Paul Campos points us to this discussion of the record of the Detroit professional basketball team:

The Detroit Pistons broke the NBA record for most consecutive losses in a season last night, with their 27th loss in a row. . . . A team’s record is, roughly speaking, a function of two factors:

(1) The team’s quality. By “quality” I mean everything about the team”s performance that isn’t an outcome of random factors, aka luck — the ability of the players, individually and collectively, the quality of the coaching, and the quality of the team’s management, for example.

(2) Random factors, aka luck.

**The relative importance of luck and skill?**

The above-linked post continues:

How do we disentangle the relative importance of these two factors when evaluating a team’s performance to some point in the season? . . . The best predictor ex ante of team performance is the evaluation of people who gamble on that performance. I realize that occasionally gambling odds include significant inefficiencies, in the form of the betting public making sentimental rather than coldly rational wagers, but this is very much the exception rather than the rule. . . . the even money over/under for Detroit’s eventual winning percentage this season was, before the first game was played, a winning percentage of .340. To this point, a little more than third of the way through the season, Detroit’s winning percentage has been .0666. . . .

To the extent that the team has had unusually bad luck, then one would expect the team’s final record to be better. But how much better? Here we can again turn to the savants of Las Vegas et. al., who currently set the even money odds of the team’s final record on the basis of the assumption that it will have a .170 winning percentage in its remaining games.

Campos shares a purported Bayesian analysis and summarizes, “if we have just two pieces of information — a prior assumption of a .340 team, and the subsequent information of a .066 performance through thirty games — the combination of these two pieces of information yields a posterior prediction of a .170 winning percentage going forward, which remarkably enough is exactly what the current gambling odds predict! . . . it appears that the estimate being made by professional gamblers is that about two-thirds of Detroit’s worse than expected record is a product of an ex ante overestimate of the team’s quality, while the other third is assumed to be accounted for by bad luck.”

I think that last statement is coming from the fact that (1/3)*0.340 + (2/3)*0.067 is approximately 0.170.

**I don’t quite follow his Bayesian logic. But never mind about that for now.**

As I said, I didn’t quite follow the Bayesian logic shared by Campos. Here’s my problem. He posts this graph:

I think I understand the “No_Prior_Info” curve in the graph: that’s the y ~ binomial(n, p) likelihood for p, given the data n=30, y=2. But I don’t understand where the “Prior” and “Posterior” curves come from. I guess the Prior distribution has a mean of 0.340 and the Posterior distribution has a mean of 0.170, but where are the widths of these curves coming from?

Part of the confusion here is that we’re dealing with inference for p (the team’s “quality,” as summarized by the probability that they’d win against a randomly-chosen opponent on a random day) and also with predictions of outcomes. For the posterior mean, there’s no difference: under the basic model, the posterior expected proportion of future games won is equal to the posterior mean of p. It gets trickier when we talk about uncertainty in p.

How, then, could we take the beginning-of-season and current betting lines–which we will, for the purposes of our discussion here, identify as the prior and posterior *means* of p, ignoring systematic biases of bettors–and extract implied prior and posterior *distributions*? There’s surely enough information here to do this, if we use information from all 30 teams and calibrate properly.

**Exploratory analysis**

I started by going to the internet, finding various sources on betting odds, team records, and score differentials, and entering the data into this file. The latest Vegas odds I could find on season records were from 19 Dec; everything else came from 27 Dec.

Next step was to make some graphs. First, I looked at point differential and team records so far:

