Sports Illustrated Jinx : Regression to the Mean

[kimberley]

"Sports Illustrated Jinx": Regression to the Mean

GENERAL BACKGROUND

A few weeks ago, my uncles and others were discussing the so-called "Sports Illustrated Jinx", "Sophomore Jinx", and "Heisman Jinx".

Statisticians have said that the Sports Illustrated Jinx, in particular, is not a jinx at all, but rather an issue of Central Tendency and Regression to the Mean. I found the issue interesting and I told the guys I'd look into it. I would love to be able to discuss it with them on Memorial Day, but after looking at it for the last week or so, I'm not sure of my methods of analysis.

Quoting from elsewhere in relevant part:

"Professional athletes and sportswriters sometimes refer to the 'Sports Illustrated [J]inx', in which bad things happen with a player right after he is featured on the cover of the magazine. Now, a player appears on the cover for spectacular performance, particularly unexpected spectacular performance. ... [A] player who is performing unexpectedly well is probably being lucky, and his luck is not going to last long. When the SI cover is announced, the player's run of luck is probably over, and his performance lapses back to his norm. (The same thing is true of All-Stars, who are frequently selected for being hot in the first half [of the season])." "http://www.visi.com/~thornley/david/philosophy/thinking/mean.html"

CAVEAT

With this definition of the "Sports Illustrated Jinx" in mind, and before stating my question(s), a basic caveat. My degree is in political science. I've taken enough courses and performed some independent study work, however, to at least have a fair understanding of the Central Limit Theorem and, with some help from MS Excel/OpenOffice , calculating mean, median, mode, standard deviations, confidence intervals about the mean, prediction intervals for individual outcomes (outliers), standard error of the mean, standard error of the estimate, skew, kurtosis, general Linear Models, Jarque-Bera test for relative "normality"--the latter courtesy of EnumaElish) etc.

CONSTRUCTING THE ISSUE

I've looked at a couple of different examples of players' average performances and exceptional peformances, and I have tried to incorporate some of the common things and anomalies that I've witnessed into the hypothetical example below. Okay, so, here's the issue I'm trying to analyze.

Let's say that a player in the NBA with six years of playing experience (approximately 500 games) has a career scoring average of 15 points per game. Therefore, for sake of simplicity, he's scored 7500 total points over 500 games.

During the last 30 games, however, the player's scoring prowess has far exceeded his career average of 15 points per game. In fact, over the last 30 games he's averaged nearly 25 points per game. Over the last 60 games, his average is 22 points per game. Over his last 100 games, his average is 19 points per game. Assume that nothing has really changed (his playing minutes are pretty much the same, the teams they are playing are those they typically play and cycle through etc.) In other words, our suspicion is that these performances are more than likely individual "outliers" as they are soaring above our typical prediction intervals of 1.96*SD, 2.326*SD, 2.58*SD etc., no matter over what period of time, save the last 30 days, that our Mean is calculated.

Now, when you look at the shape of this player's game-by-game scoring distribution over his entire career, it does not appear to be "normal". In fact, the skew & kurtosis are not even close to being 0 respectively over the 500 game period. The player's scoring production has steadily risen on average, with intermittent peaks and valleys. This seems to be consistent with an individual player "cycling" (i.e., being on a scoring streak and then regressing back to SOME mean). Curiously, when you look at his scoring production over the last 152 games, the skew & kurtosis are very close to 0. The average and median over the last 152 games are also very close to being the same--18.1 and 18.19 respectively.

PLAYER MAKES THE S.I. COVER

Our player has performed so far above his career scoring average in the last 30 games that he's been named "Player of the Week" for 3 of the last 4 weeks and made the cover of "Sports Illustrated".

A SIMPLE AVERAGE MODEL (QUESTION 1)

When analyzing this player's scoring production to try to determine how he might perform in his next game (or over some other period), from which average do we construct things like "Prediction Intervals"?

In other words, is his scoring average and standard deviation over the last 30 games, N=30: Average (30), Standard Deviation (30) the best? I would think not since this looks like a period of exceptional performance (the "Sports Illustrated Jinx"). Alternatively, would we use his career average, N=500: Average (500), Standard Deviation (500)? I would think not also because although a greater sampling period would typically produce the best result if his scoring was "normally distributed", his scoring does not appear to be following a normal distribution over his six year career. In fact, the standard deviation is actually much greater where N=500 as compared to shorter periods of time, the career average is relatively low compared to more recent averages, and prediction intervals about the mean are so wide as to be practically useless.

