The objectives of this paper are to demonstrate the algorithms employed by three statistical software programs (R, Real Statistics using Excel, and SPSS) for calculating the exact two-tailed probability of the Wald-Wo...The objectives of this paper are to demonstrate the algorithms employed by three statistical software programs (R, Real Statistics using Excel, and SPSS) for calculating the exact two-tailed probability of the Wald-Wolfowitz one-sample runs test for randomness, to present a novel approach for computing this probability, and to compare the four procedures by generating samples of 10 and 11 data points, varying the parameters n<sub>0</sub> (number of zeros) and n<sub>1</sub> (number of ones), as well as the number of runs. Fifty-nine samples are created to replicate the behavior of the distribution of the number of runs with 10 and 11 data points. The exact two-tailed probabilities for the four procedures were compared using Friedman’s test. Given the significant difference in central tendency, post-hoc comparisons were conducted using Conover’s test with Benjamini-Yekutielli correction. It is concluded that the procedures of Real Statistics using Excel and R exhibit some inadequacies in the calculation of the exact two-tailed probability, whereas the new proposal and the SPSS procedure are deemed more suitable. The proposed robust algorithm has a more transparent rationale than the SPSS one, albeit being somewhat more conservative. We recommend its implementation for this test and its application to others, such as the binomial and sign test.展开更多
This paper, comparison of two sample tests, is motivated by the fact that in the test of significant difference between two independent samples, numerous methods can be adopted;each may lead to significant different r...This paper, comparison of two sample tests, is motivated by the fact that in the test of significant difference between two independent samples, numerous methods can be adopted;each may lead to significant different results;this implies that wrong choice of test statistic could lead to erroneous conclusion. To prevent misleading information, there is a need for proper investigation of some selected methods for test of significant difference between variables/subjects most especially, independent samples. The paper examines the efficiency and sensitivity of four test statistics to ascertain which test performs better. Based on the results, the relative efficiency favours median test as being more efficient than modified median test for both symmetric and asymmetric distributions. In terms of power of test, median test is more sensitive than Modified Median (MMED) test since it has higher power irrespective of the sample sizes for both symmetric and asymmetric distribution. In terms of relative efficiency for asymmetric distribution Modified Mann-Whitney U test is more efficient than Mann-Whitney U test (MMWU), and then for symmetric distribution, Mann-Whitney U test (MMWU) is more efficient than Modified Mann-Whitney in sample size of 5;but for other sample sizes considered Modified Mann-Whitney U test (MMWU) is better than Mann-Whitney. Using power of test for both symmetric and asymmetric distributions, Mann-Whitney is more sensitive than Modified Mann-Whitney U test (MMWU) because it has higher power.展开更多
Unlike height-diameter equations for standing trees commonly used in forest resources modelling,tree height models for cut-to-length(CTL)stems tend to produce prediction errors whose distributions are not conditionall...Unlike height-diameter equations for standing trees commonly used in forest resources modelling,tree height models for cut-to-length(CTL)stems tend to produce prediction errors whose distributions are not conditionally normal but are rather leptokurtic and heavy-tailed.This feature was merely noticed in previous studies but never thoroughly investigated.This study characterized the prediction error distribution of a newly developed such tree height model for Pin us radiata(D.Don)through the three-parameter Burr TypeⅫ(BⅫ)distribution.The model’s prediction errors(ε)exhibited heteroskedasticity conditional mainly on the small end relative diameter of the top log and also on DBH to a minor extent.Structured serial correlations were also present in the data.A total of 14 candidate weighting functions were compared to select the best two for weightingεin order to reduce its conditional heteroskedasticity.The weighted prediction errors(εw)were shifted by a constant to the positive range supported by the BXII distribution.Then the distribution of weighted and shifted prediction errors(εw+)was characterized by the BⅫdistribution using maximum likelihood estimation through 1000 times of repeated random sampling,fitting and