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

Chapter 12 - Math3200 Intermediate Probability and...

This preview shows page 1 - 11 out of 33 pages.

Math3200 Intermediate Probability and Statistics Prof. Nan Lin Department of Mathematics Washington University
Image of page 1

Subscribe to view the full document.

Outline Completely randomized design (One-way ANOVA) Design Model Multiple comparison Randomized completely block design (two-way ANOVA) Nan Lin, Washington University 2
Image of page 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
Image of page 3

Subscribe to view the full document.

Graphical display Side-by-side boxplot Nan Lin, Washington University 4
Image of page 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
Image of page 5

Subscribe to view the full document.

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
Image of page 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
Image of page 7

Subscribe to view the full document.

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
Image of page 8
One-way ANOVA table Nan Lin, Washington University 9 Reject 𝐻 0 : 𝜏 1 = 𝜏 2 = ⋯ = 𝜏 ? = 0 if ? 0 > ? ?−1,?−?,𝛼
Image of page 9

Subscribe to view the full document.

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
Image of page 10
Image of page 11

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes