ABSTRACT: The global experiments in the presence of repeated measurements are represented in the case
of three overlapping factors and the third factor is represented by experimental units (Subjects). Repeated
measurements or treatments are taken for experimental units and these treatments are treated as a fourth factor.
This type of experiment has been analyzed by the parameterized methods represented by the F test. If the
conditions for variance analysis are available for repeated measurement experiments, and if the conditions are
not met, we use the non-parametric methods of converting to the ranks.
The purpose of this research is an analytical study of this type of experiment by non-parametric and nonparametric methods and the application of this experiment to thalassemia in DhiQar
This document presents a nonparametric approach to multiple regression that uses ranks instead of raw values for both the dependent and independent variables. The key points are:
1. It develops a nonparametric multiple regression model using the ranks of observations on the dependent variable and ranks of observations on the independent variables.
2. The method of least squares is applied to the rank-based model to obtain estimates of the regression coefficients.
3. Prediction equations are presented that allow predicting dependent variable ranks based on independent variable ranks.
An alternative approach to estimation of populationAlexander Decker
This document summarizes an alternative approach to estimating population mean in two-stage sampling. It proposes ratio, product, and regression estimators when the population mean of the auxiliary variable is unknown. The estimators are defined and their mean squared errors are derived. It is shown that under certain conditions, the suggested estimators are more efficient than existing estimators. Numerical examples are provided to support the theoretical results. In summary, this paper presents new estimators for population mean in two-stage sampling that make use of auxiliary information when the auxiliary population mean is unknown.
Initial and Boundary Value Problems Involving the Inhomogeneous Weber Equatio...theijes
Initial and boundary value problems of the inhomogeneous Weber differential equation are treated in this work. General solutions are expressed in terms of the parametric Nield-Kuznetsov functions of the first and second kinds, and are computed when the forcing function is a constant or a variable function of the independent variable
Analysis of the Essential Features in the Tasks of IdentificationIJERA Editor
This paper proposes a new approach to defining the essential features in the task of identification based on the use of topological methods. To extend the scope of analysis and reducing its complexity applies fuzzy representation of the original data. The obtained results show the efficiency of the proposed approach.
MANOVA stands for Multivariate Analysis of Variance. It is used to compare means across multiple populations and variables simultaneously. The F test is used to compare means across k populations. It assumes the populations have equal variances and tests if the population means are equal against the alternative that at least one pair of means is different. The F statistic is calculated and compared to a critical value from the F distribution to determine if there are statistically significant differences between population means.
The document discusses estimation of multi-Granger network causal models from time series data. It proposes a joint modeling approach to estimate vector autoregressive (VAR) models for multiple time series datasets simultaneously. The key steps are:
1. Estimate the inverse covariance matrices for each dataset using a factor model approach.
2. Use the estimated inverse covariance matrices in a generalized fused lasso optimization to jointly estimate the VAR coefficient matrices for each dataset.
Simulation results show the joint modeling approach improves estimation of the VAR coefficients and reduces forecasting error compared to estimating the models separately, especially when the number of time points is small. The factor modeling approach also provides a better estimate of the inverse covariance than using the empirical estimate.
Comparison of Bayesian and non-Bayesian estimations for Type-II censored Gen...IqraHussain31
Conference Research Article
Presented By
Iqra Sardar
16th International Conference on Statistical Sciences:
At Department of Statistics
Islamia College, Peshawar Khyber Pakhtunkhwa, Pakistan
Auto Regressive Process (1) with Change Point: Bayesian ApprochIJRESJOURNAL
Abstract : Here we consider first order autoregressive process with changing autoregressive coefficient at some point of time m. This is called change point inference problem. For Bayes estimation of m and autoregressive coefficient we used MHRW (Metropolis Hasting Random Walk) algorithm and Gibbs sampling. The effects of prior information on the Bayes estimates are also studied.
This document presents a nonparametric approach to multiple regression that uses ranks instead of raw values for both the dependent and independent variables. The key points are:
1. It develops a nonparametric multiple regression model using the ranks of observations on the dependent variable and ranks of observations on the independent variables.
2. The method of least squares is applied to the rank-based model to obtain estimates of the regression coefficients.
3. Prediction equations are presented that allow predicting dependent variable ranks based on independent variable ranks.
An alternative approach to estimation of populationAlexander Decker
This document summarizes an alternative approach to estimating population mean in two-stage sampling. It proposes ratio, product, and regression estimators when the population mean of the auxiliary variable is unknown. The estimators are defined and their mean squared errors are derived. It is shown that under certain conditions, the suggested estimators are more efficient than existing estimators. Numerical examples are provided to support the theoretical results. In summary, this paper presents new estimators for population mean in two-stage sampling that make use of auxiliary information when the auxiliary population mean is unknown.
Initial and Boundary Value Problems Involving the Inhomogeneous Weber Equatio...theijes
Initial and boundary value problems of the inhomogeneous Weber differential equation are treated in this work. General solutions are expressed in terms of the parametric Nield-Kuznetsov functions of the first and second kinds, and are computed when the forcing function is a constant or a variable function of the independent variable
Analysis of the Essential Features in the Tasks of IdentificationIJERA Editor
This paper proposes a new approach to defining the essential features in the task of identification based on the use of topological methods. To extend the scope of analysis and reducing its complexity applies fuzzy representation of the original data. The obtained results show the efficiency of the proposed approach.
MANOVA stands for Multivariate Analysis of Variance. It is used to compare means across multiple populations and variables simultaneously. The F test is used to compare means across k populations. It assumes the populations have equal variances and tests if the population means are equal against the alternative that at least one pair of means is different. The F statistic is calculated and compared to a critical value from the F distribution to determine if there are statistically significant differences between population means.
The document discusses estimation of multi-Granger network causal models from time series data. It proposes a joint modeling approach to estimate vector autoregressive (VAR) models for multiple time series datasets simultaneously. The key steps are:
1. Estimate the inverse covariance matrices for each dataset using a factor model approach.
2. Use the estimated inverse covariance matrices in a generalized fused lasso optimization to jointly estimate the VAR coefficient matrices for each dataset.
Simulation results show the joint modeling approach improves estimation of the VAR coefficients and reduces forecasting error compared to estimating the models separately, especially when the number of time points is small. The factor modeling approach also provides a better estimate of the inverse covariance than using the empirical estimate.
Comparison of Bayesian and non-Bayesian estimations for Type-II censored Gen...IqraHussain31
Conference Research Article
Presented By
Iqra Sardar
16th International Conference on Statistical Sciences:
At Department of Statistics
Islamia College, Peshawar Khyber Pakhtunkhwa, Pakistan
Auto Regressive Process (1) with Change Point: Bayesian ApprochIJRESJOURNAL
Abstract : Here we consider first order autoregressive process with changing autoregressive coefficient at some point of time m. This is called change point inference problem. For Bayes estimation of m and autoregressive coefficient we used MHRW (Metropolis Hasting Random Walk) algorithm and Gibbs sampling. The effects of prior information on the Bayes estimates are also studied.
RESIDUALS AND INFLUENCE IN NONLINEAR REGRESSION FOR REPEATED MEASUREMENT DATAorajjournal
All observations don’t have equal significance in regression analysis. Diagnostics of observations is an important aspect of model building. In this paper, we use diagnostics method to detect residuals and influential points in nonlinear regression for repeated measurement data. Cook distance and Gauss newton method have been proposed to identify the outliers in nonlinear regression analysis and parameter estimation. Most of these techniques based on graphical representations of residuals, hat matrix and case deletion measures. The results
show us detection of single and multiple outliers cases in repeated measurement data. We use these techniques
to explore performance of residuals and influence in nonlinear regression model.
International journal of engineering and mathematical modelling vol2 no1_2015_1IJEMM
Our efforts are mostly concentrated on improving the convergence rate of the numerical procedures both from the viewpoint of cost-efficiency and accuracy by handling the parametrization of the shape to be optimized. We employ nested parameterization supports of either shape, or shape deformation, and the classical process of degree elevation resulting in exact geometrical data transfer from coarse to fine representations. The algorithms mimick classical multigrid strategies and are found very effective in terms of convergence acceleration. In this paper, we analyse and demonstrate the efficiency of the two-level correction algorithm which is the basic block of a more general miltilevel strategy.
