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Inference after ANOVA, Multiple Comparisons 3/21/12 • Inference after ANOVA • The problem of multiple comparisons • Bonferroni’s Correction Section 8.2 Professor Kari Lock Morgan Duke University
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Page 1: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

Inference after ANOVA, Multiple Comparisons

3/21/12

• Inference after ANOVA• The problem of multiple comparisons• Bonferroni’s Correction

Section 8.2 Professor Kari Lock MorganDuke University

Page 2: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

• Clicker: overwhelmingly positive

• Textbook: positive

• Homework: surprisingly positive

• Lecture: mostly positive, some complaints• most common complaint: too fast• not always open to questions• full lecture slides not posted in advance

• Lab: less positive, varied complaints• TA too busy, TA not helpful• can do at home• too long

Midterm Evaluation

Page 3: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

Midterm Evaluation

Page 4: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

• Project 1 (due TOMORROW, 5pm)

• Homework 7 (due Monday, 3/26)NO LATE HOMEWORK ACCEPTED!Turn in by Friday, 3/23, 5pm to get it graded

before Exam 2.

• Start preparing for Exam 2 (next Wednesday and Thursday)

To Do

Page 5: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

• Conclusion: What have you learned? Make sure to answer the research question posed in your introduction.

• This is a paper, there should be text. Do not just give R output, formulas, and numbers.

Project 1 Comments

Page 6: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

• Exam 2:• In-class portion: Wednesday, 3/28• Lab portion: Thursday, 3/29

• In-class portion: (75%)• Open only to a calculator and two double sided

pages of notes prepared by you

• Lab portion: (25%)• Open to everything except communication of any

form with other humans

Exam 2

Page 7: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

• The emphasis will be on material we have learned since the first exam, although you are still responsible for everything we learned prior to the first exam

Exam 2

Page 8: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

• On the course website, under documents:• Last semester’s in-class exam, and solutions• Last semester’s labs exam, and solutions• Full solutions to all essential synthesis

problems from Unit 3• Full solutions to all review problems from

Unit 3• Full solutions to all odd problems from

Chapters 7 and 8

• Doing problems is the key to success!!!

Practice

Page 9: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

• Work lots of practice problems! • Take last year’s exams under realistic

conditions (time yourself, do it all before looking at the solutions, etc.)

• Prepare a good cheat sheet and use it when working problems

• Read the corresponding sections in the book if there are concepts you are still confused about

Keys to In-Class Exam Success

Page 10: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

• Primarily, make sure you know how to summarize, visualize, create an interval, and conduct a test for any one variable or relationship between two variables

• For practice, try doing both intervals and tests for any one or two variables in our class survey

• Beyond that, make sure you are comfortable with the content from the labs

• Open-book does NOT mean you don’t have to study. You will not have time to look up every command you need during the exam.

Keys to Lab Exam Success

Page 11: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

You have LOTS of opportunities for help!

• Today, 3 – 5pm (Prof Morgan)• Today, 6 – 8 pm (Michael)• Friday, 1:30 – 3 pm (Prof Morgan)• Sunday, 5 – 7 pm (Jessica)• Sunday, 7 – 9 pm (Michael)• Monday, 3 – 4 pm (Prof Morgan)• Monday, 4 – 6 pm (Christine)• Tuesday, 3 – 6 pm (Prof Morgan)• Tuesday, 6 – 8 pm (Yue)

Office Hours before Exam

Page 12: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

Cuckoo Birds•Cuckoo birds lay their eggs in the nests of other birds (typically small birds).

•When the cuckoo baby hatches, it kicks out all the original eggs/babies

•If the cuckoo is lucky, the mother will raise the cuckoo as if it were her own

•Do cuckoo birds found in nests of different species differ?

http://opinionator.blogs.nytimes.com/2010/06/01/cuckoo-cuckoo/

Page 13: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

Length of Cuckoo Eggs

Page 14: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

Length of Cuckoo EggsBird Sample

MeanSample

SDSample

Size

Pied Wagtail 22.90 1.07 15

Pipit 22.50 0.97 60

Robin 22.58 0.68 16

Sparrow 23.12 1.07 14

Wren 21.13 0.74 15

Overall 22.46 1.07 120

35.90

101.29

137.19

SSG

SSE

SST

Is there a significant difference between the groups?

