## Wednesday 23 October 2013

### Are wickets more likely on hat-trick balls?

A student of mine has emailed me asking if I know anything about whether in cricket a wicket is more or less likely on a hat-trick ball then on any other ball. (Note for Eric and others similarly challenged in the finer nuances of cricket, a hat-trick occurs when a bowler takes a wicket with each of three consecutive deliveries. Note for followers of other sports: this is the original use of the term "hat-trick" in sport.)  The student and his flatmate have surmised that taking a wicket on a hat-trick ball is more likely than on any other randomly chosen ball. I don’t know what the data say on this, but I think the students are almost certainly right, mostly for statistical reasons. It is fun to think about how to formalise the hypothesis, and then how to test the effect of different forces. Maybe it could be a future Honours project to take this theory to the data.

Take a set of games in a particular format (say test cricket), and find the total number of deliveries and the fraction of those that resulted in the bowler being credited with a wicket. Then find the total number of deliveries in all those matches where, if the bowler had taken a wicket he would have achieved a hat-trick, and find the fraction of those deliveries where a wicket was in fact taken. Our guess is that this latter fraction will be higher than the general fraction of deliveries with wickets, and that that difference would be statistically significant. I am fairly confident about this purely because of sample selection:
• Pitches vary considerably across matches; if a bowler has already taken two wickets in two balls, it is likely that the pitch for that game (and that point in the game) is an easier one for taking wickets than the average.
• Bowlers (and their supporting fielders) vary in ability; if a bowler has already taken two wickets in two balls it is likely that he is a better bowler (with better supporting fielders) than the average.
• Batsmen vary in ability and batter ability is both correlated within the batting order and correlated within teams; if a bowler has taken two wickets in two balls it is likely that the batting team has below average quality batsmen and that it is one of the weaker batsmen in the team who is facing the hat-trick ball.
• Statistically (I can confirm this from test-cricket data), batsmen are more at risk at being dismissed early in their innings than later on; there is a high likelihood that the batsmen facing the hat-trick ball is facing his first ball of the innings.

So let’s control for these sample selection issues and consider instead a conditional probability question: Given the ability of the bowler and fielders, the batsman, how early it is in the batsman’s innings and the state of the pitch, does being on a hat-trick change the probability of a wicket? The question here becomes whether the unusual situation leads players to change their behaviour in some way. On the bowling side, the captain might set more aggressive wicket-taking fields on a hat-trick ball, but the bowler might try too hard and lose his rhythm. Similarly, the batsmen might be more conscious about not giving his wicket away, but at the same time the pressure of the situation might lead to his having leaden feet.

I would expect that the psychological effect would be greater on a batsman new to the crease than a bowler who has had a chance to find his rhythm. And in test cricket, I think that batsmen are always concentrating only on wicket preservation on the first ball they face. So If I had to guess, I would say that in test cricket the net result would still be that wickets are more likely on balls where the bowler is on a hat-trick, but the effect would be very small (and probably not discernible with statistical significance in the data). In limited overs cricket, I would expect the effect to be much smaller or even zero.

Now, if only I had ball-by-ball data for the entire history of test cricket!

1. Hmmm. I spent 6-7 years umpiring the top level of club cricket in Auckland, so here are some experience-based thoughts on your conditional probability question. Yes, it's undoubtedly true that behaviours change, especially setting more aggressive fields (usually an ultra-aggressive one for the hat trick ball). So it's ntirely an emprical issue on which (changing) incentives will prevail. But I suspect the changing behaviour of the batsman will dominate the outcome. It's hard to get batsmen out even when they're playing their shots, and an order of magnitude harder again when the batsman shuts up shop (rather like trying to win at chess when one player is playing for a draw). My guess would be, greater batting defensiveness will make a third-ball wicket less likely

2. As a club umpire, Donal, you will have witnessed a lot more hat-trick balls than I have. I wonder, however, if there is a difference between club cricket and test cricket. In particular, are club cricketers more likely than test cricketers to be looking for runs from the first ball they face and hence have a margin on which they can adjust to being more defensive should they be on a hat-trick ball. My sense is that batsmen are looking purely to get a feel of the pitch on the first ball they face, and so there is no additional conservatism possible for hat-trick balls.

3. I think you could still find statistical significance because
- Bowler has good rhythm (has just taken two wickets)
- More aggressive field set
- Fielders concentration high (less likely to drop a catch)
- Batsman under immense pressure and nervous (feet not moving etc.)
- Batsman possibly is rushed to get ready and hasn't fully prepared as they would (mentally and physically)
-Umpire possibly caught up in hype and more likely to raise finger

4. Nice point about the endogenous response of umpires and the fact that the batsman (f it is a within-over hat- trick) won't have been padded up when the 1st wicket fell. I'm not sure about the fielder effect. It could go either way; they will be more alert but possibly also have the same effect as a nervous batsman losing fluency of movement.