Chebyshev Collocation Approach for a Continuous Formulation of Implicit Hybri...IOSR Journals
In this paper, an implicit one-step method for numerical solution of second order Initial Value
Problems of Ordinary Differential Equations has been developed by collocation and interpolation technique.
The one-step method was developed using Chebyshev polynomial as basis function and, the method was
augmented by the introduction of offstep points in order to bring about zero stability and upgrade the order of
consistency of the new method. An advantage of the derived continuous scheme is that it can produce several
outputs of solution at the off-grid points without requiring additional interpolation. Numerical examples are
presented to portray the applicability and the efficiency of the method.
Concepts in order statistics and bayesian estimationAlexander Decker
This academic article discusses concepts in order statistics and Bayesian estimation. It provides definitions and formulas related to order statistics, including probability density functions for order statistics. It also defines survival functions, life probability functions, life probability density functions, and failure rate functions. Additionally, it covers concepts in Bayesian statistics such as loss functions, risk functions, and prior distribution functions.
Bayesian Estimation for Missing Values in Latin Square Designinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
A novel numerical approach for odd higher order boundary value problemsAlexander Decker
This document presents a novel numerical approach for solving odd higher order boundary value problems (BVPs), specifically fifth, seventh, and ninth order linear and nonlinear BVPs. The method uses the Galerkin weighted residual method with Legendre polynomials as basis functions to satisfy the boundary conditions. Matrix formulations are derived for the fifth, seventh, and ninth order cases. Several examples are presented and the results are compared to existing methods to demonstrate the reliability and efficiency of the proposed method.
A ( )-Stable Order Ten Second Derivative Block Multistep Method for Stiff I...inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online
This document discusses applications of first order ordinary differential equations (ODEs) as mathematical models. It provides examples of using first order ODEs to model population growth and decay, predator-prey interactions, and mixing problems. The modeling of logistic population growth with a first order ODE is shown to be more powerful than exponential modeling. Basic principles for modeling like mass action and conservation of mass are also outlined.
The International Journal of Engineering and Science (The IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Money, income, and prices in bangladesh a cointegration and causality analysisAlexander Decker
This document summarizes a study that examines the causal relationship between money, income, and prices in Bangladesh from 1972/73 to 2009/10. It finds:
1) Money, income, and prices are cointegrated, indicating a long-run relationship.
2) Bivariate analysis finds bidirectional causality between money and income, supporting neither Keynesians nor Monetarists. It finds unidirectional causality from money to prices, supporting Monetarists.
3) Trivariate analysis also finds bidirectional causality between money and income conditional on prices, and unidirectional causality from money to prices conditional on income.
So monetary policy should consider feedback effects between money and income
Certain Algebraic Procedures for the Aperiodic Stability Analysis and Countin...Waqas Tariq
To evaluate the performance of a linear time-invariant system, various measures are available. In this paper employing Routh’s table, two geometrical criteria for the aperiodic stability analysis of linear time-invariant systems having real coefficients are formulated. The proposed algebraic stability criteria check whether the given linear system is aperiodically stable or not.The additional significance of the two criteria is , it can be used to count the number of complex roots of a system having real coefficients which is not possible by the use of original Routh’s Table. These procedures can also be used for the design of linear systems. In the proposed methods , the characteristic equation having real coefficients are first converted to complex coefficient equations using Romonov’s transformation. These complex coefficients are used in two different ways to form the Modified Routh’s tables for the two schemes named as Sign Pair Criterion I (SPC I) and Sign Pair Criterion II (SPC II). It is found that the proposed algorithms offer computational simplicity compared to other algebraic methods and is illustrated with suitable examples. The developed MATLAB program make the analysis most simple.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
The paper deals with a generalized estimator of population mean which includes several estimators as its particular cases. Under certain conditions, the proposed estimator is more efficient than existing estimators. Results are supported by numerical illustration.
This document provides a midterm exam for a differential equations course. It includes 5 questions testing skills like finding integrating factors, determining linear dependence, using methods like variation of parameters, and applying concepts like damping to solve differential equations modeling physical systems. Students are instructed to complete the exam without notes in 75 minutes and solutions will be posted in a few days.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The document discusses analysis of variance (ANOVA). It defines ANOVA and describes its basic purpose as testing the homogeneity of several means. The document outlines the assumptions and mathematical models of ANOVA for one-way and two-way classifications. For one-way classification, the total variation is separated into variation between classes and variation within classes. An example problem and solution is provided to illustrate one-way ANOVA.
Using QR Decomposition to calculate the sum of squares of a model has a limitation that the number of rows,
which is also the number of observations or responses, has to be greater than the total number of parameters used in the
model. The main goal in the experimental design model, as a part of the Linear Model, is to analyze the estimable function
of the parameters used in the model. In order not to deal with generalized invers, partitioned design matrix may be used
instead. This partitioned design matrix method may be used to calculate the sum of squares of the models whenever the total
number of parameters is greater than the number of observations. It can also be used to find the degrees of freedom of each
source of variation components. This method is discussed in a Balanced Nested-Factorial Experimental Design.
On Some Double Integrals of H -Function of Two Variables and Their ApplicationsIJERA Editor
This paper deals with the evaluation of four integrals of H -function of two variables proposed by Singh and
Mandia [7] and their applications in deriving double half-range Fourier series for the H -function of two
variables. A multiple integral and a multiple half-range Fourier series of the H -function of two variables are
derived analogous to the double integral and double half-range Fourier series of the H -function of two
variables.
RESIDUALS AND INFLUENCE IN NONLINEAR REGRESSION FOR REPEATED MEASUREMENT DATAorajjournal
All observations don’t have equal significance in regression analysis. Diagnostics of observations is an important aspect of model building. In this paper, we use diagnostics method to detect residuals and influential points in nonlinear regression for repeated measurement data. Cook distance and Gauss newton method have been proposed to identify the outliers in nonlinear regression analysis and parameter estimation. Most of these techniques based on graphical representations of residuals, hat matrix and case deletion measures. The results
show us detection of single and multiple outliers cases in repeated measurement data. We use these techniques
to explore performance of residuals and influence in nonlinear regression model.
International journal of engineering and mathematical modelling vol2 no1_2015_1IJEMM
Our efforts are mostly concentrated on improving the convergence rate of the numerical procedures both from the viewpoint of cost-efficiency and accuracy by handling the parametrization of the shape to be optimized. We employ nested parameterization supports of either shape, or shape deformation, and the classical process of degree elevation resulting in exact geometrical data transfer from coarse to fine representations. The algorithms mimick classical multigrid strategies and are found very effective in terms of convergence acceleration. In this paper, we analyse and demonstrate the efficiency of the two-level correction algorithm which is the basic block of a more general miltilevel strategy.
Chebyshev Collocation Approach for a Continuous Formulation of Implicit Hybri...IOSR Journals
In this paper, an implicit one-step method for numerical solution of second order Initial Value
Problems of Ordinary Differential Equations has been developed by collocation and interpolation technique.
The one-step method was developed using Chebyshev polynomial as basis function and, the method was
augmented by the introduction of offstep points in order to bring about zero stability and upgrade the order of
consistency of the new method. An advantage of the derived continuous scheme is that it can produce several
outputs of solution at the off-grid points without requiring additional interpolation. Numerical examples are
presented to portray the applicability and the efficiency of the method.
Concepts in order statistics and bayesian estimationAlexander Decker
This academic article discusses concepts in order statistics and Bayesian estimation. It provides definitions and formulas related to order statistics, including probability density functions for order statistics. It also defines survival functions, life probability functions, life probability density functions, and failure rate functions. Additionally, it covers concepts in Bayesian statistics such as loss functions, risk functions, and prior distribution functions.
Bayesian Estimation for Missing Values in Latin Square Designinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
A novel numerical approach for odd higher order boundary value problemsAlexander Decker
This document presents a novel numerical approach for solving odd higher order boundary value problems (BVPs), specifically fifth, seventh, and ninth order linear and nonlinear BVPs. The method uses the Galerkin weighted residual method with Legendre polynomials as basis functions to satisfy the boundary conditions. Matrix formulations are derived for the fifth, seventh, and ninth order cases. Several examples are presented and the results are compared to existing methods to demonstrate the reliability and efficiency of the proposed method.