nba <- read.table("nba2023.txt", header=TRUE, skip=1) nba$ppg <- nba$avg_points nba$ppg_a <- nba$avg_points_opponent nba$ppg_diff <- nba$ppg - nba$ppg_a nba$record <- nba$win_fraction nba$start_odds <- nba$over_under_beginning/82 nba$dec_odds <- nba$over_under_as_of_dec/82 nba$sched <- - (nba$schedule_strength - mean(nba$schedule_strength)) # signed so that positive value implies a more difficult schedule so far in season nba$future_odds <- (82*nba$dec_odds - 30*nba$record)/52 pdf("nba2023_1.pdf", height=3.5, width=10) par(mfrow=c(1,2), oma=c(0,0,2,0)) par(mar=c(3,3,1,1), mgp=c(1.5,.5,0), tck=-.01) # par(pty="s") rng <- range(nba$ppg_a, nba$ppg) plot(rng, rng, xlab="Points per game allowed", ylab="Points per game scored", bty="l", type="n") abline(0, 1, lwd=.5, col="gray") text(nba$ppg_a, nba$ppg, nba$team, col="blue") # par(pty="m") plot(nba$ppg_diff, nba$record, xlab="Point differential", ylab="Won/lost record so far", bty="l", type="n") text(nba$ppg_diff, nba$record, nba$team, col="blue") # mtext("Points per game and won-lost record as of 27 Dec", line=.5, side=3, outer=TRUE) dev.off()

**Here's a question you should always ask yourself: What do you expect to see?**

Before performing any statistical analysis it's good practice to anticipate the results. So what do you think these graphs will look like?

- Ppg scored vs. ppg allowed. What do you expect to see? Before making the graph, I could have imagined it going either way: you might expect a negative correlation, with some teams doing the run-and-gun and others the physical game, or you might expect a positive correlation, because some teams are just much better than others. My impression is that team styles don't vary as much as they used to, so I was guessing a positive correlation.

- Won/lost record vs. point differential. What do you expect to see? Before making the graph, I was expecting a high correlation. Indeed, if I could only use one of these two metrics to estimate a team's ability, I'd be inclined to use point differential.

Aaaand, here's what we found:

Hey, my intuition worked on these! It would be interesting to see data from other years to see if I just got lucky with that first one.

**Which is a better predictor of won-loss record: ppg scored or ppg allowed?**

OK, this is a slight distraction from Campos's question, but now I'm wondering, which is a better predictor of won-loss record: ppg scored or ppg allowed? From basic principles I'm guessing they're about equally good.

Let's do a couple of graphs:

pdf("nba2023_2.pdf", height=3.5, width=10) par(mfrow=c(1,3), oma=c(0,0,2,0)) par(mar=c(3,3,1,1), mgp=c(1.5,.5,0), tck=-.01) # par(pty="m") rng <- range(nba$ppg_a, nba$ppg) plot(rng, range(nba$record), xlab="Points per game scored", ylab="Won/lost record so far", bty="l", type="n") abline(0, 1, lwd=.5, col="gray") text(nba$ppg, nba$record, nba$team, col="blue") # par(pty="m") plot(rng, range(nba$record), xlab="Points per game allowed", ylab="Won/lost record so far", bty="l", type="n") abline(0, 1, lwd=.5, col="gray") text(nba$ppg_a, nba$record, nba$team, col="blue") # par(pty="m") plot(range(nba$ppg_diff), range(nba$record), xlab="Avg score differential", ylab="Won/lost record so far", bty="l", type="n") abline(0, 1, lwd=.5, col="gray") text(nba$ppg_diff, nba$record, nba$team, col="blue") # mtext("Predicting won-loss record from ppg, ppg allowed, and differential", line=.5, side=3, outer=TRUE) dev.off()

Which yields:

So, about what we expected. To round it out, let's try some regressions:

library("rstanarm") print(stan_glm(record ~ ppg, data=nba, refresh=0), digits=3) print(stan_glm(record ~ ppg_a, data=nba, refresh=0), digits=3) print(stan_glm(record ~ ppg + ppg_a, data=nba, refresh=0), digits=3)

The results:

Median MAD_SD (Intercept) -1.848 0.727 ppg 0.020 0.006 Auxiliary parameter(s): Median MAD_SD sigma 0.162 0.021 ------ Median MAD_SD (Intercept) 3.192 0.597 ppg_a -0.023 0.005 Auxiliary parameter(s): Median MAD_SD sigma 0.146 0.019 ------ Median MAD_SD (Intercept) 0.691 0.335 ppg 0.029 0.002 ppg_a -0.030 0.002 Auxiliary parameter(s): Median MAD_SD sigma 0.061 0.008

