How would you go about modelling a sport with a fixed maximum score line (e.g. volleyball or tennis?)

In short, I don't have a great maths background but have always mucked about with modelling sports and making predictions (as I enjoy it).

I have always used the Hal Stern "The Probability of Winning an American Football Game" American Statistician (August 1991) - as a basis for this, assuming things were normally distributed (possibly incorrectly) and adding other little modifiers (e.g. homefield advantage) and then solving it in a least squares puzzle in Excel.

Standard deviation of Game Results about prediction from a Ratings system (in this case least squares regression with a few tweaks) equals the stochastic element in a normal distribution function where: x = the margin of victory that you are trying to get a % against (e.g. it could be a point spread e.g. % to cover) the mean = the predicted margin of victory derived in this case from my least squares regression to create power ratings for teams sigma = standard deviation of error on the difference between the predicted margin of victory and actual margin of victory based on past results.

Based on the above I would treat in Excel terms a win % for team A as: 1-NORMDIST(0.5,mean,sigma,true) - with the mean and sigma having values obviously.

This when solved (in Excel - I think you could do it with matrices as well) derives me ratings which can then be broken down further to work out an offensive and defensive component so you can work out scores.

This seems to work OK (as a rough approximation) for sports with no fixed score (e.g. NFL, NBA, Australian Rules Football, Rugby). I have a new interest in Vollyball however which has a fixed maximum score of 3 sets wins the game. I ran this process on the current Italian Volleyball league and whilst I was trusting the ratings that came out the scores didn't make sense (e.g. one game was predicted 1.5 vs 1.8).

Is there a work around that could be applied here (as I believe it is possible e.g. the Ken Massey, Massey ratings site models Volleyball) as I couldn't see anything obvious.

If you are trying to predict the scores of different tennis matches, say, perhaps a censored regression (also called 'Tobit model') would be useful. In sports with capped scoring systems (you can't go above 3 sets in men's tennis), you get ceiling effects. A ceiling effect occurs when “information regarding true differences between individuals scoring at the highest possible value is lost” (McBee, 2010; p. 314) due to a measurement being at the maximum of a scale, or at a scale’s ‘censoring point’. The censoring point for a men's tennis match is 3. For example, I could beat my dad (I think) in straight sets, and so too (I think) could Novak Djokovic, but our scores would not reflect the difference in skill between myself and Novak - we would both get a score of 3. Our scores are in this sense censored, and there is no information available to distinguish between scores of 3.

More generally, a player's distribution of sets won in each match might look like this:

This particular player has 557 wins (score of 3 sets) and 880 losses (score of less than 3 sets).

Tobit models have been shown to produce unbiased estimates (i.e., estimates approximately equal to regression estimates had the outcome variable not been censored) on data where ceiling effects are present (Long, 1997; McBee, 2010). Note that the model will give you fitted (predicted) values that are greater than 3 (in the case of men's tennis), and I suppose you could compare predicted values for each player to predict winners.

McBee, M. (2010). Modeling outcomes with floor or ceiling effects: An introduction to the Tobit model. Gifted Child Quarterly, 54(4), 314-320.

Long, S. J. (1997). Regression models for categorical and limited dependent variables. Thousand Oaks, CA.

• Thanks. Will have a definate look into this before marking it as an answer. A friend has also suggested the binomial distribution route as a possibility but haven't read up on that enough to be honest. – user8812 Nov 1 '18 at 16:57
• Yeah, of course, I am sure that there is more than one way to do it! Tobit models should solve some of the problems you mentioned, though. Let me know what you decide, I'd be curious to know. Good luck! – arranjdavis Nov 1 '18 at 17:00

so a simple brute force solution would be to predict probability of gaining a point.then just run simulations to get overall distributions of winning games or sets.....