Ensemble Based Box-Cox Transformation via Meta Analysis

BackgroundMany nursing and health related research studies have continuous outcome measures that are inherently non-normal in distribution. The Box-Cox transformation provides a powerful tool for developing a parsimonious model for data representation and interpretation when the distribution of the dependent variable, or outcome measure, of interest deviates from the normal distribution. The objectives of this study was to contrast the effect of obtaining the Box-Cox power transformation parameter and subsequent analysis of variance with or without a priori knowledge of predictor variables under the classic linear or linear mixed model settings.MethodsSimulation data from a 3 × 4 factorial treatments design, along with the Patient Falls and Patient Injury Falls from the National Database of Nursing Quality Indicators (NDNQI®) for the 3rd quarter of 2007 from a convenience sample of over one thousand US hospitals were analyzed. The effect of the nonlinear monotonic transformation was contrasted in two ways: a) estimating the transformation parameter along with factors with potential structural effects, and b) estimating the transformation parameter first and then conducting analysis of variance for the structural effect.ResultsLinear model ANOVA with Monte Carlo simulation and mixed models with correlated error terms with NDNQI examples showed no substantial differences on statistical tests for structural effects if the factors with structural effects were omitted during the estimation of the transformation parameter.ConclusionsThe Box-Cox power transformation can still be an effective tool for validating statistical inferences with large observational, cross-sectional, and hierarchical or repeated measure studies under the linear or the mixed model settings without prior knowledge of all the factors with potential structural effects.

This study aims to compare normality tests in different sample sizes in data with normal distribution under different kurtosis and skewness coefficients obtained simulatively. To this end, firstly, simulative data were produced using the MATLAB program for different skewness/kurtosis coefficients and different sample sizes. The normality analysis of each data type was conducted using the MATLAB program and ten different normality tests; namely, (Kolmogorov Smirnov (KS) Test, KS Stephens Modification, KS Marsaglia, KS Lilliefors Modification, Anderson-Darling Test, Cramer- Von Mises Test, Shapiro-Wilk Test, Shapiro-Francia Test, Jarque-Bera Test, and D’Agostino & Pearson Test). As a result of the analyses conducted according to ten different normality tests, it was found that normality tests were not affected by the sample size when the skewness and kurtosis coefficients were equal to or close to zero; however, in cases where the skewness and kurtosis coefficients moved away from zero, it was found that normality tests are affected by the sample size, and such tests tend to give significant results. Therefore, in large samples, it may be suggested that critical values for skewness and kurtosis coefficients’ z-scores as proposed by Kim (2013) and Mayers (2013) or the histogram graphs be used.

Some Thoughts About Data Type, Distribution, and Statistical Significance

Within the fields of risk management and banking, the normality condition is one of the basic assumptions to apply value at risk, capital asset pricing or linear regression models on credit risk assessment. However, banking sector data related to loans may not be normally distributed. Hence, it needs to be put through scientific tests. For this purpose, firstly, Anderson-Darling, Jarque-Bera, Kolmogorov-Smirnov, Shapiro-Wilk, and Shapiro-Francia tests are applied to ninety-two banking sector loan variables and it is demonstrated that most of the variables are not normally distributed. Additionally, the parameters of Normal, Birnbaum-Saunders, Exponential, Extreme Value, Gamma, Generalized Extreme Value, Inverse Gaussian, Log-Logistic, Logistic, Lognormal, Nakagami, Negative Binomial, Non-Parametric, Poisson, Rayleigh, Rician, t- Location-Scale, and Weibull distributions are estimated for loan variables. Thirdly, when the data are not normally distributed, it is necessary to examine the other test results. Therefore, Kolmogorov-Smirnov, Anderson Darling, and Chi-square test results are employed for sixty-one distributions related to the variables and best fitted distribution per variable is aimed at. The results indicate that different computer codes and programs may give different outcomes in connection with the normality and best fitting distribution. Therefore, the use of different strategies may also be adopted in risk management courses along with the traditional ones since the normality assumption is an essential first step for the application of such techniques. Finally, pedagogically speaking, it should be noted that teaching the essence of mathematical background and computer codes could be strategically useful for students in internalizing these distribution concepts. Keywords: distribution, banking sector loans, risk management

Background: In medical research, statistical tests have become more important. Several parametric procedures are available but each of them requires normality assumption. As normality violation may affect interpretation and inferences reliability and validity, so importance of normal distribution is undeniable. Although several normality tests available for software users but the power of each test in specified situation is not clear. Methods: The aim of this study is to compare the power of nine normality tests. In this paper power of Jarque-Bera test, D’Agostino and Pearson test, Chi-square test, Kolmogrov-Smirnov test, Lilliefors test, Cramer-Von Mises test, Anderson-Darling test, Shapiro-Wilk test and Shapiro-Francia test compared via Monte Carlo simulation of sample data generated from alternative distributions that follow symmetric, skewed, skewed & heavy tailed, highly skewed and highly skewed & heavy tailed distributions. Results: Simulation study shows that Shapiro-Francia test under symmetric, skewed, skewed& heavy tailed and highly skewed& heavy tailed distributions perform better than others. Also, Shapiro-Wilk performs better when underlying distribution is skewed. Conclusion: Although, Shapiro-Francia and Shapiro-Wilk have greater power than their Original Research Article British Journal of Applied Science & Technology, 4(18): 2646-2660, 2014 2647 competitors but their powers are still low for small sample size. So their singly use is not recommended.