So, yeah, points scored and points allowed are about equal as predictors of won-loss record. Given that, it makes sense to recode as ppg differential and total points:

print(stan_glm(record ~ ppg_diff + I(ppg + ppg_a), data=nba, refresh=0), digits=3)

Here's what we get:

Median MAD_SD (Intercept) 0.695 0.346 ppg_diff 0.029 0.002 I(ppg + ppg_a) -0.001 0.001 Auxiliary parameter(s): Median MAD_SD sigma 0.062 0.009

Check. Once we include ppg_diff as a predictor, the average total points doesn't do much of anything. Again, it would be good to check with data from other seasons, as 30 games per team does not supply much of a sample.

**Now on to the betting lines**

Let's now include the Vegas over-unders in our analysis. First, some graphs:

pdf("nba2023_3.pdf", height=3.5, width=10) par(mfrow=c(1,3), oma=c(0,0,2,0)) par(mar=c(3,3,1,1), mgp=c(1.5,.5,0), tck=-.01) # par(pty="s") rng <- range(nba$start_odds, nba$record) plot(rng, rng, xlab="Betting line at start", ylab="Won/lost record so far", bty="l", type="n") abline(0, 1, lwd=.5, col="gray") text(nba$start_odds, nba$record, nba$team, col="blue") # par(pty="s") rng <- range(nba$record, nba$dec_odds) plot(rng, rng, xlab="Won/lost record so far", ylab="Betting line in Dec", bty="l", type="n") abline(0, 1, lwd=.5, col="gray") text(nba$record, nba$dec_odds, nba$team, col="blue") # par(pty="s") rng <- range(nba$start_odds, nba$dec_odds) plot(rng, rng, xlab="Betting line at start", ylab="Betting line in Dec", bty="l", type="n") abline(0, 1, lwd=.5, col="gray") text(nba$start_odds, nba$dec_odds, nba$team, col="blue") # mtext("Won-lost record and over-under at start and in Dec", line=.5, side=3, outer=TRUE) dev.off()

Which yields:

Oops--I forgot to make some predictions before looking. In any case, the first graph is kinda surprising. You'd expect to see an approximate pattern of E(y|x) = x, and we do see that--but not at the low end. The teams that were predicted to do the worst this year are doing even worse than expected. It would be interesting to see the corresponding graph for earlier years. My guess is that this year is special, not only in the worst teams doing so bad, but in them underperforming their low expectations.

The second graph is as one might anticipate: Betters are predicting some regression toward the mean. Not much, though! And the third graph doesn't tell us much beyond the first graph.

Upon reflection, I'm finding the second graph difficult to interpret. The trouble is that "Betting line in Dec" is the forecast win percentage for the year, but 30/82 of that is the existing win percentage. (OK, not every team has played exactly 30 games, but close enough.) What I want to do is just look at the forecast for their win percentage for the rest of the season:

pdf("nba2023_4.pdf", height=3.5, width=10) par(mfrow=c(1,3), oma=c(0,0,2,0)) par(mar=c(3,3,1,1), mgp=c(1.5,.5,0), tck=-.01) # par(pty="s") rng <- range(nba$record, nba$dec_odds) plot(rng, rng, xlab="Won/lost record so far", ylab="Betting line of record for rest of season", bty="l", type="n") abline(0, 1, lwd=.5, col="gray") fit <- coef(stan_glm(future_odds ~ record, data=nba, refresh=0)) print(fit) abline(fit, lwd=.5, col="blue") text(nba$record, nba$future_odds, nba$team, col="blue") # dev.off()

Here's the graph:

The fitted regression line has a slope of 0.66:

Median MAD_SD (Intercept) 0.17 0.03 record 0.66 0.05 Auxiliary parameter(s): Median MAD_SD sigma 0.05 0.01

Next step is to predict the Vegas prediction for the rest of the season given the initial prediction and the team's record so far:

print(stan_glm(future_odds ~ start_odds + record, data=nba, refresh=0), digits=2) Median MAD_SD (Intercept) -0.02 0.03 start_odds 0.66 0.10 record 0.37 0.06 Auxiliary parameter(s): Median MAD_SD sigma 0.03 0.00

It's funny--everywhere we look, we see this 0.66. And 30 games is 37% of the season!

