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ANOVA. Analysis of Variance. Introduction. We’re making paper bags and want to see what we can do to improve the tensile strength of the bags. Conduct an experiment to determine if the level of hardwood in the bags affects tensile strength. Investigate 4 levels of hardwood.

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Anova

ANOVA

Analysis of Variance


Introduction

Introduction

  • We’re making paper bags and want to see what we can do to improve the tensile strength of the bags.

  • Conduct an experiment to determine if the level of hardwood in the bags affects tensile strength.

  • Investigate 4 levels of hardwood.

    • Select 6 samples at each level

    • 5%10%15%20%


Introduction1

Introduction

Source: Dr. G. Baker, Statistics Dept., University of South Carolina


Introduction2

Introduction

  • Study of pain threshold / color of a person’s hair

  • Small sample size

  • Means differ

    • Dark brunette: 37.4

    • Light brunette: 42.5

    • Dark blonde: 51.2

    • Light blonde: 59.2

  • Are these different or is this caused by random variation?


Blonde v brunette

Blonde v. Brunette

Source: Family Weekly, Gainesville, Sun, Gainesville, Florida, February 5, 1978


Definitions

Definitions

  • Experiment

    • A study designed to investigate the effect one variable has on the value of another variable

  • Dependent variable

    • The variable of interest that we are measuring.

    • Sometimes called the response variable.

    • Dependent variable is quantitative.

  • Independent variable

    • The observed (or controlled) variable we use to detect its effect on the independent variable.

    • Independent variable can be quantitative or qualitative.


Definitions1

Definitions

  • Independent variable

    • Sometimes referred to as a factor or explanatory variable.

      • One or more factors may be involved in a study.

    • An experiment may involve different factor levels.

    • Each specific level of a factor (or the intersection of multiple factors) is referred to as a treatment.

    • When there’s only one factor level in an experiment, the terms factor levels and treatments are used interchangeably.


Definitions2

Definitions


Experiments

Experiments

  • Observational study

    • No control over the factors

  • Completely randomized design

    • Select random independent samples for each treatment OR

    • Randomly assign treatments to selected people

  • Complete randomization is not possible in observational studies


Hypothesis testing

Hypothesis Testing

  • H0: µ1 = µ2 = ... = µt *

    • µt * = the total number of treatments in the experiment

  • Ha: At least one of the treatment group means differs from the rest. OR At least two of the population means are not equal.

  • We will be comparing the variation between groups and the variation within groups

    • Between>within indicates a difference in the means


Anova terminology

ANOVA Terminology

  • Labels for individual responses

    • = the individual response for the jth observation receiving the ith treatment

    • = the average of all the observations receiving a treatment

    • = the average for all the observations in the experiment


Definitions3

Definitions

Variation within groups…

Variation between groups…

d

a

b

c


Variation within v between

Variation: Within v. Between

a

b

c


Variation within v between1

Variation: Within v. Between

a

c

b


Paper bag example

Paper Bag Example


Paper bag hypothesis

Paper Bag Hypothesis

  • H0: µ.05= µ.10= µ.15 = µ.20

  • Ha: At least one mean differs from the other 3

    OR At least 2 of the means are not equal


Anova

Data


Anova table

ANOVA Table


Anova table1

ANOVA Table


Anova table from excel

ANOVA Table from Excel


Anova table from minitab

ANOVA Table from Minitab

  • One-way ANOVA: C16 versus C15

  • Source DF SS MS F P

  • C15 3 382.79 127.60 19.61 0.000

  • Error 20 130.17 6.51

  • Total 23 512.96

  • S = 2.551 R-Sq = 74.62% R-Sq(adj) = 70.82%

  • Individual 95% CIs For Mean Based on Pooled StDev

  • Level N Mean StDev +---------+---------+---------+---------

  • 5.00% 6 10.000 2.828 (----*----)

  • 10.00% 6 15.667 2.805 (----*-----)

  • 15.00% 6 17.000 1.789 (----*-----)

  • 20.00% 6 21.167 2.639 (-----*----)

  • +---------+---------+---------+---------

  • 8.0 12.0 16.0 20.0

  • Pooled StDev = 2.551


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