A ( )-Stable Order Ten Second Derivative Block Multistep Method for Stiff I...inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online
This document discusses applications of first order ordinary differential equations (ODEs) as mathematical models. It provides examples of using first order ODEs to model population growth and decay, predator-prey interactions, and mixing problems. The modeling of logistic population growth with a first order ODE is shown to be more powerful than exponential modeling. Basic principles for modeling like mass action and conservation of mass are also outlined.
The International Journal of Engineering and Science (The IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Money, income, and prices in bangladesh a cointegration and causality analysisAlexander Decker
This document summarizes a study that examines the causal relationship between money, income, and prices in Bangladesh from 1972/73 to 2009/10. It finds:
1) Money, income, and prices are cointegrated, indicating a long-run relationship.
2) Bivariate analysis finds bidirectional causality between money and income, supporting neither Keynesians nor Monetarists. It finds unidirectional causality from money to prices, supporting Monetarists.
3) Trivariate analysis also finds bidirectional causality between money and income conditional on prices, and unidirectional causality from money to prices conditional on income.
So monetary policy should consider feedback effects between money and income
Certain Algebraic Procedures for the Aperiodic Stability Analysis and Countin...Waqas Tariq
To evaluate the performance of a linear time-invariant system, various measures are available. In this paper employing Routh’s table, two geometrical criteria for the aperiodic stability analysis of linear time-invariant systems having real coefficients are formulated. The proposed algebraic stability criteria check whether the given linear system is aperiodically stable or not.The additional significance of the two criteria is , it can be used to count the number of complex roots of a system having real coefficients which is not possible by the use of original Routh’s Table. These procedures can also be used for the design of linear systems. In the proposed methods , the characteristic equation having real coefficients are first converted to complex coefficient equations using Romonov’s transformation. These complex coefficients are used in two different ways to form the Modified Routh’s tables for the two schemes named as Sign Pair Criterion I (SPC I) and Sign Pair Criterion II (SPC II). It is found that the proposed algorithms offer computational simplicity compared to other algebraic methods and is illustrated with suitable examples. The developed MATLAB program make the analysis most simple.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
The paper deals with a generalized estimator of population mean which includes several estimators as its particular cases. Under certain conditions, the proposed estimator is more efficient than existing estimators. Results are supported by numerical illustration.
This document provides a midterm exam for a differential equations course. It includes 5 questions testing skills like finding integrating factors, determining linear dependence, using methods like variation of parameters, and applying concepts like damping to solve differential equations modeling physical systems. Students are instructed to complete the exam without notes in 75 minutes and solutions will be posted in a few days.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The document discusses analysis of variance (ANOVA). It defines ANOVA and describes its basic purpose as testing the homogeneity of several means. The document outlines the assumptions and mathematical models of ANOVA for one-way and two-way classifications. For one-way classification, the total variation is separated into variation between classes and variation within classes. An example problem and solution is provided to illustrate one-way ANOVA.
Using QR Decomposition to calculate the sum of squares of a model has a limitation that the number of rows,
which is also the number of observations or responses, has to be greater than the total number of parameters used in the
model. The main goal in the experimental design model, as a part of the Linear Model, is to analyze the estimable function
of the parameters used in the model. In order not to deal with generalized invers, partitioned design matrix may be used
instead. This partitioned design matrix method may be used to calculate the sum of squares of the models whenever the total
number of parameters is greater than the number of observations. It can also be used to find the degrees of freedom of each
source of variation components. This method is discussed in a Balanced Nested-Factorial Experimental Design.
On Some Double Integrals of H -Function of Two Variables and Their ApplicationsIJERA Editor
This paper deals with the evaluation of four integrals of H -function of two variables proposed by Singh and
Mandia [7] and their applications in deriving double half-range Fourier series for the H -function of two
variables. A multiple integral and a multiple half-range Fourier series of the H -function of two variables are
derived analogous to the double integral and double half-range Fourier series of the H -function of two
variables.
Dimensional analysis is a technique to reduce the number of variables in a physical problem by expressing them as dimensionless parameters. It enables scaling between experiments of different physical dimensions. The document discusses dimensional analysis methods including the Buckingham Pi Theorem and exponent method. It provides an example application to a hydraulic jump, identifying the relevant variables and deriving the dimensionless parameters of Reynolds number, Froude number, and depth ratio that the problem depends on.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document presents an analysis of the exponential distribution under an adaptive type-I progressive hybrid censoring scheme for competing risks data. Maximum likelihood and Bayesian estimation methods are used to estimate the distribution parameter. Specifically, maximum likelihood estimators are derived for the exponential distribution parameter. Bayesian estimators are also obtained for the parameter based on squared error and LINEX loss functions using gamma priors. Asymptotic confidence intervals and Bayesian credible intervals are proposed. A simulation study is conducted to evaluate the performance of the estimators.
Modul fizik cakna kelantan spm 2014 k3 set 2 dan skemaCikgu Pejal
This document contains the questions and diagrams for a physics exam on optics and electricity. In Section A, the questions assess students' understanding of an experiment on the relationship between the angle of incidence and angle of refraction for light passing through a glass block. Students are asked to identify variables, record angle measurements, tabulate results, and plot a graph. In Section B, there are two multi-part questions about experiments on the elastic properties of rubber bands and the operation of a bicycle dynamo. Students must describe hypothetical experiments to investigate given hypotheses. The document contains the exam questions, spaces for students to show their work, and diagrams related to the questions.
International Journal of Engineering Research and Applications (IJERA) is a team of researchers not publication services or private publications running the journals for monetary benefits, we are association of scientists and academia who focus only on supporting authors who want to publish their work. The articles published in our journal can be accessed online, all the articles will be archived for real time access.
Our journal system primarily aims to bring out the research talent and the works done by sciaentists, academia, engineers, practitioners, scholars, post graduate students of engineering and science. This journal aims to cover the scientific research in a broader sense and not publishing a niche area of research facilitating researchers from various verticals to publish their papers. It is also aimed to provide a platform for the researchers to publish in a shorter of time, enabling them to continue further All articles published are freely available to scientific researchers in the Government agencies,educators and the general public. We are taking serious efforts to promote our journal across the globe in various ways, we are sure that our journal will act as a scientific platform for all researchers to publish their works online.
This document describes a single factor experiment investigating the effect of cotton percentage on tensile strength of a synthetic fiber. Five cotton percentage levels (15, 20, 25, 30, 35%) were tested with multiple replicates at each level. Analysis of variance (ANOVA) was used to determine if changing the cotton percentage significantly affected tensile strength by comparing the variation between treatments to the variation within treatments. If the between treatment variation is significantly higher, then cotton percentage affects tensile strength.
Mixed Spectra for Stable Signals from Discrete Observationssipij
This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the
modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral
measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to
estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we
propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial
kernel to build a periodogram which we then smooth by two spectral windows taking into account the
width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing
often encountered in the case of estimation from discrete observations of a continuous time process.
Mixed Spectra for Stable Signals from Discrete Observationssipij
This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the
modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral
measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to
estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we
propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial
kernel to build a periodogram which we then smooth by two spectral windows taking into account the
width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing
often encountered in the case of estimation from discrete observations of a continuous time process.
MIXED SPECTRA FOR STABLE SIGNALS FROM DISCRETE OBSERVATIONSsipij
This paper proposes a method to estimate the spectral density of a continuous-time stable alpha symmetric process from discrete observations of the process. Specifically, it considers when the spectral measurement is a mixture of a continuous component and discrete jumps. It samples the process at periodic times to create a periodogram, which is shown to be an asymptotically unbiased but inconsistent estimator. The periodogram is then smoothed using two spectral windows to account for the bandwidth of the spectral density, providing a consistent estimator of the spectral density at the jump points.
Mixed Spectra for Stable Signals from Discrete Observationssipij
This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the
modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral
measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to
estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we
propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial
kernel to build a periodogram which we then smooth by two spectral windows taking into account the
width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing
often encountered in the case of estimation from discrete observations of a continuous time process.
