Win Probability Graphs and Regressions

Earlier in this blog I wrote a post on Win Probability in every possible game situation. I posted the excel files but they aren't as informative as a graph. I made up graphs for home and away and +2, +1, 0, -1, and -2 goal differentials for every minute. I didn't make up graphs for GD's bigger than that because there is basically no point. The fact that a team has a .999% win probability when they are up 4-0 isn't that exciting.

Each graph has the line of best fit and a scatter plot of the data. The equations for those lines are also on the graph along with the r^2 value for correlation. The graphs are below to look at.

Some interesting things I noticed:

-Most graphs show a very strong relationship between minute and win probability. The only ones that don't really are when teams are away and are tied, when teams are home and up by 2, and when teams are away and down by 2. Not really sure why these three stick out.

-Some of the graphs have linear relationships, while others are quadratic. Again, not really sure why this is. Why is the win probability when you are at home and tied follow a quadratic curve while the win probability of a team at home and down by 1 is linear? Maybe people have ideas as to why this happens.

-For some of the scenarios (the +2 and -2 GD's for home and away) I didn't start the graph at minute 1 because the data points were a little all over the place. This happens because there are so few data points so the win probabilities are screwed. Example: There aren't many times when a team has a 2-0 lead in the 5th minute.

-I added the graphs of all the goal differentials together for comparison, one for home and one for away. They're interesting to look at.

-Finally, because of this we now have some basic equations to model a team's chance of winning a game. Feel free to use them and check them out.

WPA and AGW: Van Persie is overrated

Well, maybe the title is a little exaggerated. What I really mean is the value of Van Persie's goals last season are overweighted. On the other hand, Darren Bent's goals were undervalued. The explanation comes from WPA, or "win probability added".

If you read the last post, I explained win probability. If not, check it out here. Because we have a probability for every game situation, I was able to weight goals by the added win probability a team has from that goal. In soccer, is a little more complicated because teams can tie. To solve this, I use win percentages instead of win probability. To get a team's win percentage you weight a win as 1 point, a draw as 1/3 of a point, and a loss as 0. The sum of these divided by the number of games a team has played gives us the win percentage. I guess in this case it should be win percentage added instead.

The added part comes in by calculating how much a goal adds to a teams win percentage. Here are a couple of examples:
-A goal in the 95th minute to put the home team up by a goal would have a WPA of .666666. A tie game in the 95th minute gives the home team a win percentage of .33333 (almost every time they will draw the game). However, in this case the home team scored. Now the score is 1-0 in the 95th minute. Now the home team's win percentage is almost 1 (almost every time they will win the game). To get the WPA of the goal we subtract the win percentage before the goal (.3333) from the win percentage after the goal (1). This gives us a WPA of .666666

Basically what WPA does is values goals that are more important to the team. In the example above, that goal is obviously very important to the team. However, a goal in the 90th minute to put a team up by 6 would be worthless to the team. That goal would have a WPA of 0.

I calculated the WPA of the top scorers in the EPL last season (players with more than 10 goals). Interestingly enough, the list shook up a bit. The table is below.

Notably, Darren Bent moves up to first on the list, and Van Persie moves down to 8th. Beyond this, I wanted to know which players tend to score more important goals and which players score non-important goals. Obviously, Van Persie has a higher WPA than most of these players because he scored a lot more goals than them. 

The way I did this was to calculate the average WPA of a goal by a player. I called this the Average Goal Weight, or AGW. The list of the AGW versus goals is below.

Not surprisingly, Van Persie moves to the bottom of the list, and Bent stays at the top. So what does all this mean? I don't think its a good idea to jump to the conclusion that Van Persie is not a good goal scorer. Despite everything, he scored 18 goals last season, which is good no matter how you score them.  However, I think AGW is a good supplement to the top goal scorers list. Last season, Bent was consistently scoring goals that added a whole 10 points to the winning percentage than Van Persie on average.

You shouldn't base your entire assessment of a goal scorer only on AGW. However, I think its something to take in to account.

Win Probability Added in Soccer

Everyone hears it all the time: A 2-0 lead is the most dangerous lead in soccer. But is it really? Thinking about the led me to wonder how exactly dangerous leads were in soccer. In fact, I wanted to find out what win, loss and draw percentages a team had in all situations. The best way to find this out is to analyze a lot of games and calculate the win, loss, and draw percentages in every possible game situation. To do this, I took in to account the venue of the game (home versus away), the goal differential between the teams (team is up by 2, team is up by 1, game is tied, team is down 1, team is down by 2 etc) and the minute of the game. I took goal differentials of -5 to 5 and minutes 1-90. I thought these were probably really the most important factors. You could maybe take in to account cards too, but this is hard and makes it pretty complicated. Overall, there are 2*11*90 = 1980 combinations of game situations.