Now let's add into the regression the points-per-game differential, as this should include additional information beyond what was in the won-loss so far:

print(stan_glm(future_odds ~ start_odds + record + ppg_diff, data=nba, refresh=0), digits=2) Median MAD_SD (Intercept) 0.06 0.06 start_odds 0.67 0.09 record 0.20 0.11 ppg_diff 0.01 0.00 Auxiliary parameter(s): Median MAD_SD sigma 0.03 0.00

Hard to interpret this one, as ppg_diff is on a different scale from the rest. Let's quickly standardize it to be on the same scale as the won-lost record so far:

nba$ppg_diff_std <- nba$ppg_diff * sd(nba$ppg_record) / sd(nba$ppg_diff) print(stan_glm(future_odds ~ start_odds + record + ppg_diff_std, data=nba, refresh=0), digits=2) Median MAD_SD (Intercept) 0.06 0.06 start_odds 0.67 0.09 record 0.20 0.11 ppg_diff_std 0.17 0.10 Auxiliary parameter(s): Median MAD_SD sigma 0.03 0.00

OK, not enough data to cleanly disentangle won-lost record and point differential as predictors here. My intuition would be that, once you have point differential, the won-lost record tells you very little about what will happen in the future, and the above fitted model is consistent with that intuition, but it's also consistent with the two predictors being equally important, indeed it's consistent with point differential being irrelevant conditional on won-lost record.

What we'd want to do here--and I know I'm repeating myself--is to repeat the analysis using data from previous years.

**Interpreting the implied Vegas prediction for the rest of the season as an approximate weighted average of the preseason prediction and the current won-lost record**

In any case, the weighting seems clear: approx two-thirds from starting odds and one-third from the record so far, which at least on a naive level seems reasonable, given that the season is about one-third over.

Just for laffs, we can also throw in difficulty of schedule, as that could alter our interpretation of the teams' records so far.

nba$sched_std <- nba$sched * sd(nba$record) / sd(nba$sched) print(stan_glm(future_odds ~ start_odds + record + ppg_diff_std + sched_std, data=nba, refresh=0), digits=2) Median MAD_SD (Intercept) 0.06 0.06 start_odds 0.68 0.09 record 0.21 0.11 ppg_diff_std 0.17 0.10 sched_std 0.04 0.03

So, strength of schedule does not supply much information. This makes sense, given that 30 games is enough for the teams' schedules to mostly average out.

**The residuals**

Now that I've fit the regression, I'm curious about the residuals. Let's look:

fit_5 <- stan_glm(future_odds ~ start_odds + record + ppg_diff_std + sched_std, data=nba, refresh=0) fitted_5 <- fitted(fit_5) resid_5 <- resid(fit_5) # pdf("nba2023_5.pdf", height=5, width=8) par(mar=c(3,3,1,1), mgp=c(1.5,.5,0), tck=-.01) # par(pty="m") plot(fitted_5, resid_5, xlab="Vegas prediction of rest-of-season record", ylab="Residual from fitted model", bty="l", type="n") abline(0, 0, lwd=.5, col="gray") text(fitted_5, resid_5, nba$team, col="blue") # dev.off()

And here's the graph:

The residual for Detroit is negative (-0.05*52 = -2.6, so the Pistons are expected to win about 3 games less than their regression prediction based on prior odds and outcome of first 30 games). Cleveland and Boston are also expected to do a bit worse than the model would predict. On the other direction, Vegas is predicting that Memphis will win about 4 games more than predicted from the regression model.

I have no idea whassup with Memphis. The quick generic answer is that the regression model is crude, and bettors have other information not included in the regression.

**Reverse engineering an implicit Bayesian prior**

OK, now for the Bayesian analysis. As noted above, we aren't given a prior for team j's average win probability, p_j; we're just given a prior point estimate of each p_j.