Mixed Spectra for Stable Signals from Discrete Observationssipij
This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial kernel to build a periodogram which we then smooth by two spectral windows taking into account the width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing often encountered in the case of estimation from discrete observations of a continuous time process.
This document discusses regression analysis techniques. Regression analysis is used to model the relationship between a dependent variable (Y) and one or more independent variables (X1, X2, etc). Simple linear regression involves one independent variable, while multiple linear regression involves two or more independent variables. The key assumptions of linear regression are outlined. Methods for estimating regression coefficients using least squares and testing the significance of regression coefficients and the overall regression model are also described. An example application involving modeling personal pollutant exposure (Y) based on hours outdoors (X1) and home pollutant levels (X2) is provided.
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Analysis of variance of global experiments with repeated measurements with practical application
1. International Journal of Multidisciplinary and Current
Educational Research (IJMCER)
ISSN: 2581-7027 ||Volume|| 2 ||Issue|| 1 ||Pages|| 09-21 ||2020||
9Pagewww.ijmcer.com|1| Issue2Volume|
Analysis of variance of global experiments with repeated
measurements with practical application
Zaineb Salih Hameed , Salam Abdulhussein Sehen, Sara Mahdi Abooud ,
Marwi Adil Mutlag, Yusra Rasim Jabbar
University of AL-Qadisiyah, Iraq
ABSTRACT: The global experiments in the presence of repeated measurements are represented in the case
of three overlapping factors and the third factor is represented by experimental units (Subjects). Repeated
measurements or treatments are taken for experimental units and these treatments are treated as a fourth factor.
This type of experiment has been analyzed by the parameterized methods represented by the F test. If the
conditions for variance analysis are available for repeated measurement experiments, and if the conditions are
not met, we use the non-parametric methods of converting to the ranks.
The purpose of this research is an analytical study of this type of experiment by non-parametric and non-
parametric methods and the application of this experiment to thalassemia in DhiQar
I. INTRODUCTION AND GOAL
Designs of repeated measurements is a method used in designs of experiments that are taken measurements
repeatedly for each experimental unit and there will usually be a link between observations within the
experimental unit. Repetitive measurements designs are used to increase the accuracy of experiments by
omitting or neglecting the differences between experimental units to estimate the effects of treatment and
experimental error. This type of designs is very useful when the experimental units are limited and this type of
analysis in the design of experiments is widespread, especially in psychology and in analytical experiences and
educational research.
In this paper, we assume that we have a nested experience of two factors, that is, the first factor (A) has (b)
levels and the second factor (B) and has (q) levels since the levels of the second factor (B) are nested within the
levels of the first factor (A) And then this relationship is denoted by the symbol (A) B and the second factor (B)
is called the Nested Factor and the first factor (A) by the factor Factor Nest then the experimental units are
taken for each level of the interfering factor (B) where these experimental units are considered as a third factor
(C) ) It has (n) levels, and the levels of the third factor are intertwined within the levels of both the first (A) and
the second (B) levels. (AB) It is called the three-stage cross-design, and then takes the responses for each
experimental unit with different time periods and this process is known as the repeated measurements where the
responses of the same experimental unit are combined and these repeated measurements will be considered as a
fourth factor (D) and has (r) from the levels where the levels of the fourth factor Intersecting with the levels of
the first factor (A) as well as intersecting with the levels of the second factor (B) within the levels of the first
factor (A) as well as intersecting with the levels of the third factor (C) within the levels of both the first factors
(A) and the second (B) therefore we will have an experiment Interfering operation with repeated measurements.
As shown in the following figure, where X means an intersection relationship and o means an interference
relationship
3. Analysis of variance of global experiments with repeated measurements with practical application
11Pagewww.ijmcer.com|1| Issue2Volume|
Table (1) is a chart of two experiments with repeated measurements
The aim of this research is an analytical study of interferential globalization experiments with the presence of
repeated measurements with the parameter methods represented by the F test as well as the nonparametric
methods by converting data to the ranks.
the theoretical side
This topic deals with the mathematical model and the variance analysis test (F) for this experiment
Mathematical model
A mixed linear model for this experiment can be written as follows:
Y_ijkL = μ + A_i + B_j (i) + C_ (k (ij)) + D_L + AD_ (iL) + DB_Lj (i) + 〖E〗 _kL (ij) ... (1)
i = 1,2,. . . , p; j = 1,2,. . ., q;
k = 1,2,. . ., n; L = 1,2,. . ., r;
whereas:
Y_ijkL: the value of the observation below the level (i) of the first factor (A), the level (j) of the second factor
(B) interfering within the level (i) of the first factor (A) and the level (k) of the third factor (C) interfering
within The levels (j, i) of the first factors (A) and the second (B), respectively, and the level (L) of the fourth
factor (D.)
:تمثل represents the average mean effect and is constant and unknown.
A_i: Effect of Level (i) from Factor One (A.)
B_ (j (i)): effect of the level (j) of the second factor (B) interfering within the level (i) of the first factor (A.)
C_k (ij): effect of the level (k) of the third factor (C) intertwined within both levels (j, i) of the first factors (A)
and the second B)), respectively, which is a random variable
That is to say :-
〖C〗 _ (k (ij)) ~ N (0, σ_c ^ 2)
:D_L effect of level (L) of the fourth factor (D) representing repeated measurements.
AD_iL: effect of the interaction between level (i) of first factor (A) and level (L) of fourth factor (D.)
DB_ (Lj (i)): effect of the interaction between level (L) of the fourth factor D)) and the level (j) of the second
factor (B) interfering within the level (i) of the first factor (A.)
〖E〗 _ (kL (ij)): the effect of a random error resulting from the effect of the interaction between the level (L) of
the fourth factor (D) and the level (k) of the third factor C)) interfering under both levels (j, i) of the factor First
(A) and second factor (B)
ANOVA Table
The methods or mathematical formulas appropriate for calculating the sums of squares are obtained as follows:
SST
= ∑ ∑ ∑ ∑(YijkL – Y̅….)
2
r
L=1
n
k=1
q
j=1
p
i=1
= ∑ ∑ ∑ ∑(YijkL – Y̅ijk. + Y̅ijk. – Y̅….)
2
r
L=1
n
k=1
q
j=1
p
i=1
= ∑ ∑ ∑ ∑(YijkL − Y̅ijk.)
2
r
L=1
n
k=1
q
j=1
p
i=1
+ r ∑ ∑ ∑( Y̅ijk. − Y̅….)
2
n
k=1
q
j=1
p
i=1
+ 2 ∑ ∑ ∑ ∑(YijkL − Y̅ijk.)( Y̅ijk. − Y̅….)
r
L=1
n
k=1
q
j=1
p
i=1
. . .
. . .
. . .
.
.
.
.
.
.
Ypq11Ypq12 . . . Ypq1r
Ypq21Ypq22 . . . Ypq2r
. . .
. . .
. . .
Ypqnbq1Ypqnbq2 . . . Ypqnbqr
1
2
.
.
.
𝐧 𝐛𝐪
𝐛 𝐪
4. Analysis of variance of global experiments with repeated measurements with practical application
12Pagewww.ijmcer.com|1| Issue2Volume|
= ∑ ∑ ∑ ∑(YijkL − Y̅ijk.)
2
r
L=1
n
k=1
q
j=1
p
i=1
+ r ∑ ∑ ∑( Y̅ijk. − Y̅….)
2
n
k=1
q
j=1
p
i=1
+ 2( 0 )
= SSWithin + SSBetween
أن بسبب وذلك
∑ ∑ ∑ ∑(YijkL − Y̅ijk.)( Y̅ijk. − Y̅….)
r
L=1
n
k=1
q
j=1
p
i=1
= ∑ ∑ ∑( Y̅ijk. − Y̅….)[∑(YijkL − Y̅ijk.)
r
L=1
]
n
k=1
q
j=1
p
i=1
= 0
SSW = ∑ ∑ ∑ ∑(YijkL − Y̅ijk.)
2
r
L=1
n
k=1
q
j=1
p
i=1
= ∑ ∑ ∑ ∑(YijkL − Y̅ijk. − Y̅…. + Y̅…. − Y̅i..L + Y̅i..L − Y̅ij.L + Y̅ij.L − Y̅ij.. + Y̅ij.. − Y̅…L
r
L=1
n
k=1
q
j=1
p
i=1
+ Y̅…L − Y̅i… + Y̅i…)
2
= ∑ ∑ ∑ ∑(YijkL − Y̅ijk. − Y̅ij.L + Y̅ij..)
