Background

This all started with a nice post from Anindya Mozumdar on the R Bloggers feed. The topic material was fun for me (analyzing the performance of male 100m sprinters and the fastest man on earth), as well as exploring bayesian methods.

Last post in this series I made use of one of the nice features about a Bayesian approach – we don’t have to worry nearly as much about the multiple comparisons issue [Gelman, Hill, Yajima (2012)]. But, quite frankly, the code was very ugly with a lot of repetition and cutting and pasting. In this post I want to clean that all up. So let’s load the necessary libraries.

library(rvest) # to ha"rvest" the web page
library(tidyverse) # using readr, dplyr, and purrr
library(ggstatsplot)
library(BayesFactor)

Next let’s duplicate Anindya’s earlier work and scrape the Track and Field All-Time Performances webpage to get the data. One change I’m making is to remove n_max = 3263 which is unnecessary and was preventing grabbing the newer race results from summer 2019.

male_100_html <-
  read_html("http://www.alltime-athletics.com/m_100ok.htm")
male_100_pres <- male_100_html %>%
  html_nodes(xpath = "//pre")
male_100_htext <- male_100_pres %>%
  html_text()
male_100_htext <- male_100_htext[[1]]

male_100 <- read_fwf(
  male_100_htext,
  skip = 1,
  #  n_max = 3263, # n_max removed to cpture newer races
  col_types = cols(.default = col_character()),
  col_positions = fwf_positions(
    c(1, 16, 27, 35, 66, 74, 86, 93, 123),
    c(15, 26, 34, 65, 73, 85, 92, 122, NA)
  )
)

male_100 <- male_100 %>%
  select(X2, X4) %>%
  transmute(timing = X2, runner = X4) %>%
  mutate(
    timing = gsub("A", "", timing),
    timing = as.numeric(timing)
  )

# 3267 as of July 8, 2019
nrow(male_100) # if you're cautious you can check against the webpage
## [1] 3267

Let’s focus on the top 6 runners who have more than 40 race results recorded. We’ll make an effort throughout this post to capture the parameters we use and store them as variables and use the names. If you choose to replicate this post on your own you should be able to change the parameters below and see how the results vary based upon your choices (for example the top 5 or 10 runners or more than 30 races).

numbraces <- 40
howmanyrunners <- 6

Having made our selections let’s use a series a dplyr commands piped %>% together to create a character vector called orderbymean which contains the names of the 6 runners who meet our criteria. We can use this vector to filter our dataframe down to just the 6 we want with a filter(runner %in% orderbymean) statement as well as force the factor levels of runner to be in mean order with factor(male_100$runner, levels = orderbymean).

orderbymean <- male_100 %>%
  group_by(runner) %>%
  summarise(avgtime = mean(timing), races = n()) %>%
  arrange(avgtime) %>%
  filter(races >= numbraces) %>%
  top_n(-howmanyrunners, avgtime) %>%
  pull(runner) %>%
  as.character()

orderbymean
## [1] "Usain Bolt"     "Asafa Powell"   "Tyson Gay"      "Justin Gatlin"  "Yohan Blake"    "Maurice Greene"
male_100 <- male_100 %>%
  filter(runner %in% orderbymean) %>%
  mutate_if(is.character, as.factor) %>%
  droplevels()

male_100$runner <-
  factor(
    male_100$runner,
    levels = orderbymean
  )

glimpse(male_100)
## Observations: 530
## Variables: 2
## $ timing <dbl> 9.58, 9.63, 9.69, 9.69, 9.69, 9.71, 9.72, 9.72, 9.74, 9.74, 9.75, 9.75, 9.75, 9.75, 9.76, 9.…
## $ runner <fct> Usain Bolt, Usain Bolt, Usain Bolt, Tyson Gay, Yohan Blake, Tyson Gay, Usain Bolt, Asafa Pow…
levels(male_100$runner)
## [1] "Usain Bolt"     "Asafa Powell"   "Tyson Gay"      "Justin Gatlin"  "Yohan Blake"    "Maurice Greene"

Okay we now have the 530 relevant times for the 6 runners we’re focusing on. The focus of this post is to build the bayesian equivalent of a frequentist’s pairwise comparisons test across all the unique runner pairings pairwise.t.test(x = male_100$timing, g = male_100$runner, p.adjust.method = "holm"). As I mentioned in an earlier post this would be the logical next step after conducting a oneway ANOVA of timing ~ runner or it’s bayesian equivalent BayesFactor::anovaBF(timing ~ runner, male_100)

