1. Designs with One Source of Variation PhD seminar 31/01/2014

2. Introduction Randomization Model for a Completely Randomized Design Estimation of parameters One-Way Analysis of Variance Sample Sizes Contents 2

3. 3

4. Treatment: a level of a factor in a single factor experiment (or a combination in a factorial experiment). Introduction (1) Algorithms (Treatments) Dataset (Experimental Unit) 4

5. Introudction (2) Experimental Design – the rule that determines the assignment of the experimental units to treatments. – The simplest: completely randomized design One Source of Variation – The variation of the factor of interest (levels). 5

6. Randomization Design : – Consists of assigning the experimental units to the treatments in a random manner. – Reduces (and with a bit of luck, removes) the selection bias and/or accidental bias and tends to produce comparable groups [Suresh KP 2011] – It is best used when relatively homogeneous experimental units are available – This design will normally be analysed using a one- way analysis of variance [Michael FW Festing] Randomized Design 6

7. Model for a Completely Randomized Design Cause X “  ” Effect Y Observed Y = X + ε Controlled variable Independent Variable Factor Treatment Independent Variable Factor Treatment Dependent Variable Observations Response variable Dependent Variable Observations Response variable External factors Not controlled Ignored External factors Not controlled Ignored 7

8. Model for a Completely Randomized Design Experimental Unit 1 y1y1 y2y2 yryr Observations Treatment 1 ε Experimental Unit 2 y1y1 y2y2 yryr Observations Treatment 2 ε 8

9. Linear statistical model Y it -is the observation of the t-th repetition of the i-th treatment μ i - is the mean of the i-th treatment ϵ it - experimental error Model for a Completely Randomized Design 9

10. Full model Each population defined by the treatment has his own mean Reduced model There is no difference between the means of the populations Figure from: [http://www.dpye.iimas.unam.mx/patricia/indexer/completamente_al_azar.pdf] 10

11. Model for a Completely Randomized Design Linear statistical model μ - is the general mean (common to all experimental units - homogeneous) τ i - effect of the i-th treatment 11

12. Knows as: One-way analysis of variance model – The model includes only one major source of variation (treatment) – The analysis of data involves a comparition of measures of variation. Model for a Completely Randomized Design 12

13. Estimation of parameters Figure from: [Dean, A. Voss, D.,Design and Analysis of Experiments] Least Squares (balanced model) Maximum-Likelihood Estimation (MLE) (unbalanced model) 13

14. In an experiment involving several treatments the treatments differ at all in terms of their effects on the response variable? H 0 :{τ 1 = τ 2 =…= τ v } H A :{at least two of the τ i ’s differ} One-Way Analysis of Variance 14

15. Compare the sum of squares for errors (ssE) between the full and reduced model. H 0 is false if ssE(fullModel) << ssE(reducedModel) H 0 is true if ssE(fullModel) ≈ ssE(reducedModel) ANOVA One-Way Analysis of Variance 15

16. Before an experiment can be run, it is necessary to determine the number of observations that should be taken on each treatment. Consider time and money Two methods: – Specifying the desired length of confidence intervals – Specifying the power required of the analysis of variance. Sample sizes 16

17. Sample Sizes using power of a test – The power of a test at Δ, denoted π(Δ), is the probability of rejecting H 0 when the effects of two treatments differ by Δ. Sample sizes 17

18. Dean, A.M., Voss, D., Design and Analysis of Experiments, Spring-Verlag, 1999 http://www.3rs-reduction.co.uk/html/about.html http://www.ugr.es/~bioestad/guiaspss/practica7/Arc hivosAdjuntos/EfectosFijos.pdf http://www.dpye.iimas.unam.mx/patricia/indexer/co mpletamente_al_azar.pdf http://www.ugr.es/~bioestad/guiaspss/practica7/Arc hivosAdjuntos/EfectosFijos.pdf 18