Basic Concepts of Standard Experimental Designs ( Statistics )
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This document outlines key concepts in standard experimental design. It defines experimental design as assigning experimental units to treatment conditions to measure and compare treatment effects. Sample design selects units for measurement from a population. The document discusses necessary steps like replication and randomization. It presents linear statistical models including fixed, random, and mixed effects models. It also explains analysis of variance and standard designs like completely randomized design, randomized block design, and Latin square design, including their analysis of variance tables. The conclusion compares the efficiency of these standard designs.
Basic Concepts of Standard Experimental Designs ( Statistics )
- 1. Basic Concepts of Standard Experimental Designs Presented by: Hasnat Israq Islamic University , Bangladesh On behalf of group E 1
- 2. OUTLINE 1. Concept of experimental design 2. Experimental design vs sample design 3. Necessary steps of an experiment 4. Basic principles of experimental design 5. Linear statistical model 6. Analysis of Variance (ANOVA) 7. Standard designs 8. Conclusion 2
- 3. EXPERIMENTAL DESIGN Experimental design refers to a powerful statistical plan for assigning experimental units to treatments condition where treatment is the procedure whose effects is to be measured and compare with similar others in an experiment. A releated point, Sample design methods are the technique used to select sample units for measurement. 3
- 4. Experimental Design VS Sample Design: Sample design concerns with collection of sample from finite and existent population with absolute experiment, where experiment design is concerned with data from nonexistent and infinite population with comparative experiment.
- 5. NECESSARY STEPS OF AN EXPERIMENT
- 6. BASIC OF PRINCIPLES EXPERIMENTAL DESIGN Three basic principles: (i) Replication: Replication means the repeatition of basic treatments on several experimental units under investigation. (ii) Randomisation: Randomisation is the process of distributing the treatments to experimental units by chance mechanism. (iii) Local control: Local control is the procedure of reducing and controlling error variation by blocking the subjects. 6
- 7. LINEAR STATISTICAL MODEL An equation displaying the relation of the effects with response variable along with error term in any experiment 7 Classifying this we get: Fixed effects model Random effects model Mixed effects model
- 8. FIXED EFFECTS MODEL Fixed effect model is model in which all the assignable factors have fixed effects and only the error effect is random. The model is, 𝑦𝑖=𝛼 𝑖 + 𝛽𝑗 𝑥𝑖+ 𝜀𝑖 Here, 𝑦𝑖= dependent variable 𝛼𝑖= intercept (fixed) 𝛽𝑗 = slope coefficient (fixed) 𝑥𝑖= independent variable 𝜀𝑖= random error 𝜀𝑖~ iid N(0, 𝜎2) 8
- 9. RANDOM EFFECTS MODEL In statistics, a random effects model is a statistical model where the model parameters are random variables. The model is, 𝑦𝑖= 𝛽0 + 𝛽𝑗 𝑥𝑖+ 𝜀𝑖 Here, 𝑦𝑖= dependent variable 𝛼𝑖= intercept (random) 𝛽𝑗=slope coefficient (random) 𝑥𝑖= independent variable 𝜀𝑖= random error 𝛼𝑖~ iid N(0, 𝜎 𝛼 2) 𝛽𝑗 ~ iid N(0, 𝜎𝛽 2)
- 10. MIXED EFFECTS MODEL A model in which some factors fixed effects while others have random effects is mixed effect model. The model is, 𝑦𝑖=𝛼 𝑖 + 𝛽𝑗 𝑥𝑖+ 𝜀𝑖 Here, 𝑦𝑖= dependent variable 𝛼𝑖= intercept 𝛽𝑗=slope coefficient 𝑥𝑖= independent variable 𝜀𝑖= random error Where 𝛽𝑗 are random variable while at least one 𝛽𝑗 is a constant. 10
- 11. ANALYSIS OF VARIANCE(ANOVA) According to R.A Fisher “ANOVA is the separation of variance ascribable to one group of causes from the variance ascribable to other group. Analysis of variance is classified by two types ,they are 1.One-way classification and 2.Two-way classification
- 12. CONTINUE… One-way classification Arrangement the observation in various classes on the basis of a single factor. Two-way classification Arrangement of the observation according to two factors.
