment. Again, using .390 as a baseline, the team's probability of hitting 50.4% of 460 three-point attempts is much tinier than before, .0000004, about 4 in 10 million.

Another potentially relevant concept that I'd like to mention briefly is regression toward the mean, which Lady Raider basketball announcer Ryan Hyatt sometimes invokes in his radio broadcasts. Regression toward the mean simply refers to the tendency for extreme values in the early rounds of performance -- either extremely high or extremely low -- to be followed by values more in the center of the distribution.

In conclusion, the statistical phenomena of small samples and regression toward the mean both suggest that the Texas Tech men will suffer some drop-off from their current 50.4% three-point shooting percentage. You probably don't need to have a statistics teacher tell you a 50% three-point shooting clip is unlikely to be maintained for a full season, any more than you need one to tell you that baseball players batting over .400 for the first month of the season will almost certainly fall off in their averages. If, however, you have some interest in the statistical concepts associated with teams' and players' fall-off after hot starts, you've visited the right place!

In about a half-hour, the Utah Jazz will attempt to improve upon its 12-1 start to the current NBA season (game-by-game log; ignore the pre-season games that are listed first). The Jazz finished exactly at .500 last year (41-41), so such a torrid start this season comes as a surprise to most observers. This article provides some ideas of why Utah appears to be so improved.

Happy Thanksgiving!

There was a men's college basketball game televised earlier today, in which Southern Illinois went scoreless in overtime in losing to Arkansas. Going scoreless in OT seems like an interesting type of cold hand.

Going scoreless for a five-minute stretch at any point in a game is probably fairly unusual. Further, if a game goes to overtime, that would seem to suggest the teams are pretty evenly matched (at least on that day or night). Therefore, one team shouldn't be able to shut out the opponent by sheer intimidation, for example by continually pressing and stealing the ball.

Shot clocks range from giving teams 24 seconds per possession to shoot in the NBA to 35 seconds in men's NCAA play (women's college ball uses 30 seconds, whereas the WNBA switched last season to 24 seconds from 30). Thus, unless both teams exhaust their full allotments of time to shoot, it would seem that teams could get about two possessions per minute, or 10 for an entire overtime. That's a lot of shots to miss (although a team could have fewer, due to turnovers), not to mention possible free throws.

One mechanism by which a team could go scoreless in OT -- of which you'll see some apparent evidence below -- is that it could get desperate after falling behind early in the extra period and then start jacking up threes.

I naturally wondered how often overtime shutouts have occurred. To get an estimate, I did some web searching using keywords such as overtime, scoreless, shut out and, to exclude other sports, basketball. It might not be the most scientific way to approach the problem, but it should provide a ballpark (or in this case, arena) figure. Below is a list of games I found from 2000 onward, complete with web links to game articles and box scores.

Men's College

Southern Illinois (vs. Arkansas), November 23, 2006
(SIU was 0-3 from the field in OT, no FT attempts)

George Mason (vs. James Madison), February 7, 2004
("The Patriots missed all six of their field goal tries, four from behind the arc, and went 0-for-2 at the free throw line in overtime.")

Women's College

Indiana (vs. Michigan State), February 29, 2004
(IU was 0-6 in OT field goal attempts, all from three-point land, no FT attempts)

Men's Pro

Boston (vs. Indiana), April 29, 2003
(first OT shutout in NBA play-off history; Celtics missed six shots from field and two FT attempts)

Vancouver (vs. Indiana), December 2, 2000

Women's Pro

None found.

With yesterday's 17-16 win over the Buffalo Bills, the Indianapolis Colts have gotten off to a 9-0 start this season. This makes them the only team in NFL history to start out 9-0 in two consecutive seasons. Last year, in fact, the Colts won their first 13 games of the season.

Obviously, winning regular-season games is not the problem for Indy - it's getting to the Super Bowl. We'll see if things are any better this season, come play-off time.

Some players in a given sport seem to perform at the same level night after night, whereas others show more variability from good to bad in how they do. Is it more advantageous to have one of type of player than the other? Sal Baxamusa investigates this question with regard to selected MLB starting pitchers, in The Hardball Times.

