Introduction

The collection of serial measurements on athletes across a season (or multiple seasons) is one of the more common types of data being generated in the applied sport science environment. The question that coaches and practitioners often have is, “Is this player outside of their ‘normal range’?”

The best approach for establishing a reference range of ‘normal’ values is a frequently discussed topic in sport science. One common strategy is to use z-scores and represent the reference range as 1 standard deviation above or below the mean (Figure A) or plot the raw values and set the reference range 1 standard deviation above or below the raw mean (Figure B), for practitioners who might have a difficult time understanding standardized scores. Of course, the mean and standard deviation will now be related to all prior values. As such, if the athletes go through a training phase with substantially higher values than other phases (e.g., training camp) it could skew your reference ranges. To alleviate this issue, some choose to use a rolling mean and standard deviation, to represent the normal range of values relative to more recent training sessions (Figure C).

A problem with the approaches above is that they require a number of training sessions to allow a mean and standard deviation to be determined for the individual athlete. One solution to this issue is to base our initial normal reference ranges off of prior knowledge that we have from collecting data on players in previous seasons (or prior knowledge from research papers, if we don’t have data of our own yet). This type of Bayesian updating approach has been applied in WADA’s drug testing practices¹. More recently, Hecksteden et al., used this approach to evaluate the CK levels of team-sport athletes in both fatigued and non-fatigued states².

The mathematics of the approach was presented in the paper but might look intimidating to those not used to looking at mathematical equations in this manner.

The author’s provided a nice excel sheet where you can input your own data and get the updated reference ranges. However, the sheet is a protected sheet, which doesn’t afford the opportunity of seeing how the underlying equations work and you can’t alter the sheet to make appropriate for your data (for example, the data in the sheet log transforms the raw data automatically). Thus, I’ve decided to code the analysis out, both in excel and R, to help practitioners looking to adopt this approach.

Setting Priors

To apply this type of analysis, we need to first establish some prior values for three parameters: Prior Mean (mu), Prior Standard Deviation (tau), and a Prior Repeated-Measures Standard Deviation (sigmaRM). These values represent our current knowledge of the variable we are measuring before seeing any new data. As new data is collected, we can update these priors to get an individual (posterior) estimate for the athlete. I’ll use the priors set by Hecksteden and colleagues for CK levels of Male athletes:

• Mu = 5.527
• Tau = 0.661
• sigmaRM = 0.504

Once we have established our prior parameters, we are ready to update them, using the math equations above, as new data comes in.

Bayesian Updating in Excel

The excel sheet is available at my GitHub page. It looks like this:

All of the heavy lifting occurs in the two columns under the header Bayesian Updating (Log Scale). The equation in the first row (Test 1) is different than the other equations below it because requires the prior information to get going. After that first test, the updated data become the prior for the next test and this continues for all tests forward. You can download the excel sheet and see how the equations work, so I won’t go through them here. Instead, I’ll show them more clearly in the R script, below.

Bayesian Updating in R

We first need to convert the math equations provided in the paper (posted above) into R code. Rather that leaving things to mathematical notation, I’ll plug in the variables in plain English:

To be clear, here are the definitions for the variables above:

Now that we know the variables we need for each equation we can begin the process of updating or reference ranges.

First create a data set of the test observations and their log values. This will be the same data we observed in our excel sheet:

Then we set our priors (in log format):

## priors
prior_mu <- 5.527
prior_sd <- 0.661
prior_repeated_measure_sd <- 0.504

We will start by seeing how the updating works for the mean and standard deviation parameters after the first test. To do this, we will create a function for each parameter (mean and standard deviation) that updates the priors with the observed values based on the above equations:

posterior_mu <- function(prior_mu, prior_sd, prior_repeated_measure_sd, obs_value){
  
  numerator <- prior_repeated_measure_sd^2 * prior_mu + prior_sd^2 * obs_value
  denominator <- prior_repeated_measure_sd^2 + prior_sd^2
  
  post_mu <- numerator / denominator
  return(post_mu)
  
  }

posterior_sd <- function(prior_repeated_measure_sd, prior_sd, test_num){
  
  post_var <- 1 / ((test_num - 1 + 1) * 1/prior_repeated_measure_sd^2 + 1/prior_sd^2) 
  post_sd <- sqrt(post_var)
  return(post_sd)
  
}

After running the functions on the observations of our first test, our updated mean and standard deviation are:

Notice that we obtain the same values that we see following test_1 in our excel workbook. We can also calculate 95% confidence intervals and take the exponent (since the data is on a log scale) to get the individual athlete’s updated reference range on the raw scale:

Again, these results confirm the values we see in our excel workbook.