- 13. LAYOUT OF ONE-WAY CLASSIFICATION 1 2 … … … … … … k 𝑦11 𝑦21 … … … … … … … … 𝑦 𝑘1 𝑦12 𝑦22 … … … … … … … … 𝑦 𝑘2 . . . 𝑦1𝑟1 𝑦2𝑟2 … … … … … … … … 𝑦 𝑘𝑟𝑘 Total: 𝑇1 𝑇2 … … … … … … … … 𝑇𝑘 Mean:𝑦1 𝑦2 … …. …. … … … … … 𝑦 𝑘
- 14. LAYOUT OF TWO WAY CLASSIFICATION 𝑩 𝟏 𝑩 𝟐 … … … 𝑩 𝒌 Total mean 𝐴1 𝑦11 𝑦12 … … … 𝑦1𝑘 𝑅1 𝐴2 𝑦21 𝑦22 … … … 𝑦2𝑘 𝑅2 𝑦2 . . . 𝐴 𝑟 𝑦 𝑟1 𝑦 𝑟2 … … … 𝑦 𝑟𝑘 𝑅 𝑟 𝑦 𝑟 Total : 𝐶1 𝐶2 … … … 𝐶 𝑘 Mean: 𝑦.1 𝑦.2 … … … 𝑦.𝑘
- 15. STANDARD DESIGNS 15 Commonly used experimental designs are : I. Completely Randomized Design (CRD) II. Randomized (Complete) Block Design (RBD) III. Latin Square Design (LSD)
- 16. COMPLETELY RANDOMIZED DESIGN(CRD) A CRD is a design in which the selected treatments are allocated or distributed to the experimental units completely at random. Fixed effect model for CRD: 𝑦𝑖𝑗= 𝜇 + 𝜏𝑖+ 𝜀𝑖𝑗 𝑖 = 1,2,……,k 𝑗 =1,2,…,𝑟𝑖 Where, 𝑦𝑖𝑗= observation in the 𝑗 𝑡ℎ unit under 𝑖 𝑡ℎ treatment 𝜇 = general mean 𝜏𝑖= fixed effect of 𝑖 𝑡ℎ 𝜀𝑖𝑗= random error component in the (i.𝑗)th unit Total SS = Treatment SS+ Error SS16
- 17. ANOVA TABLE FOR CRD Source of Variation DF SS MS 𝒇 𝒄 Treatment s Error K-1 n-k 𝑖=1 𝑘 𝑟𝑖(𝑦𝑖-y̅)² 𝑖=1 𝑘 𝑗=1 𝑟 𝑖 (𝑦𝑖𝑗-𝑦𝑖 )² 𝑖=1 𝑘 𝑟 𝑖( 𝑦̅ 𝑖−y̅) ² 𝑘−1 = 𝑠𝑡 2 𝑖=1 𝑘 𝑗=1 𝑟 𝑖 (𝑦𝑖𝑗−𝑦𝑖 )² 𝑛 − 𝑘 = 𝑠 𝑒 2 𝑠 𝑡 2 𝑠 𝑒 2 Total n-1
- 18. RANDOMISED BLOCK DESIGN (RBD) A RBD is a design in which the subjects are arranged in several blocks which are internally homogeneous and externally heterogeneous. Fixed effect model for RBD: 𝑦𝑖𝑗= 𝜇 + 𝛽𝑖 +𝜏𝑖+ 𝜀𝑖𝑗 𝑖 = 1,2,……,r 𝑗 =1,2,…,𝑘 Here, 𝑦𝑖𝑗= observation in the 𝑗 𝑡ℎ treatment under 𝑖 𝑡ℎ block 𝜇 = general mean 𝛽𝑗= fixed effect of the 𝑖 𝑡ℎ block 𝜏𝑖= fixed effect of the 𝑗 𝑡ℎ treatment 𝜀𝑖𝑗= random error component in the (i.𝑗)th unit
- 19. ANOVA TABLE FOR RBD Source of variatio n DF SS MS F Blocks Treatme nt Error (r-1) (k-1) (r-1)(k- 1) k 𝑖=1 𝑟 (𝑦𝑖.−y̅)² =𝑠 𝑏 2 r 𝑗=1 𝑘 (𝑦.𝑗 −y̅)² =𝑠𝑡 2 𝑖=1 𝑟 𝑗=1 𝑘 (𝑦𝑖𝑗 − 𝑦𝑖. -𝑦.𝑗 + y)²=𝑠 𝑒 2 𝑆 𝑏 2 𝑟−1 = 𝑠 𝑏 2 𝑠 𝑡 2 𝑘−1 = 𝑠𝑡 2 𝑠 𝑒 2 (𝑟−1)(𝑘−1) =𝑠𝑡 2 𝑠𝑡 2 𝑠 𝑒 2 Total rk-1
- 20. LATIN SQUARE DESIGN (LSD) LSD IS A DESIGN IN WHICH EXPERIMENTAL UNITS ARE ARRANGED IN COMPLETE BLOCKS IN ROWS AND COLUMNS. Latin square design is a design in which experimental units are arranged in complete blocks in rows and columns. Linear model for LSD: 𝑦𝑖𝑗𝑘= 𝜇 + 𝛼𝑖 + 𝛽𝑗 + 𝜏 𝑘+ 𝜀𝑖𝑗𝑘 𝑖, 𝑗, 𝑘 = 1,2,……,r Where, 𝑦𝑖𝑗= observation in the 𝑖 𝑡ℎ row, 𝑗 𝑡ℎ column, 𝑘 𝑡ℎ treatment 𝜇 = general mean 𝛼𝑖= fixed effect of 𝑖 𝑡ℎ row 𝛽𝑗= fixed effect of the 𝑗 𝑡ℎ column 𝜏 𝑘= fixed effect of the 𝑘 𝑡ℎ treatment 𝜀𝑖𝑗= random error term
- 21. ANOVA TABLE FOR LSD Source of variation DF SS MS F Rows Columns Treatments Error (r-1) (r-1) (r-1) (r-1)(r- 2) R C T E 𝑅 𝑟−1 = 𝑠𝑟 2 𝑅 𝑟−1 =𝑠𝑐 2 𝑇 𝑟−1 = 𝑠𝑡 2 𝐸 (𝑟−1)(𝑟−2) =𝑠𝑒 2 𝑠𝑡 2 𝑠 𝑒 2 Total 𝑟2 − 1
- 22. Conclusion: CRD is the basic and simplest design. RBD is more efficient than CRD and thus provides more accurate and precise results than CRD. LSD is more efficient than RBD and CRD.
- 23. THANK YOU TO ALL For More Email : www.hasnat.rbm@gmail.com LinkedIn : hasnatrbm