Abstract
The shoulder tip pain study of Lumley [13] is re-investigated. It is shown that the new logarithmic quantile estimation (LQE) technique in [9] applies and behaves well under singular covariance structure and small sample sizes as in the shoulder tip pain study. The findings in [6] can be assured under weaker assumptions using a combination of LQE and an ANOVA type statistic.
Highlights
This note adds to this endeavor and introduces a new method of hypothesis testing for longitudinal data motivated by the study of Lumley [13] on a ’shoulder tip pain’ data set, and subsequentially re-investigated in Brunner, Domhof and Langer [6]
It turned out that the corresponding Wald type test statistics was badly approximated by its limiting distribution while the ANOVA type statistics gave good results in connection with the BoxWelch-Satterhwaite approximation
Logarithmic quantile estimation is based on almost sure central limit theorems and has, for the first time, been set up in [9] in the general framework of simple linear rank statistics
Summary
The model for the longitudinal data we have in mind is a nonparametric factorial design with two factors (A and B), each of them having two groups. It is known that under the assumptions below and under the null hypothesis H0F : MF = 0 ⇔ CF = 0, the statistic Qn(M) has, asymptotically, the same distribution as the random variable. In [6] the statistic in (2.4) can be used for testing if the asymptotic distribution is approximated by an F distribution whose degrees of freedom need to be estimated. Using the logarithmic quantile estimation developed in [9] we were able to obtain approximations for the quantiles of the test statistic defined in (2.4). In order to perform the almost sure quantile estimation for the statistics defined in (2.4) we need to verify the almost sure weak convergence of (3.1) towards the limiting distribution.
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