ssrs 2012 barcode font Nonparametric Tests in Software

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CHAPTER 10 Nonparametric Tests
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Remark 3 A value corresponding to sample 2 is given by the statistic U N1N2 N2(N2 2 1) R2 (5)
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and has the same sampling distribution as statistic (2), with the mean and variance of formulas (3). Statistic (5) is related to statistic (2), for if U1 and U2 are the values corresponding to statistics (2) and (5), respectively, then we have the result U1 We also have R1 where N N1 R2 N(N 2 1) (7) U2 N1N2 (6)
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N2. Result (7) can provide a check for calculations.
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Remark 4 The statistic U in equation (2) is the total number of times that sample 1 values precede sample 2 values when all sample values are arranged in increasing order of magnitude. This provides an alternative counting method for finding U.
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The Kruskal Wallis H Test
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The U test is a nonparametric test for deciding whether or not two samples come from the same population. A generalization of this for k samples is provided by the Kruskal Wallis H test, or briefly the H test. This test may be described thus: Suppose that we have k samples of sizes N1, N2, . . . , Nk, with the total size of all samples taken together being given by N N1 N2 c Nk. Suppose further that the data from all the samples taken together are ranked and that the sums of the ranks for the k samples are R1, R2, . . . , Rk, respectively. If we define the statistic H
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k R2 j 12 N(N 1) ja Nj 1
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then it can be shown that the sampling distribution of H is very nearly a chi-square distribution with k 1 degrees of freedom, provided that N1, N2, . . . , Nk are all at least 5. The H test provides a nonparametric method in the analysis of variance for one-way classification, or onefactor experiments, and generalizations can be made.
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The H Test Corrected for Ties
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In case there are too many ties among the observations in the sample data, the value of H given by statistic (8) is smaller than it should be. The corrected value of H, denoted by Hc, is obtained by dividing the value given in statistic (8) by the correction factor 1 a (T N3
T) N
where T is the number of ties corresponding to each observation and where the sum is taken over all the observations. If there are no ties, then T 0 and factor (9) reduces to 1, so that no correction is needed. In practice, the correction is usually negligible (i.e., it is not enough to warrant a change in the decision).
The Runs Test for Randomness
Although the word random has been used many times in this book (such as in random sampling and tossing a coin at random ), no previous chapter has given any test for randomness. A non-parametric test for randomness is provided by the theory of runs. To understand what a run is, consider a sequence made up of two symbols, a and b, such as a a
(10)
CHAPTER 10 Nonparametric Tests
In tossing a coin, for example, a could represent heads and b could represent tails. Or in sampling the bolts produced by a machine, a could represent defective and b could represent nondefective. A run is defined as a set of identical (or related) symbols contained between two different symbols or no symbol (such as at the beginning or end of the sequence). Proceeding from left to right in sequence (10), the first run, indicated by a vertical bar, consists of two a s; similarly, the second run consists of three b s, the third run consists of one a, etc. There are seven runs in all. It seems clear that some relationship exists between randomness and the number of runs. Thus for the sequence a
(11)
there is a cyclic pattern, in which we go from a to b, back to a again, etc., which we could hardly believe to be random. In that case we have too many runs (in fact, we have the maximum number possible for the given number of a s and b s). On the other hand, for the sequence a a a a a a
(12)
there seems to be a trend pattern, in which the a s and b s are grouped (or clustered) together. In such case there are too few runs, and we could not consider the sequence to be random. Thus a sequence would be considered nonrandom if there are either too many or too few runs, and random otherwise. To quantify this idea, suppose that we form all possible sequences consisting of N1 a s and N2 b s, for a total of N symbols in all (N1 N2 N ). The collection of all these sequences provides us with a sampling distribution. Each sequence has an associated number of runs, denoted by V. In this way we are led to the sampling distribution of the statistic V. It can be shown that this sampling distribution has a mean and variance given, respectively, by the formulas mV 2N1N2 N1 N2 1 s2 V 2N1N2(2N1N2 (N1 N2)2(N1 N1 N2 N2) 1) (13)
By using formulas (13), we can test the hypothesis of randomness at appropriate levels of significance. It turns out that if both N1 and N2 are at least equal to 8, then the sampling distribution of V is very nearly a normal distribution. Thus Z V mV sV (14)
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