By J.C.W. Rayner
"The equipment complicated during this booklet have their genesis in conventional nonparametrics. They include the facility of contemporary pcs to make the method extra entire and extra legitimate than formerly attainable. The authors' unified therapy and readable kind make the topic effortless to stick to and the recommendations simply carried out, even if you're a fledgling or a pro researcher."--BOOK JACKET.
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Extra resources for A contingency table approach to nonparametric testing
J ) /( ) = ZTZ. n .. n .. We now confirm that our model leads to this test statistic. Suppose H is orthogonal and diagonalises cov(Z). Asymptotically we then have HTcov(Z)H = I(r - 1)(c - 1) ⊕ 0(r + c - 1), where ⊕ means direct or Kronecker sum. Define Y = HTZ. Now ZTZ = Y T Y, in which Y, by the multivariate Central Limit Theorem, is asymptotically Nrc(0, I(r - 1)(c - 1) ⊕ 0(r + c - 1) ) under the null hypothesis of simple random sampling. It follows that under the null hypothesis, X 2P = 2 ZTZ = YTY asymptotically has the χ(r - 1)(c - 1) distribution.
First recall that X 2P has a fixed value, (t - 1)n, for all tables, and so is not available as a test statistic. 3, the multivariate Central Limit Theorem shows that each Vs is asymptotically N t (0, I(t - 1) ⊕ 0). Moreover consideration of any pair Vs, Vt shows that they are asymptotically jointly multivariate normal, and since their cov(Vs, Vt) = 0, any pair Vs and Vt are asymptotically independent. The VTs Vs still partition n - 1 X 2P. It is the pairwise independence and convenient χt2 - 1 n distribution of each VTs Vs that makes data analysis so informative and convenient.
Define Y = HTZ. Now ZTZ = Y T Y, in which Y, by the multivariate Central Limit Theorem, is asymptotically Nrc(0, I(r - 1)(c - 1) ⊕ 0(r + c - 1) ) under the null hypothesis of simple random sampling. It follows that under the null hypothesis, X 2P = 2 ZTZ = YTY asymptotically has the χ(r - 1)(c - 1) distribution. 3 Partitioning Pearson’s Statistic We now show that X 2P may be partitioned into components, the sth of which may be interpreted as detecting the departure in the sth moment in what may be expected under the null hypothesis of similarly distributed rows (treatments).
A contingency table approach to nonparametric testing by J.C.W. Rayner