That’s cool and all, but we need to be able to iteratively update the data set as new data comes in. Let’s write a for() loop!

First, we create a new column in the data that provides us with the updated standard deviation after observing the results in each test. This is a necessary first step as we will use this value to then update the mean value.

## Calculate the updated SD based on sample size
df2 <- df %>%
  mutate(bayes_sd = sqrt(1 / ((test - 1 + 1) * 1 / prior_repeated_measure_sd^2 + 1 / prior_sd^2))) 

df2

Next, we need a for() loop. This is a bit tricky because test_1 is updating based solely on the priors while all other tests (test_2 to test_N) will be updating based on the mean and standard deviation in the row above them. Thus, we need to have our for() loop look back at the previous row above it once those values are calculated. This sort of thing is easy to set up in excel but in R (or Python) we need to think about how to code our row indexes. I covered this sort of iterative row computing in two previous articles HERE and HERE.

Within the loop we first set up vectors for the prior variance (sd^2), the denominator in our equation, and the log transformed observations from our data set. Then, we calculate the updated posterior for the mean (mu) on each pass through the loop, each time using the value preceding it in the vector, [i – 1], to allow us to iteratively update the data.

Once we run the loop, we add the results to our data set (removing the first observation in the vector since that was the original prior before seeing any data):

# Create a vector to store results
N <- length(df2$ln_value) + 1
bayes_mu <- c(prior_mu, rep(NA, N - 1))


## For loop
for(i in 2:N){
  
  ## Set up vectors for the variance, denominator, and newly observed values
  prior_var <- c(prior_sd^2, df2$bayes_sd^2)
  denominator <- prior_repeated_measure_sd^2 + prior_var
  vals <- df2$ln_value
  
  ## calculate bayesian updated mu
  bayes_mu[i] <- (prior_repeated_measure_sd^2 * bayes_mu[i-1] + prior_var[i-1] * vals[i-1]) / denominator[i-1]
    
}

df2$bayes_mean <- bayes_mu[-1]
df2

The two columns, bayes_sd and bayes_mean, contain our updated prior values and they are the exact same results we obtained in our excel workbook.

To use these updated parameters for creating individual athlete reference ranges, we calculate the 95% Confidence Intervals:

NOTE: I added a row at the start of data frame to establish the priors, before seeing the data, so that they could also be plotted as part of the reference ranges.

### Confidence Intervals
first_prior <- data.frame(test = 0, value = NA, ln_value = NA, bayes_sd = prior_sd, bayes_mean = prior_mu)

df2 <- df2 %>%
  bind_rows(first_prior) %>%
  arrange(test)

## Exponentiate back to get the reference range
df2$low95 <- exp(df2$bayes_mean - 1.96*df2$bayes_sd)
df2$high95 <- exp(df2$bayes_mean + 1.96*df2$bayes_sd)
df2

Finally, we plot the observations along with the continually updated references ranges. You can clearly see how large the normal range is before seeing any data (test_0) and then how quickly this range begins to shrink down once we start observing data from the individual.

To access the R code and the excel workbook please visit my GitHub page.

References

• Sottas PE et al. (2007). Bayesian detection of abnormal values in longitudinal biomarkers with application to T/E ratio. Biostatistics; 8(2): 285-296.
• Hecksteden et al. (2017). A new method to individualize monitoring of muscle recovery in athletes. Int J Sport Phys Perf; 12: 1137-1142.

When looking at sports statistics it’s important to keep in mind that performance is a blend of both skill and luck. As such, recognizing that all athletes exhibit some level of regression to the mean helps us put into perspective that observed performance is not necessarily where that individual’s true performance lies. For example, we wouldn’t believe that a baseball player who starts the MLB season going 6 for 10 has a true .600 batting average that they will carry throughout the season. Eventually, they will regress back down to something more normal (more average). Conversely, if a player starts the season 1 for 10 we would expect them to eventually move back up to something more normal.