## 
##  Pairwise comparisons using t tests with pooled SD 
## 
## data:  male_100$timing and male_100$runner 
## 
##                Usain Bolt Asafa Powell Tyson Gay Justin Gatlin Yohan Blake
## Asafa Powell   0.0460     -            -         -             -          
## Tyson Gay      0.0250     1.0000       -         -             -          
## Justin Gatlin  0.0024     1.0000       1.0000    -             -          
## Yohan Blake    0.0052     1.0000       1.0000    1.0000        -          
## Maurice Greene 6.9e-05    0.1649       1.0000    1.0000        1.0000     
## 
## P value adjustment method: holm

Our Bayes equivalent to this matrix won’t report “p values” but rather the Bayes Factor associated with the pairing. Unlike the frequentist’s “reject/don’t reject” criteria, the BF we report will indicate what the data provide as odds that our hypothesis is correct. We’ll build it methodically and with an eye towards code that is easily reused in the future.

So, quick quiz, how many unique pair combinations are there for our 6 runners? The order of the pair doesn’t matter at this juncture we’re looking for the number of possible head to head races among these 6 runners. If you’re like me you don’t know the answer to that off of the top of your head and it can be tedious figuring it out, so let’s let the computer always calculate it for us, and tell us. For this case it’s 15. Then we can use combn to take the 6 names and show us what those pairings are e.g., “Usain Bolt, Asafa Powell”. The result is a matrix of 15 columns (all the pairings) and two rows (the names of the runners for the pairings). Just to make it easier to see I’ve added a t() so the display is vertical and you can see the pairs.

numberofpairings <- factorial(howmanyrunners) / 
                    (factorial(2) * factorial(howmanyrunners - 2))
numberofpairings
## [1] 15
t(combn(orderbymean, 2))
##       [,1]            [,2]            
##  [1,] "Usain Bolt"    "Asafa Powell"  
##  [2,] "Usain Bolt"    "Tyson Gay"     
##  [3,] "Usain Bolt"    "Justin Gatlin" 
##  [4,] "Usain Bolt"    "Yohan Blake"   
##  [5,] "Usain Bolt"    "Maurice Greene"
##  [6,] "Asafa Powell"  "Tyson Gay"     
##  [7,] "Asafa Powell"  "Justin Gatlin" 
##  [8,] "Asafa Powell"  "Yohan Blake"   
##  [9,] "Asafa Powell"  "Maurice Greene"
## [10,] "Tyson Gay"     "Justin Gatlin" 
## [11,] "Tyson Gay"     "Yohan Blake"   
## [12,] "Tyson Gay"     "Maurice Greene"
## [13,] "Justin Gatlin" "Yohan Blake"   
## [14,] "Justin Gatlin" "Maurice Greene"
## [15,] "Yohan Blake"   "Maurice Greene"

Purrring right along

Now we can grab the output from combn and create two separate vectors, runner1 and runner2. We’ll take those vectors and create a series of purrr statements using pipes.

• The first is a map2 which takes runner1 and runner2 and creates a list of 15 dataframes, one for each pairing. The anonymous function(a, b) is simply an organized way of working our way through the 15 pairings. Filtering and dropping levels and explicitly converting to a dataframe because BayesFactor::ttestBF will generate a warning if you pass it a tibble.

• Next we purrr::map the list we just created (.x = .) and call the ttestBF function. For each item in the list of 15 dataframes (one for each runner pairing) it runs with formula = timing ~ runner, the dataframe we passed data = ., and in this case we have deliberately specified a directional hypothesis nullInterval = c(-Inf, 0) (it’s “-Inf” because the faster runner has a smaller timing). See the excellent BayesFactor documentation here for a more complete explanation of directional hypothesis testing.

• We started with two vectors runner1 and runner2 after map2 we had a list of 15 dataframes. Now after the first map pipe we have a list of 15 Bayes Factor objects (that’s what ttestBF generates). We’ll immediately pipe (%>%) that list (.x = .) into another map where we’ll invoke the extractBF function. extractBF produces a dataframe, in this case with 2 rows, one with the rowname “Alt., r=0.707 -Infwhich we don’t need.

