I was wondering if anybody could give a concise rundown as to the definitions and uses of p-values, significance level and type I error.

I understand that p-values are defined as "the probability of obtaining a test statistic at least as extreme as the one we actually observed", while a significance level is just an arbitrary cutoff value to gauge if the p-value is significant or not. Type I error is the error of rejected a null hypothesis that was true. However, I am unsure regarding the difference between significance level and the type I error, are they not the same concept?

For example, assume a very simple experiment where I flip a coin 1000 times and count the number of times it lands on 'heads'. My null hypothesis, H0, is that heads = 500 (unbiased coin). I then set my significance level at alpha = 0.05.

I flip the coin 1000 times and then I calculate the p-value, if the p-value is > 0.05 then I fail to reject the null hypothesis and if the p-value is < 0.05 then I reject the null hypothesis.

Now if I did this experiment repeatedly, each time calculating the p-value and either rejecting or failing to reject the null hypothesis and keeping a count of how many I rejected/failed to reject, then I would end up rejecting 5% of null hypotheses which were in actuality true, is that correct? This is the definition of type I error. Therefore, the significance level in Fisher significance testing is essentially the type I error from Neyman-Pearson hypothesis testing if you performed repeated experiments.

Now as for p-values, if I had gotten a p-value of 0.06 from my last experiment and I did multiple experiments and counted all the ones that I got a p-value from 0 to 0.06, then would I also not have a 6% chance of rejecting a true null hypothesis?

up vote 16 down vote accepted

The question looks simple, but your reflection around it shows that it is not that simple.

Actually, p-values are a relatively late addition to the theory of statistics. Computing a p-value without a computer is very tedious; this is why the only way to perform a statistical test until recently was to use tables of statistical tests, as I explain in this blog post. Because those tables were computed for fixed $\alpha$ levels (typically 0.05, 0.01 and 0.001) you could only perform a test with those levels.

Computers made those tables useless, but the logic of testing is still the same. You should:

  1. Formulate a null hypothesis.
  2. Formulate an alternative hypothesis.
  3. Decide a maximum type I error (the probability of falsely rejecting the null hypothesis) error you are ready to accept.
  4. Design a rejection region. The probability that the test statistic falls in the rejection region given that the null hypothesis is your level $\alpha$. As @MånsT explains, this should be no smaller than your acceptable type I error, and in many cases use asymptotic approximations.
  5. Carry out the random experiment, compute the test statistic and see whether it falls in the rejection region.

In theory, there is a strict equivalence between the events "the statistic falls in the rejection region" and "the p-value is less than $\alpha$", which is why it is felt that you can report the p-value instead. In practice, it allows you to skip step 3. and evaluate the type I error after the test is done.

To come back to your post, the statement of the null hypothesis is incorrect. The null hypothesis is that the probability of flipping a head is $1/2$ (the null hypothesis cannot pertain to the results of the random experiment).

If you repeat the experiment again and again with a threshold p-value of 0.05, yes, you should have approximately 5% rejection. And if you set a p-value cut-off of 0.06, you should end up with roughly 6% rejection. More generally, for continuous tests, by definition of the p-value $p$

$$ Prob(p < x) = x, \, (0 < x < 1), $$

which is only approximately true for discrete tests.

Here is some R code that I hope can clarify this a bit. The binomial test is relatively slow, so I do only 10,000 random experiments in which I flip 1000 coins. I perform a binomial test and collect the 10,000 p-values.

set.seed(123)
# Generate 10,000 random experiments of each 1000 coin flipping
rexperiments <- rbinom(n=10000, size=1000, prob=0.5)
all_p_values <- rep(NA, 10000)
for (i in 1:10000) {
    all_p_values[i] <- binom.test(rexperiments[i], 1000)$p.value
}
# Plot the cumulative density of p-values.
plot(ecdf(all_p_values))
# How many are less than 0.05?
mean(all_p_values < 0.05)
# [1] 0.0425
# How many are less than 0.06?
mean(all_p_values < 0.06)
# 0.0491

You can see that the proportions are not exact, because the sample size is not infinite and the test is discrete, but there is still an increase of roughly 1% between the two.

  • @MånsT Thanks! +1 to you for the distinction between continuous and discrete tests (that I would have honestly completely overlooked). – gui11aume Aug 1 '12 at 10:23
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    @gui11aume, thanks for you input! However, your statement "p-values are a relatively late addition to the theory of statistics" is strange. From what I have read, Fisher's 'significance testing' with p-values originated around 1925. While the Neyman-Pearson's 'hypothesis testing' came about as an 'improvement' to Fisher's work a few years later. While it is true that p-values were difficult to compute (hence why standard levels of significance were used), his work was monumental. In fact, he is referred to as 'the father of statistics' because he formed the basis of much of modern statistics. – BYS2 Aug 1 '12 at 11:41
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    @BYS2 Absolutely right (+1). The theory of p-values dates back from the origin of statistics. It is their pervasive use that is recent. Thanks for noticing ;-) – gui11aume Aug 1 '12 at 18:55
  • @guillaume thanks for that, I have another quick question though. You say that my null hypothesis Cannot be H 0=500 but I have seem numerous texts use for example: nul l hypothesis is that the mean will be 0 or that the difference in means will be 10.. I've never had any problems doing it like that :s.. The t distribution essentially just scales if I used H0=500 instead of H0=0.5 – BYS2 Aug 2 '12 at 0:41
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    @gui11aume: Maybe it could be interesting to take a look at my answer: stats.stackexchange.com/questions/166323/… – user83346 Aug 15 '15 at 15:30

You are getting good answers here from @MansT & @gui11aume (+1 to each). Let me see if I can get more explicitly at something in both of their answers.

