STAT 135: Anova

111 Pages·2015·1.06 MB·English

by  Joan Bruna

STAT 135: Anova
Joan Bruna
DepartmentofStatistics
UC,Berkeley
April 10, 2015
JoanBruna STAT135:Anova
The color? The size? The brand? The material?
Relevant groups should help me explain the variance.
Motivation
Say you want to buy a pair of shoes. You go to a store and
observe a large sample of shoes with varying price. Where does the
variance in the price come from?
JoanBruna STAT135:Anova
Relevant groups should help me explain the variance.
Motivation
Say you want to buy a pair of shoes. You go to a store and
observe a large sample of shoes with varying price. Where does the
variance in the price come from?
The color? The size? The brand? The material?
JoanBruna STAT135:Anova
Motivation
Say you want to buy a pair of shoes. You go to a store and
observe a large sample of shoes with varying price. Where does the
variance in the price come from?
The color? The size? The brand? The material?
Relevant groups should help me explain the variance.
JoanBruna STAT135:Anova
We will introduce the statistical framework for such problems.
Motivation
We want to compare the effect of several treatments or several
groups. Also, analyze other factors simultaneously.
Examples:
Movie Recommendations.
Testing Several Drugs.
Include other factors, such as Age, Sex, etc.
JoanBruna STAT135:Anova
Motivation
We want to compare the effect of several treatments or several
groups. Also, analyze other factors simultaneously.
Examples:
Movie Recommendations.
Testing Several Drugs.
Include other factors, such as Age, Sex, etc.
We will introduce the statistical framework for such problems.
JoanBruna STAT135:Anova
For each setting, two different approaches:
Parametric: Methods based on the Normal distribution.
Non-parametric: Mostly based on rank statistics.
Different Settings
One-way Layout: Independent Measurements under several
Treatments. This generalizes the methods from previous
Chapter.
Two-way Layout: Analyze two factors simultaneously (ex:
several Treatments + several age ranges).
JoanBruna STAT135:Anova
Different Settings
One-way Layout: Independent Measurements under several
Treatments. This generalizes the methods from previous
Chapter.
Two-way Layout: Analyze two factors simultaneously (ex:
several Treatments + several age ranges).
For each setting, two different approaches:
Parametric: Methods based on the Normal distribution.
Non-parametric: Mostly based on rank statistics.
JoanBruna STAT135:Anova
One-Way Layout
Suppose we have I groups G ,...,G , with measurements
1 I
G G G
1 2 I
Y Y Y
1,1 2,1 I,1
Y Y Y
1,2 2,2 I,2
... ... ...
Y Y Y
1,J 2,J I,J
We assume for now the same # of measurements in each group.
Question: Are the means of each group the same?
JoanBruna STAT135:Anova
Main assumption: Errors (cid:15) are iid N(0,σ2).
i,j
I
(cid:88)
We define µ such that α = 0 .
i
i=1
Testing equality of means: Null hypothesis is
H : ∀i , α = 0 .
0 i
One-Way Layout Model
Model observations as
Y = µ+α +(cid:15) , where
i,j i i,j
µ: global mean across all groups.
α: differential effect of i-th group.
i
(cid:15) : random error of each observation.
i,j
JoanBruna STAT135:Anova