Thus, should we use the period of N=152, where his scoring takes on the shape of a normal distribution with a skew & kurtosis very close to 0 respectively, and where the Mean and Median are almost identical? THIS IS WHY I ASKED ENUMAELISH ABOUT THE JARQUE-BERA TEST. That is, I was trying to incorporate sample size (games), mean, standard deviation, skew & kurtosis all into one formula to find out over which period the players' scoring is most normally distributed so that I could construct useful (i.e., relevant) Prediction Intervals AND to find the Mean to which his scoring was most likely to regress tommorow night, or over the next week etc.

PREDICTION INTERVAL EQUATION I USED FOR DETECTION OF POSSIBLE SCORING OUTLIERS (where N is normally distributed)

Xbar+/-ZSCORE(1.96,2.326,2.58,3.291 etc.)*STANDARD DEVIATION*SQRT(1+1/152).

A SIMPLE LINEAR MODEL (QUESTION 2)

In the few examples I tried, the Linear Model seemed to be the least consistent in predicting individual outcomes. Sometimes it was completely unreliable and other times it was right on the mark. For instance, when I found the period of time with the highest R-squared value, and then constructed Prediction Intervals for single outcomes using a Linear Regression Line and Standard Error of the Estimate for that time period, often times the model would completely collapse. That is, the player's scoring in a single game, where he scored just 4 points, would of course drop as much as 5 or 6 Standard Errors from the Linear Regression Line. Other times, it would be dead-on and the players scoring would hit 2 Standard Errors of the Estimate above the Linear Regression Line or 2 Standard Errors below it, and then regress right back to the Linear Regression Line.

What's the problem with my Linear Model?

PREDICTION INTERVAL EQUATION ABOUT THE LINEAR REGRESSION LINE WITH HIGHEST R-SQUARED VALUE

LINEAR REGRESSION VALUE+/-ZSCORE(1.96,2.326,2.58,3.291 etc.)*STANDARD ERROR OF THE ESTIMATE*SQRT(1+1/N).THE SI JINX: REGRESSION TO THE MEAN GENERALLY (QUESTION 3)

Assuming for the sake of argument that the Simple Average Model, where N is normally distributed, is the best way to analyze this issue, where N=152, Mean=18.1, SD=2.6, and this player has been posting up points for many of the last 30 games that are AT LEAST (1.96*2.6SD) above the Mean of 18.1, how can I determine when his exceptional performance of averaging 25 points over the last 30 days is going to meet up with the "Jinx", and revert back to more games that average 18.1 points?

CLOSING

I've had fun with this, and learned quite a bit, but am I even on the right track? If not, what's the best way to statistically look at this sort of issue? Thanks a bunch in advance and Happy Memorial Day!

Kimberley

 

[EnumaElish]

The mean and the conf. int. are related but separate issues. Regardless of how the distribution "looks" (normal or otherwise), the sample average is the "best unbiased estimator" of the population mean. Same goes for the values of the regression coefficients. Normality comes into play only when you are constructing confidence intervals around the parameters (the sample average or the regression coefficient).

But, in a given sample, "outlier" observations can throw off the estimated value of the parameter (e.g. periods --games-- of especially high or esp. low performance). You can try filtering out those performances by restricting your sample to include only those observations that are within 2 or 3 standard dev.s around the overall mean (where the stand. dev. and the mean are calculated based on the player's entire history).

What are the variables in your regression model? (Left-hand side vs. right-hand side?)

Have a nice Memorial Day.

 

[EnumaElish]

And I am not sure about the term 1+1/N inside the SQRT; can you provide a reference?

 

[kimberley]

Prediction Interval

And I am not sure about the term 1+1/N inside the SQRT; can you provide a reference?

Enuma,

Thanks again. Here's a link to the formula for a Predication Interval for a single outcome--(i.e., predicting the statistical boundaries for individual residuals). http://en.wikipedia.org/wiki/Prediction_interval" Here's the ultimate formula set forth in that article: Xbar(n)+/-T(a)*S(n)*Sqrt(1+1/n)--where (n) is the sample size, T(a) is the z-score from the Student's T-distribution table at (n-1) degrees of freedom, and S is your Standard Deviation.

Of course, the Prediction Interval formula above is simply an extension of the Confidence Interval formula for the Mean with (1+...) added in the last part of the formula. The formula for a Confidence Interval for the mean in the related CI article is Xbar(n)+/-T(a)*S(n)*Sqrt(1/n).

