Test of Proportions: Twin Beta Distribution
Post #1058
According to research, base sliding is the activity that led to 71% of all softball injuries. Standard bases in the baseball diamond are held in place firmly with a post sticking into the ground. But, by being immovable, they invite injury as you slide into them.
This led Roger Hall to invent Break-away bases (bases that detach when great force is applied to them). But to prove if these new bases are better, David Janda et al. recorded injuries in over 2,000 games played with either type of base. In the original head-to-head study, just over 600 games were played on each.
But after witnessing a dramatic drop in base injuries with Break-away bases, Janda continued recording injuries in games using them: for a grand total of 1,668 games. You can visualize the difference in probability of injury by super-imposing beta distributions built up from the existing evidence.
Beta distributions show you a probability distribution of probabilities.
[the graph was created in R, using published data from PMID: 2126674]
As you can see, there is no contest. The evidence led to a distribution of possible injury probabilities for standard bases (red) that reached down to about 0.04 (4 injuries per 100 games) and reached up to about 0.10 (10 injuries per 100 games). The width reveals that the amount of evidence was not high enough for a very precise estimate.
But with Break-away bases (blue; peak not shown) the narrow distribution of possible injury probabilities shows us that the level of evidence was sufficient to rule out any injury probabilities that lie above 0.02 (2 injuries per 100 games). The probability that the injury rate is less than 2 per 100 games is: 0.999 999 999 984.
That’s equivalent to just 16 chances in a trillion that true injury rates are higher. In other words, if scientific evidence exists for anything, then it exists for the superiority of Break-away bases to the standard ones. In court cases, the level of evidence that is deemed “jury convincing” is 1 chance in a million, or 0.000 001.
The evidence for Break-away bases is 62,000 times stronger than what it takes for a guilty verdict in a courtroom.
NOTES
A 99% Confidence Interval was formed around the injury rates (in Columns G and H below) and the boundaries of the simultaneous 99% Confident Intervals do not even come close to overlapping each other:
While one-sided tests based off of a single 99% Confidence Interval are not enough to be called “conclusive” (they still have a 1-in-200 chance of failure), when twin 99% Confidence Intervals do not overlap, then the evidence can be said to be conclusive.
Reference
[Roger Hall invented Break-away bases] — https://www.latimes.com/archives/la-xpm-1989-02-26-sp-932-story.html
[Janda et al. proved they work] — Janda DH, Wojtys EM, Hankin FM, Benedict ME, Hensinger RN. A three-phase analysis of the prevention of recreational softball injuries. Am J Sports Med. 1990 Nov-Dec;18(6):632-5. doi: 10.1177/036354659001800613. PMID: 2126674. https://pubmed.ncbi.nlm.nih.gov/2126674/