But we can use the empirical prior-to-posterior transformation, along with the known likelihood function, under the simplifying assumption the 30 win-loss outcomes for each team j are independent with constant probability p_j for team j. This assumption that is obviously wrong, given that teams are playing each other, but let's just go with it here, recognizing that with full data it would be straightforward to extend to an item-response model with an ability parameter for each team (as here).

To continue, the above regression models show that the Vegas "posterior Bayesian" prediction of p_j after 30 games is approximately a weighted average of 0.65*(prior prediction) + 0.35*(data won-loss record). From basic Bayesian algebra (see, for example, chapter 2 of BDA), this tells us that the prior has about 65/35 as much information as data from 30 games. So, informationally, the prior is equivalent to the information from (65/35)*30 = 56 games, about two-thirds of a season worth of information.

**Hey--what happened??**

But, wait! That approximate 2/3 weighting for the prior and 1/3 weighting of the data from 30 games is the *opposite* of what Campos reported, which was a 1/3 weighting of the prior and 2/3 of the data. Recall: prior estimated win probability of 0.340, data win rate of 0.067, take (1/3)*0.340 + (2/3)*0.067 and you get 0.158, which isn't far from the implied posterior estimate of 0.170.

What happened here is that the Pistons are an unusual case, partly because the Vegas over-under for their season win record is a few percentage points lower than the linear model predicted, and partly because when the probability is low, a small percentage-point change in the probability corresponds to a big change in the implicit weights.

Again, it would be good to check all this with data from other years.

**Skill and luck**

There's one more loose end, and that's Campos taking the weights assigned to data and prior and characterizing them as "skill" and "luck" in prediction errors. I didn't follow that part of the reasoning at all so I'll just let it go for now. Part of the problem here is in one place Campos seems to be talking about skill and luck as contributors to the team's record, and in another place he seems to considering them as contributors to the difference between preseason predictions and actual outcomes.

One way to think about skill and luck in a way that makes sense to me is within an item-response-style model in which the game outcome is a stochastic function of team abilities and predictable factors. For example, in the model,

score differential = ability of home team - ability of away team + home-field advantage + error,

the team abilities are in the "skill" category and the error is in the "luck" category, and, ummm, I guess home-field advantage counts as "skill" too? OK, it's not so clear that the error in the model should all be called "luck." If a team plays better against a specific opponent by devising a specific offensive/defensive plan, that's skill, but it would pop up in the error term above.

In any case, once we've defined what is skill and what is luck, we can partition the variance of the total to assign percentages to each.

Another way of looking at this is to consider the extreme case of pure luck. If outcomes determined only by luck, then each game is a coin flip, and we'd see this in the data because the team win proportions after 30 games would follow a binomial distribution with n=30 and p=0.5. The actual team win proportions have mean 0.5 (of course) and sd 0.18, as compared to the theoretical mean of 0.5 and sd of 0.5/sqrt(30) = 0.09. That simple calculation suggests that skill is (0.18/0.09)^2 = 4 times as important as luck when determining the outcome of 30 games.

And maybe I'm getting just getting this all tangled myself. The first shot at any statistical analysis often will have some mix of errors in data, modeling, computing, and general understanding, with that last bit corresponding to the challenge of mapping from substantive concepts to mathematical and statistical models. Some mixture of skill and luck, I guess.

**Summary**

1. Data are king. In the immortal words of Hal Stern, the most important aspect of a statistical analysis is not what you do with the data, it’s what data you use. I could do more than Campos did, not so much because of my knowledge of Bayesian statistics but because I was using data from all 30 teams.

2. To continue with that point, you can do lots better than me by including data from other years.

3. Transparency is good. All my data and code are above. I might well have made some mistakes in my analyses, and, in any case, many loose ends remain.

4. Basketball isn't so important (hot hand aside). The idea of backing out an effective prior by looking at information updating, that's a more general idea worth studying further. This little example is a good entry point into the potential challenge of such studies.

