Name: P值問題。統計速成班#22 (P-Value Problems: Crash Course Statistics #22)
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Hi, I'm Adriene Hill, and Welcome back to Crash Course, Statistics.

To recap from last time, P-values tell us how “rare” something is.

So far, we've been using that information to decide whether or not our hypotheses are

reasonable, and using P-values to reject or fail to reject an idea.

Today, we're going to explore p-values a little more and talk about the logic of p-values

Remember, to calculate a p-value, we first assume that the null distribution is the true

Then we calculate how often we'd see a value that is at least as extreme as our observed value.

So in probability terms, the p-value is the probability of getting a sample as or more

extreme than ours, given that the null hypothesis is true:

So all the values that we see in the sampling distribution are means we could actually get

For example, let's say the average cat weigh 10lbs (or 4.5 kg).

We might want to calculate the probability of getting a group of 30 randomly selected

calico cats who have an average weight of 11 lbs (or 5 kg) if calico cats have the same

average weight as the whole population of cats.

The first issue is if, in real life, there is no connection between two things like fur

color and weight --we still might get samples of calicos, mackerel tabbies, or tortoise

shells that are different enough to cause us to “reject” the null hypothesis that

Our alpha tells us how often this will happen.

Let's say our hypothesis is that the reaction time of older professional chess players is

different from the reaction time of the general population of professional chess players.

Even if older chess players are the same as their colleagues, if we ran this study over

and over, we'd expect that 5% of the time, we'd mistakenly reject the null if it were true.

This is one reason why p-values are pretty controversial in the statistical community right now.

Not everyone agrees that a p-value less than 0.05 is sufficient evidence to reject the

In fact, some studies that look at incredibly important things like new medications, have

already decided that an alpha of 0.05 isn't low enough.

They want p-values lower than 0.01 so that if the null hypothesis is true, they'll

only mistakenly reject it 1% of the time.

Still others argue that 0.005 is the better cutoff.

As you can see, the standard cutoff is arbitrary.

Null Hypothesis Significance Testing requires that we draw a line in the sand somewhere,

Arguments have been made that we can have different p-value cutoffs--our alphas--depending

on the situation, and that scientists should be allowed to justify their reasons for picking

But on the whole, many fields that regularly use p-values have some sort of “official”

The second, related issue is that a p-value tells you how “extreme” your data would

be if you assume the null hypothesis is true.

But when you really think about it...that's not what we want to know.

We want to know whether the null is correct, or at least probably correct.

In other words, the probability of the null, given that we've seen our data.

A p-value of 0.02 in a study on cancer rates in mice tells you that if your new drug didn't

work and there was no difference between the cancer rates of mice on and off the drug,

then you'd only expect 2% of identically run studies to produce a difference in cancer

rates that's as or more extreme than the one you just observed.

But we can't use these p-values alone to tell us about the probability of the null

being true or false, even though it can be tempting to think we can.

One common misinterpretation of a p-value is that it can tell you the probability that

For example, if a random sample of tuna has a 10% higher mercury content than a random

sample of mahi-mahi, it would be incorrect to say that a p-value of 0.02 in this case

means there's only a 2% chance that the null hypothesis is true.

This is an especially tempting misinterpretation because it feels like it maybe should be true,

but again, when we calculate our p-value, we've already assumed for a moment that

the null hypothesis is true and that any sample differences we see are actually due to just

If our p-value for the chess study was 0.01, that means that we already assumed older chess

players were the same as the general population of chess players, so 0.01 can't tell us

much about the probability that older chess players are the same as their colleagues.

That would be like saying “assuming that grass is green, what's the probability that

Similarly, p-values can't tell you the probability that you've made an error, given that you

Again, this is because p-values don't tell you about the probability of the null being

If you've rejected the null hypothesis--like that drinking orange juice is not associated

with higher levels of cavities than drinking coffee--either you did so correctly, because

there really is a difference between cavities in OJ and coffee drinkers, or you did so mistakenly

because there really is no discernible difference.

But p-values--since they assume the null is true--don't tell you how likely either of

Ronald Fisher--one of the first proponents of Null Hypothesis Significance Testing wrote

that: “ In general tests of significance are based on hypothetical probabilities calculated

They do not generally lead to any probability statements about the real world, but to a

rational and well-defined measure of reluctance to the acceptance of the hypotheses they test."

In other words, getting a p-value of 0.04 doesn't mean that there's a 4% chance

The probability we want to know is the opposite conditional probability from what a p-value

We want to know the probability of the null hypothesis given that we got this data.

From the p-value we get the Probability of the data given the null.

|older chess players are the same as population of chess players ) but we wish we could calculate

P(older chess players are the same as population of chess players | data).

And while all the same pieces are there, they're not the same.

This is made even more clear when you realize the probability of being a child, given that

you're at Chuck E Cheese is NOT the same as the probability of being at Chuck E Cheese,

This is one reason why p-values are so perplexing.

They don't give us the probability that we truly want.

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P值問題。統計速成班#22 (P-Value Problems: Crash Course Statistics #22)

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