Using Randomization Methods to Build Conceptual Understanding in Statistical Inference: Day 1 Lock, Lock, Lock, Lock, and Lock Minicourse – Joint Mathematics Meetings Boston, MA January 2012 WiFi: marriotconference, password: 1134ams
Introductions: Name Institution
Schedule: Day 1 Wednesday, 1/4, 4:45 – 6:45 pm 1. Introductions and Overview 2. Bootstrap Confidence Intervals • What is a bootstrap distribution? • How do we use bootstrap distributions to build understanding of confidence intervals? • How do we assess student understanding when using this approach? 3. Getting Started on Randomization Tests • What is a randomization distribution? • How do we use randomization distributions to build understanding of p-values? • How do these methods fit with traditional methods? 4. Minute Papers
Schedule: Day 2 Friday, 1/6, 3:30 – 5:30 pm 5. More on Randomization Tests • How do we generate randomization distributions for various statistical tests? • How do we assess student understanding when using this approach? 6. Connecting Intervals and Tests 7. Technology Options • Short software demonstrations (Minitab, Fathom, R, Excel, ...) – pick two! 8. Wrap-up • How has this worked in the classroom? • Participant comments and questions 9. Evaluations
Why use Randomization Methods?
These methods are great for teaching statistics… (the methods tie directly to the key ideas of statistical inference so help build conceptual understanding)
And these methods are becoming increasingly important for doing statistics. (Attend the Gibbs Lecture tonight!)
It is the way of the past… "Actually, the statistician does not carry out this very simple and very tedious process [the randomization test], but his conclusions have no justification beyond the fact that they agree with those which could have been arrived at by this elementary method." -- Sir R. A. Fisher, 1936
… and the way of the future “... the consensus curriculum is still an unwitting prisoner of history. What we teach is largely the technical machinery of numerical approximations based on the normal distribution and its many subsidiary cogs. This machinery was once necessary, because the conceptually simpler alternative based on permutations was computationally beyond our reach. Before computers statisticians had no choice. These days we have no excuse. Randomization-based inference makes a direct connection between data production and the logic of inference that deserves to be at the core of every introductory course.” -- Professor George Cobb, 2007
Question Can you use your clicker? A. Yes B. No C. Not sure D. I don’t have a clicker
Question How frequently do you teach Intro Stat? A. Regularly B. Occasionally C. Rarely/Never
Question How familiar are you with simulation methods such as bootstrap confidence intervals and randomization tests? A. Very B. Somewhat C. A Little / Not at all
Question Have you used randomization methods in any statistics class that you teach? A. Yes, as a significant part of the course B. Yes, as a minor part of the course C. No
Question Have you used randomization methods in Intro Stat? A. Yes, as a significant part of the course B. Yes, as a minor part of the course C. No
Intro Stat – Revise the Topics • • •• • • • •
Descriptive Statistics – one and two samples Normal distributions Bootstrap confidence intervals Data production (samples/experiments) Randomization-based hypothesis tests Sampling distributions (mean/proportion) Normal/sampling distributions Confidence intervals (means/proportions)
• Hypothesis tests (means/proportions) • ANOVA for several means, Inference for regression, Chi-square tests
It’s close to 5 pm; We need a snack!
What proportion of Reese’s Pieces are Orange? Find the proportion that are orange for your box.
Bootstrap Distributions Or: How do we get a sense of a sampling distribution when we only have ONE sample?
Suppose we have a random sample of 6 people:
Original Sample
Create a “sampling distribution” using this as our simulated population
Bootstrap Sample: Sample with replacement from the original sample, using the same sample size.
Original Sample
Bootstrap Sample
Simulated Reese’s Population
Sample from this “population”
Create a bootstrap sample by sampling with replacement from the original sample. Compute the relevant statistic for the bootstrap sample. Do this many times!! Gather the bootstrap statistics all together to form a bootstrap distribution.
Original Sample Sample Statistic
Bootstrap Sample
Bootstrap Statistic
Bootstrap Sample
Bootstrap Statistic
. . . Bootstrap Sample
. . . Bootstrap Statistic
Bootstrap Distribution
We need technology! Introducing
StatKey.
Example: Atlanta Commutes What’s the mean commute time for workers in metropolitan Atlanta? Data: The American Housing Survey (AHS) collected data from Atlanta in 2004.
