In many biomedical studies a difference in is of specific interest since the upper quantile represents the upper range of biomarkers and/or is used as the cut-off value for a disease classification. we employ the developed methods to test the differences in upper quantiles in two different studies the oral colonization of pneumonia pathogens for intensive care unit patients treated by two different oral treatments and the biomarker expressions of normal and abnormal bronchial epithelial cells. et al= 1 2 there are Rabbit polyclonal to PECI. experimental units. Let denote the random variable of the with the continuous distribution function have at least two times differentiable in some neighborhood of the < 1). Also let and the pooled data respectively and is the distribution function of the pooled data. We Ac-LEHD-AFC are interested in comparison of the is is Ac-LEHD-AFC represent the empirical Ac-LEHD-AFC probabilities replacing are obtained to maximize the EL function (3) subject to relevant constraints with respect to the hypothesis (1). Using the definition of the quantile and the empirical probabilities for each group (= 1 2 we can establish the empirical equality as under < 0 and ≥ 0. Following the classical EL approach we have the empirical constraints consistent with (1) in a form of is the EL estimator of θ that maximizes (3). According to the constraints (4) and (5) the log of the likelihood function is maximized subject to = 1 2 Maximization can be achieved based on the Lagrange multiplier method through the function (= 1 2 are Lagrange Ac-LEHD-AFC multipliers. Let Δ(in (8) converges in distribution to χ12 distribution as → ∞ = 1 2 Proposition 1 can be proven by applying the results from Lopez is a nonnegative differentiable function satisfying and is Ac-LEHD-AFC a bandwidth. It has been shown that the performance of the smoothed version of the ELR test can be improved in terms of the Type I error and power comparing the ELR test based on the identity function (e.g. Zhou and Jing 2003 Yu is commonly a function of the sample size and other parameters that are estimated based on the sample (e.g. Altman and Léger 1995 An extensive Monte-Carlo study Ac-LEHD-AFC demonstrated that the proposed tests are robust to the choice of different bandwidths; however in the context of the approach of Hyndman and Yao (2002) we chose a bandwidth of for group for actual applications and simulation studies which showed empirically reasonable performances among many available methodologies. We now propose a test statistic based on the plug-in EL approach. The plug-in EL approach simply replaces the EL estimator by the sample based on the pooled sample and subsequently we define → ∞ = 1 2 and / ν based on (8) with in (9) converges in distribution to χ12 distribution where and indicate sample and the pooled sample with the pooled sample. Based on the asymptotic distribution of (Serfling 1980 based on various underlying distributions. The significance level is 0.05. Note that the distributions in each scenario have a matching 0.95-quantile. Note that the second parameters in the normal … Table 2 The Monte Carlo Type I errors to compare based on various underlying distributions. Note that the distributions in each scenario have a matching 0.9-quantile. The significance level is 0.05. Note that the second parameters in the normal … Table 3 The Monte Carlo powers to compare based on various underlying distributions. The significance level is 0.05. Note that the second parameters in the normal and lognormal distributions are the variance or the variance of the logarithmic value. … Table 4 The Monte Carlo powers to compare based on various underlying distributions. The significance level is 0.05. Note that the second parameters in the normal and lognormal distributions are the variance or the variance of the logarithmic value. … With the relatively small sample sizes (i.e. for some α > 0 and (Winter 1979 and (Serfling 1980 under by (? θ) with ((θ)) then (A-5) can be expressed as