We now know how to fit models (in two ways!). Let's do experiments now!
Let's say we all go out and test our own favorite method for avoiding COVID: vitamin C, vaccination, lots of coffee, lucky charms, chloroquine... Everyone comes back with an estimate of the relative risk of getting COVID using their treatment. How useful is this?
Some people have much bigger studies than others: whose methods will have the largest estimates: large studies or small ones?
We need to also report a measure of uncertainty and/or the strength of statistical support for our results.
Me: I have an amazing new treatment! I had ten random people soak their feet in ice water three times a day for a month, and none of them got sick!
You: Okay, sure, but they might not have got sick anyhow?
Me: I got that covered, ten other random people who didn't do this, and three of them got sick.
You: Hm, okay, but isn't it kinda likely that you'd get such a big difference even if the ice water doesn't do anything? Just by random chance?
Me: Gee, I dunno, how do we find out?
You: Well, this isn't perfect, but
ice = rng.binomial(n=10, p=3/20, size=10000)
not_ice = rng.binomial(n=10, p=3/20, size=10000)
np.mean(ice - not_ice >= 3)
0.0521
I think that writing affects the ligaments in your fingers, so that the index finger on peoples' writing hands tends to be a different length than the index finger on the other hand.
Let's collect some data!
num_longer = 14
num_shorter = 5
It's maybe more natural to suppose there isn't a difference. Okay, let's see how good my evidence is against this idea: what's the probability that we'd see such a big difference between those numbers even if we all just flipped coins to decide instead?
# rng.choice(['h', 't'])
num_heads = 7
num_tails = 12
Exercise: find the "probability that we'd see such a big difference between those numbers even if we all just flipped coins", by simulation.
Conclusion: if we were flipping coins, the chance of getting at least 9 more of one thing than the other is about 2%.
# np.sum(rng.choice(["H", "T"], size=19) == "H")
num_heads = rng.binomial(n=19, p=1/2, size=10000)
num_tails = 19 - num_heads
np.mean(np.abs(num_heads - num_tails) > np.abs(num_longer - num_shorter))
0.0192
The $p$-value is
the probability of seeing a result at least as surprising as what we observed in the data, if the null hypothesis is true.
The parts of this are:
the probability ... if the null hypothesis is true: we need a concrete model we can compute probabilities with
a result: a statistic summarizing how strongly our data suggest that model is not right
at least as surprising: usually, the statistic is chosen so that larger values are more surprising
In the finger exercise, we found the probability that
if we all flipped fair coins,
the difference
between the number of people with Heads and with Tails
was at least as big as
Why's this a $p$-value?
Recall that the variance of a random variable $X$ with mean $\mu$ is $$ \text{var}[X] = \mathbb{E}[ (X - \mu)^2 ] . $$ If we have samples $x_1, \ldots, x_n$, with sample mean $\bar x = (x_1 + \cdots + x_n)/n$, this suggests estimating the variance of the distribution they are drawn from as $$ \sigma^2(x) = \frac{1}{n} \sum_{i=1}^n (x_i - \bar x)^2 . $$
However: the estimator for $\var[X]$ you want to use has an $n-1$ instead of $n$, and is called the sample variance: $$ s^2(x) = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar x)^2 . $$
It is an unbiased estimator: if $X_1, \ldots, X_n$ are independent draws from $X$, then $$ \E[s^2(X_1, \ldots, X_n)] = \var[X], $$ while $\sigma^2(X)$ gives an underestimate: $$ \E[\sigma^2(X_1, \ldots, X_n)] = \frac{n-1}{n} \var[X]. $$
The sample SD, as returned for instance by np.std( ), is $s(x)$.
Checking this with math is straightforward but tedious; here's a quick simulation check:
def sigma2(x):
return np.sum((x - np.mean(x))**2)/len(x)
def s2(x):
return np.sum((x - np.mean(x))**2)/(len(x) - 1)
n = 4
sig_vals = [sigma2(rng.exponential(size=n)) for _ in range(1000)]
s_vals = [s2(rng.exponential(size=n)) for _ in range(1000)]
true_var = np.var(rng.exponential(size=10000))
print(f"The true variance is {true_var}.")
print(f"The mean sigma^2 is {np.mean(sig_vals)}.")
print(f"The mean sample variance is {np.mean(s_vals)}.")
The true variance is 1.002170012000653. The mean sigma^2 is 0.7774769057906117. The mean sample variance is 0.9952134357948298.
It turns out that thanks to the Central Limit Theorem, if
then the difference between the sample mean and $\mu$, in units of $S$: $$ T = \frac{\bar X - \mu}{S/\sqrt{n}} $$ has, approximately$^*$, Student's t distribution with $n-2$ degrees of freedom.
$^*$ (the approximation is better the bigger $n$ is and the closer $X_i$ is to Normal)
So: let's suppose we have measurements $x_1, \ldots, x_n$.
How do we estimate the mean? With the sample mean, $(x_1 + \cdots + x_n)/n$. And, the SD? With the sample SD, $s$.
How sure are we the mean is not zero? Well, if it was zero, then $t = \bar x / (s/\sqrt{n})$ would be Student's $t(n-1)$ distributed, so if $t$ is bigger than we'd expect then we can be pretty sure: the $p$-value is $$ \P\{ T > t \} , $$ where $T$ is Student's t with $n-1$ degrees of freedom.
In other words:
Let's say we've measured how typing speed of 100 people changes after drinking coffee: the average change is 3 wpm and the SD is 10 wpm. Is this good evidence that coffee increases typing speed?
Well, this has a $t$ statistic of $$ t = \frac{3}{10/\sqrt{100}} = 3, $$ and the probability a $t$ distribution with 98 degrees of freedom is at least 3 is
from scipy.stats import t
2 * (1 - t.cdf(3, df=98))
0.003423264463729536
... so, it does seem like good evidence? ($p = 0.003$, $t$-test with $\text{df}=98$)
We found an average of 3 wpm higher after drinking coffee; this is good evidence that coffee increases typing speed ($p = 0.003$, by $t$-test with df=98).
If there was no mean difference in typing speed, the chance that we would see a difference as big as what we saw is only 0.3%.
If I do 1000 different studies and none of the things I'm looking for are real, then I only expect statistical evidence that is this strong in about 3 of them.
However: this does not answer the questions: "What is the range of likely increases in typing speed?" (see: confidence intervals) or "Is this difference in typing speed big enough to care about?" (just think about the problem)