Sample of n=500 Atlanta Commutes Dot Plot
CommuteAtlanta
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Time
Where might the “true” μ be?
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StatKey can be found at
www.lock5stat.com
How can we get a confidence interval from a bootstrap distribution?
Method #1: Use the standard deviation of the bootstrap statistics as a “yardstick”
Using the Bootstrap Distribution to Get a Confidence Interval – Version #1 The standard deviation of the bootstrap statistics estimates the standard error of the sample statistic.
Quick interval estimate :
For the mean Atlanta commute time:
Using the Bootstrap Distribution to Get a Confidence Interval – Version #2 95% CI=(27.35,30.96) Chop 2.5% in each tail
Keep 95% in middle
Chop 2.5% in each tail
For a 95% CI, find the 2.5%-tile and 97.5%-tile in the bootstrap distribution
90% CI for Mean Atlanta Commute 90% CI=(27.64,30.65) Chop 5% in each tail
Keep 90% in middle
Chop 5% in each tail
For a 90% CI, find the 5%-tile and 95%-tile in the bootstrap distribution
Bootstrap Confidence Intervals Version 1 (Statistic ± 2 SE): Great preparation for moving to traditional methods Version 2 (Percentiles): Great at building understanding of confidence intervals
Playing with StatKey! See the purple pages in the folder.
We want to collect some data from you. What should we ask you for our one quantitative question and our one categorical question?
What quantitative data should we collect from you? A. What was the class size of the Intro Stat course you taught most recently?
What categorical data should we collect from you? A. Did you fly or drive to these meetings? B. Have you attended any previous JMM meetings?
How do we assess student understanding of these methods (even on in-class exams without computers)? See the green pages in the folder.
Paul the Octopus
http://www.youtube.com/watch?v=3ESGpRUMj9E http://www.cnn.com/2010/SPORT/football/07/08/germany.octopus.explainer/index.html
Paul the Octopus • Paul the Octopus predicted 8 World Cup games, and predicted them all correctly • Is this evidence that Paul actually has psychic powers? • How unusual would this be if he were just randomly guessing (with a 50% chance of guessing correctly)? • How could we figure this out?
Simulate! • Each coin flip = a guess between two teams • Heads = correct, Tails = incorrect • Flip a coin 8 times and count the number of heads. Remember this number! Did you get all 8 heads? (a) Yes (b) No
Hypotheses Let p denote the proportion of games that Paul guesses correctly (of all games he may have predicted) H0 : p = 1/2 Ha : p > 1/2
Randomization Distribution • A randomization distribution is the
distribution of sample statistics we would observe, just by random chance, if the null hypothesis were true • A randomization distribution is created by simulating many samples, assuming H0 is true, and calculating the sample statistic each time
Randomization Distribution • Let’s create a randomization distribution for Paul the Octopus! • On a piece of paper, set up an axis for a dotplot, going from 0 to 8 • Create a randomization distribution using each other’s simulated statistics • For more simulations, we use StatKey
p-value • The p-value is the probability of getting a statistic as extreme (or more extreme) as that observed, just by random chance, if the null hypothesis is true • This can be calculated directly from the randomization distribution!
StatKey
( 2) 1
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Randomization Test • Create a randomization distribution by simulating assuming the null hypothesis is true • The p-value is the proportion of simulated statistics as extreme as the original sample statistic
Coming Attractions - Friday • How do we create randomization distributions for other parameters? • How do we assess student understanding? • Connecting intervals and tests • Technology for using simulation methods • Experiences in the classroom
Using Randomization Methods to Build Conceptual Understanding of Statistical Inference: Day 2 Lock, Lock, Lock, Lock, and Lock Minicourse- Joint Mathematics Meetings Boston, MA January 2012
Cocaine Addiction • In a randomized experiment on treating cocaine addiction, 48 people were randomly assigned to take either Desipramine (a new drug), or Lithium (an existing drug) • The outcome variable is whether or not a patient relapsed • Is Desipramine significantly better than Lithium at treating cocaine addiction?