This study evaluated the performance of various normality tests including Shapiro–Wilk, Shapiro–Francia, Anderson–Darling, Lilliefors, Cramer–von Mises, and Jarque–Bera under different conditions, both with and without the presence of outliers. Monte Carlo simulations were conducted to calculate the type I error rates, power, and the Kappa–Fleiss agreement coefficient, which measured the concordance among the tests. For normally distributed data without outliers, the Shapiro–Wilk and Shapiro–Francia tests showed the best control over the type I error rate. In contrast, with the introduction of outliers, the Lilliefors and Cramer–von Mises tests performed better. In terms of test power, the Shapiro–Wilk and Shapiro–Francia tests performed best for distributions without outliers, while the Jarque–Bera test was more robust in the presence of outliers. Overall, the results highlight the sensitivity of these tests to sample size and the presence of outliers, suggesting that Shapiro–Wilk and Shapiro–Francia are suitable for data without outliers, while Jarque–Bera may be preferred in contaminated samples. The tests showed higher concordance for exponential and lognormal distributions but lower concordance for beta, χ², and t-Student distributions, illustrating the complexity of normality identification across various contexts.

Maclean et al. (1976) applied a specific Box-Cox transformation to test for mixtures of distributions against a single distribution. Their null hypothesis is that a sample of n observations is from a normal distribution with unknown mean and variance after a restricted Box-Cox transformation. The alternative is that the sample is from a mixture of two normal distributions, each with unknown mean and unknown, but equal, variance after another restricted Box-Cox transformation. We developed a computer program that calculated the maximum likelihood estimates (MLEs) and likelihood ratio test (LRT) statistic for the above. Our algorithm for the calculation of the MLEs of the unknown parameters used multiple starting points to protect against convergence to a local rather than global maximum. We then simulated the distribution of the LRT for samples drawn from a normal distribution and five Box-Cox transformations of a normal distribution. The null distribution appeared to be the same for the Box-Cox transformations studied and appeared to be distributed as a chi-square random variable for samples of 25 or more. The degrees of freedom parameter appeared to be a monotonically decreasing function of the sample size. The null distribution of this LRT appeared to converge to a chi-square distribution with 2.5 degrees of freedom. We estimated the critical values for the 0.10, 0.05, and 0.01 levels of significance.

In this study, statistical analysis of High performance concrete (HPC) with steel fiber addition at 0.50, 0.75, 1.00, 1.25, & 1.50 % was evaluated each with a sample size (N) of 50 using 100 mm side cubes. Normality test and screening was done on the data set, frequency histogram with superimposed normal distribution curve plotted, as well as P-value compared with the significance level. Other parameters investigated were confidence interval and probability plot. It was seen that the data does not follow the normal distribution because the data float around the ideal normal curve, P-value is less than the significance level and Anderson-darling is high. An attempt was made to transform the data set for Goodness of fit using Johnson Transformation, and it was seen that the P-value significantly improved. Overall, an improvement in the mean compressive strength is observed with increase in fiber addition by utilizing large sample size.

The small sample powers of two statistics, the likelihood ratio test, and a test based on the asymptotic normality of maximum likelihood estimators (z-test) were compared in a simulation experiment. Two models were specified, one containing the Box-Cox transformation on the dependent variable only, and one containing the Box Cox transformation on both the dependent and independent variables. The transformation parameter,λ was estimated 200 times, for each of six different values of z in each of three sample sizes foi both models. At each replication. 17 hypotheses were tested using both a likelihood ratio test and a z-test. Results indicate that w hiic both likelihood ratio tests and z-tests are unbiased, in small samples the z-test is generally preferable to the likelihood ratio test.

This paper considers various estimation methods to estimate the unknown parameters of the DUS Inverse Weibull (DIW) distribution using the maximum likelihood (ML), least squares (LS), weighted least squares (WLS), Cramer-von Mises (CVM) and the Anderson-Darling (AD) estimators. A Monte-Carlo simulation study is conducted to determine the most preferable estimators in terms of their efficiencies. Furthermore, the distribution of the error terms in the simple linear regression is assumed to be DIW to show the implementation of it to the linear models. We also carry out a simulation study for comparing the performances of the estimators of the unknown regression parameters.

The correct application of a statistical test is directly connected with information related to the distribution of data. Anderson–Darling is one alternative used to test if the distribution of experimental data follows a theoretical distribution. The conclusion of the Anderson–Darling test is usually drawn by comparing the obtained statistic with the available critical value, which did not give any weight to the same size. This study aimed to provide a formula for calculation of p-value associated with the Anderson–Darling statistic considering the size of the sample. A Monte Carlo simulation study was conducted for sample sizes starting from 2 to 61, and based on the obtained results, a formula able to give reliable probabilities associated to the Anderson–Darling statistic is reported.