5. Models can be useful, not just for prediction but also for understanding, as we saw for the problem of partitioning outcomes into skill and luck.

**P.S.** Last week, when the Pistons were 2-25 or something like that, I was taking with someone who's a big sports fan but not into analytics, the kind of person who Bill James talked about when he said that people interpret statistics as words that describe a situation rather than as numbers that can be added, subtracted, multiplied, and divided. The person I was talking with predicted that the Pistons would win no more than 6 games this year. I gave the statistical argument why this was unlikely: (a) historically there's been regression to the mean, with an improving record among the teams that have been doing the worst and an average decline among the teams at the top of the standings, (b) if a team does unexpectedly poorly, you can attribute some of that to luck. Also, 2/30 = 0.067, and 5/82 = 0.061, so if you bet that the Pistons will win no more than 6 games this season, you're actually predicting they might do *worse* in the rest of the season. All they need to do is get lucky in 5 of the remaining games. He said, yeah, sure, but they don't look like they can do it. Also, now all the other teams are trying extra hard because nobody wants to be the team that loses to the Pistons. OK, maybe. . . .

Following Campos, I'll just go with the current Vegas odds and give a point prediction the Pistons will end the season with about 11 wins.

**P.P.S.** Also related is a post from a few years back, “The Warriors suck”: A Bayesian exploration.

**P.P.P.S.** Unrelatedly, except for the Michigan connection, I recommend these two posts from a couple years ago:

and

Enjoy.

You do realize that in many cases, statistics *are* just words that describe a situation?

You are a Keynesian!

Great post.

Re:

“I have no idea whassup with Memphis. The quick generic answer is that the regression model is crude, and bettors have other information not included in the regression.”

Yes, other information: Their best player spent the first part of the season serving a 25 game suspension.

And they’ve been without Steven Adams for a while too.

“Also, 2/30 = 0.067, and 6/82 = 0.073, so if you bet that the Pistons will win no more than 6 games this season, you’re actually predicting they’ll do worse in the rest of the season.”

2/30 is the current record, right? And 6/82 the prospective full season record? If so… I may be missing something, but that seems to imply superior performance over the remainder of the season (since 0.073 > 0.067, and 4/52 > 2/30).

Um, yeah, I just went in and fixed that. Apparently I can multiply, divide, and do square roots, but I have trouble with subtraction!

Actually, I thought the math was correct, because a bet of no more than 6 games won includes the possibilities of 2 through 5. Under any reasonable distribution of probabilities, the expected number of games won given no more than 6 won yields an expected decrease in performance.

John:

Yes, that’s what I was thinking in my revision (“if you bet that the Pistons will win no more than 6 games this season, you’re actually predicting they might do worse in the rest of the season”) using the word “might.”

The residual chart is very interesting and fits a stylized model of how teams behave at the end of the season.

At the end of the season bad teams engage in a ‘tank-off’ where they try to lose as many games as possible to improve their draft odds.

At the same time the best teams, begin resting their players to have everyone healthy and in peak condition for the playoffs, as they are already easily classified to the playoffs.

The teams in the middle on the other hand have an incentive to continue playing hard as they are all jockeying for position to make the playoffs and improve their seed or avoid the dreaded play in tournament.

So you might expect the middle of the road teams to ‘steal’ wins from the best and worst teams leading to the inverted U shape of the residuals

I just want to say that Cade Cunningham has played with style, grace, and passion and deserves recognition. I propose that we create an award for a good player on a crappy team. I’m torn between calling it the Robin Roberts or the Archie Manning award. Manning might be better since there is another person named Robin Roberts. Of course, there are other Mannings in sports, even an Arch Manning. Other nominees?

I hope that today’s discussion is remembered this spring. It will be interesting to review.

Oncodoc:

What about Frank Yates? He worked with style, grace, and passion on R. A. Fisher’s team, which must have been quite a struggle.

