How to [pretend to] be a better coach using bad statistics

Here is a simple scenario from practice: Coach A uses YOYOIRL1 test and Coach B uses 30-15IFT (for more info see recent paper my Martin Buchheit, which also stimulated me to write this blog) to gauge improvements in endurance

Coach A: We have improved distance covered in YOYOIRL1 test from 1750m to 2250m in four weeks. That is 500m improvement or ~28%

Coach B: We have improved velocity reached in 30-15IFT from 19km/h to 21km/h in four weeks . That is 2km/h improvement or ~10%

If you present those to someone who is not statisticaly educated he/she might conclude the following:

• Coach A did a better job, since the improvement is 28% compared to 10% of Coach B
• YOYOIRL1 test is more sensitive to changes than 30-15IFT

As a coaches, we needs to report to a manager(s), so which one would you prefer reporting? 28% or 10%? Be honest here!

Unfortunately, we cannot conclude who did a better job (Coach A or Coach B), nor which test is more sensitive (YOYOIRL1 or 30-15IFT) from percent change data. A lot of managers and coaches don't get this. At least I haven't until recently.

What we need is Effect Size statistics, or Cohen's D. But for that we need to know variability in the groups, expressed as SD (standard deviation). Let's simulate the data and use usual SDs for YOYOIRL1 and 30-15IFT

require(ggplot2, quietly = TRUE)
require(reshape2, quietly = TRUE)
require(plyr, quietly = TRUE)
require(randomNames, quietly = TRUE)
require(xtable, quietly = TRUE)
require(ggthemes, quietly = TRUE)
require(gridExtra, quietly = TRUE)

set.seed(1)
numberOfPlayers <- 150
playerNames <- randomNames(numberOfPlayers)

# Create YOYOIRL1 Pre- and Post- data using 300m as SD
YOYOIRL1.Pre <- rnorm(mean = 1750, sd = 300, n = numberOfPlayers)
YOYOIRL1.Post <- rnorm(mean = 2250, sd = 300, n = numberOfPlayers)

# We need to round YOYOIRL1 score to nearest 40m, since those are the
# increments of the scores
YOYOIRL1.Pre <- round_any(YOYOIRL1.Pre, 40)
YOYOIRL1.Post <- round_any(YOYOIRL1.Post, 40)

# Create 30-15IFT Pre- and Post- data using 1km/h as SD
v3015IFT.Pre <- rnorm(mean = 19, sd = 1, n = numberOfPlayers)
v3015IFT.Post <- rnorm(mean = 21, sd = 1, n = numberOfPlayers)

# We need to round 30-15IFT to nearest 0.5km/h, since those are the
# increments of the scores
v3015IFT.Pre <- round_any(v3015IFT.Pre, 0.5)
v3015IFT.Post <- round_any(v3015IFT.Post, 0.5)

# Put those test into data.frame
testDataWide <- data.frame(Athlete = playerNames, YOYOIRL1.Pre, YOYOIRL1.Post, 
    v3015IFT.Pre, v3015IFT.Post)

# And print first 15 athletes
print(xtable(head(testDataWide, 15), border = T), type = "html")

	Athlete	YOYOIRL1.Pre	YOYOIRL1.Post	v3015IFT.Pre	v3015IFT.Post
1	Shrestha, Ezell	2000.00	2520.00	18.50	19.50
2	Cha, Gequan	1440.00	2080.00	20.50	20.00
3	Brown, Hindav	2360.00	2040.00	19.50	19.50
4	Venegas Delarosa, Destinee	1640.00	1800.00	19.50	21.50
5	Simon, Barrington	2240.00	1800.00	19.00	19.50
6	Williams, Hyeju	2200.00	2280.00	18.00	19.00
7	Gutierrez, Sabrina	1760.00	2400.00	17.50	23.00
8	Wilder, Johannah	1920.00	1640.00	19.00	22.00
9	Martin Dean, Jillian	1440.00	1960.00	21.00	22.00
10	Thomas, Neil	1840.00	2400.00	19.00	20.50
11	Nosker, Andrew	2080.00	2120.00	19.00	21.00
12	Blackford, Matthew	1760.00	2880.00	19.50	21.50
13	Mata, Rachel	1600.00	2640.00	18.50	19.50
14	Cheng, Ryan	1560.00	2440.00	17.50	21.00
15	True, Ashley	1720.00	2240.00	21.50	21.00

To plot the data and to do simple descriptive stats we need to reshape the data from wide format to long format using reshape2 package by Hadley Wickham

# Reshape the data
testData <- melt(testDataWide, id.vars = "Athlete", variable.name = "Test", 
    value.name = "Score")