When working with discrete data, there are only certain p-values possible, and the problem is worse with fewer possibilities / smaller data sets. For example, imagine flipping a coin $n$ times. The probability of getting a particular number of heads, $k$, is: $$ p(k)=\frac{n!}{k!(n-k)!}p^k(1-p)^{n-k} $$ Let's say a researcher want's to test a given coin (which actually is fair) for fairness by flipping it 10 times and recording the number of heads. That is, the null hypothesis is true here. Our researcher sets $\alpha=.05$, by convention and because that's what's necessary for acceptance by the larger community. Now, ignoring the conventional alpha for a moment, let's consider the 2-tailed p-values (type I error rates) that are possible in this situation:

number of heads:           0    1    2    3    4    5    6    7    8    9   10
individual probability:  .001 .010 .044 .117 .205 .246 .205 .117 .044 .010 .001
type I error rate:       .002 .021 .109 .344 .754   1  .754 .344 .109 .021 .002

What this demonstrates is that using $\alpha=.05$ will lead to a long-run 2-tailed type I error rate of $.021$. So this is clearly a case where $\alpha\ne\text{type I error}$, however, if $\alpha$ were set to one of the above values (instead of $.05$) then the significance level would equal the type I error rate. Despite that problem, the p-value does equal the type I error rate in this case. Note that there is no issue here with a mismatch between a discrete inferential statistic and a continuous reference distribution, because I used the exact binomial probabilities. Note further that situations like this have prompted the development of the mid p-value to help minimize the discrepancy between the p-value and the significance level.

There can be cases where the calculated p-value does not equal the long-run type I error rate, in addition to the fact that the type I error rate doesn't necessarily equal the significance level. Consider a 2x2 contingency table with these observed counts:

     col1 col2
row1   2    4   
row2   4    2

Now, how should I calculate the p-value for the independence of the rows and columns? There are many options (which I discuss here). I will start by calculating the $\chi^2$ statistic and comparing it to it's reference distribution; that yields $\chi^2_{1}=1.3, p=.248$. The $\chi^2$ reference distribution is continuous, though, and so only an approximation to how this particular (discrete) $\chi^2$ statistic will behave. We can use Fisher's exact test to determine the true type I error rate; then I get $p=.5671$. When the p-value is calculated this way, it does equal the type I error rate, although we still have the question of whether one of the possible p-values is exactly 5%. Let me admit that I cheated a little, if I had used Yates's correction for continuity, I would have gotten a better approximation of the true type I error rate, but it still wouldn't have been quite right ($.5637\ne .5671$).

Thus, the issues here are that, with discrete data:

  • your preferred significance level may not be one of the possible type I error rates, &
  • using (conventional) approximations to continuous statistics will yield inaccurate calculated p-values.

These problems are exacerbated the smaller your $N$. So far as I know, these problems don't exist with continuous data.

(Although the question doesn't ask about solutions to these problems) there are there are things that mitigate these issues:

  • larger $N$ means more possible values, making things more continuous-ish,
  • there are often corrections (such as Yates's correction for continuity) that will bring calculated values closer to correct values,
  • exact tests (if tractable, i.e., if $N$ is small enough) will yield correct p-values
  • the mid p-value offers the possibility of getting your type I error rate closer to your chosen confidence level,
  • you can explicitly use one of the type I error rates that exist (or note what it would be).
  • Great that you went into the detail that we left on the side (+1). – gui11aume Aug 1 '12 at 19:55
  • Great post!!! thank you for this! – BYS2 Aug 2 '12 at 13:41
  • @gung - could you comment on how you got the type I error rates for the first table? – stats134711 Sep 4 '15 at 17:39
  • @stats134711, it's just the sum of the individual probabilities for the options that are as extreme or more extreme (2-tailed). – gung Sep 4 '15 at 19:14

The concepts are indeed intimately linked to each other.

The significance level is the probability of a type I error, or rather, the presumed probability of such an event. ${\rm P}({\rm type~I~error})= \alpha$ can generally only be obtained when working with continuous distributions, so in classic test theory a test is said to have significance level $\alpha$ if ${\rm P}({\rm type~I~error})\leq \alpha$, meaning that the probability of a type I error is bounded by $\alpha$. However, tests that use approximations of one kind or another actually tend to have ${\rm P}({\rm type~I~error})\approx \alpha$, in which case the probability of a type I error can be larger than the nominal $\alpha$.

The p-value is the lowest significance level at which the null hypothesis would be accepted. Thus it tells us "how significant" the result is.

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