The Buffalo Sabres lost tonight to the Atlanta Thrashers, 5-4 in an overtime shoot-out. The loss, Buffalo's first of the season, means the Sabres must share the spot in the NHL record book with the 1993-94 Toronto Maple Leafs for best start to a season, at 10-0.

The 2006 World Series has just ended -- literally minutes ago -- with the St. Louis Cardinals closing out the Detroit Tigers in five games. Some notes from the perspective of streakiness:

Detroit pitcher Kenny Rogers, who was scheduled to go in a potential Game Six, will now have to wait for another year (and at age 41, he may not have many left in the game) to see if he can extend his current streak of 23 consecutive post-season scoreless innings. As shown in an ESPN graphic a few nights ago, these shut-out innings were compiled in three starts, one in each round of this year's play-offs: 7 and 2/3 against the Yankees, 7 and 1/3 against the A's, and 8 in the Tigers' lone win against the Cardinals. As shown in this article, the streak Rogers put together this post-season ranks among the best of all time. Should Detroit (or any other team Rogers pitches for) make the play-offs next season, all eyes will be on Rogers's throwing hand -- I mean, streak.

The Cards' David Eckstein, who got only two hits in his first 22 at-bats of this post-season, went 8-for-13 in the final three games against Detroit, en route to winning the World Series MVP award. For the Tigers, it was Sean Casey who had the hot bat.

Lastly, three stretches illustrate the apparent limitations of momentum. Both St. Louis and Detroit, of course, ended the regular season in slumps, yet made the World Series (see two articles in the links section on the right, addressing the issue of carryover from the end of the regular season to the play-offs). Also, the Tigers came into the World Series on a seven-game winning streak (three over the Yankees, four over the A's), yet took only one game from the Cards. It's not just that the Tigers lost the Fall Classic; numerous crucial fielding errors showed they were not at the top of their game against St. Louis.

Buffalo won again tonight in NHL action (3-0 over the New York Islanders), giving the Sabres a perfect 10-0 ledger. Buffalo's hot start ties the league record for most wins to begin the season.

Just one of those streaky days...

Michigan State took a 3-0 lead over Northwestern in their Big 10 football match-up, then fell behind 38-3 as the Wildcats scored 38 consecutive points. The Spartans then answered with 38 straight of their own, to win 41-38. In doing so, MSU set a new NCAA Division I-A record for size of deficit overcome to win (35 points).

If you look at the lists of the teams' drives (underneath the graphical diagram of scoring drives), you can see vividly what has to happen to produce a comeback like this -- one team gets amazingly hot and the other, similarly cold.

Prior to starting its comeback, Michigan State's last five possessions (excluding a brief one at the end of the first half) had consisted exclusively of punts and turnovers -- hardly a sign that the Spartans could put together four sustained touchdown drives (the other points came on a punt return TD and a field goal).

For Northwestern, the pattern was just the opposite. The Wildcats went from being able to move the ball up and down the field with apparent ease (at least that's what's implied by five touchdown drives) to their own stretch of entirely punts and turnovers -- including three straight "three and outs."

***

The Buffalo Sabres have opened up the National Hockey League season with a perfect 8-0 record, with tonight's win over the Boston Bruins. The Sabres are now two more wins away from tying the NHL record for most consecutive wins to start a season.

***

Finally, in tonight's World Series opener, won by St. Louis 7-2, Cardinal rookie Anthony Reyes retired 17 straight Tigers (i.e., nearly six straight innings of perfect baseball) after giving up a first-inning run. A graphic on the television broadcast noted that Reyes's streak was the best in World Series play since 1990, when Cincinnati's Jose Rijo put down 20 straight Oakland battersin the Reds' series-clinching Game 4 victory (see little game-by-game summaries on the right-hand side of the linked document).

I don't believe I've ever before drawn a hot-hand example from the Canadian Football League. Thanks to Phil Birnbaum's blog, however, there's always a first time.

Despite a nearly 200ドル million payroll and virtual all-star team for their starting line-up, the New York Yankees have once again stumbled in the play-offs, losing three games to one to the Detroit Tigers in the first round. The Yanks have not won the World Series since 2000.