2
r
L=1
n
k=1
q
j=1
p
i=1
+ ∑ ∑ ∑ ∑(Y̅ij.L − Y̅ij.. − Y̅i..L + Y̅i…)
2
r
L=1
n
k=1
q
j=1
p
i=1
+ ∑ ∑ ∑ ∑(Y̅i..L − Y̅i… − Y̅…L + Y̅….)2
r
L=1
n
k=1
q
j=1
p
i=1
+ ∑ ∑ ∑ ∑( Y̅…L − Y̅….)2
r
L=1
n
k=1
q
j=1
p
i=1
= SSD + SSAD + SSDB(A) + SSDC(AB)
SSB = r ∑ ∑ ∑( Y̅ijk. − Y̅….)
2
n
k=1
q
j=1
p
i=1
= r ∑ ∑ ∑( Y̅ijk. − Y̅…. − Y̅i… + Y̅i… − Y̅ij.. + Y̅ij..)
2
n
k=1
q
j=1
p
i=1
= r ∑ ∑ ∑( Y̅i… − Y̅….)2
n
k=1
q
j=1
p
i=1
+ r ∑ ∑ ∑( Y̅ij.. − Y̅i…)
2
n
k=1
q
j=1
p
i=1
+ r ∑ ∑ ∑( Y̅ijk. − Y̅ij..)
2
n
k=1
q
j=1
p
i=1
+ 2 ( 0 ) = SSA + SSB(A) + SSC(AB)
[1] =
Y….
2
pqnr
= C. F [2] = ∑ ∑ ∑ ∑ YijkL
2
r
L=1
n
k=1
q
j=1
p
i=1
[3] =
∑ Yi…
2p
i=1
nqr
[4] =
∑ Y.j..
2q
j=1
npr
[5] =
∑ Y..k.
2n
k=1
pqr
[6] =
∑ Y...L
2r
L=1
npq
[7] =
∑ ∑ Yij..
2q
j=1
p
i=1
nr
[8] =
∑ ∑ Yi.k.
2n
k=1
p
i=1
qr
[9] =
∑ ∑ Yi..L
2r
L=1
p
i=1
nq
[10] =
∑ ∑ Y.jk.
2n
k=1
q
j=1
pr
[11] =
∑ ∑ Y.j.L
2r
L=1
q
j=1
np
[12] =
∑ ∑ Y..kL
2r
L=1
n
k=1
pq
[13] =
∑ ∑ ∑ Yijk.
2n
k=1
q
j=1
p
i=1
r
[14] =
∑ ∑ ∑ Yij.L
2r
L=1
q
j=1
p
i=1
n
[15] =
∑ ∑ ∑ Y.jkL
2r
L=1
n
k=1
q
j=1
p
[16] =
∑ ∑ ∑ Yi.kL
2r
L=1
n
k=1
p
i=1
q
The table of variance analysis for the following design is shown in the table below:
The sum of the total squares will be:
SST = 2− 1 = SSB + SSW… (2)
5. Analysis of variance of global experiments with repeated measurements with practical application
13Pagewww.ijmcer.com|1| Issue2Volume|
Sum of squares between experimental units:
SSB = 13− 1 = SSA + SSB (A) + SSCAB… (3)
Sum of squares inside experimental units:
SSW = 2− 13 = SSD + SSAD + SSDB (A) + SSDC (AB) ... (4)
The sum of the squares of the first factor A will be:
SSA = 3− 1… (5)
The sum of squares of the third factor (experimental units) that are nested within the first factor (A) and the
second factor (B:)
SSC (AB) = 13− 7… (6)
The sum of the squares of the second factor (B) that overlap within the first factor (A) will be:
SSB (A) = 7− 3… (7)
The sum of the squares of the fourth factor (repeated measurements) (D) will be:
SSD = 6− 1… 8
The sum of the squares of the interaction between the first factor (A) and the fourth factor (D) is:
SSAD = 9− 3− 6+ 1… (9)
The sum of the squares of the interaction between the fourth factor (D) and the second factor (B) in the first
factor (A)
SSDB (A) = 14− 7−9 + 3… (10)
The sum of the squares of the error is:
SSError = SSDC (AB) = 2−13−14 + 7… (11)
As these interconnected global experiences are considered to be three balanced stages, whereas balanced
interrelated trials mean the number of levels of interfering factor equal within each level of interfering factor,
meaning that the levels of interfering factor (B) are equal within each level of interfering factor (A) as well as
factor levels The overlap (C) is equal within each level of the interference factor (B.)
Likewise:
𝟏 − SSB(A) = SSB + SSAB … (12)
∗ SSB =
∑ Y.j..
2q
j
pnr
−
Y….
2
pqnr
= [4] − [1]
∗ SSAB =
∑ ∑ Yij..
2q
j
p
i
nr
−
∑ Yi…
2p
i
qnr
−
∑ Y.j..
2q
j
pnr
+
Y….
2
pqnr
= [7] − [3] − [4] + [1]
∗ SSB(A) =
∑ Y.j..
2q
j
pnr
–
Y….
2
pqnr
+
∑ ∑ Yij..
2q
j
p
i
nr
−
∑ Yi…
2p
i
qnr
−
∑ Y.j..
2q
j
pnr
+
Y….
2
pqnr
= [4] − [1] + [7] − [3] − [4] + [1]
=
∑ ∑ Yij..
2q
j
p
i
nr
–
∑ Yi…
2p
i
qnr
= [7] − [3]
𝟐 − SSC(AB) = SSC + SSAC + SSBC + SSABC … (13)
∗ SSC =
∑ Y..k.
2n
k
pqr
−
Y….
2
pqnr
= [5] − [1]
∗ SSAC =
∑ ∑ Yi.k.
2n
k
p
i
qr
−
∑ Yi…
2p
i
qnr
−
∑ Y..k.
2n
k
pqr
+
Y….
2
pqnr
= [8] − [3] − [5] + [1]
∗ SSBC =
∑ ∑ Y.jk.
2n
k
q
j
pr
−
∑ Y.j..
2q
j
pnr
−
∑ Y..k.
2n
k
pqr
+
Y….
2
pqnr
= [10] − [4] − [5] + [1]
∗ SSABC =
∑ ∑ ∑ Yijk.
2n
k
q
j
p
i
r
+
∑ Yi…
2p
i
qnr
+
∑ Y.j..
2q
j
pnr
+
∑ Y..k.
2n
k
pqr
−
∑ ∑ Yij..
2q
j
p
i
nr
–
∑ ∑ Yi.k.
2n
k
p
i
qr
–
∑ ∑ Y.jk.
2n
k
q
j
pr
–
Y….
2
pqnr
= [13] + [3] + [4] + [5] − [7] − [8] − [10] − [1]
6. Analysis of variance of global experiments with repeated measurements with practical application
14Pagewww.ijmcer.com|1| Issue2Volume|
∴ SSC(AB) =
∑ Y..k.
2n
k
pqr
−
Y….
2
pqnr
+
∑ ∑ Yi.k.
2n
k
p
i
qr
−
∑ Yi…
2p
i
qnr
−
∑ Y..k.
2n
k
pqr
+
Y….
2
pqnr
+
∑ ∑ Y.jk.
2n
k
q
j
pr
−
∑ Y.j..
2q
j
pnr
−
∑ Y..k.
2n
k
pqr
+
Y….
2
pqnr
+
∑ ∑ ∑ Yijk.
2n
k
q
j
p
i
r
+
∑ Yi…
2p
i
qnr
+
∑ Y.j..
2q
j
pnr
+
∑ Y..k.
2n
k
pqr
−
∑ ∑ Yij..
2q
j
p
i
nr
–
∑ ∑ Yi.k.
2n
k
p
i
qr
–
∑ ∑ Y.jk.
2n
k
q
j
pr
–
Y….
2
pqnr
= [5] − [1] + [8] − [3] − [5] + [1] + [10] − [4] − [5] + [1] + [13] + [3] + [4]
+ [5] − [7] − [8] − [10] − [1]
=
∑ ∑ ∑ Yijk.
