The answer to this question is, of course, the familiar p-value. Or – by asking, “if the null hypothesis really were true, how likely is it that I’d see a value for my test statistic that’s as extreme (or more extreme) than what I’ve actually observed here (again, taking into account the assumptions about the underlying population and the sampling method that was used). If not, we would not reject the null hypothesis. If it is surprising, we would reject the null hypothesis.Then, we ask – “if in fact the null hypothesis were actually true, is the value of our test statistic “surprising”, or not?. Typically, this estimator would have to be transformed ( e.g., “standardized”) to make it “pivotal” – that is, having a sampling distribution that does not depend on any other unknown parameters. Usually, this would be a statistic that had already been found to be a “good” estimator of the parameter under test. We combine the sample values into a single statistic.For example, we might use simple random sampling, so that all sample values are mutually independent of each other. Then we take a carefully constructed sample from the population of interest.We call this the “alternative hypothesis”. We also state clearly what situation will prevail if the hypothesis to be tested is not true.We want to test the validity of statement (“null hypothesis”) about a parameter associated with a well-defined underlying population.The whole procedure would have been presented, more or less, along the following lines: It would have just been “statistical hypothesis testing”. It probably wasn’t described to you in so many words. I’m sure that the first exposure that you had to this was actually in terms of “classical”, Neyman-Pearson, testing. When you took your first course in economic statistics, or econometrics, no doubt you encountered some of the basic concepts associated with testing hypotheses. This might seem a bit redundant, but it will help us to see how permutation tests differ from the sort of tests that we usually use in econometrics. Let’s begin with some background discussion to set the scene. The above means that there are 120 ways that we could select the 5 marbles where order matters and where repetition is not allowed.Permutation tests, which I’ll be discussing in this post, aren’t that widely used by econometricians. Refer to the factorials page for a refresher on factorials if necessary. Where n is the number of objects in the set, in this case 5 marbles. If we were selecting all 5 marbles, we would choose from 5 the first time, 4, the next, 3 after that, and so on, or: For example, given that we have 5 different colored marbles (blue, green, red, yellow, and purple), if we choose 2 marbles at a time, once we pick the blue marble, the next marble cannot be blue. We can confirm this by listing all the possibilities: 11įor permutations without repetition, we need to reduce the number of objects that we can choose from the set each time. For example, given the set of numbers, 1, 2, and 3, how many ways can we choose two numbers? P(n, r) = P(3, 2) = 3 2 = 9. Where n is the number of distinct objects in a set, and r is the number of objects chosen from set n. When a permutation can repeat, we just need to raise n to the power of however many objects from n we are choosing, so Like combinations, there are two types of permutations: permutations with repetition, and permutations without repetition. Permutations can be denoted in a number of ways: nP r, nP r, P(n, r), and more. In cases where the order doesn't matter, we call it a combination instead. To unlock a phone using a passcode, it is necessary to enter the exact combination of letters, numbers, symbols, etc., in an exact order. Another example of a permutation we encounter in our everyday lives is a passcode or password. A phone number is an example of a ten number permutation it is drawn from the set of the integers 0-9, and the order in which they are arranged in matters. Home / probability and statistics / inferential statistics / permutation PermutationĪ permutation refers to a selection of objects from a set of objects in which order matters.
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