Chapter 12 - Math3200 Intermediate Probability and Statistics Prof Nan Lin Department of Mathematics Washington University Outline Completely randomized

# Chapter 12 - Math3200 Intermediate Probability and...

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Math3200 Intermediate Probability and Statistics Prof. Nan Lin Department of Mathematics Washington University

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Outline Completely randomized design (One-way ANOVA) Design Model Multiple comparison Randomized completely block design (two-way ANOVA) Nan Lin, Washington University 2
Completely randomized (CR) design ? ≥ 2 groups: E. g. treatment1 vs treatment 2 vs control (placebo) Sample ? : 𝑦 ?1 , … , 𝑦 ?? ? , ? = 1, … , ? Total sample size 𝑁 = 𝑛 ? ? ?=1 One-way layout Nan Lin, Washington University 3

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Graphical display Side-by-side boxplot Nan Lin, Washington University 4
Model 𝑦 ?? = ? ? + 𝜖 ?? , ? = 1, … , ?; ? = 1, … 𝑛 ? , 𝜖 ?? ~𝑁 0, 𝜎 2 iid 𝑦 ?? ∼ 𝑁(? ? , 𝜎 2 ) independently One may also write the model as 𝑦 ?? = ? + 𝜏 ? + 𝜖 ?? by letting ? ? = ? + 𝜏 ? . 𝜏 ? is then called the treatment effect . A constraint is necessary, e.g. 𝜏 ? = 0 ? ?=1 . Special case if ? = 2 : Two-sample t-test with equal variance Nan Lin, Washington University 5

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Parameter estimation Least square estimates ? = 𝑦 the overall average 𝜏 ? = 𝑦 ? − 𝑦 ? ? = 𝑦 ? the group average 𝜎 2 = ? 2 = ? ?? −? ? 2 ? ? ?=1 𝑎 ?=1 ?−? = ? 1 −1 𝑠 1 2 +⋯+ ? 𝑎 −1 𝑠 𝑎 2 ? 1 −1 +⋯+(? 𝑎 −1) A pooled estimate 𝑑𝑓 = ? = 𝑁 − ? Nan Lin, Washington University 6
Distribution results For every ? = 1, … , ? , 𝑦 ? − ? ? ?/ 𝑛 ? ∼ ? ?−? 100 1 − ? % confidence interval of ? ? is 𝑦 ? − ? ?−?, 𝛼 2 ? 𝑛 ? ≤ ? ? ≤ 𝑦 ? + ? ?−?, 𝛼 2 ? 𝑛 ? Nan Lin, Washington University 7

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One-way ANOVA Hypotheses 𝐻 0 : ? 1 = ? 2 = ⋯ = ? ? vs 𝐻 1 : otherwise This is equivalent to 𝐻 0 : 𝜏 1 = 𝜏 2 = ⋯ = 𝜏 ? = 0 ??? = ??? + ??? 𝑦 ?? 𝑦 2 ? ? ?=1 ? ?=1 = 𝑛 ? 𝑦 ? − 𝑦 2 ? ? + 𝑦 ?? − 𝑦 ? 2 ? ? ?=1 ? ?=1 Degrees of freedom: 𝑁 − 1 = (? − 1) + (𝑁 − ?) Nan Lin, Washington University 8
One-way ANOVA table Nan Lin, Washington University 9 Reject 𝐻 0 : 𝜏 1 = 𝜏 2 = ⋯ = 𝜏 ? = 0 if ? 0 > ? ?−1,?−?,𝛼

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Diagnostics Residual plots Plot residuals against fitted values may suggest violation of equal variance assumption qq-plot of residuals may suggest non-normality Transformation may help to correct these problems Nan Lin, Washington University 10

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