scholarly journals Mathematical Analysis of a Five Periods Crossover Design for Two Treatments

2021 ◽  
Author(s):  
Cornelious Omwando Nyakundi ◽  
Joseph Kipsigei Koske ◽  
John Muindi Mutiso ◽  
Isaac Kipkosgei Tum
Keyword(s):  
Estimation Method ◽  
Simulated Data ◽  
Higher Order ◽  
Repeated Measurements ◽  
Unbiased Estimation ◽  
Experimental Unit ◽  
Best Linear Unbiased Estimation ◽  
Best Linear Unbiased ◽  
Carry Over

Introduction: A cross-over design is a repeated measurements design such that each experimental unit receives different treatments during different time periods. Lower order cross-over designs such as the two treatments, two periods and two sequences C (2, 2, 2) design have been discovered to be inefficient and erroneous in their analysis of treatments efficacy. In this regard, higher order cross-over designs have been recommended and developed like: the two treatments, three periods and four sequence C (2, 3, 4) design; and the two treatments, four periods and four sequence C (2, 4, 4) designs. However, there still exists more efficient higher order cross-over designs for two treatments which can be used in bioequivalence experiments. This study gives a new design and analysis for two treatments, five periods and four sequence C (2, 5, 4) cross-over design that gives more precise estimates and provides estimates for intra subject variability. Method: A hypothetical case study was considered on 160 experimental units which are assumed to be randomly selected from a given population. A cross over design of two treatments (A, B) in five periods whose sequences are given by BABAA, ABABB, BAABA and ABBAB were used. Each of the experimental units was used as its own control. The estimates for both direct treatments and treatments carry-over effects were obtained using best linear unbiased estimation method (BLUE). We simulated data for two treatments in five periods and four sequences and used it to test the null hypotheses of no significant differences in both the direct treatments and treatments carry-over effects using the

scholarly journals Bayesian Analysis of an Optimal Five Period Cross-over Design

2021 ◽  
pp. 123-130
Author(s):  
Cornelious Omwando Nyakundi ◽  
Isaac Kipkosgei Tum
Keyword(s):  
Bayesian Analysis ◽  
Bayesian Method ◽  
Repeated Measurements ◽  
Unbiased Estimation ◽  
Experimental Unit ◽  
Carryover Effects ◽  
Significant Treatment ◽  
Analysis Methods ◽  
Carry Over ◽  
Highest Posterior Density

A crossover design is a repeated measurements design such that each experimental unit receives different treatments during the different time periods. In a majority of bioequivalence studies, design and analysis of cross-over using classical methods such as analysis of variance (ANOVA) and test are normally associated with erroneous results. The Bayesian method is desirable in the analysis of crossover designs to eliminate errors associated with carryover effects. The objective of this study was to compare the Bayesian and the - test analysis methods on treatments and carryover effects for an optimal two treatments, five periods and four sequence C (2, 5, 4) design. The treatments and residual estimates were obtained using Best Linear Unbiased Estimation (BLUE) method. In the Bayesian method of analysis, the posterior quantities were obtained for the mean intervals of treatments and carry-over effects and the highest posterior density (HPD) graphs were plotted and interpreted using conditional probability statements. For validation purposes, the Bayesian method results were compared with the existing -tests results. From the Bayesian analysis, the probability of significant treatment difference in the presence of carryover effects was 1, while from the -test, the calculated value of 11.73 was greater than the two sided tabulated value at 95  level of significance. The two analysis methods implied significant differences in the treatment effects. In conclusion, it was established that Bayesian method of analysis can be used for bioequivalence analysis even when the carry-over effects are present and hence it is highly recommended for bioequivalence studies.

scholarly journals Best linear unbiased estimation for varying probability with and without replacement sampling

2019 ◽  
Vol 7 (1) ◽  
pp. 78-91
Author(s):  
Stephen Haslett
Keyword(s):  
Parameter Estimation ◽  
Covariance Matrix ◽  
Linear Models ◽  
Unbiased Estimation ◽  
Error Covariance Matrix ◽  
Best Linear Unbiased Estimation ◽  
Error Covariance ◽  
Inclusion Probabilities ◽  
Best Linear Unbiased ◽  
Linear Unbiased Estimation

Abstract When sample survey data with complex design (stratification, clustering, unequal selection or inclusion probabilities, and weighting) are used for linear models, estimation of model parameters and their covariance matrices becomes complicated. Standard fitting techniques for sample surveys either model conditional on survey design variables, or use only design weights based on inclusion probabilities essentially assuming zero error covariance between all pairs of population elements. Design properties that link two units are not used. However, if population error structure is correlated, an unbiased estimate of the linear model error covariance matrix for the sample is needed for efficient parameter estimation. By making simultaneous use of sampling structure and design-unbiased estimates of the population error covariance matrix, the paper develops best linear unbiased estimation (BLUE) type extensions to standard design-based and joint design and model based estimation methods for linear models. The analysis covers both with and without replacement sample designs. It recognises that estimation for with replacement designs requires generalized inverses when any unit is selected more than once. This and the use of Hadamard products to link sampling and population error covariance matrix properties are central topics of the paper. Model-based linear model parameter estimation is also discussed.

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