For those that were taught that p-values evaluation of a hypotheses test statistics do demand to be known: ___ Its analytical expression and that of its inverse, shall be surprised a lot how simple and straightforward Monte Carlo simulations deal with the problem.
Data: A seventy values sample (rats weight, below). Objective: see if it could be originated from a normal population: sample mean = 112.3, stdev.= 9.06
Taking in account Tables of critical values one have: D. Wurtz, H.G. Katzberger: arxiv.org/abs/math/0509423 ______n= 50_____4.9757 ______n= 75_____5.2777 Then we should decide that H0, normality, is not rejected.
HOWEVER: What about the Excess Kurtosis = 1.067 compared with the expected value = 0? It seems too HIGH . . . The p values (10 000 samples each, repeated 10 tines) had shown: __0.032, 0.032, 0.031, 0.035, 0.032, 0.034, 0.034, 0.033, 0.034, 0.032: all lesser than 0.05. and we became suspicious that the sample could be leptokurtic.
It´s time to test S0 = -0.066. Value : 0.577
pK = 0.034, pS=0.577
Using the Fisher´s formula H= -2* (ln pK + ln pS) = 7.863 Because we are dealing with a 4df Chi square ____p(chi<=7.863| 4df) = 0.9033 then alpha=9.67%
Is this 2% compatible to the Sidak analysis? Let see: Alpha = 0.05 leads to alpha comp.= 0.025320565 for the composite test._____ From Wikipedia: < A related correction, called the ?idák correction (or Dunn- Sidak correction) that is often used is 1 - (1 - ?)^(1 / n) . This correction is often confused with the Bonferroni correction. The ?idák correction is derived by assuming that the individual tests are independent >. Conclusion: Because 0.0253?< 0.0967 the Sidak critical value is located at right the observed alpha. We should conclude that H0 no rejection is more likely. **** Critics: My say (pardon me): Classical Statisticians concerning Significance Tests are compared to sec. XIX physicians: in total (or almost) absence of clinical analysis, illnesses were detected throughout SYMPTOMS and PERSONAL EXPERIENCE . . . In fact we don?t know fundamental things as: ___is Sidak appropriated in this context? I don?t know. ___and are they independent? Unlikely!
Luis A. Afonso
DATA 120,116,094,120,112,112,106,102,118,112 DATA 116,098,116,114,120,124,112,122,110,084 DATA 106,122,124,112,118,128,108,120,110,106 DATA 140,102,122,112,110,130,112,106,102,114 DATA 108,110,116,118,118,108,102,110,104,112 DATA 122,112,116,110,112,118,098,104,120,106 DATA 108,110,102,110,120,126,114,098,116,100