The idea relates to WPA in baseball. Basically, WPA is a measurement of how much a play adds to the chance the team wins a game. For example, how much does a 2 run home run help the team's chances in the 6th inning? In soccer, a question would be how much does a goal at home to give you a 2 goal lead in the 67th minute change your winning percentage in the game? Pretty simple concept.

To get the percentages for all of these situations I imported game data from the past 10 years of the EPL in to Excel. My Excel skills are not the best but with some help I was able to eventually get these to convert in to percentages for each game situation mentioned above. The basic idea is this: how often do teams with a 1 goal lead in the 40th minute at home win? How often do they draw? How often do they lose? This was done for every minute and every goal differential both home and away. The results truly tell us how dangerous a variety of leads are.

Here's an example: The team is away, the game is tied, and it's the 67th minute. Any guesses on the win, draw and loss percentages? Well turns out the team has about a 19% chance of winning, a 51% chance of drawing, and a 30% chance of losing.

We can also test the "2 goal leads are the most dangerous leads theory". Let's say the team is home and it's the 35th minute. Here are the percentages for 1 and 2 goal leads:

1 goal lead: win: 78%, draw: 16%, loss: 6%
2 goal lead: win: 96%, draw: 2%, loss: 2%

The same holds true for all minutes and both home and away teams. A 2 goal lead is in fact not the most dangerous lead in soccer.

I'm also in the process of making a Java Applet to post here that lets you input the goal differential, venue, and minute, and spits out the win, loss and draw percentages. Again, my Java programming talents are not the best, so no promises on anything getting finished or uploaded soon. I uploaded the actual excel files to a google sites page though if you're curious to look at other percentages. If you want to download the files click here and type in the search bar ".htm" without the quotes to find the files.

Next, I'm planning on relating this more to how WPA is used in baseball by using it to analyze specific players by calculating how much percentage they add to their team winning by scoring goals. Not sure how useful this statistic will actually be, but it's worth a shot.

Can a Formula for Points Predict the Final Standings?

Based on the post below determining a formula for points based on goals for and goals against, I naturally wondered if this formula was in any way projective. At the half way point of the season, could the goals for and goals against of a team predict the final standings?

The simplest way to do this is to compare the points formula standings accuracy with the accuracy of the standings by just doubling the points of teams at the half way point of the season. At first glance, it seems that the points formula is projective. The average points error for the MLS is 4.26, which is not off by much. However, if we compare it with the accuracy of simply doubling the midyear points, it is not as projective. The average points error for the MLS with this approach is only 2.75. Clearly, the points formula is not very good at predicting the final standings. You'd be better off just looking at the mid year standings.

Below is the chart for the 2009 season showing what is explained above.

An Accurate Formula for Points Using GD and GA

It may seem pretty obvious that goals for and goals against should be a good predictor of success in soccer. Teams that score a lot and concede a little should win more than teams that don't score often and concede a lot of goals. But how predictive is it? How accurately can we determine the standings based solely on the goals for and goals against of a team in a season?

Apparently, if the formula is tweaked enough, we can get it down to an average error of just above 3 points across all teams. Considering 3 points equates to only one win in most leagues, goals for and goals against can narrow down the error to only one win. Not bad.

The equation I used to do this is based on Bill James' Pythagorean Expectation formula. The formula is pretty simple: (Runs Scored)^2/((Runs Scored)^2 + (Runs Allowed)^2). Basically, Bill James calculates the winning ratio using the formula above.

Of course, soccer is not the same as baseball and a few adjustments have to be made. First, there are draws in soccer. How can the formula be adjusted to take in to account draws? If you think about it, a draw is basically equal to 1/3 of a win (a draw counts for 1 point, and a win counts for 3 points). Therefore, we can calculate the winning percentage of a team as (wins + draws/3)/(total number of games * 3). This formula is used to calculate winning percentage. We can then convert the winning percentage back in to points by multiplying it by 3.

The next change I made was to change the exponent. Using data from the tables from 2000-2010 for the MLS and 97-98 to 09-10 for the EPL, the exponent that minimized the average error was 1.4

Next, I added a coefficient to further minimize the average error. After fooling around with different exponents and coefficients, the combination of 1.4 for the exponent and .9 for the coefficient got the average error to right around 3 for both the MLS and the EPL. This gives an equation of .9*(GF^1.4)/((GF^1.4)+(GA^1.4))

Overall, the average error for the EPL was 3.21 points and the average error for the MLS was 2.85 points.

Interestingly enough, the equation works for both the EPL and the MLS. Doesn't seem to matter across these two leagues. I haven't looked, but I assume it would work for other leagues across the world.

Below is an example of a one of the charts I created. This one is using the tables for one year in the MLS.