A goal in sports analytics is to try and estimate the true performance of a player given some observed data. One of my favorite papers on this topic is from Efron and Morris (1977), Stein’s Paradox in Statistics. The paper discusses ways of using observed outcomes to make future forecasts using the James-Stein Estimator and then a Bayes Estimation approach.

Using 2019 MLB data, I’ve put together some R code to walk through the methods proposed in the paper.

Getting Data

First, we need to load Bill Petti’s baseballr package, which is a handy package for scraping MLB data. We will also load the ggplot2 package for data visualizations

library(baseballr)
library(ggplot2)

The aim of this analysis will be to look at hitting performance (batting average) over the first 30 days of the 2019 MLB season, make a forecast of the player’s true batting average, based on the observations of those first 30 days, and then test that forecast on the next 30 days.

We will obtain two data sets, a first 30 days data set and a second 30 days data set.

### Get first 30 days of 2019 MLB and Second 30 days

dat_first30 <- daily_batter_bref(t1 = "2019-03-28", t2 = "2019-04-28")
dat_second30 <- daily_batter_bref(t1 = "2019-04-29", t2 = "2019-05-29")

## Explore the data frames

head(dat_first30)
head(dat_second30)

dim(dat_first30) # 595 x 29
dim(dat_second30) # 611 x 29

Evaluating The Data

Let’s now reduce the two data frame’s down to the columns we need (Name, AB, and BA).

dat_first30 <- dat_first30[, c("Name", "AB", "BA")]
dat_second30 <- dat_second30[, c("Name", "AB", "BA")]

Let’s look at the number of observations (AB) we see for the players in the two data sets.

quantile(dat_first30$AB)
quantile(dat_second30$AB)

par(mfrow = c(1,2))
hist(dat_first30$AB, col = "grey", main = "First 30 AB")
rug(dat_first30$AB, col = "red", lwd = 2)
hist(dat_second30$AB, col = "grey", main = "Second 30 AB")
rug(dat_second30$AB, col = "red", lwd = 2)

We see a considerable right skew in the data with a large number of players observing a small amount of at bats over the first 30 days and a few players observing a large number observations (the everyday players).

Let’s see what Batting Average looks like.

quantile(dat_first30$BA, na.rm = T)
quantile(dat_second30$BA, na.rm = T)

par(mfrow = c(1,2))
hist(dat_first30$BA, col = "grey", main = "First 30 BA")
rug(dat_first30$BA, col = "red", lwd = 2)
hist(dat_second30$BA, col = "grey", main = "Second 30 BA")
rug(dat_second30$BA, col = "red", lwd = 2)

We see the median batting average for MLB players over the 60 days ranges from .224, in the first 30 days, to .234, in the second 30 days. We also see some players with a batting average of 1.0, which is probably because they batted only one time and got a hit. We also see that there are a bunch of players with a batting average of 0.

This broad range of at bats and batting average values is actually going to be interesting when we attempt to forecast future performance as small sample sizes make it difficult to have faith in a player’s true ability. This is one area in which the James-Stein Estimator and Bayes Estimation approaches may be useful, as they allow for shrinkage of the observed batting averages towards the mean. In this way, the forecast isn’t too overly confident about the player who went 6 for 10 and it isn’t too under confident about the player who went 1 for 10. Additionally, for the player who never got an at bat, the forecast will suggest that the player is an average player until future data/observations can be gathered to prove otherwise.

Estimating Performance

We will use 3 approaches to forecast performance in the second 30 days:

1. Use the batting average of the player in the first 30 days and assume that the next 30 days would be similar. This will server as our benchmark for which the other two approaches need to improve upon if we are to use them. Note, however, that this approach doesn’t help us at all for players who had 0 at bats.
2. Use a James-Stein Estimator to forecast the second 30 days by taking the observed batting averages and applying a level of shrinkage to account for some regression to the mean.
3. Account for regression to the mean by using some sort of prior assumption of the batting average mean and standard deviation of MLB players. This is a type of Bayes approach to handling the problem and is discussed in the last section of Effron and Moriss’s article.

Before we start making our forecasts, I’m going to take a random sample of 50 players from the first 30 days data set. This will allow us to work with a subset of the data for the sake of the example. Additionally, rather than cleaning up the data or setting an inclusion criteria based on number of at bats, by taking a random sample, I’ll get a good mix of players that had no at bats, very little at bats, an average number of at bats, and a large number of at bats. This will allow us to see how well the three forecast approaches handle a large variability in observations. (Technical Note: If you are going to follow along with the r-script below, make sure you use the same set.seed() as I do to ensure reproducibility).

