# Is winning a soccer match independent of previous wins\losses?

$\quad$ I have a friend of mine who is a bit of a gambler ask me this question. He is of poor mathematical background, but has a sense of logic and will probably accept a logical answer in the natural language.

$\quad$ So my friend asked me if the probability of an outcome would change in consecutive trials of playing roulette. I said that it will not, and he needed no further verification. He only asked if it is because the roulette has no memory. I said he could put it that way if he liked.

$\quad$ Then the person asked me if the same is true for soccer matches. I told him that, despite the influence on soccer by other factors, the same should hold. My friend insisted on the contrary, and I didn't know what to say to him.

$\quad$ I would like to be able to explain to him why soccer matches are independent events. If this is not true, what is the dependence? Would considering a particular team change anything. Furthermore, what restrictions can we apply to ensure independence? Having the same opponent team?

p.p.s. I asked the exact same question on math.SE, where I am told it does not belong. I hope this is the appropriate place to ask.

• I vaguely remember having read that someone did an analysis for some sport. There weren't any short term memory effects (apart from what is mentioned in the answers; injuries etc). Only one thing was found. If a team lost 3 times in a row, it was more likely to lose again (psychological?). To your friend you could show a genuine random walk plot and ask if he sees patterns.
– Gere
Feb 28, 2013 at 8:05

It is often the case in sports analytics that people ask questions about more ethereal concepts like momentum, clutch, or home-field advantage. At the surface it would sound silly to say that these things don't exist. However, whether or not they exist is a separate question from whether or not we can meaningfully use them in any sort of predictive analysis. Sometimes it takes so much data to separate the signal from the noise that the signal is mostly gone by the time it can be detected (this is largely, but not entirely, the case with clutch hitting in Major League Baseball for instance). Home-field advantage, on the other hand, is quite a significant predictor over a range of sports (often at least partially due to subconscious biases in officials). In the case of successive soccer matches, since the players are human beings with memory, of course matches will not be completely independent, but I don't think that's a very interesting question to ask. I think it's more interesting to wonder what meaningful predictive value might be found.

This question is basically a variant of the so-called "hot hand effect." That is, does success in the recent past lead to more success in the future (after controlling for quality of teams, etc.)? I can't speak specifically about soccer, but this has been looked at numerous times across various sports (perhaps most notably in free-throw shooting in basketball where conditions are more homogeneous and outside forces are minimized). Typically any tiny effect that might be found in these studies tends to be too small to do much with. Perhaps soccer is different, but I wouldn't bet on it.

If you wanted to investigate for yourself, you'd want to make sure you account for the skill and health of each team. If Team A won yesterday, I'd guess they are more likely to win tomorrow as well. This isn't because of a memory but because the knowledge that Team A won increases our belief that they are an above-average, healthy team. Also, there might be a fair amount of non-randomness in scheduling that you might need to account for. In Major League Baseball for instance, a large number of very good teams happened to be clustered on the East Coast, and teams from the West Coast tend to have their games against them clustered together. What might at the surface appear to be the team having a memory of their losses could just be a scheduling artifact.

I think most people will agree that successive outcomes of soccer matches (of the same team?!) are not independent of each other. Clearly there are factors, such as injured players, making matches that are close in time dependent.

The exact nature of these invisible ties is nearly impossible to state correctly and in particular completely. People very involved in the sport are probably able to make more educated guesses about the outcome of coming events.

I don't think you can do anything to ensure independence between the results of soccer matches. Basically nothing on this earth is independent of anything else. True independence only occurs in idealized thought experiments. But the degree of correlation varies greatly.

You can begin to control for the most influential factors (injuries, home play, change of players). But this list is infinite, although the effect of the factors on it decreases rapidly. So practically you can control for a large portion of dependencies. The remaining correlations will at some point become too small to detect - they will become empirically zero, but they will never become provably zero.

By the way, successive real world roulette outcomes are also dependent on each other. But real world roulette resembles its idealistic counterpart very closely. So without considering any factors, they are very, very independent of each other.

I am a beginner in sports analyses but perhaps a quick empirical example: There is a package "vcd" which contains all soccer games in the German Bundesliga from 1963 to 2008. We can use this dataset to have a look whether we see some (preliminary) evidence for a correlation between the performance across three consecutive games (it is already sorted). For simplicity reasons let us examine only one team (my favorite team Borussia Dortmund or BVB)

install.packages("vcd")
library("vcd")
data("Bundesliga")
bvb <- subset(Bundesliga, HomeTeam == "Borussia Dortmund" | AwayTeam == "Borussia Dortmund")
bvb$$Points <- 0 bvb$$Home <- 0
# ...
reg <- lm(Points ~ as.factor(Year) + Home + lag_Points + lag_Points2, data = bvb)
summary(reg)


and we get the following regression results (standard errors are not corrected for autocorrelation or potential intra-season correlation and OLS is probably not a very well-suited estimator for a dependent variable which is either zero, one or three, but this should be only an example):

Call:
lm(formula = Points ~ as.factor(Year) + Home + lag_Points + lag_Points2, data = bvb)
...
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)          1.414743   0.241077   5.868 5.58e-09
as.factor(Year)1964  0.091076   0.323087   0.282  0.77807
...
as.factor(Year)2008  0.086770   0.312788   0.277  0.78151
Home                 0.884637   0.070048  12.629  < 2e-16
lag_Points          -0.078543   0.027344  -2.872  0.00414
lag_Points2         -0.052414   0.026662  -1.966  0.04952


We see (as expected) a strong relationship between playing at home and points obtained, and $$--$$ perhaps surprisingly $$--$$ negative coefficients for the points in the last two games prior to the current match. This could be just regression towards the mean but it is some anecdotical evidence for possible correlation over time.

• I can't say that I am familiar with this notation. Is this a programming language? Which one is it? What is vcd? Feb 28, 2013 at 7:39
• Sorry I forgot to mention this: It is R a statistical programming language. Feb 28, 2013 at 7:50

I found an article related to this. There the runs test and the chi squared goodness of fit test is used to test if the number of winning streaks is in line with the theoretical expectation under independence. Google: Winning Streaks in Sports and the Misperception of Momentum.

• Please add the complete reference for the article. Mar 15, 2013 at 12:30
• Vergin, R. (2000). Winning streaks in sports and the misperception of momentum. Journal of Sport Behavior, 23, 181-197. Mar 19, 2013 at 6:56
• I was able to read the article at thefreelibrary.com. Not sure what kind of page that is though. Mar 19, 2013 at 7:00