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

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

Project 1 (due TOMORROW, 5pm) Project 1 Homework 7 (due Monday, 3/26) Homework 7 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

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

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

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

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

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

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

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

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?

Length of Cuckoo Eggs

BirdSample Mean Sample SD Sample Size Pied Wagtail Pipit Robin Sparrow Wren Overall Is there a significant difference between the groups? (a) Yes (b) No

Source Groups Error Total df Sum of Squares Mean Square F Statistic p-value 4.3 × ANOVA Table We have very strong evidence that average length of cuckoo eggs differs for nests of different species Equal variability Normal(ish) data

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…

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:

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)No BirdSample MeanSample SDSample Size Pied Wagtail Pipit Robin Sparrow Wren Overall (22.19, 22.97)

Cuckoo Birds How long are cuckoo bird eggs found in robins’ nests? Give a 90% confidence interval. t*t* We are 90% confident that the average length of Cuckoo eggs found in Robins’ nests is between and mm.

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

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

Cuckoo Birds

Length of Cuckoo Eggs

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

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-test b)t-test c)Chi-square test d)ANOVA

Extrasensory Perception Based on the available data, how would you test whether belief in ESP differs between 1990 and 2001? a)z-test b)t-test c)Chi-square test d)ANOVA

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