## Get a sample of 50 players from the first 30

set.seed(1657)
N <- nrow(dat_first30)
samp_size <- 50
samp <- sample(x = N, size = samp_size, replace = F)

first30_samp <- dat_first30[samp, ]
head(first30_samp)

We need to locate these same players in the second 30 days data set so that we can test our forecasts.

## Find the same players in the second 30 days data set

second30_samp <- subset(dat_second30, Name %in% unique(first30_samp$Name))

nrow(second30_samp) # 37

Only 37 players from the first set of players (the initial 50) are available in the second 30 days. This could be due to a number of reasons such as injury, getting sent down to triple A, getting benched for a player that was performing better, etc. In any event, we will work with these 37 players from here on out. So, we need to go into the sample from the first 30 days and find those players. Then we merge the two sample sets together

## Subset out the 37 players in the first 30 days sample set

first30_samp <- subset(first30_samp, Name %in% second30_samp$Name)

nrow(first30_samp) # 37

## Merge the two samples together

df <- merge(first30_samp, second30_samp, by = "Name")
head(df)

The first 30 days are denoted as AB.x and BA.x while the second are AB.y and BA.y.

First 30 Day Batting Average to Forecast Second 30 Day Batting Average

Using the two columns, BA.x and BA.y, we can subtract one from the other to obtain the difference in our forecast had we simply assumed the first 30 day’s performance (BA.x) would be similar to the second (BA.y). From there, we can calculate the mean absolute error (MAE) and the root mean square error (RMSE), which we will use to compare the other two methods.

## Difference between first 30 day BA and second 30 day BA

df$Proj_Diff_Avg <- with(df, BA.x - BA.y)
mae <- mean(abs(df$Proj_Diff_Avg), na.rm = T)
rmse <- sqrt(mean(df$Proj_Diff_Avg^2, na.rm = T))

Using the James-Stein Estimator

The forumla for the James-Stein Estimator is as follows:

JS = group_mean + C(obs_BA – group_mean)

Where:

• group_mean = the average of all players in the first 30 days
• obs_BA = the observed batting average for an individual player in the first 30 days
• C = a constant that represents the shrinkage factor. C is calculated as:
  • C = 1 – ((k – 3)*σ²)/(Σ(y – ŷ)²)
    • k = the number of unknown means we are trying to forecast (in this case the sample size of our first 30 days)
    • σ² = the group variance observed during the first 30 days
    • (Σ(y – ŷ)²) = the sum of the squared differences between each player’s observed batting average and the group average during the first 30 days

First we will calculate our group mean, group SD, and squared differences from the first 30 day sample.

# Calculate grand mean and sd

group_mean <- round(mean(df$BA.x, na.rm = T), 3)
group_mean # .214

group_sd <- round(sd(df$BA.x, na.rm = T), 3)
group_sd # .114

## Calculate sum of squared differences from the group average

sq.diff <- sum((df$BA.x - group_mean)^2)
sq.diff # .467

Next we calculate our shrinkage factor, C.

## Calculate shirinkage factor

k <- nrow(df)
c <- 1 - ((k - 3) * group_sd^2) / sq.diff
c # .053

Now we are ready to make a forecast of batting average for the second 30 days using the James-Stein Estimator and calculate the MAE and RMSE.

df$JS <- group_mean + c*(df$BA.x - group_mean)

df$Proj_Diff_JS <- with(df, JS - BA.y)

mae_JS <- mean(abs(df$Proj_Diff_JS), na.rm = T)
rmse_JS <- sqrt(mean(df$Proj_Diff_JS^2, na.rm = T))

Using a Prior Assumption for MLB Batting Average

In this example, the prior batting average I’m going to use will be the mean and standard deviation from the entire first 30 day data set (the original data set, not the sample). I could, of course, use historic data to build my prior assumption but I figured I’ll just start with this approach since I have the data readily accessible.