goodness-of-fit testing,each time by randomly taking only one observation from each tree to circumvent the potential adverse impact of serial correlation in the data on parameter estimation and inferences.The nonparametric two sample Kolmogorov-Smirnov(KS)goodness-of-fit test and its closely related Kuiper’s(KU)test showed the fitted BⅫdistributions provided a good fit to the highly leptokurtic and heavy-tailed distribution ofε.Random samples generated from the fitted BⅫdistributions ofεw+derived from using the best two weighting functions,when back-shifted and unweighted,exhibited distributions that were,in about97 and 95%of the 1000 cases respectively,not statistically different from the distribution ofε.Our results for cut-tolength P.radiata stems represented the first case of any tree species where a non-normal error distribution in tree height prediction was described by an underlying probability distribution.The fitted BXII prediction error distribution will help to unlock the full potential of the new tree height model in forest resources modelling of P.radiata plantations,particularly when uncertainty assessments,statistical inferences and error propagations are needed in research and practical applications through harvester data analytics.展开更多
Lake Michigan, the sixth largest freshwater lake in the world by surface area, was utilized as a water body for assessment. Field data collected at sampling sites throughout the lake in an intensive monitoring effort ...Lake Michigan, the sixth largest freshwater lake in the world by surface area, was utilized as a water body for assessment. Field data collected at sampling sites throughout the lake in an intensive monitoring effort were utilized for evaluation of the distribution of sediment measurements. An assessment of sediment nutrient and carbon measurements within Lake Michigan was completed to recognize strata resulting from the hydrodynamics of the system. Nonparametric comparison tests revealed that significant differences exist between measurements of sediment nutrients and organic carbon in the lake using strata based upon water column depth (all results demon-strated a p < 0.05, α = 0.05). Cross-validation analysis was applied to the field-collected samples, revealing that large errors occur when estimating sediment flux of carbon or nutrients at a given location in the lake without considering stratification of the distributions of these measurements. Errors in estimating sediment concentrations of nutrients and carbon specific to a location in the lake demonstrated a statistically significant increase when stratification of sediment measurements wasn’t employed among sites. For example, distributions of errors in estimating all nutrients and organic carbon concentrations, whereby distance squared inverse interpolation methods were applied, demonstrated a statistically significant increase in absence of stratification (all p < 0.001, α = 0.05). These results have implications for characterization, monitoring, and modeling sediment and water interaction as related to eutrophication, as well as to contaminant exposure and bioaccumulation for chemicals within Lake Michigan and large water bodies where stratification of the sediment based upon physics of the system exists.展开更多
We analyze ten of the longest (127 to 230 year-long) time series of European daily temperatures available from five different Köppen-Geiger climate classes. We split these according to the level of solar cycl...We analyze ten of the longest (127 to 230 year-long) time series of European daily temperatures available from five different Köppen-Geiger climate classes. We split these according to the level of solar cycle activity (H for “higher than median” and L for “lower than median”). This reveals coherent patterns in the temperature differences: when TH-TL?are stacked according to their calendar date, the daily averages from January 1 to December 31st disclose characteristic features in addition to the dominant annual seasonal wave, namely variations up to 2°C lasting for about 1.5 to 3 months. The five observatories at intermediate latitudes in a band from Oxford in the West to Prague in the East (same climate class) have very similar signatures. These similarities are most unlikely to be due to pure chance (confirmed by confidence levels in excess of 99% with the Kolmogorov-Smirnov and Kuiper nonparametric tests). The TH-TL patterns carry a regional signature, modulated by a more local response function. On the other hand, northern European observatories (St Petersburg and Arkhangelsk), those south of the Alps (Milan and Bologna), and the easternmost one in Astrakhan, corresponding to different climate classes, have different signatures. Similarly, preliminary study of long air pressure recordings confirms what emerges from the analysis of temperatures. These new observations lead us to conclude that the climate in different regions presents different responses to variations in solar activity. Moreover, the distributions of the lower, middle, and higher quartiles of the temperature and pressure indices in solar cycles with high versus low activity are significantly different, providing further robust statistical confirmation to this conclusion (confidence level higher to much higher than 99% using the Kuiper test).展开更多