2n
k
q
j
p
i
r
−
∑ ∑ Yij..
2q
j
p
i
nr
= [13] − [7]
𝟑 − SSDB(A) = SSDB + SSDBA … (14)
∗ SSDB =
∑ ∑ Y.j.L
2r
L
q
j
np
−
∑ Y.j..
2q
j
pnr
−
∑ Y...L
2r
L
npq
+
Y….
2
pqnr
= [11] − [4] – [6] + [1]
∗ SSDBA =
∑ ∑ ∑ Yij.L
2r
L
q
j
p
i
n
+
∑ Yi…
2p
i
qnr
+
∑ Y.j..
2q
j
pnr
+
∑ Y…L
2r
L
npq
−
∑ ∑ Yij..
2q
j
p
i
nr
−
∑ ∑ Yi..L
2r
L
p
i
nq
−
∑ ∑ Y.j.L
2r
L
q
j
np
–
Y….
2
pqnr
= [14] + [3] + [4] + [6] − [7] − [9] − [11] − [1]
∴ SSDB(A) =
∑ ∑ Y.j.L
2r
L
q
j
np
−
∑ Y.j..
2q
j
pnr
−
∑ Y...L
2r
L
npq
+
Y….
2
pqnr
+
∑ ∑ ∑ Yij.L
2r
L
q
j
p
i
n
+
∑ Yi…
2p
i
qnr
+
∑ Y.j..
2q
j
pnr
+
∑ Y...L
2r
L
npq
−
∑ ∑ Yij..
2q
j
p
i
nr
−
∑ ∑ Yi..L
2r
L
p
i
nq
−
∑ ∑ Y.j.L
2r
L
q
j
np
−
Y….
2
pqnr
= [11] − [4] − [6] + [1] + [14] + [3] + [4] + [6] − [7] − [9] − [11] − [1]
=
∑ ∑ ∑ Yij.L
2r
L
q
j
p
i
n
+
∑ Yi…
2p
i
qnr
−
∑ ∑ Yij..
2q
j
p
i
nr
−
∑ ∑ Yi..L
2r
L
p
i
nq
= [14] + [3] − [7] − [9]
𝟒 − SSDC(AB) = SSDC + SSACD + SSBCD + SSABCD … (15)
∗ SSDC =
∑ ∑ Y..kL
2r
L
n
k
pq
−
∑ Y..k.
2n
k
pqr
−
∑ Y...L
2r
L
npq
+
Y….
2
pqnr
= [12] − [5] − [6] + [1]
∗ SSACD =
∑ ∑ ∑ Yi.kL
2r
L
n
k
p
i
q
+
∑ Yi…
2p
i
qnr
+
∑ Y..k.
2n
k
pqr
+
∑ Y...L
2r
L
npq
−
∑ ∑ Yi.k.
2n
k
p
i
qr
−
∑ ∑ Yi..L
2r
L
p
i
nq
−
∑ ∑ Y..kL
2r
L
n
k
pq
−
Y….
2
pqnr
= [16] + [3] + [5] + [6] − [8] − [9] − [12] − [1]
∗ SSBCD =
∑ ∑ ∑ Y.jkL
2r
L
n
k
q
j
p
+
∑ Y.j..
2q
j
pnr
+
∑ Y..k.
2n
k
pqr
+
∑ Y...L
2r
L
npq
–
∑ ∑ Y.jk.
2n
k
q
j
pr
−
∑ ∑ Y.j.L
2r
L
q
j
np
−
∑ ∑ Y..kL
2r
L
n
k
pq
−
Y….
2
pqnr
= [15] + [4] + [5] + [6] − [10] − [11] − [12] − [1]
∗ SSABCD = ∑ ∑ ∑ ∑ YijkL
2
−
r
L
n
k
q
j
p
i
∑ ∑ ∑ Yijk.
2n
k
q
j
p
i
r
−
∑ ∑ ∑ Yij.L
2r
L
q
j
p
i
n
−
∑ ∑ ∑ Yi.kL
2r
L
n
k
p
i
q
−
∑ ∑ ∑ Y.jkL
2r
L
n
k
q
j
p
−
∑ Yi…
2p
i
qnr
−
∑ Y.j..
2q
j
pnr
−
∑ Y..k.
2n
k
pqr
−
∑ Y...L
2r
L
npq
+
∑ ∑ Yij..
2q
j
p
i
nr
+
∑ ∑ Yi.k.
2n
k
p
i
qr
+
∑ ∑ Yi..L
2r
L
p
i
nq
+
∑ ∑ Y.jk.
2n
k
q
j
pr
+
∑ ∑ Y.j.L
2r
L
q
j
np
+
∑ ∑ Y..kL
2r
L
n
k
pq
+
Y….
2
pqnr
= [2] − [13] − [14] − [16] − [15] − [3] − [4] – [5] − [6] + [7] + [8] + [9] + [10]
+ [11] + [12] + [1]
7. Analysis of variance of global experiments with repeated measurements with practical application
15Pagewww.ijmcer.com|1| Issue2Volume|
∴ SSDC(AB) =
∑ ∑ Y..kL
2r
L
n
k
pq
−
∑ Y..k.
2n
k
pqr
−
∑ Y...L
2r
L
npq
+
Y….
2
pqnr
+
∑ ∑ ∑ Yi.kL
2r
L
n
k
p
i
q
+
∑ Yi…
2p
i
qnr
+
∑ Y..k.
2n
k
pqr
+
∑ Y...L
2r
L
npq
−
∑ ∑ Yi.k.
2n
k
p
i
qr
−
∑ ∑ Yi..L
2r
L
p
i
nq
−
∑ ∑ Y..kL
2r
L
n
k
pq
−
Y….
2
pqnr
+
∑ ∑ ∑ Y.jkL
2r
L
n
k
q
j
p
+
∑ Y.j..
2q
j
pnr
+
∑ Y..k.
2n
k
pqr
+
∑ Y...L
2r
L
npq
–
∑ ∑ Y.jk.
2n
k
q
j
pr
−
∑ ∑ Y.j.L
2r
L
q
j
np
−
∑ ∑ Y..kL
2r
L
n
k
pq
−
Y….
2
pqnr
+ ∑ ∑ ∑ ∑ YijkL
2
−
r
L
n
k
q
j
p
i
∑ ∑ ∑ Yijk.
2n
k
q
j
p
i
r
−
∑ ∑ ∑ Yij.L
2r
L
q
j
p
i
n
−
∑ ∑ ∑ Yi.kL
2r
L
n
k
p
i
q
−
∑ ∑ ∑ Y.jkL
2r
L
n
k
q
j
p
−
∑ Yi…
2p
i
qnr
−
∑ Y.j..
2q
j
pnr
−
∑ Y..k.
2n
k
pqr
−
∑ Y...L
2r
L
npq
+
∑ ∑ Yij..
2q
j
p
i
nr
+
∑ ∑ Yi.k.
2n
k
p
i
qr
+
∑ ∑ Yi..L
2r
L
p
i
nq
+
∑ ∑ Y.jk.
2n
k
q
j
pr
+
∑ ∑ Y.j.L
2r
L
q
j
np
+
∑ ∑ Y..kL
2r
L
n
k
pq
+
Y….
2
pqnr
= [12] − [5] − [6] + [1] + [16] + [3] + [5] + [6] − [8] − [9] − [12] − [1] + [15]
+ [4] + [5] + [6] − [10] − [11] − [12] – [1] + [2] − [13] − [14] − [16] − [15]
− [3] − [4] – [5] − [6] + [7] + [8] + [9] + [10] + [11] + [12] + [1]
= ∑ ∑ ∑ ∑ YijkL
2
r
L
n
k
q
j
p
i
+
∑ ∑ Yij..
2q
j
p
i
nr
−
∑ ∑ ∑ Yijk.
2n
k
q
j
p
i
r
−
∑ ∑ ∑ Yij.L
2r
L
q
j
p
i
n
= [2] + [7] − [13] − [14]
S .O . V dF S .S M . S E M S F
Between(sub.
)
A
B(A)
Error(Bet.)
= C(AB)
Within(sub.)