• Now we have a list of 15 dataframes, this time containing the results of our ttestBF. We want the row in each of them that is named “Alt., r=0.707 -Infmap_dbl to let it know we want a list of 15 numbers, map_dbl(.x = ., ~ .["Alt., r=0.707 -Inf<d<0", "bf"]).

• The final pipe simply does some trivial rounding.

runner1 <- combn(orderbymean, 2)[1, ]
runner1
##  [1] "Usain Bolt"    "Usain Bolt"    "Usain Bolt"    "Usain Bolt"    "Usain Bolt"    "Asafa Powell" 
##  [7] "Asafa Powell"  "Asafa Powell"  "Asafa Powell"  "Tyson Gay"     "Tyson Gay"     "Tyson Gay"    
## [13] "Justin Gatlin" "Justin Gatlin" "Yohan Blake"
runner2 <- combn(orderbymean, 2)[2, ]
runner2
##  [1] "Asafa Powell"   "Tyson Gay"      "Justin Gatlin"  "Yohan Blake"    "Maurice Greene" "Tyson Gay"     
##  [7] "Justin Gatlin"  "Yohan Blake"    "Maurice Greene" "Justin Gatlin"  "Yohan Blake"    "Maurice Greene"
## [13] "Yohan Blake"    "Maurice Greene" "Maurice Greene"
bfresults <- map2(
  runner1,
  runner2,
  function(a, b)
    male_100 %>%
      filter(runner %in% c(a, b)) %>%
      droplevels() %>%
      as.data.frame()
) %>%
  map(.x = ., ~ ttestBF(
    formula = timing ~ runner,
    data = .,
    nullInterval = c(-Inf, 0)
  )) %>%
  map(.x = ., ~ extractBF(x = .)) %>%
  map_dbl(.x = ., ~ .["Alt., r=0.707 -Inf<d<0", "bf"]) %>%
  round(., digits = 4)
bfresults
##  [1]   10.2865    9.4539   59.3688   36.4792 2709.0974    0.3217    0.5782    0.7394    8.5068    0.2271
## [11]    0.2902    0.8574    0.2033    0.5368    0.4184

To some, the complex set of steps that leads to bfresults may look daunting. I’d be a liar if I tried to say I wrote all those lines in one pass and got everything right. My suggestion is that as you build the pipeline you work step by step producing intermediate objects. Once you get the individual steps correct it’s trivial to join them using %>% and .x = ..

Now that we have our 15 bayes factors for each of the 15 pairings of runners we should probably join them together into one neat dataframe resultsdf that lays everything out for us. Based on the data available we would read line #5 as the odds are 2709:1 that Usain is faster than Maurice.

resultsdf <-
  data.frame(
    Runner1 = runner1,
    Runner2 = runner2,
    oddsfaster = bfresults
  )
resultsdf
##          Runner1        Runner2 oddsfaster
## 1     Usain Bolt   Asafa Powell    10.2865
## 2     Usain Bolt      Tyson Gay     9.4539
## 3     Usain Bolt  Justin Gatlin    59.3688
## 4     Usain Bolt    Yohan Blake    36.4792
## 5     Usain Bolt Maurice Greene  2709.0974
## 6   Asafa Powell      Tyson Gay     0.3217
## 7   Asafa Powell  Justin Gatlin     0.5782
## 8   Asafa Powell    Yohan Blake     0.7394
## 9   Asafa Powell Maurice Greene     8.5068
## 10     Tyson Gay  Justin Gatlin     0.2271
## 11     Tyson Gay    Yohan Blake     0.2902
## 12     Tyson Gay Maurice Greene     0.8574
## 13 Justin Gatlin    Yohan Blake     0.2033
## 14 Justin Gatlin Maurice Greene     0.5368
## 15   Yohan Blake Maurice Greene     0.4184

The Matrix reloaded (still waiting for #4)

Now that we have our resultsdf we can continue about the business of comparing the frequentist results of paired t-tests with their bayesian counterparts. Imagine that we have just completed a Oneway ANOVA of timing ~ runner (I’ll show the results in a bit). Given significant results of the omnibuds F test our next step is likely to run all the pairwise comparisons with some sort of correction for multiple comparisons like pairwise.t.test. (see this post for a review) The results are almost always given as a matrix often without repeating one of the diagonals. The results tell us that we can reject the null hypothesis that the runners have the same time for Usain Bolt versus all the other competitors. But it doesn’t allow us to make any statements about how different (despite the temptation inherent in the very different p values). It supplies almost no information about the other pairings, just that we can not reject the null.