Andrew, thank you – this was a fun read. Why is it necessarily the case that “you can do lots better than me by including data from other years”? It seems to me that NBA teams have a habit of changing dramatically between seasons; taking together league-wide rule changes, team changes (player mobility, coaching regime changes, scheme changes), and player differences (maturation, decline, and critically, injuries), I would guess that on a scale of 0%-100%, the season-to-season consistency of the median NBA team in recent years is around 40%. That is to say, data from prior seasons is giving you information about a (mostly) different team. Might just be me!

Pavel:

By “including data from other years,” I’m not talking about pooling multi-year data for individual teams. Rather, I’m saying we could just apply the secret weapon and perform the same analyses I did above, but using data from other years, to see if the same patterns show up.

Ignoring the pre-season odds, as well as current point differential, and going ONLY on their current W-L record, their rest of season win% can be calculated as follows:

( Wins + 7 ) / (Games + 14)

For a team that is currently 2/30, that yields: 9/44 = .205 rest-of-season win%

You can of course make this more complex by including their pre-season forecast, as well as their current point differential. But in terms of back-of-enveloper, one-sentence forecast, it’d hard to do better than the above.

So, NBA is 14 games of regression, MLB is 70, NHL is 36, and NFL is 12, though all this was determined from 20 years ago.

Tangotiger:

Thanks—very helpful! And it checks out with our data, if we take the Vegas over-under as the outcome (so we don’t have to wait till the season ends):

Your model is y = (x+7)/(n+14) = 7/(n+14) + (n/(n+14))*x.

With n=30, this is y = 0.16 + 0.68*x, which lines up essentially perfectly with the regression model (no surprise if bettors know the rule to which you’re referring).

In a Bayesian context, this is a beta(7,7) prior, corresponding to a distribution of true win probabilities (under the impossible approximate model in which a team’s game outcomes are independent with equal win probabilities) with mean 0.50 and sd 0.13, which is not completely consistent with the empirical data from 30 games (win proportions have an sd of 0.18 with sampling sd 0.09, thus the simple estimate of the true underlying sd of win probs is sqrt(0.18^2 – 0.09^2) = 0.16. The difference between 0.13 and 0.16 corresponds to the first games of this season being a weird time with a few teams holding historically bad records.

http://www.insidethebook.com/ee/index.php/site/comments/true_talent_levels_for_sports_leagues/

I used many years of data, some 20 years ago. I’m sure it’s still similar these days. Here’s a snippet as it relates to NBA:

“NBA var(obs) is around .145^2. var(rand) just like hockey, or .055^2. So var(true) = .134^2 . To get an r of .5, you need only 14 NBA games! Sheesh. This is a huge problem here. 14 NBA games tells you as much as 36 NHL games.”

As an aside, I abhor the Beta(alpha,beta) construction and MUCH prefer it presented as Beta(N,mean). It makes things far more obvious. For NBA, that’s Beta(14,.500). Naturally, EVERY sport is going to have Beta(N,.500). So, we’re just down to showing “N”, which is the point at which correlation is r=.5. So obviously simple and straightforward as a presentation.

Tangotiger:

Thanks for the link, and I agree about the beta parameterization.

I can’t hope to have any idea what you’re talking about in this post.

But I will say that Detroit’s winless streak seems all that much more of an improbable event to me, looking at previous winless streaks historically, because of an increasing variance resulting from the 3-point shot (the increasing number of 3-point shots taken in a game since it was first introduced).

I don’t know it it’s actually true but I would guess that more and more wins/loses can be explained by variance in 3-point percentages in a given game. A good team, seems to me, would be more likely to lose any particular game because in that particular game they had an outlying poor performance from behind the line. Contrarywise, a bad team would be more likely to win any particular game because of an outlying strong performance on 3’s.

So long wining or losing streaks today would be in a sense, less likely than 25 years again or even 10 years ago.

Does that make any sense statistically?

20 years ago in Basketball on Paper, Dean Oliver produced an estimate of a team’s winning percentage from its ppg and ppg allowed along with variance and covariance of these. It also uses the Norm of the distribution of the data

There’s always a lot of luck involved in a record streak.