D
AD
DB(A)
Error(Within
)
= C(AB
pq(i)n(ij) − 1
p − 1
P(q(i)– 1)
q(i)(n(ij) − 1)
pq(i)n(ij)(r−1)
r−1
(p−1)(r − 1)
P(r−1)(q(i) −
1)
pq(i)(r −
1)(nij − 1)
SSB
SSA
SSB(A)
SSE(B)
SSW
SSD
SSAD
SSDB(A)
SSError
SSA
P − 1
SSB(A)
P(q(i)– 1)
SSE(B)
pq(i)(n(ij) − 1)
SSD
r − 1
SSAD
(p − 1)(r − 1)
SSDB(A)
P(r − 1)(q(i) − 1)
SSE(W)
pq(i)(r − 1)(n(ij) − 1)
rσc
2
+
qnr∅A
rσc
2
+ nr∅B
rσc
2
σDC
2
+ pqn∅D
σDC
2
+ qn∅AD
σDC
2
+ n∅DB
MSA
MSE(B)
MSB(A)
MSE(B)
MSD
MSE(W)
MSAD
MSE(W)
MSDB(A)
MSE(W)
Total pq(i)n(ij)r−1 SST
8. Analysis of variance of global experiments with repeated measurements with practical application
16Pagewww.ijmcer.com|1| Issue2Volume|
Table (2) shows ANOVA for its experiments with the presence of repeated measurementsThis analysis is used
to test the hypothesis:
H_0 = T_1 = T_2 = ... = T_r ... (16)
Analysis of variance for repeated measurement experiments requires the following conditions to be met:
The main influences are aggregate: It means that the factors influence add to each other to determine the values
of observations
The random and independent random distribution of the experimental error: This condition assumes that the
errors are randomly and independently distributed with an average of zero and its value varies (σ ^ 2), meaning
that:
e_ij ~ NID (0, σ ^ 2) ... (17)
Homogeneity of variance: This condition means that random differences are equal within groups homogeneous
and therefore random differences are equal for different samples, which helps to obtain a reduced variance of all
groups.
There is no correlation between the mean and the contrast
Sphericality: This condition means the order of the experimental units does not change in the results of the
experiment as well as the association of any two treatments is the same for each pair of treatments and the
spherical condition indicates that there is no interaction between the factor of the experimental units and the
treatments (repeated measurements).
Analytical methods for this experiment
There are several statistical methods for analyzing the designs of repeated measurements, including parameter
and non-parametric methods and multivariate methods and each of the mentioned methods is implemented
according to certain conditions. In this research we will use the parameter methods represented by testing F and
nonparametric methods by means of Rank Transformation.
The practical side:
This study was applied to data collected from Al-Hussein Hospital _ Thalassemia Center in DhiQar from
patients with beta anemia of the Mediterranean type of beta or the so-called Thalassemia major, and the number
of observations (160) was represented by two groups (two medications) and each group included (20) Patient
(10 males, 10 females) and given treatment at four equal time periods each 30-day time period
Figure (2) Box-and-Whisker Plot plot of Thalassemia data, where no abnormal values are shown
The hypothesis that errors are distributed naturally is tested with an average (0) and a variance (σ2) that are:
H_0 = ε_ij ~ N (0, σ_ε ^ 2). . . (18)
H_1 = ε_ij≁ N (0, σ_ε ^ 2). . . (19)
First, we find the error values (ε_ij) according to the linear model of the design:
Y_ijkL = μ + A_i + B_j (i) + C_ (k (ij)) + D_L + AD_iL + DB_Lj (i) + 〖E〗 _kL (ij) ... (20)
ε_ (Lk (ij)) = Y_ijkL- μ_ (ijk.) - μ_ (ij.L) + μ_ (ij ..). . . (21)
σDC
2
P-Value
Test ValueTest
Box-and-Whisker Plot
0 2 4 6 8
(X 1000)
Col_5
9. Analysis of variance of global experiments with repeated measurements with practical application
17Pagewww.ijmcer.com|1| Issue2Volume|
Table (5) Natural Distribution Test.
Since the p-value of the Chi-Squared test is greater than α = 0.01, the null hypothesis is accepted, meaning that
errors are naturally distributed with mean (0) and variance (σ2.)
2- Test of Homogeneity of Variances
The homogeneity test for treatments is the second condition of the analysis of variance. This condition can be
formulated as follows:
H_0 ∶σ_1 ^ 2 = σ_2 ^ 2 = σ_3 ^ 2 = σ_4 ^ 2 V.S H_1 leastat least two (σ ^ 2) not equal. . . (22)
Using the Bartlett and Cochrane tests to check this condition as:
P-Value
Test valuetest
0.6888
0.50551
1.00567
0.30676
Bartlett's test
Cochran's test
Table (6): Bartlett and Cochran tests for homogeneity of variations
From the above table it appears that Bartlett and Cochran values where the value of p-value is greater than α =
0.01 This leads to acceptance of the null hypothesis as the variations are homogeneous.
3- Test of Correlation
The third condition of the variance analysis is the absence of an association between the mean and the variance
of the treatments
H_0: ρ = 0 VS H_1: ≠ ≠ 0… (23)
It was found that the p-value of the Pearson correlation coefficient is equal to (0.532) which is greater than α =
0.01, thus accepting the null hypothesis that there is no correlation.
Test of Sphericity - Spherical Test 4
To test the fourth condition of the variance analysis conditions for repeated measurements experiments is
H_0 ∶Σ is sphericity. . . (24)
H_1 ∶Σ is not sphericity. . . (25)
To test this hypothesis, we use:
Mauchlys test
This test is based on the Eigen value of the contrast and contrast matrix common to repeated measurements and
the test statistics are:
W = (∏▒λ_i) / [1 / (A-1) ∑▒λ_i] ^ (A-1) = 0.7987236078
χ _ ((w)) ^ 2 = - (1-f) (s-1) ln 〖( w)〗 = 18.465218557
f = (2 〖(A-1)〗 ^ 2 + A + 2) / (6 (A-1) (S-1)) = 0.03418803419
Tabular value χ _ ((α, v)) ^ 2 where:
V = A (A-1) = 6; α = 0.01
〖Χ〗 _table ^ 2 = χ _ ((α, v)) ^ 2 = χ _ ((0.01,6)) ^ 2 = 16.81
Since the calculated value of χ2 is greater than the tabular value of لذلك2, it therefore rejects the null hypothesis
that the contrast and contrast matrix is spherical. Since the spherical condition is not fulfilled, the F test
becomes inaccurate and therefore the degrees of freedom must be adjusted for the F test as follows:
Green house - Geisser Correction - A, which is a value calculated using double Central, and this value is
denoted by (ε) ̂ and is calculated as follows:
ε ̂ = (∑_a▒S_ (a, a)) ^ 2 / ((A-1) ∑_ (a, a ') _S_ (a, a') ^ 2) = 0.888081355
Huynh - Feldt Correction -B
It is the most used formula for modifying degrees of freedom Test F because it is more powerful and it is based
or dependent on the value calculated from Green house - Geisser and symbolized by the symbol (ε ̃:)
ε ̃ = (S (A-1) ε ̂-2) / (A-1) [S-1- (A-1) ε ̂] = 0.9592916574
0.0229
0.1297
0.8111
0.0181
42.2750
0.9743
0.2389
2.3624
Chi-Squared
Shapiro –Wilk W
Skewness Z-Score
Kurtosis Z-Score
10. Analysis of variance of global experiments with repeated measurements with practical application
18Pagewww.ijmcer.com|1| Issue2Volume|
We now find a table of variance analysis for the data as follows:
The number of levels of factor I A (pharmacokinetics) p = 2
The number of levels of the second factor B (gender) q = 2
The number of levels of the third factor D (times) r = 4
The number of levels of the fourth factor C (experimental units) n = 10
S .O . V dF S .S M . S FC Ftable
Between(su
b.)
A
B(A)
Error(Bet.)
= C(AB)
Within(sub.