pairwise.t.test(
  x = male_100$timing,
  g = male_100$runner,
  p.adjust.method = "holm"
)
## 
##  Pairwise comparisons using t tests with pooled SD 
## 
## data:  male_100$timing and male_100$runner 
## 
##                Usain Bolt Asafa Powell Tyson Gay Justin Gatlin Yohan Blake
## Asafa Powell   0.0460     -            -         -             -          
## Tyson Gay      0.0250     1.0000       -         -             -          
## Justin Gatlin  0.0024     1.0000       1.0000    -             -          
## Yohan Blake    0.0052     1.0000       1.0000    1.0000        -          
## Maurice Greene 6.9e-05    0.1649       1.0000    1.0000        1.0000     
## 
## P value adjustment method: holm

Another typical way of displaying the information is graphically as demonstrated here by ggstatsplot::ggbetweenstats.

ggbetweenstats(
  data = male_100,
  x = runner,
  y = timing,
  type = "p",
  var.equal = TRUE,
  pairwise.comparisons = TRUE,
  pairwise.display = "all",
  partial = FALSE,
  effsize.type = "unbiased",
  sort = "ascending",
  point.jitter.height = 0,
  messages = FALSE
)

Let’s see if we can’t at least produce a similar matrix to what pairwise.t.test yields. I’d like us to be able to do a sort of side by side comparison of the frequentists versus bayesian results.

Step by step the process we’ll follow is:

1. Use diag to create a matrix with ones in the diagonal we’ll set the size to howmanyrunners
2. Grab the runners names from orderbymean and populate the rownames and colnames
3. Use combn again this time populating it with numbers (one & two) rather than the runners names
4. Feed those vectors into a for loop to populate the bfmatrix with the data from resultsdf
5. To be consistent with pairwise.t.test remove the first row bfmatrix[-1, ] and the last column bfmatrix[, -howmanyrunners]
6. Finally populate the upper triangle part of the matrix with NA bfmatrix[upper.tri(bfmatrix)] <- NA

bfmatrix <- diag(nrow = howmanyrunners)
rownames(bfmatrix) <- orderbymean
colnames(bfmatrix) <- orderbymean
bfmatrix
##                Usain Bolt Asafa Powell Tyson Gay Justin Gatlin Yohan Blake Maurice Greene
## Usain Bolt              1            0         0             0           0              0
## Asafa Powell            0            1         0             0           0              0
## Tyson Gay               0            0         1             0           0              0
## Justin Gatlin           0            0         0             1           0              0
## Yohan Blake             0            0         0             0           1              0
## Maurice Greene          0            0         0             0           0              1
combn(howmanyrunners, 2)
##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14] [,15]
## [1,]    1    1    1    1    1    2    2    2    2     3     3     3     4     4     5
## [2,]    2    3    4    5    6    3    4    5    6     4     5     6     5     6     6
one <- combn(howmanyrunners, 2)[2, ]
two <- combn(howmanyrunners, 2)[1, ]
for (i in 1:numberofpairings) {
  bfmatrix[one[i], two[i]] <- resultsdf[i, 3] # row i, column 3 which is the BF value
}
bfmatrix
##                Usain Bolt Asafa Powell Tyson Gay Justin Gatlin Yohan Blake Maurice Greene
## Usain Bolt         1.0000       0.0000    0.0000        0.0000      0.0000              0
## Asafa Powell      10.2865       1.0000    0.0000        0.0000      0.0000              0
## Tyson Gay          9.4539       0.3217    1.0000        0.0000      0.0000              0
## Justin Gatlin     59.3688       0.5782    0.2271        1.0000      0.0000              0
## Yohan Blake       36.4792       0.7394    0.2902        0.2033      1.0000              0
## Maurice Greene  2709.0974       8.5068    0.8574        0.5368      0.4184              1
bfmatrix <- bfmatrix[-1, ]
bfmatrix <- bfmatrix[, -howmanyrunners]
bfmatrix
##                Usain Bolt Asafa Powell Tyson Gay Justin Gatlin Yohan Blake
## Asafa Powell      10.2865       1.0000    0.0000        0.0000      0.0000
## Tyson Gay          9.4539       0.3217    1.0000        0.0000      0.0000
## Justin Gatlin     59.3688       0.5782    0.2271        1.0000      0.0000
## Yohan Blake       36.4792       0.7394    0.2902        0.2033      1.0000
## Maurice Greene  2709.0974       8.5068    0.8574        0.5368      0.4184
bfmatrix[upper.tri(bfmatrix)] <- NA
bfmatrix
##                Usain Bolt Asafa Powell Tyson Gay Justin Gatlin Yohan Blake
## Asafa Powell      10.2865           NA        NA            NA          NA
## Tyson Gay          9.4539       0.3217        NA            NA          NA
## Justin Gatlin     59.3688       0.5782    0.2271            NA          NA
## Yohan Blake       36.4792       0.7394    0.2902        0.2033          NA
## Maurice Greene  2709.0974       8.5068    0.8574        0.5368      0.4184