This article is concerned with the high-dimensional location testing problem.For highdimensional settings,traditional multivariate-sign-based tests perform poorly or become infeasible since their Type I error rates ar...This article is concerned with the high-dimensional location testing problem.For highdimensional settings,traditional multivariate-sign-based tests perform poorly or become infeasible since their Type I error rates are far away from nominal levels.Several modifications have been proposed to address this challenging issue and shown to perform well.However,most of modified sign-based tests abandon all the correlation information,and this results in power loss in certain cases.We propose a projection weighted sign test to utilize the correlation information.Under mild conditions,we derive the optimal direction and weights with which the proposed projection test possesses asymptotically and locally best power under alternatives.Benefiting from using the sample-splitting idea for estimating the optimal direction,the proposed test is able to retain type-I error rates pretty well with asymptotic distributions,while it can be also highly competitive in terms of robustness.Its advantage relative to existing methods is demonstrated in numerical simulations and a real data example.展开更多
Distance-based regression model,as a nonparametric multivariate method,has been widely used to detect the association between variations in a distance or dissimilarity matrix for outcomes and predictor variables of in...Distance-based regression model,as a nonparametric multivariate method,has been widely used to detect the association between variations in a distance or dissimilarity matrix for outcomes and predictor variables of interest in genetic association studies,genomic analyses,and many other research areas.Based on it,a pseudo-F statistic which partitions the variation in distance matrices is often constructed to achieve the aim.To the best of our knowledge,the statistical properties of the pseudo-F statistic has not yet been well established in the literature.To fill this gap,the authors study the asymptotic null distribution of the pseudo-F statistic and show that it is asymptotically equivalent to a mixture of chi-squared random variables.Given that the pseudo-F test statistic has unsatisfactory power when the correlations of the response variables are large,the authors propose a square-root F-type test statistic which replaces the similarity matrix with its square root.The asymptotic null distribution of the new test statistic and power of both tests are also investigated.Simulation studies are conducted to validate the asymptotic distributions of the tests and demonstrate that the proposed test has more robust power than the pseudo-F test.Both test statistics are exemplified with a gene expression dataset for a prostate cancer pathway.展开更多
文摘The objectives of this paper are to demonstrate the algorithms employed by three statistical software programs (R, Real Statistics using Excel, and SPSS) for calculating the exact two-tailed probability of the Wald-Wolfowitz one-sample runs test for randomness, to present a novel approach for computing this probability, and to compare the four procedures by generating samples of 10 and 11 data points, varying the parameters n<sub>0</sub> (number of zeros) and n<sub>1</sub> (number of ones), as well as the number of runs. Fifty-nine samples are created to replicate the behavior of the distribution of the number of runs with 10 and 11 data points. The exact two-tailed probabilities for the four procedures were compared using Friedman’s test. Given the significant difference in central tendency, post-hoc comparisons were conducted using Conover’s test with Benjamini-Yekutielli correction. It is concluded that the procedures of Real Statistics using Excel and R exhibit some inadequacies in the calculation of the exact two-tailed probability, whereas the new proposal and the SPSS procedure are deemed more suitable. The proposed robust algorithm has a more transparent rationale than the SPSS one, albeit being somewhat more conservative. We recommend its implementation for this test and its application to others, such as the binomial and sign test.