(a) Yes(b) No

Page 15: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

Source

Groups

Error

Total

df

4

115

119

Sum ofSquares35.90

101.29

137.19

MeanSquare

8.97

0.88

FStatistic10.19

p-value

4.3 × 10-7

ANOVA Table

We have very strong evidence that average length of cuckoo eggs differs for nests of different species

Equal variability Normal(ish) data

Page 16: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

Cuckoo Birds•How long are cuckoo bird eggs found in robins’ nests?

•Is there a significant difference between the average length of eggs found in robins’ nests and the average length of eggs found in sparrows’ nests?

•While we could proceed with formulas from Chapter 6 or simulation methods from Chapters 3 and 4, there are special ways of doing inference after ANOVA…

Page 17: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

Inferences after ANOVA•If the ANOVA assumption of equal variability across groups is satisfied, we can use the data from all groups to estimate variability:

1. Estimate any within-group standard deviation with MSE

2. Use the error degrees of freedom, , for -distributionsn k t

1

11

Single Mean:

SM

nnE

SE

2 21 2

1 2 1 2

Difference in Two Means:

1 1

n nSE MSE

n n

Page 18: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

Cuckoo Birds•How long are cuckoo bird eggs found in robins’ nests? Give a 90% confidence interval.

•Is there a significant difference between the average length of eggs found in robins’ nests and the average length of eggs found in sparrows’ nests?

(a) Yes

(b) NoBird Sample Mean Sample SD Sample Size

Pied Wagtail 22.90 1.07 15

Pipit 22.50 0.97 60

Robin 22.58 0.68 16

Sparrow 23.12 1.07 14

Wren 21.13 0.74 15

Overall 22.46 1.07 120

(22.19, 22.97)

Page 19: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

Cuckoo Birds•How long are cuckoo bird eggs found in robins’ nests? Give a 90% confidence interval.

1.66

22.58 0.3

0

8

.8

9

822.58

16

22.19,22.97

*statistic t SE

*R

R

MSEX t

n

t*

We are 90% confident that the average length of Cuckoo eggs found in Robins’ nests is between 22.19 and 22.97 mm.

Page 20: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

Cuckoo Birds•Is there a significant difference between the average length of eggs found in robins’ nests and the average length of eggs found in sparrows’ nests?

statistic nullt

SE

22.58 23.12

1 10.88

1

0.541.57

0.3416

3114

R S

R S

X Xt

MSEn n

This study does not provide evidence for a difference in average mean length of Cuckoo eggs between those found in Robins and Sparrows nests.

0 :

:R S

a R S

H

H

Page 21: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

Pairwise Comparisons•Pairwise comparisons test for a difference in means between each pair of groups

•Only do pairwise comparisons if the overall ANOVA is significant

•If there are lots of categories, the number of possible pairwise comparisons grows quickly

•Automate the process with RStudio

Page 22: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

Cuckoo Birds

Page 23: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

Length of Cuckoo Eggs

Page 24: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

Extrasensory Perception•Is there such a thing as extrasensory perception (ESP), or a “sixth sense”?

•Do you believe in ESP or a sixth sense?

(a) Yes (b) No (c) Not sure

Page 25: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

Extrasensory Perception

Page 26: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

Extrasensory Perception

How would you test whether American belief in ESP differs between current Duke students who take STAT 101 and all Americans in 2001?

a) z-testb) t-testc) Chi-square testd) ANOVA

Page 27: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

Extrasensory Perception

Based on the available data, how would you test whether belief in ESP differs between 1990 and 2001?

a) z-testb) t-testc) Chi-square testd) ANOVA

Page 28: Inference after ANOVA, Multiple Comparisons 3/21/12 Inference after ANOVA The problem of multiple comparisons Bonferroni’s Correction Section 8.2 Professor.

Summary

• When performing inference after ANOVA, use √MSE as an estimate for standard deviation within groups, and use n – k as the degrees of freedom for the t-distribution


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