Success! bfmatrix now looks a lot like the results of pairwise.t.test. But we can do better. Looking at the object produced by pairwise.t.test using str() we can see it is a list with 4 items. $method contains a simple text string explaining what it is. $data.name is another text string that tells us where the data came from. $p.value contains the actual p values and finally $p.adjust.method contains which method of p adjustment (holm) we used. Since there is no analog “adjustment method” for a bayes factor we can ignore it. Let’s make our own list called bfpairs that mimics that structure.

str(pairwise.t.test(
  x = male_100$timing,
  g = male_100$runner,
  p.adjust.method = "holm"
))
## List of 4
##  $ method         : chr "t tests with pooled SD"
##  $ data.name      : chr "male_100$timing and male_100$runner"
##  $ p.value        : num [1:5, 1:5] 4.60e-02 2.50e-02 2.44e-03 5.15e-03 6.89e-05 ...
##   ..- attr(*, "dimnames")=List of 2
##   .. ..$ : chr [1:5] "Asafa Powell" "Tyson Gay" "Justin Gatlin" "Yohan Blake" ...
##   .. ..$ : chr [1:5] "Usain Bolt" "Asafa Powell" "Tyson Gay" "Justin Gatlin" ...
##  $ p.adjust.method: chr "holm"
##  - attr(*, "class")= chr "pairwise.htest"
bfpairs <- list(
  method = " r = 0.707 Alt Hyp = -Inf<d<0",
  data.name = "male_100$timing and male_100$runner",
  p.value = bfmatrix
)
str(bfpairs)
## List of 3
##  $ method   : chr " r = 0.707 Alt Hyp = -Inf<d<0"
##  $ data.name: chr "male_100$timing and male_100$runner"
##  $ p.value  : num [1:5, 1:5] 10.29 9.45 59.37 36.48 2709.1 ...
##   ..- attr(*, "dimnames")=List of 2
##   .. ..$ : chr [1:5] "Asafa Powell" "Tyson Gay" "Justin Gatlin" "Yohan Blake" ...
##   .. ..$ : chr [1:5] "Usain Bolt" "Asafa Powell" "Tyson Gay" "Justin Gatlin" ...

What we’d like to do is mimic the print method for pairwise.t.test to include little things like substituting in an em dash instead of the NAs. To do that we need to peek at the print method with getAnywhere(print.pairwise.htest). getAnywhere() is a life saver if you want to be able to inspect a function.

getAnywhere(print.pairwise.htest)
## A single object matching 'print.pairwise.htest' was found
## It was found in the following places
##   registered S3 method for print from namespace stats
##   namespace:stats
## with value
## 
## function (x, digits = max(1L, getOption("digits") - 5L), ...) 
## {
##     cat("\n\tPairwise comparisons using", x$method, "\n\n")
##     cat("data: ", x$data.name, "\n\n")
##     pp <- format.pval(x$p.value, digits = digits, na.form = "-")
##     attributes(pp) <- attributes(x$p.value)
##     print(pp, quote = FALSE, ...)
##     cat("\nP value adjustment method:", x$p.adjust.method, "\n")
##     invisible(x)
## }
## <bytecode: 0x7fa1687942a8>
## <environment: namespace:stats>

Hmmmmm. Okay so we pass it a list object (the bfpairs we just created) and it takes the sub components and puts them into the right places on the screen. Spoiler alert, I won’t go into all the details but suffice it to say that format.pval() is problematic for us. It does a very nice job working with p values but p values have a different set of characteristics than bayes factors.