Andrew, I remember when some baseball player twenty or twenty-five years ago had a streak of getting on base…I think they eventually made it past 28 games, and when they got to 28 some pundit said “they’re halfway to the all-time record”; you correctly pointed out that that although that is true in one sense — yes, 28 is half of 56 — there’s another sense in which it’s extremely misleading. Maybe most readers of this blog already realize this but…well, suppose you pay me a dollar, and you’re going to flip a coin 10 times and if it comes up heads every time then I pay you $1000. You flip the coin…heads! Flip it again…heads again! Flip, flip, flip, heads, heads, heads. You’ve flipped the coin five times and it’s been heads every time. You’re halfway there! But your probability of completing the streak is still far, far less than 50%. If you look at it in terms of the odds, your chance of making it doesn’t reach 50% until you’ve flipped heads 9 times in a row. NOW you’re halfway there.

Oh, I said “getting on base”, I should have said “getting on base via a hit”.

Dec 30, 2023: Pistons win! Pistons win!

The jinx (recently discredited): in-progress streaks, no-hitters, hot-hands etc. will end as soon as you talk about them.

Oh wait!

Gib:

People have been talking about the Pistons’ in-progress streak for a few weeks now, so if there is any jinx operating here, it is slow-acting!

1. I love this class of blog post. Reading them is like listening to Jazz standards: the beginning is usually quite familiar, but once you get deep into the piece, you can encounter something that is surprising or makes you re-evaluate how you think about things.

2. For me, these posts are most valuable when you play around with the data yourself. Here is the boilerplate to get started for those that (like me) are more comfortable in Python than R:

“`

# %%

import numpy as np

import pandas as pd

import matplotlib.pyplot as plt

import seaborn as sns

sns.set_theme(style=”darkgrid”)

# %%

# Load data

nba_df = pd.read_csv(“nba2023.txt”, skiprows=1, sep=” “)

nba_df = nba_df.rename(

columns={

“avg_points”: “ppg”,

“avg_points_opponent”: “ppg_a”,

“win_fraction”: “record”,

}

)

nba_df = nba_df.assign(

ppg_diff=(nba_df.ppg – nba_df.ppg_a),

start_odds=nba_df.over_under_beginning / 82,

dec_odds=nba_df.over_under_as_of_dec / 82,

sched=-(

nba_df.schedule_strength – nba_df.schedule_strength.mean()

), # signed so that positive value implies a more difficult schedule so far in season

)

nba_df = nba_df.assign(future_odds=(82 * nba_df.dec_odds – 30 * nba_df.record) / 52)

nba_df.head()

# %%

ax = nba_df.plot(

x=”ppg_a”,

y=”ppg”,

kind=”scatter”,

xlabel=”Points per game allowed”,

ylabel=”Points per game scored”,

)

# Draw a line at the x=y diagonal

ax.plot(ax.get_xlim(), ax.get_ylim(), alpha=0.2, color=”k”)

# %%

ax = nba_df.plot(

x=”ppg_diff”,

y=”record”,

kind=”scatter”,

xlabel=”Point differential”,

ylabel=”Won/lost record so far”,

)

ax.axvline(0, color=”k”, alpha=0.2)

ax.axhline(0.5, color=”k”, alpha=0.2)

# %%

“`

3. The header for the data table says `Data from teamratings.com and bleacherreport.com as of 27 Dec 2023, win totals odds from 19 Dec 2020`. I’m assuming that `2020` is a typo, and it should be `19 Dec 2023`.

4. There are some interesting things happening around the “Avg score diff == 0” boundary. For example, Sac has a negative avg score differential, but has a win rate of almost 60% nonetheless. And Atl has the same avg score differential, but has a win rate of 40%. I personally would like to see the game-level results, rather than aggregated data, to get a better understanding of some of the details.

5. Do you have a follow-up scheduled to look back on this once the basketball season is over?

Asking “what do you expect to see” is particularly useful in designing an experiment. It helps emphasize what you learned from the experiment. Not asking it will, in many cased, lead to the impression that the experiment was a waste of time…

I often tried to convince the client that confirmation is also very useful, but what they mostly value is new knowledge.