# Get a prior for BA

prior_BA_avg <- mean(dat_first30$BA, na.rm = T)
prior_BA_sd <- sd(dat_first30$BA, na.rm = T)

prior_BA_avg # .203
prior_BA_sd # .136

For our Bayes Estimator, we will use the following approach:

BE = prior_BA_avg + prior_BA_sd(obs_BA – prior_BA_avg)

df$BE <- prior_BA_avg + prior_BA_sd*(df$BA.x - prior_BA_avg)
mae_BE <- mean(abs(df$Proj_Diff_BE), na.rm = T)
rmse_BE <- sqrt(mean(df$Proj_Diff_BE^2, na.rm = T))

Looking at Our Forecasts

Let’s put the MAE and RMSE of our 3 approaches into a data frame so we can see how they performed.

Comparisons <- c("First30_Avg", "James_Stein", "Bayes")
mae_grp <- c(mae_avg, mae_JS, mae_BE)
rmse_grp <- c(rmse_avg, rmse_JS, rmse_BE)

model_comps <- data.frame(Comparisons, MAE = mae_grp, RMSE = rmse_grp)
model_comps[order(model_comps$RMSE), ]

It looks like the lowest RMSE is the Bayes Estimator. The James-Stein Estimator is not far behind. Both approaches out performed just using the player’s first 30 day average as a naive forecast for future performance.

We can visualizes the differences in our projections for the three approaches as well.

We can see that making projections based off of first 30 day average (green) is wildly spread out and the mean value appears to over project a player’s true ability (the peak is greater than 0, the dashed red line). Conversly, the James-Stein Estimator (blue) has a pretty strong peak that is just below 0, meaning it may be under projecting players and pulling some players down too far towards the group average. Finally, the Bayes Estimator (grey) resides in the middle of the two projections with its peak just above 0.

The last thing I’ll do is put some confidence intervals around the Bayes Estimator for each player and take a look at how the sample size influences the forecast and where the observed batting average during the second 30 days was in relationship to that forecast.

df$Bayes_SE <- with(df, sqrt((bayes * (1-bayes))/AB.x))
df$Low_CI <- df$bayes - 1.96*df$Bayes_SE
df$High_CI <- df$bayes + 1.96*df$Bayes_SE

We can then plot the results. (Technical Note: To keep the plot from being too busy, I used only plot the first 15 rows of the data set).

ggplot(df[1:15, ], aes(x = reorder(Name, AB.x), y = bayes)) +
	geom_point(color = "blue") +
	geom_point(aes(y = BA.y), color = "red") +
	geom_errorbar(aes(ymin = Low_CI, ymax = High_CI), width = 0) +
	geom_text(aes(label = paste(AB.x, "ABs", sep = " ")), vjust = -1, size = 3) +
	geom_hline(aes(yintercept = 0), linetype = "dashed") +	
	coord_flip() +
	theme_classic() +
	ggtitle("Bayes Estimator", 
		subtitle = "Blue = Bayes Estimation \nRed = Observed BA during second 30 day \nLabeled ABs = Individual Sample Size the Estimation was built on (First 30 days ABs)")

The at bats labelled in the plot are specific to the first 30 days of data, as they represent the number of observations for each individual that the forecast was built on. We can see that when we have more observations, the forecast does better at identifying the player’s true performance based on their first 30 days. Rizzo and Igelsias were the two that really beat their forecast in the second 30 days of the 2019 season (Rizzo in particular). The guys at the bottom (Butera, Wynns, Duplantier, and Fedde) are much harder to forecast given they had such few observations in the first 30 days. In the second 30 days, Duplantier only had 1 AB while Butera and Fedde only had 2.

Wrapping Up

The aim of this post was to work through the approaches to forecasting performance used in a 1977 paper from Efron and Morris. Dealing with small samples is a problem in sport and what we saw was that if we naively just use the average performance of a player during those small number of observations we may be missing the boat on their underlying true potential due to regression to the mean. As such, things like the James-Stein Estimator or Bayes Estimator can help us obtain better estimates by using a prior assumption about the average player in the population.

There are other ways to handle this problem, of course. For example, an Empirical Bayes Approach could be used by assuming a beta distribution for our data and making our forecasts from there. Finally, alternative approaches to modeling could account for different variables that might influence a player during the first 30 days (injury, park factors, strength of opponent, etc). However, the simple approaches presented by Efron and Morris are a nice start.

References

1. Efron, B., Morris, CN. (1977). Stein’s Paradox in Statistics. Scientific American, 236(5): 119-127.