Rather than modify format.pval I simply decided to use the generic format function instead. That way the end user can specify all sorts of parameters like the number of digits, the symbol to replace NA, and the justification etc..

Here’s what I came up with after a little bit of work. Hopefully you’ll agree it does a reasonably good job of replicating the functionality of print.pairwise.htest?

print.pairwise.bftest <- function(x,
                                  digits = 2,
                                  nsmall = 0,
                                  width = 9,
                                  justify = "right",
                                  scientific = FALSE,
                                  nareplace = "-") {
  cat("\nPairwise comparisons of bayes factors with", x$method, "\n\n")
  cat("data: ", x$data.name, "\n\n")
  pp <- format(x$p.value,
    digits = digits,
    nsmall = nsmall,
    width = width,
    justify = justify,
    scientific = scientific
  )
  pp <- gsub("NA", nareplace, pp)
  print(pp, quote = FALSE)
  cat("\n\nAnalyzed using BayesFactor::ttestBF\n")
  invisible(x)
}
print.pairwise.bftest(bfpairs, digits = 1)
## 
## Pairwise comparisons of bayes factors with  r = 0.707 Alt Hyp = -Inf<d<0 
## 
## data:  male_100$timing and male_100$runner 
## 
##                Usain Bolt Asafa Powell Tyson Gay Justin Gatlin Yohan Blake
## Asafa Powell        10.3         -            -         -             -   
## Tyson Gay            9.5        0.3           -         -             -   
## Justin Gatlin       59.4        0.6          0.2        -             -   
## Yohan Blake         36.5        0.7          0.3       0.2            -   
## Maurice Greene    2709.1        8.5          0.9       0.5           0.4  
## 
## 
## Analyzed using BayesFactor::ttestBF

Notice that bayes factors aren’t shockingly dissimilar than the conclusions you would draw from a frequentist’s perspective. I still think they are a better choice because you can talk about odds and probabilities cleanly without falling into the frequentist “traps” surrounding what rejection of the null hypothesis is. With our “new” perspective we are safe in making statements that out data strongly support some of the pairwise differences (odds of 2709 to 1 are pretty convincing) and in other cases we can now quantify that odds are it’s “anyone’s race.”

Play it again Sam

As I wrote this post I wanted to ensure that I could run the analysis on a different set of runners with minimal effort. What follows is the code minus all of the intermediate printing and explanation. The difference is this time we’ll look at the top 7 fastest sprinters and widen our analysis to anyone with at least 20 races.

male_100_html <-
  read_html("http://www.alltime-athletics.com/m_100ok.htm")
male_100_pres <- male_100_html %>%
  html_nodes(xpath = "//pre")
male_100_htext <- male_100_pres %>%
  html_text()
male_100_htext <- male_100_htext[[1]]

male_100 <- read_fwf(
  male_100_htext,
  skip = 1,
  col_types = cols(.default = col_character()),
  col_positions = fwf_positions(
    c(1, 16, 27, 35, 66, 74, 86, 93, 123),
    c(15, 26, 34, 65, 73, 85, 92, 122, NA)
  )
)

male_100 <- male_100 %>%
  select(X2, X4) %>%
  transmute(timing = X2, runner = X4) %>%
  mutate(
    timing = gsub("A", "", timing),
    timing = as.numeric(timing)
  )

numbraces <- 20
howmanyrunners <- 7

orderbymean <- male_100 %>%
  group_by(runner) %>%
  summarise(avgtime = mean(timing), races = n()) %>%
  arrange(avgtime) %>%
  filter(races >= numbraces) %>%
  top_n(-howmanyrunners, avgtime) %>%
  pull(runner) %>%
  as.character()

male_100 <- male_100 %>%
  filter(runner %in% orderbymean) %>%
  mutate_if(is.character, as.factor) %>%
  droplevels()

male_100$runner <-
  factor(
    male_100$runner,
    levels = orderbymean
  )

numberofpairings <- factorial(howmanyrunners) / 
                    (factorial(2) * factorial(howmanyrunners - 2))

runner1 <- combn(orderbymean, 2)[1, ]
runner2 <- combn(orderbymean, 2)[2, ]