)
D
AD
DB(A)
Error(With
in)
= C(AB)
39
1
2
36
120
3
3
6
108
681492955
161885522.9
113997336.1
405610096
98061199
19081539
3122776
7601402
68255482
161885522.9
56998668.05
11266947.11
6360513
1040925.333
1266900.333
631995.203
14.368
5.058
10.064
1.647
2.005
F(0.01,1,36) = 7.31
F(0.01,2,36) = 5.18
F(0.01,3,104) = 3.95
F(0.01,3,104) = 3.95
F(0.01,6,104) = 2.96
Total 159 779554154
Table (7) Table of analysis of variance of the global experiment in the presence of repeated
measurementsThrough the results of the ANOVA table, we note that:
There are significant differences in levels between factor A (pharmacists.)
There were no significant differences between levels of factor II B (gender) within factor I (pharmacists.)
There are significant differences between factor D levels (repeated measurements.)
There were no significant differences for interaction between factor A (pharmacokinetics) and factor III D
(repeated measurements.)
-There were no significant differences for interaction between factor II B (gender) and factor IIID (repeated
measurements) within factor IA (pharmacokinetics.)
We are now taking the data rank, finding the table of variance analysis and studying the conditions of variance
analysis for repeated measurements experiments.
5- Test of Normality for errors
To test the hypothesis that errors are naturally distributed (18) using the Chi-Squared test after taking error
estimates(21)
P-ValueTest valuetest
0.20109
0.37615
0.96351
0.00037
31.7625
0.9797
0.0457
3.5566
Chi-Squared
Shapiro –Wilk W
Skewness Z-Score
Kurtosis Z-Score
Table (8) testing the normal distribution of data types.
Since the p-value of the Chi-Squared test is greater than α = 0.01, the null hypothesis is accepted, meaning that
errors are naturally distributed with mean (0) and variance (σ2.)
6- Test of Homogeneity of Variances
To test this condition of the conditions of analysis of variance, which is homogeneity of variations (22), by
using the Bartlet test and Cochran test we conclude:
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P-Value
Test value
Test
0.93525
1.0
1.00275
0.26725
Bartlett's test
Cochran's test
Table (9): Bartlett and Cochran tests for homogenization of variations of data types.From the above table it
appears that Bartlett and Cochran values where the value of p-value is greater than α = 0.01 This leads to
acceptance of the null hypothesis as the variations are homogeneous.
7- Test of Correlation
To test the third condition of the conditions of the analysis of variance, which is the absence of a correlation
between the arithmetic mean and the variance as in the hypotheses (23) through the value of p-value of the
Pearson correlation coefficient where it is equal to (0.588) which is greater than 0.01 = α, therefore the null
hypothesis is not rejected, that is, there is no Correlation between mean and variance.
Test of Sphericity Spherical Test - 8
To test the spherical hypothesis (24), which is one of the conditions for analyzing variance for experiments,
repeat measurements:
Mauchlys test
This test is based on the roots of the Eigen value of the coefficient of variance and co-contrast of the treatments
and the test statistic:
W = (∏▒λ_i) / [1 / (A-1) ∑▒λ_i] ^ (A-1) = 0.7399671452
χ _ ((w)) ^ 2 = - (1-f) (s-1) ln 〖( w)〗 = 11.34329754
f = (2 〖(A-1)〗 ^ 2 + A + 2) / (6 (A-1) (S-1)) = 0.03418803419
The test statistic is compared to the tabular value of χ2 with the degree of freedom v where:
V = A (A-1) = 6 and α = 0.01
〖Χ〗 _table ^ 2 = χ _ ((α, v)) ^ 2 = χ _ ((0.01,6)) ^ 2 = 16.81
Since the computed value of χ2 is less than the tabular value of لذلك2, the null hypothesis that the contrast and
contrast matrix is common is spherical. Therefore, the spherical condition has been met, and the F test becomes
accurate.
After checking the conditions of variance analysis, we find a table of variance analysis for the data types for
comparison with the table of variance analysis for the original data as:
The number of levels of factor I A (pharmacokinetics) p = 2
The number of levels of the second factor B (gender) q = 2
The number of levels of the third factor D (times) r = 4
The number of levels of the fourth factor C (experimental units) n = 10
S .O . V dF S .S M . S FC Ftable
Between(sub.
)
A
B(A)
Error(Bet.)
= C(AB)
Within(sub.)
D
AD
DB(A)
Error(Withi
n)
= C(AB)
39
1
2
36
120
3
3
6
108
297801.375
62805.625
53354.925
181640.825
43517.125
10087.7125
1018.587
2901
29509.825
62805.625
26677.4625
5045.57847
3362.5708
339.529
483.5
273.23912
12.447
5.287
12.306
1.242
1.769
F(0.01,1,36) = 7.31
F(0.01,2,36) = 5.18
F(0.01,3,108) = 3.95
F(0.01,3,108) = 3.95
F(0.01,6,108) = 2.96
Total 159 341318.5
Table (10): Table of variance analysis for the data typesThrough the results of the ANOVA table, we note that:
There are significant differences in levels between factor A (pharmacists).
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There are significant differences between levels of factor II B (gender) within factor I (pharmacists).
There are significant differences between factor D levels (repeated measurements).
There were no significant differences for interaction between factor A (pharmacokinetics) and factor III D
(repeated measurements).
There were no significant differences for interaction between factor B (sex) and factor D (repeated
measurements) within factor I (pharmacokinetics).
• Conclusions
1- It was found that converting data from parameterized to non-parametric methods by grade resulted in
providing conditions for variance analysis for repeated measurements experiments such as normal distribution
of errors, homogeneity of variations, correlation between mean and variance, as well as the spherical condition.
2- Where it was observed that the conditions for normal distribution and homogeneity of variations, as well as
the condition for the absence of correlation between the mean and variance, were achieved and after a transfer
to the ranks, it improved the value of value - P for the condition of normal distribution from (0.0229) to
(0.2210) as well as for homogeneity of variance for Bartlett's test ( 0.688) to (0.935) and likewise for the
Cochran test, it changed from (0.505) to (1.0), and the correlation condition was a value equal to (0.532) and
changed to (0.588). As for the spherical condition, unfulfilled by the value of the test (18.465), which led to
adjusting the degrees of freedom for the F test From (3,108) to (3,104), after a conversion to the data levels, the
spherical condition was met and the test value was equal to (11.343). As for the results of the F test, it may
change. It ranges for the first factor (A) from (14.368) to (12.447), for the second factor (B) from (5.058) to
(5.287) and the fourth factor (D) from (10.069) to (12.306) and for the interaction between the first factor and
the fourth factor (AD From (1.647) to (1.212), and also with regard to the interaction between the second factor
and the fourth factor (DB), it has changed from (2.05) to (1.769).
3- In the practical application of overlapping factor experiments with the presence of repeated measurements
and there were no alternative methods for unparalleled parameters such as Friedman or others so the method we
used was the alternative method for these cases.
4- Among the medical conclusions, the majority of patients with blood types (A +) and (B +), as well as the
majority of patients from his parents who carry the disease and are relatives and it was also observed with poor
or uneducated academic achievement for one of the spouses (housewives, earners) and it was found that the
majority of patients have His residence is rural.
5- Through the four applications, we see that giving the dose of the first drug to patients led to an increase in the
amounts of iron for patients over time, and this type of treatment is used when the amounts of iron are high.
Likewise, when administering the dose of the second medication to patients, it led to increased amounts of iron
for patients over time, but with lower proportions than the first treatment, and this type of treatment is used
when the iron is weak in the patient’s blood.
• Recommendation
Through this research, we reached the following recommendations that they should be taken into consideration
and serve the scientific and human aspect and preserve the life of the individual, we recommend the following:
1. Applying the conditions of variance analysis for repeated measurements in the case of three or more factors,
and in their different situations and experiments.
2. Application of unbalanced overlapping globalization experiments with repeated measurements. Or missing
values.
3. Transferring data to non-parametric methods by converting ranks as one of the solutions in the absence of
conditions for variance analysis for repeated measurements experiments.
4. The use of multivariate methods for analyzing experiments, repeated measurements, and clarifying the
conditions for this type of analysis.
5. Applying repeated measurements in the case of medical data of all kinds in the service of the scientific
process, including psychological data and time-dependent psychology.
6. Giving the dose to patients resulted in increasing the amounts of iron for patients over time, so it is necessary
to carry out a study on different drugs or doses in order to lead to reducing the amounts of iron as well as
reducing complications for patients.
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