文摘This paper, comparison of two sample tests, is motivated by the fact that in the test of significant difference between two independent samples, numerous methods can be adopted;each may lead to significant different results;this implies that wrong choice of test statistic could lead to erroneous conclusion. To prevent misleading information, there is a need for proper investigation of some selected methods for test of significant difference between variables/subjects most especially, independent samples. The paper examines the efficiency and sensitivity of four test statistics to ascertain which test performs better. Based on the results, the relative efficiency favours median test as being more efficient than modified median test for both symmetric and asymmetric distributions. In terms of power of test, median test is more sensitive than Modified Median (MMED) test since it has higher power irrespective of the sample sizes for both symmetric and asymmetric distribution. In terms of relative efficiency for asymmetric distribution Modified Mann-Whitney U test is more efficient than Mann-Whitney U test (MMWU), and then for symmetric distribution, Mann-Whitney U test (MMWU) is more efficient than Modified Mann-Whitney in sample size of 5;but for other sample sizes considered Modified Mann-Whitney U test (MMWU) is better than Mann-Whitney. Using power of test for both symmetric and asymmetric distributions, Mann-Whitney is more sensitive than Modified Mann-Whitney U test (MMWU) because it has higher power.
文摘Unlike height-diameter equations for standing trees commonly used in forest resources modelling,tree height models for cut-to-length(CTL)stems tend to produce prediction errors whose distributions are not conditionally normal but are rather leptokurtic and heavy-tailed.This feature was merely noticed in previous studies but never thoroughly investigated.This study characterized the prediction error distribution of a newly developed such tree height model for Pin us radiata(D.Don)through the three-parameter Burr TypeⅫ(BⅫ)distribution.The model’s prediction errors(ε)exhibited heteroskedasticity conditional mainly on the small end relative diameter of the top log and also on DBH to a minor extent.Structured serial correlations were also present in the data.A total of 14 candidate weighting functions were compared to select the best two for weightingεin order to reduce its conditional heteroskedasticity.The weighted prediction errors(εw)were shifted by a constant to the positive range supported by the BXII distribution.Then the distribution of weighted and shifted prediction errors(εw+)was characterized by the BⅫdistribution using maximum likelihood estimation through 1000 times of repeated random sampling,fitting and goodness-of-fit testing,each time by randomly taking only one observation from each tree to circumvent the potential adverse impact of serial correlation in the data on parameter estimation and inferences.The nonparametric two sample Kolmogorov-Smirnov(KS)goodness-of-fit test and its closely related Kuiper’s(KU)test showed the fitted BⅫdistributions provided a good fit to the highly leptokurtic and heavy-tailed distribution ofε.Random samples generated from the fitted BⅫdistributions ofεw+derived from using the best two weighting functions,when back-shifted and unweighted,exhibited distributions that were,in about97 and 95%of the 1000 cases respectively,not statistically different from the distribution ofε.Our results for cut-tolength P.radiata stems represented the first case of any tree species where a non-normal error distribution in tree height prediction was described by an underlying probability distribution.The fitted BXII prediction error distribution will help to unlock the full potential of the new tree height model in forest resources modelling of P.radiata plantations,particularly when uncertainty assessments,statistical inferences and error propagations are needed in research and practical applications through harvester data analytics.
文摘Lake Michigan, the sixth largest freshwater lake in the world by surface area, was utilized as a water body for assessment. Field data collected at sampling sites throughout the lake in an intensive monitoring effort were utilized for evaluation of the distribution of sediment measurements. An assessment of sediment nutrient and carbon measurements within Lake Michigan was completed to recognize strata resulting from the hydrodynamics of the system. Nonparametric comparison tests revealed that significant differences exist between measurements of sediment nutrients and organic carbon in the lake using strata based upon water column depth (all results demon-strated a p < 0.05, α = 0.05). Cross-validation analysis was applied to the field-collected samples, revealing that large errors occur when estimating sediment flux of carbon or nutrients at a given location in the lake without considering stratification of the distributions of these measurements. Errors in estimating sediment concentrations of nutrients and carbon specific to a location in the lake demonstrated a statistically significant increase when stratification of sediment measurements wasn’t employed among sites. For example, distributions of errors in estimating all nutrients and organic carbon concentrations, whereby distance squared inverse interpolation methods were applied, demonstrated a statistically significant increase in absence of stratification (all p < 0.001, α = 0.05). These results have implications for characterization, monitoring, and modeling sediment and water interaction as related to eutrophication, as well as to contaminant exposure and bioaccumulation for chemicals within Lake Michigan and large water bodies where stratification of the sediment based upon physics of the system exists.
