# Large-sample tests for comparing Likert-type scale data

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### "Large-sample tests for comparing Li..." refers background in this paper

...Gibbons and Chakraborti (2011) further give conditions under which ARE of tests based on asymptotically normal statistics are comparable....

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...…K12K 122 K012 q ¼ ffiffiffiffiffiffiffiffiffiffi K11:2 p > 0 is 1ffiffiffimp times the square root of the efficacy (see Gibbons and Chakraborti 2011) of the test based on Tð1Þ: For the test under “a priori” model recall that Tð2Þm, n ¼ ĥ ð2Þ m, n…...

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...1 dEðTð2Þm, nÞ=dh h i h¼0ffiffiffiffi m p VðTð2Þm, nÞjh¼0 ¼ kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1þ kÞPiPj sijq > 0 is 1ffiffiffimp times the square root of the efficacy (see Gibbons and Chakraborti 2011) of the test based on Tð2Þ:...

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### "Large-sample tests for comparing Li..." refers background in this paper

...…hrFðhÞ , 1 : Hence, APFFðhÞ 1 U sa hrFðhÞ ¼ U sa hrFðhÞ For testing H0 : h 0 against H1 : h > 0 it can be noted that the approximate expression for APFFðhÞ exceeds a for h > 0 and lies below a for h 0: Hence the test is asymptotically unbiased at significance level a (see Lehmann and Romano 2008)....

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### Additional excerpts

...An iterative procedure is needed to solve these equations, with each iteration further consisting of two steps: successively updating estimates of h and x ¼ ðx1, x2, :::, xkÞ: The estimate of h for given x1, x2, :::, xk for the first part of each step of iteration, as well as, for the second part, those of the xj’s for given h can be obtained by the scoring method: @2lðhÞ @h2 ¼ Xkþ1 j¼1 gj 1 dj @ 2dj @h2 1 d2j @dj @h 2" # , so i11ðh, xÞ ¼ E @ 2lðhÞ @h2 ¼ Xkþ1 j¼1 mdj 1 dj @ 2dj @h2 1 d2j @dj @h 2" # ¼ Xkþ1 j¼1 m dj @dj @h 2 : while the matrix I22ðh, xÞ has entries iij, 22ðh, xÞ ¼ nf 2ðxiÞ 1pi þ 1 piþ1 þmf 2ðxi hÞ 1di þ 1 diþ1 , j ¼ i nf ðxiÞf ðxi 1Þ pi mf ðxi hÞf ðxi 1 hÞ di , j ¼ i 1 nf ðxiÞf ðxiþ1Þ piþ1 mf ðxi hÞf ðxiþ1 hÞ diþ1 , j ¼ iþ 1 0 otherwise 8>>>>< >>>>: Also, the entries of the vector i12ðh, xÞ are given by E @ 2lðh, xÞ @h@xj " # ¼ mf ðxj hÞ f ðxjþ1 hÞ f ðxj hÞ djþ1 f ðxj hÞ f ðxj 1 hÞ dj " # for j ¼ 1, 2, :::, k: Now, we can conclude from Serfling (1980) or Ferguson (1996) that the regularity conditions imply, for the solution of the likelihood equations, the following: ðĥM , x̂1, x̂2, :::, x̂kÞ a Nkþ1ððh, x1, x2, :::, xkÞ,I 1m, nÞ: Hence, ðĥMm, nÞ a N 0, ið11Þððh, xÞÞ as m, n !...

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...…multinomial pmfs, is given by lððh, xÞÞ ¼ c1 þ Xkþ1 j¼1 fj log pj þ c2 þ Xkþ1 j¼1 gj log dj where c1, c2 > 0: Standard regularity conditions are given, for example, in Serfling (1980) or Ferguson (1996), guaranteeing efficient and consistent asymptotic normality of the maximum likelihood estimator....

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...…1 0 otherwise 8>>>>< >>>>: Also, the entries of the vector i12ðh, xÞ are given by E @ 2lðh, xÞ @h@xj " # ¼ mf ðxj hÞ f ðxjþ1 hÞ f ðxj hÞ djþ1 f ðxj hÞ f ðxj 1 hÞ dj " # for j ¼ 1, 2, :::, k: Now, we can conclude from Serfling (1980) or Ferguson (1996) that the regularity conditions imply, for…...

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...The loglikelihood, arising from the product of two independent multinomial pmfs, is given by lððh, xÞÞ ¼ c1 þ Xkþ1 j¼1 fj log pj þ c2 þ Xkþ1 j¼1 gj log dj where c1, c2 > 0: Standard regularity conditions are given, for example, in Serfling (1980) or Ferguson (1996), guaranteeing efficient and consistent asymptotic normality of the maximum likelihood estimator....

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