In a comment on my post on Significance, I discovered that our rate of published Type-1 error in science would probably be higher if humans had eight fingers instead of ten. Type-1 error is when we wrongly reject the null hypothesis – when our studies seem to give us evidence of an effect or link that in reality is false in the population at large. It's a false positive, a finding that isn't a true finding. Setting our threshold of statistical significance at p = 0.05 is one way we try to reduce Type-1 error.
People have had trouble understanding that comment – it's too brief, skips some steps. I'll lay it out more clearly here.
In the post on Significance, I pointed out that one reason why we use the p = 0.05 threshold for statistical significance in many of our tests is that humans have ten fingers.
Because we have ten fingers, we use a base-10 number system, and we tend to prefer numbers that are multiples of ten or five for many purposes. It's probably intuitive for most of you that scientists would have been unlikely to choose 0.04 or 0.061 as our significance threshold. We needed something in that ballpark, something sufficiently stringent, and it's not surprising that we chose 0.05 – nor would it be surprising if we had chosen 0.10 or 0.01. We see those numbers as nice clean, "round" numbers, not 0.03 or 0.04.
I noted that if we had eight fingers, and thus had ultimately settled on a base-8 number system, we might use 0.04 as our threshold for significance. Holding everything else about human nature and psychology constant, it seems likely that in that scenario, we'd prefer numbers that were multiples of eight and four, just as we prefer tens and fives in our universe.
In a comment, Jonathan Jones pointed out that 0.04 in base-8 is actually 0.0625 base-10.
What does that mean?
It means the numeral, the string of symbols 0.04 would represent a different value in a base-8 number system than it does in base-10. It means that when people in a base-8 civilization write or say 0.04, it represents a different quantity of stuff (or of probability) than it does when people in our society write or say 0.04.
It's difficult to think in different number base systems, because we're so conditioned to certain symbols corresponding to certain values. It's similar to the Stroop task, where you might have to identify the color of the word green when it's displayed in orange-colored text (either identifying the named color or the displayed color, depending on the task.)
To understand different bases it's helpful to distinguish a numerical value and the symbols we use to represent those values. You can get this just by noting that we could any symbol we wanted to represent the number 4, e.g. we could treat @ as 4, but we've long settled on the symbol 4.
A base-10 system has ten numerical digits, ten unique graphemes that represent the first ten integers (including zero): 0 1 2 3 4 5 6 7 8 9. A grapheme is an elemental visual symbol of written languages, what in computing we might call a character (see Unicode). Every letter you're currently reading is a grapheme – i.e. the letters of the alphabet are graphemes. The symbols for numerical digits are also graphemes (Note that the word digit comes from the Latin digitus, which means finger or toe, which helps illustrate how our number system is based on our finger count.)
There is no digit or grapheme to represent the value ten (10) because we've used one of the ten digits to represent zero. Once we hit ten, we need multiple digits to represent numbers.
A base-8 system has eight numerical digits, eight unique graphemes that represent the first eight integers (including zero): 0 through 7.
The integers 0 1 2 3 4 5 6 7 – just those single digit integers – represent the same values in base-8 and base-10.
Things change once we get past the number 7, or into multiple digits. The numerals 8 and 9 do not exist in a base-8 system. Once we get past 7, we need multiple digits to represent numbers, just like in base-10 we need multiple digits to go higher than 9. In base-8, the value 8 is represented as 10. The value 9 is represented as 11.
And 0.04 in base-8 represents the value 0.0625 in base-10.
Why? How do we convert decimal, fractional values from one system to the other?
Let's start with 0.1. Using our standard positioning, 0.1 represents one-nth of the base. So in our base-10 system, 0.1 means one-tenth. In base-8, 0.1 means one-eighth.
Then let's try 0.01. This represents one-n²th of the base. So in base-10, 0.01 means one-hundredth (1/10²). In base-8, it means one-sixty-fourth (1/8²).
Therefore, in base-8, 0.04 is four-sixty-fourths, or 4/64, which is 0.0625 in base-10.
Since a p-value of 0.0625 or lower is easier to obtain (and literally more likely) than a p-value of 0.05 or lower, more Type-1 errors would be published. If we had eight fingers, it's quite plausible our threshold would be 0.0625 (base-10), which we would call 0.04, and we'd have a slightly more error-prone scientific culture. That's interesting.
This assumes the same scientific ecology where significant findings are favored over marginally significant and non-significant findings – it assumes we're holding everything else constant, which seems quite right. It also assumes that a different objective threshold doesn't impact the quality of the research, or have any other tricky dynamic effects.
A lot of the rules of thumb and threshold values we use in our civilization are ultimately grounded in the fact that we have ten fingers. This reminds me of the book I'll write on evolutionary psychology and exobiology in ten years. I think it would be very fruitful if evolutionary psychologists (and biologists) zoomed back a bit and framed human evolution by thinking of how various background factors would be different on other planets, and what the implications are. It goes much further than the number of fingers we have. For example, think about fire and its many impacts and implications, and consider that fire will be atmospherically impossible on many planets (even those in habitable zones), and how that would impact the course of life compared to Earth, the kinds of organisms that are possible and not possible, and consequences ten or twenty steps deep into the model.
José L. Duarte
Social Psychology, Scientific Validity, and Research Methods.