Breast cancer remains a major global health problem, characterized by high incidence and mortality rates. Developing accurate prediction models is essential to improving early detection and treatment outcomes. Machine learning (ML) has become a valuable resource in breast cancer prediction; however, the complexities inherent in medical data, including biases and imbalances, can hinder the effectiveness of these models. This paper explores combining the Box–Cox transformation with ML models to normalize data distributions and stabilize variance, thereby enhancing prediction accuracy. Two datasets were analyzed: a synthetic gamma-distributed dataset that simulates skewed real-world data and the Surveillance, Epidemiology, and End Results (SEER) breast cancer dataset, which displays imbalanced real-world data. Four distinct experimental scenarios were conducted on the ML models with a synthetic dataset, the SEER dataset with the Box–Cox transformation, a SEER dataset with the logarithmic transformation, and with Synthetic Minority Over-sampling Technique (SMOTE) augmentation to evaluate the impact of the Box–Cox transformation through different lambda values. The results show that the Box–Cox transformation significantly improves the performance of Artificial Intelligence (AI) models, particularly the stacking model, achieving the highest accuracy with 94.53% and 94.74% of the F1 score. This study demonstrates the importance of feature transformation in healthcare analytics, offering a scalable framework for improving breast cancer prediction and potentially applicable to other medical datasets with similar challenges.

The Indonesian Government introduced rice farm insurance scheme (Asuransi Usaha Tani Padi – AUTP program) to protect rice farm from loss caused by flood, drought and pest and disease infestations. It has been discussed the method to calculate pure premium of the rice farm insurance scheme in Indonesia with assume that the distribution of rice yield data is normal distribution, gamma distribution, or normal mixture distribution. The normal distribution can be used for the case of distribution of rice yield data in the form of symmetry. The gamma distribution can be used for the case of distribution of rice yield data in the form of skewed to the right or positively skewed. In addition, the normal and gamma distributions are categorized as unimodal distributions. It has also been discussed the method of calculating pure premium of the rice farm insurance scheme in Indonesia with the assumption that the rice yield data is normal mixture distribution. The normal mixture distribution can be categorized as multimodal distribution. Characteristic of the normal mixture distribution is suitable for rice yield data in Indonesia that contain rice yield data from several provinces. On the other hand it is also known that the 4-parameter beta mixture distribution can be applied to rice yield data in Indonesia. This distribution is more flexible than the normal mixture distribution in terms of the tail shape of the distribution. In this paper, pure premium calculation method of the AUTP program is formulated with assume that the distribution of rice yield data is the 4-parameter beta mixture distribution.Monte Carlo simulation is used to evaluate performance of the method. The Monte Carlo simulation results show that the proposed method has higher accuracy and precision when increasing sample size. The method is applied to the rice productivity data in several provinces in Indonesia for the period 1970 to 2016.

The standard Cramer-von Mises and Anderson-Darling goodness-of-fit tests require continuous underlying distributions with known parameters. In this paper, tables of critical values are generated for both tests for Weibull distributions with unknown location and scale parameters and known shape parameters. The powers of the Cramer-von Mises, Anderson-Darling, Kolmogorov-Smirnov, and Chi-Square tests for this situation are investigated. The Cramer-von Mises test has most power when the shape is 1.0 and the Anderson-Darling test has most power when the shape is 3.5. Finally, a relation between critical value and inverse shape parameter is presented.

International recommendations for determination of reference intervals have been recently updated, especially for small reference sample groups, and use of the robust method and Box-Cox transformation is now recommended. Unfortunately, these methods are not included in most software programs used for data analysis by clinical laboratories. We have created a set of macroinstructions, named Reference Value Advisor, for use in Microsoft Excel to calculate reference limits applying different methods. For any series of data, Reference Value Advisor calculates reference limits (with 90% confidence intervals [CI]) using a nonparametric method when n≥40 and by parametric and robust methods from native and Box-Cox transformed values; tests normality of distributions using the Anderson-Darling test and outliers using Tukey and Dixon-Reed tests; displays the distribution of values in dot plots and histograms and constructs Q-Q plots for visual inspection of normality; and provides minimal guidelines in the form of comments based on international recommendations. The critical steps in determination of reference intervals are correct selection of as many reference individuals as possible and analysis of specimens in controlled preanalytical and analytical conditions. Computing tools cannot compensate for flaws in selection and size of the reference sample group and handling and analysis of samples. However, if those steps are performed properly, Reference Value Advisor, available as freeware at http://www.biostat.envt.fr/spip/spip.php?article63, permits rapid assessment and comparison of results calculated using different methods, including currently unavailable methods. This allows for selection of the most appropriate method, especially as the program provides the CI of limits. It should be useful in veterinary clinical pathology when only small reference sample groups are available.