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

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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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