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!