I think many would agree that the team's breakdown occurred in two areas -- an aging and injury-prone starting pitching rotation and some shockingly poor hitting by a number of players who are typically very productive with the bat; the latter problem is discussed in this article.

The list below compares selected Yankees' 2006 regular-season batting averages (in parentheses) to their batting performances in the Detroit Series:

Robinson Cano (.342) 2-for-15

Gary Sheffield (.298) 1-for-12

Alex Rodriguez (.290) 1-for-14

Jason Giambi (.253) 1-for-8

Using an online binomial calculator, one can estimate the probability for each player of his getting as many hits (or fewer) as he did in the Detroit series purely by chance, given his (much higher) regular-season baseline batting average. Such probabilities are indeed low: Cano, .07; Sheffield, .09; Rodriguez, .06; and Giambi, .36. To reject a chance explanation, however, we typically require a probability of .05 or smaller (also known as "statistical significance").

For ARod, this latest poor performance in the postseason is nothing new. As the above-linked article notes, "Dating back to Game 4 of the 2004 American League Championship Series, Rodriguez has five hits in his last 46 postseason at-bats."

Here are some additional statistical observations from the baseball play-offs, by Elias Sports Bureau.

The St. Louis Cardinals' 8 1/2 game lead over the Houston Astros in the National League Central with only 12 games left (after the close of play on September 19) has almost completely evaporated. The Cards' lead now is only 1/2 game, with the final weekend of play remaining -- Houston at Atlanta, and St. Louis hosting Milwaukee (game-by-game logs for the Astros and Cards).

Presumably in connection with this dramatic turnaround, CNN/SI has posted its list of the greatest Pennant Race Collapses of all-time. To refer to these occurrences purely as "collapses" tells only half the story, in my view. In many (if not most) cases, one team's collapse was accompanied by another team's getting extremely hot. In some instances, the team making the comeback only won the pennant (league or divisional) by a single game at the end or in a special play-off after the seasonal standings ended in a tie. Thus, both the collapse by one team and the hot streak by the other were necessary for the latter to win out.

From CNN/SI's list, for example, the No. 2 entry involved the following from 1993: "On July 22, the Giants led Atlanta by 10 games. The Braves went 49-16 down the stretch to win the NL West by one game..."

And the No. 8 entry, the famous 1951 showdown between the Brooklyn Dodgers and New York Giants had this storyline: "On Aug. 11, the Dodgers led the Giants by 13 1/2 games, but the Giants won 16 in a row at one point and went 37-7 overall before winning a three-game playoff, which culminated in Bobby Thomson's Shot Heard 'Round the World."

In 2004, on the previous incarnation of the Hot Hand website, I did a fairly extensive analysis to mark the 35th anniversary of the 1969 NL East race, in which the New York Mets overtook the Chicago Cubs (this write-up is no longer posted here, but is available from me upon request). The following graph shows that, once again, both a hot streak by one team and a cold one by the other were implicated. The Mets ultimately won the division by eight games, so in this instance, they could have gotten by with a bit less torrid of a winning stretch.















Will this year's Cardinals join the ignominious list of teams that squandered big leads and perhaps earn a special "honor" for doing it so late in the season? Stay tuned this weekend!

Byron Nelson, one of the all-time great golfers, died today at age 94. Among Nelson's many accomplishments was his record streak of winning 11 straight tournaments, which he did in 1945.

Tiger Woods is currently on his own streak, having won the last five tournaments he's played. In the above-linked article on Nelson's passing, Woods is quoted as follows:

"In this day and age, with this competition, to win 11 in a row would be almost unheard of," Woods said after his fifth straight victory when asked how Nelson's accomplishment compared with others, like Joe DiMaggio's 56-game hitting streak.

"What Byron accomplished, that goes down as one of the great years in the history of our sport. ... DiMaggio's record, I see that being broken more than winning 11 in a row."

In yesterday's pro football action, Washington quarterback Mark Brunell set a new NFL record with 22 straight pass completions. In looking over the play-by-play sheet (which can be accessed from the linked article), it appeared that many of Brunell's completions were short passes. Ultimately, however, a record is a record, and no one else had ever completed 22 in a row.