# And print first 30 rows
print(xtable(head(testData, 30), border = T), type = "html")

	Athlete	Test	Score
1	Shrestha, Ezell	YOYOIRL1.Pre	2000.00
2	Cha, Gequan	YOYOIRL1.Pre	1440.00
3	Brown, Hindav	YOYOIRL1.Pre	2360.00
4	Venegas Delarosa, Destinee	YOYOIRL1.Pre	1640.00
5	Simon, Barrington	YOYOIRL1.Pre	2240.00
6	Williams, Hyeju	YOYOIRL1.Pre	2200.00
7	Gutierrez, Sabrina	YOYOIRL1.Pre	1760.00
8	Wilder, Johannah	YOYOIRL1.Pre	1920.00
9	Martin Dean, Jillian	YOYOIRL1.Pre	1440.00
10	Thomas, Neil	YOYOIRL1.Pre	1840.00
11	Nosker, Andrew	YOYOIRL1.Pre	2080.00
12	Blackford, Matthew	YOYOIRL1.Pre	1760.00
13	Mata, Rachel	YOYOIRL1.Pre	1600.00
14	Cheng, Ryan	YOYOIRL1.Pre	1560.00
15	True, Ashley	YOYOIRL1.Pre	1720.00
16	Inouye, Connor	YOYOIRL1.Pre	2080.00
17	Tatum, Janice	YOYOIRL1.Pre	1600.00
18	Latour, Pearl	YOYOIRL1.Pre	1720.00
19	Tripathi, Juan	YOYOIRL1.Pre	1360.00
20	Moore, Michelle	YOYOIRL1.Pre	1880.00
21	O'Sullivan, Johanna	YOYOIRL1.Pre	2160.00
22	Sharp, Gregory	YOYOIRL1.Pre	2200.00
23	Blum, Jennifer	YOYOIRL1.Pre	2000.00
24	Doering, Darius	YOYOIRL1.Pre	1200.00
25	Sohn, Kendle	YOYOIRL1.Pre	1880.00
26	Horton, Grant	YOYOIRL1.Pre	1880.00
27	Waynewood, Nicholas	YOYOIRL1.Pre	1640.00
28	Pallen, Raymundo	YOYOIRL1.Pre	1800.00
29	Montoya, Simon	YOYOIRL1.Pre	1480.00
30	Clark, Bryce	YOYOIRL1.Pre	1960.00

From the tables above it is easy to see the difference between wide and long data formats.

Let's calculate simple stats using plyr package from Hadley Wickham (yes, he is a sort of celebrity in R community) and plot them using violin plots, which is great since they show the distribution of the scores

# Subset YOYOIRL1 tets
ggYOYO <- ggplot(subset(testData, Test == "YOYOIRL1.Pre" | Test == "YOYOIRL1.Post"), 
    aes(x = Test, y = Score))

ggYOYO <- ggYOYO + geom_violin(fill = "red", alpha = 0.5) + theme_few() + stat_summary(fun.y = mean, 
    geom = "point", fill = "white", shape = 23, size = 5)


# Subset 30-15IFT tets
ggIFT <- ggplot(subset(testData, Test == "v3015IFT.Pre" | Test == "v3015IFT.Post"), 
    aes(x = Test, y = Score))

ggIFT <- ggIFT + geom_violin(fill = "steelblue", alpha = 0.5) + theme_few() + 
    stat_summary(fun.y = mean, geom = "point", fill = "white", shape = 23, size = 5)


# Plot the graphs
grid.arrange(ggYOYO, ggIFT, ncol = 2)

# Calculate the summary table
testDataSummary <- ddply(testData, "Test", summarize, N = length(Score), Mean = mean(Score), 
    SD = sd(Score))
# Print the summary table
print(xtable(testDataSummary, border = T), type = "html")

	Test	N	Mean	SD
1	YOYOIRL1.Pre	150	1749.60	309.40
2	YOYOIRL1.Post	150	2246.13	318.41
3	v3015IFT.Pre	150	18.86	1.12
4	v3015IFT.Post	150	20.98	1.09

From the table above we can calculate the percent change.

YOYOIRL1.Change <- (testDataSummary$Mean[2] - testDataSummary$Mean[1])/testDataSummary$Mean[1] * 
    100
v3015IFT.Change <- (testDataSummary$Mean[4] - testDataSummary$Mean[3])/testDataSummary$Mean[3] * 
    100

print(xtable(data.frame(YOYOIRL1.Change, v3015IFT.Change), border = T), type = "html")

	YOYOIRL1.Change	v3015IFT.Change
1	28.38	11.22

But as mentioned in the beginning of the post, percent change is not the best way to express change and sensitivity of the tests (although it is great to impress the managers or your superiors, or claim that your test is more sensitive).

What we need to do is to calculate effect size (ES). ES takes into account the difference between the means and SD (in this case of the Pre- test, but it can also use pooled SD).

YOYOIRL1.ES <- (testDataSummary$Mean[2] - testDataSummary$Mean[1])/testDataSummary$SD[1]
v3015IFT.ES <- (testDataSummary$Mean[4] - testDataSummary$Mean[3])/testDataSummary$SD[3]

print(xtable(data.frame(YOYOIRL1.ES, v3015IFT.ES), border = T), type = "html")

	YOYOIRL1.ES	v3015IFT.ES
1	1.60	1.88

From the data above we can conclude that they are pretty similar and that 30-15IFT might be a bit more sensitive (or the Coach B did a better job).

Anyway, to summarize this blog post - start reporting ES alongside with percent change. If someone claims high improvements in testing scores to show how great coach he is, or how great his program is, ask to see ES or the distribution of the change scores or Pre- and Post- tests. Besides we need to also ask for SWC and TE, but more on that later.