文摘We analyze ten of the longest (127 to 230 year-long) time series of European daily temperatures available from five different Köppen-Geiger climate classes. We split these according to the level of solar cycle activity (H for “higher than median” and L for “lower than median”). This reveals coherent patterns in the temperature differences: when TH-TL?are stacked according to their calendar date, the daily averages from January 1 to December 31st disclose characteristic features in addition to the dominant annual seasonal wave, namely variations up to 2°C lasting for about 1.5 to 3 months. The five observatories at intermediate latitudes in a band from Oxford in the West to Prague in the East (same climate class) have very similar signatures. These similarities are most unlikely to be due to pure chance (confirmed by confidence levels in excess of 99% with the Kolmogorov-Smirnov and Kuiper nonparametric tests). The TH-TL patterns carry a regional signature, modulated by a more local response function. On the other hand, northern European observatories (St Petersburg and Arkhangelsk), those south of the Alps (Milan and Bologna), and the easternmost one in Astrakhan, corresponding to different climate classes, have different signatures. Similarly, preliminary study of long air pressure recordings confirms what emerges from the analysis of temperatures. These new observations lead us to conclude that the climate in different regions presents different responses to variations in solar activity. Moreover, the distributions of the lower, middle, and higher quartiles of the temperature and pressure indices in solar cycles with high versus low activity are significantly different, providing further robust statistical confirmation to this conclusion (confidence level higher to much higher than 99% using the Kuiper test).
基金NNSF of China Grants(Grant Nos.11925106,11690015,11931001 and 11971247)NSF of Tianjin Grant(Grant Nos.18JCJQJC46000 and 18ZXZNGX00140)+1 种基金111 Project B20016National Science Foundation(Grant Nos.DMS 1820702,DMS 1953196 and DMS 2015539)。
文摘This article is concerned with the high-dimensional location testing problem.For highdimensional settings,traditional multivariate-sign-based tests perform poorly or become infeasible since their Type I error rates are far away from nominal levels.Several modifications have been proposed to address this challenging issue and shown to perform well.However,most of modified sign-based tests abandon all the correlation information,and this results in power loss in certain cases.We propose a projection weighted sign test to utilize the correlation information.Under mild conditions,we derive the optimal direction and weights with which the proposed projection test possesses asymptotically and locally best power under alternatives.Benefiting from using the sample-splitting idea for estimating the optimal direction,the proposed test is able to retain type-I error rates pretty well with asymptotic distributions,while it can be also highly competitive in terms of robustness.Its advantage relative to existing methods is demonstrated in numerical simulations and a real data example.
基金partially supported by Beijing Natural Science Foundation under Grant No.Z180006.
文摘Distance-based regression model,as a nonparametric multivariate method,has been widely used to detect the association between variations in a distance or dissimilarity matrix for outcomes and predictor variables of interest in genetic association studies,genomic analyses,and many other research areas.Based on it,a pseudo-F statistic which partitions the variation in distance matrices is often constructed to achieve the aim.To the best of our knowledge,the statistical properties of the pseudo-F statistic has not yet been well established in the literature.To fill this gap,the authors study the asymptotic null distribution of the pseudo-F statistic and show that it is asymptotically equivalent to a mixture of chi-squared random variables.Given that the pseudo-F test statistic has unsatisfactory power when the correlations of the response variables are large,the authors propose a square-root F-type test statistic which replaces the similarity matrix with its square root.The asymptotic null distribution of the new test statistic and power of both tests are also investigated.Simulation studies are conducted to validate the asymptotic distributions of the tests and demonstrate that the proposed test has more robust power than the pseudo-F test.Both test statistics are exemplified with a gene expression dataset for a prostate cancer pathway.