In line with these formulas and my previous posts, isn't it true that these formulas are only as reliable (and thus predictive) of the boundaries of the true mean in the case of CI, and residual boundaries in the case of the PI, if the distribution is a "normal distribution"? Stated otherwise, I've read repeatedly that a Confidence Interval about a mean, or the related Predication Interval for residual boundaries, only provide reliable boundaries for the mean and residuals if the distribution is normal. (Think in terms of the basketball player, again, where we are constantly recalculate his Mean regardless of whether we are using a simple average or linear model). Back to that example, often times these guys retire at or near the height of their careers and, therefore, the distribution of their scoring often resembles the distributions I remember for resonance from high school.

These formulas and this discussion leads me to a few other questions:

1. Why do we use the Z-score from the Student's T-distribution table for the two-tailed Confidence/Prediction Intervals formulae? For instance, regardless of the value of (n), it was my understanding that you use 1.96 for the .05 level (but again is that only if your distribution is normal?) For instance, the Z-score for the 95% level (for a two-tailed test) is 2.26 (not 1.96) where n=10 (i.e., 9 degrees of freedom). Is the point of the z-scores in the Student's T-distribution table meant to "normalize" things based on the size of (n)? In other words, why don't we ALWAYS just use 1.96 (95%), 2.327 (97.5%), 2.58 (99%), 3.291 (99.9%), 3.89 (99.99%), regardless of sample size? What's the point of adjusting our z-scores based on sample size?

2. Finally, if your using a Linear model as opposed to a Simple Average Model, are we still interested in normality? For instance, when looking at some data where n=60, this data has a very high R-squared value of .97--suggesting a strong linear relationship between my dependent and independent variables. However, my residuals don't appear normally distributed because all 60 residuals are below the end value of my 60 period Linear Regression Line. My understanding of Skew & Kurtosis is that if you graph something and want to see if it "looks normal", you draw a horizontal line across your graph at the end value of your Average or Linear Regression Line and, for skew, look to see how if you have an equal number of residuals above and below that Mean line.

3. What tests other than R-squared can we use to test the strength of a simple Linear Model? Let's again use our current, simple example of the player where the independent value is nothing more than a consectutive number representing the game of the season or career (.i.e. 1,2,3,4, etc.)

4 31 points (most recent game)
3 21 points
2 24 points
1 18 points (first game)

Kimberley

 

[EnumaElish]

Assuming for the sake of argument that the Simple Average Model, where N is normally distributed, is the best way to analyze this issue, where N=152, Mean=18.1, SD=2.6, and this player has been posting up points for many of the last 30 games that are AT LEAST (1.96*2.6SD) above the Mean of 18.1, how can I determine when his exceptional performance of averaging 25 points over the last 30 days is going to meet up with the "Jinx", and revert back to more games that average 18.1 points?

1. This is what is known as a natural experiment in soc sci. One way to determine this is to look for similar peaks (or troughs) in the player's past history and estimate the time it took him to get back to the previous 150-week mean (or whatever previous period you will be comparing him against when he peaked).

2. Another way is to look at other players' histories right after they made the SI cover and see how much time on average it took them to revert back to their previous means.

 

[EnumaElish]

if your using a Linear model as opposed to a Simple Average Model, are we still interested in normality?

See my previous post on this issue.

What tests other than R-squared can we use to test the strength of a simple Linear Model?

(1) The regression F statistic; (2) the t-statistics of your individual variables.

 

[mathwonk]

It is definitely (as Rainman said) a jinx. My basketball playing son was stung by this in high school. He was featured as student - athlete of the week on a tv station, not for a short term performance but his overall stats for the year and his high academic achievement.

The week it appeared he had his worst game ever, the only game in his career where he did not score a single field goal. And he was being visited by a college admissions scout too!

I think it is mental, since the unusual media attention distracts the player from his normal focus.

Or maybe when we are trying to prove a point we prepare harder, while after being praised we may relax. I have given a series of lectures in which I had a let down after a particularly good talk where I got a standing ovation. The next one of course would not be expected to top the previous one (regression to mean), but it was actually below standard.

so in a sense regression to the mean is also psychological, we try harder when we are down, and relax when we are up. the best performers are the consistent ones, not so much affected by success or failure. But even normally consistent people can be thrown off by unusual praise and attention.