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Nonparametric Comparisons of Distributions
Published in Albert Vexler, Alan D. Hutson, Xiwei Chen, Statistical Testing Strategies in the Health Sciences, 2017
Albert Vexler, Alan D. Hutson, Xiwei Chen
In order to test if a random sample X1,…, Xm, with empirical distribution fm(x), comes from a continuous population with completely specified distribution function f0(x), Anderson and Darling (1952, 1954) introduced the goodness-of-fit statistic In order to test the homogeneity of samples, Scholz and Stephens (1987) extended an Anderson–Darling rank statistic to two K-sample versions. The K-sample Anderson–Darling test is essentially a rank test, and therefore requires no restrictive parametric model assumptions. Two versions of test statistics were proposed, which are both essentially based on a doubly weighted sum of integrated squared differences between the empirical distribution functions of the individual samples and that of the pooled sample. These two versions of test statistics differ primarily in defining the empirical distribution function, while one weighting adjusts for the possibly different sample sizes and the other is inside the integration placing more weight on tail differences of the compared distributions. These tests are consistent against all alternatives. The authors derived the asymptotic null distributions for both continuous and discrete cases. The asymptotic distributions in the continuous case were shown to be the same with the (K - 1)–fold convolution of the asymptotic distribution for the Anderson–Darling one-sample statistic. Large sample approximation was investigated based on small sample Monte Carlo simulations for both versions of the statistic under different data settings and degrees of imbalances in sample size. These tests may be applied in a one-way analysis of variance to test for differences in the sampled populations without distributional assumptions or to justify the pooling of separate samples for increased sample size and power.
The rationale of applying inspiratory/expiratory muscle training within the same respiratory cycle in children with bronchial asthma: a placebo-controlled randomized clinical investigation
Published in Journal of Asthma, 2023
Ragab K. Elnaggar, Ahmad M. Osailan, Mohammed F. Elbanna
Data computations were done through the Minitab statistical software (Minitab Inc., State College PA, USA), version 19.2. The hypothesis that data comes from a normal distribution was tested using the Anderson-darling test of normality. The homogeneity regarding baseline characteristics was checked using a 1-way ANOVA test for numerical data or Pearson’s χ2 test for categorical data. The intention-to-treat analysis was considered to preserve the randomization-afforded prognostic balance and mitigate the effect of bias arising out of participant dropout, with missing data points substituted using stochastic regression imputation. The difference among study groups in dependent variables across the two measurement occasions was computed via a two-way split-plot ANOVA [one between-subject factor: groups (Placebo, In-MT, and In/Ex-MT) and one repeated-measures factor: time (pre/post-treatment)]. If a significant ANOVA interaction effect was evidenced, a post-hoc analysis was undertaken through Tukey’s honestly significance test to specify where differences occurred between groups. When the null hypothesis for split-plot ANOVA was rejected, the pre–post differences within each group were computed using paired t-test. To reveal the size of the significant between- and within-groups effects, partial eta-squared (η2Partial) and Hedges’ g formulae were applied, respectively. For all statistical tests, an α level of P < 0.05 was used to determine statistical significance.
Taguchi based Case study in the automotive industry: nonconformity decreasing with use of Six Sigma methodology
Published in Journal of Applied Statistics, 2021
Atakan Gerger, Ali Riza Firuzan
According to Pyzdek [33], the current status of the process and significant inputs and their levels causing process variance are analysed at this phase. The SS team collected data from the production line to check whether the process was under control and receive early warnings for process variation. As the measurements were conducted with destructive testing, the data were collected by five measurements for each shift, in a ten-day period. 150 measurements in total were conducted and gathered in 30 subgroups. In this study, S1, S2 and S3 indicate the day, evening and night shifts respectively. Firstly, the SS team conducted an Anderson–Darling normality test. The test of normality using the Anderson–Darling test statistic (A-squared = .37) resulted in a large p-value (0.422) indicating that the data could be considered to have a normal distribution. Thus, an 25]. The SS team examined the Figure 2 was examined to see the value of the nonconforming products in PPM. The
Classical methods of estimation on constant stress accelerated life tests under exponentiated Lindley distribution
Published in Journal of Applied Statistics, 2020
The Anderson–Darling test was developed in 1952 by T.W. Anderson and D.A. Darling [2] as an alternative to other statistical tests for detecting sample distributions departure from normality. The Anderson–Darling estimators a, b and θ are obtained by minimizing, with respect to a, b and θ, the function: 3). These estimators can also be obtained by solving the non-linear equations: 4), (5) and (6), respectively. The ADE of