bfresults <- map2(
  runner1,
  runner2,
  function(a, b)
    male_100 %>%
      filter(runner %in% c(a, b)) %>%
      droplevels() %>%
      as.data.frame()
) %>%
  map(.x = ., ~ ttestBF(
    formula = timing ~ runner,
    data = .,
    nullInterval = c(-Inf, 0)
  )) %>%
  map(.x = ., ~ extractBF(x = .)) %>%
  map_dbl(.x = ., ~ .["Alt., r=0.707 -Inf<d<0", "bf"]) %>%
  round(., digits = 4)

resultsdf <-
  data.frame(
    Runner1 = runner1,
    Runner2 = runner2,
    oddsfaster = bfresults
  )

bfmatrix <- diag(nrow = howmanyrunners)
rownames(bfmatrix) <- orderbymean
colnames(bfmatrix) <- orderbymean
one <- combn(howmanyrunners, 2)[2, ]
two <- combn(howmanyrunners, 2)[1, ]
for (i in 1:numberofpairings) {
  bfmatrix[one[i], two[i]] <- resultsdf[i, 3]
}
bfmatrix <- bfmatrix[-1, ]
bfmatrix <- bfmatrix[, -howmanyrunners]
bfmatrix[upper.tri(bfmatrix)] <- NA

bfpairs <- list(
  method = " r = 0.707 Alt Hyp = -Inf<d<0",
  data.name = "male_100$timing and male_100$runner",
  p.value = bfmatrix
)

pairwise.t.test(
  x = male_100$timing,
  g = male_100$runner,
  p.adjust.method = "holm"
)
## 
##  Pairwise comparisons using t tests with pooled SD 
## 
## data:  male_100$timing and male_100$runner 
## 
##                   Usain Bolt Asafa Powell Tyson Gay Christian Coleman Justin Gatlin Yohan Blake
## Asafa Powell      0.0682     -            -         -                 -             -          
## Tyson Gay         0.0358     1.0000       -         -                 -             -          
## Christian Coleman 0.2500     1.0000       1.0000    -                 -             -          
## Justin Gatlin     0.0033     1.0000       1.0000    1.0000            -             -          
## Yohan Blake       0.0071     1.0000       1.0000    1.0000            1.0000        -          
## Maurice Greene    8.7e-05    0.2500       1.0000    1.0000            1.0000        1.0000     
## 
## P value adjustment method: holm
print.pairwise.bftest(bfpairs, 
                      digits = 3,
                      scientific =  TRUE,
                      nareplace = ".")
## 
## Pairwise comparisons of bayes factors with  r = 0.707 Alt Hyp = -Inf<d<0 
## 
## data:  male_100$timing and male_100$runner 
## 
##                   Usain Bolt Asafa Powell Tyson Gay Christian Coleman Justin Gatlin Yohan Blake
## Asafa Powell       1.03e+01         .            .         .                 .             .   
## Tyson Gay          9.45e+00   3.22e-01           .         .                 .             .   
## Christian Coleman  3.02e+00   4.61e-01     2.93e-01        .                 .             .   
## Justin Gatlin      5.94e+01   5.78e-01     2.27e-01  2.43e-01                .             .   
## Yohan Blake        3.65e+01   7.39e-01     2.90e-01  2.83e-01          2.03e-01            .   
## Maurice Greene     2.71e+03   8.51e+00     8.57e-01  5.30e-01          5.37e-01      4.18e-01  
## 
## 
## Analyzed using BayesFactor::ttestBF

And voila! Based on our new criteria of the fastest 7 runners with at least 20 races Christian Coleman has been added to the matrix. His mean timings place him square in the middle of the pack between Tyson Gay and Justin Gatlin. But notice the BF comparing him to Usain Bolt is only about 3 which is smaller than Tyson Gay 9.5 and Justin Gatlin 59.4 or any of the other runners. This is likely because the BF always adjusts based upon the amount of evidence available and we only have 23 races of data available for Christian.

Remember that one of the nice features of bayesian methodology is that we can quantify support for both the hypothesis we have as well as it’s converse (what a frequentist would call the null hypothesis). So our hypothesis is that Justin Gatlin is faster than Yohan Blake but the bayes factor 2.03e-01 (.203) says that the evidence from the data is that the odds are 1 / 2.03e-01 or about 5:1 that Justin is NOT faster than Yohan. That’s a statement that can not be made when using frequentist methods.

Done

I’ve really enjoyed this series of posts. I am always open to comments, corrections and suggestions. Feel free to leave a comment in disqus or send me an email.

Chuck

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