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Enter data in highlighted cells JUse this page if you know values for paired alternate queries, a/a, b/b.Comments2 x 2 chi-square*Does the value of b affect the value of a?,Does the value of b affect the choice of a?+Does the value of b affect the choice of a?!chi-square with Yates' correctionlog-likelihoodBasic chi-squareWThe 2 x 2 Pearson's chi-square is one of the most under-rated of all statistical tests.Bin difference error (.05) error (.01) difference |o-e|Log-likelihood test Cramer's f Newcombe-Wilson testWilsonpz testzp'qupperlowermeanz/n probabilitiesFor these tests we also report the binomial error margin. This test is significant if the difference exceeds the margin of error (chi-square is based on the z test).1)4)2)3)Goodness of fit against IV2 x 2z test for pop proportion error = z.s 5)2-4p Wilson CCNW test, with CC z-1/n+4npq2(n+z)4Note that we carry out five different sets of tests.Yates' continuity-corrected c is recommended, and has the closest match to results obtained by Fisher's exact test. It is slightly conservative. "-WFor most purposes the 2 x 2 chi-square test is appropriate, Yates' test is recommended.With continuity correctionAdvanced different pop. z test See Wilson error bar widthse'wd+wd-d = p(a)-p(a) p1 = p(a) p2 = p(a)Binomial2-tailed (0.05)2-tailed (0.01) Binomial testExact' Binomial test6)Uses the cumulative Binomial Binomial areaBWe also include the 'exact' Binomial test for comparison purposes.Since there is only one degree of freedom, there can be only one mathematical source of any significant result. It is therefore less ambiguous to interpret than multinomial tests.s. dev. s = nP(P-1)  where P = e/N (population prob)   d+d-NWNWccdifference d = p(a)-p(a)3https://corplingstats.wordpress.com/2020/07/30/bookMWallis (2021), Statistics in Corpus Linguistics Research, New York: Routledge 8MWallis (2013), z-squared: The origin and application of c, JQL 20:4, 350-378 8 9< ?UWallis (2013), Binomial confidence intervals and contingency tests, JQL 20:3, 178-208D GC/english-usage/statspapers/binomialpoisson.pdf=/english-usage/statspapers/z-squared.pdfGraphs, from left to right: Cell frequency graph, Binomial proportion difference graph (Wilson), and repositioned difference interval (NW, NWcc)dRepositioned NW intervalud+ud- Repositioned NW interval with CC Odds ratiooddslnp1p2 odds ratiow-w+uLUCCo-o+d'z2/n zero-basedp(a)f-f+ error level a Transposed Cramer's f Fhttps://corplingstats.wordpress.com/2021/04/14/an-algebra-of-intervals7Wallis (2021), An algebra of intervals. corp.ling.statsThe first test is the conventional 2 x 2 chi-square, which tends to be the most sensitive (easiest to get a significant result) of the tests.The second and third test are single-choice 2 x 1 "goodness of fit" chi-squares for comparing a and a with the "total" column. [Despite its adherents, log-likelihood is outperformed by Yates test, so is not recommended.~Newcombe-Wilson tests perform in a closely comparable manner to 2 x 2 c. Wilson tests perform identically to goodness-of-fit.F G_ jCThe Newcombe-Wilson test with continuity correction is recommended.7BWe also compute signed Cramer's f and odds ratio, both with confidence intervals. These may be tested against 0 and 1 respectively. !delta Wallis' fp  signed fp  Wallis fp  We report effect size fp (0 = close fit), which for 2x1 tests can be reported as signed fp = p1 - P1 with confidence intervals. 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