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We will comment on the randomization principles behind a few of the tests presented in later sections, but space limitations prevent a full development. This is called an approximate randomization test.
#THEORY BASED HYPOTHESIS TEST CALCULATOR SOFTWARE#
Instead, the software will draw a large random sample of permutations, counting the number of times in the sample that the calculated test statistic is as or more extreme than that observed in the data. When the data sets become moderately large, even modern desktop computers will find the enumeration too time-consuming. Fortunately, most statistical software will carry out the enumeration for you, calculating an exact p value. Most data will contain ties, and then the tables are no longer accurate, though they can be approximately correct if the number of ties is small. These are also presented for a few of the most common nonparametric tests. For larger samples, asymptotic approximations were developed. Some of these are discussed in this chapter.
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For small samples with no ties, the early nonparametric statisticians developed tables of the distributions. In practice, the enumeration of all the possible splits of the data is too time-consuming. In fact, however, randomization tests can be developed for many test statistics whether or not they are based on ranks.
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Hence, randomization tests came to be thought of as a natural partner with the use of ranks. If those statistics only depended on the ranks, then the enumeration could be done just once for any given n 1 and n 2, because all data sets of that size with no ties would have the same ranks. Naturally, the focus was on test statistics that were quick to compute. The development of nonparametric statistics predates the advent of modern computers. The number of possible splits of the data can be calculated using the formula for combinations given in Section 2.3. For this reason, randomization tests are also known as permutation tests. Our samples would be constructed without replacement, so that we are enumerating all the ways to permute the data into two separate groups of the specified sizes. Then we would write down all the possible ways to split those values into group 1 (with n 1 values) and group 2 (with n 2 values), calculating and recording the values of the test statistic. In a randomization test, we would fill a basket with the exact values seen in our combined data set. If we do not know what the parent distribution is like, it is reasonable to use the data itself as a guess for that distribution. If it is reasonable that the values follow some other parametric distribution, for example the Poisson, then we might computerize the process described earlier. But what if the parent distribution, that is, the distribution of the values on the slips of paper, is not normal? There are several options.
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This saves us an extremely tedious process. Under the assumption of normality, the distribution of t is known mathematically. We could do this for any choice of test statistic, but the t statistic is particularly good at detecting differences in population means. Alternatively, we could calculate the proportion of values in our experiment that are as or more unusual than the observed | t |, which would give us an empirical estimate of the p value. If our observed t is in the α most unusual region of the distribution, we would claim that our t is inconsistent with the assumption that the samples came from the same basket. Based on the empirical distribution shown in the histogram, we can judge whether a value of t from an actual data set is unusual or not. Then the two samples are essentially being drawn from the same basket. The process we have described mimics the situation when the null hypothesis for the independent samples t test is correct and the underlying assumptions are valid. The result should look very like the Student’s t distribution with n 1 + n 2 − 2 degrees of freedom. Afterward, we histogram the list of t values. Then we put those slips back, shake up the basket, and repeat the process a huge number of times. For our data set, we calculate t and write down that value. We draw a random sample of n 1 + n 2 observations from the basket, and randomly assign n 1 of them to group 1, and the others to group 2. Assume further that these numbers come from a normal distribution. Īssume that we have a large basket filled with slips of paper, each with a number written on it.