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1. Randomly assign units to treatment groups
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2. Conduct experiment 3. Observe relapse counts in each group R = Relapse N = No Relapse
1. Randomly assign units to treatment groups
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10 relapse, 14 no relapse
pˆ new − pˆ old 10 18 = − 24 24 = −.333
Lithium R
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18 relapse, 6 no relapse
Randomization Test • Assume the null hypothesis is true • Simulate new randomizations • For each, calculate the statistic of interest • Find the proportion of these simulated statistics that are as extreme as your observed statistic
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10 relapse, 14 no relapse
18 relapse, 6 no relapse
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Simulate another randomization Desipramine
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16 relapse, 8 no relapse
pˆ N − pˆ O 16 12 = − 24 24 = 0.167
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12 relapse, 12 no relapse
Simulate another randomization Desipramine
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17 relapse, 7 no relapse
pˆ N − pˆ O 17 11 = − 24 24 = 0.250
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11 relapse, 13 no relapse
Simulate! • Combine everyone into one group, and
rerandomize them into the two groups • Compute your difference in proportions • Create the randomization distribution • How extreme is the observed statistic of -0.33? • Use StatKey for more simulations
The observed difference in proportions was -0.33. How unlikely is this if there is no difference in the drugs? To begin to get a sense of this: How many of you had a randomly simulated difference in proportions this extreme? Was your simulated difference in proportions more extreme than the sample statistic (that is, was it less than or equal to
StatKey
Proportion as extreme as observed statistic
observed statistic
The probability of getting results as extreme or more extreme than those observed if the null hypothesis is true, is about .02.
p-value
Cocaine Addiction You want to know what would happen • Why did you re-deal your cards? • by random chance (the random allocation to treatment groups) • Why did you leave the outcomes (relapse or no relapse) unchanged on each card? • if the null hypothesis is true (there is no difference between the drugs)
How can we do a randomization test for a mean?
Example: Mean Body Temperature Is the average body temperature really 98.6oF?
H0:μ=98.6 Ha:μ≠98.6 Data: A random sample of n=50 body temperatures. Dot Plot
BodyTemp50
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Data from Allen Shoemaker, 1996 JSE data set article
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Key idea: Generate samples that are (a) consistent with the null hypothesis (b) based on the sample data.
How to simulate samples of body temperatures to be consistent with H0: μ=98.6?
Randomization Samples How to simulate samples of body temperatures to be consistent with H0: μ=98.6?
Let’s try it on StatKey.
How can we do a randomization test for a correlation?
Is the number of penalties given to an NFL team positively correlated with the “malevolence” of the team’s uniforms?
Ex: NFL uniform “malevolence” vs. Penalty yards
r = 0.430 n = 28 Is there evidence that the population correlation is positive?
Key idea: Generate samples that are (a) consistent with the null hypothesis (b) based on the sample data.
H0 : ρ = 0 r = 0.43, n = 28
How can we use the sample data, but ensure that the correlation is zero?
Randomize one of the variables! Let’s look at StatKey.
Playing with StatKey! See the orange pages in the folder.
Choosing a Randomization Method Example: Word recall A=Sleep
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B=Caffeine
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H0: μA=μB vs. Ha: μA≠μB
Reallocate Option 1: Randomly scramble the A and B labels and assign to the 24 word recalls. Resample Option 2: Combine the 24 values, then sample (with replacement) 12 values for Group A and 12 values for Group B.
Question In Intro Stat, how critical is it for the method of randomization to reflect the way data were collected? A. Essential
How do we assess student understanding of these methods (even on in-class exams without computers)? See the blue pages in the folder.
Collecting More Data from You!
Rock-Paper-Scissors (Roshambo)
Play a game! Can we use statistics to help us win?
Rock-Paper-Scissors Which did you throw? A). Rock B). Paper C). Scissors
Rock-Paper-Scissors Are the three options thrown equally often on the first throw? In particular, is the proportion throwing Rock equal to 1/3?
What about Traditional Methods?
Intro Stat – Revised the Topics • • • • • •
Data production (samples/experiments) Descriptive Statistics – one and two samples Bootstrap confidence intervals Randomization-based hypothesis tests Normal/sampling distributions Confidence intervals (means/proportions)
• Hypothesis tests (means/proportions) • ANOVA for several means, Inference for regression, Chi-square tests
Transitioning to Traditional Inference AFTER students have seen lots of bootstrap distributions and randomization distributions… Students should be able to • Find, interpret, and understand a confidence interval • Find, interpret, and understand a p-value
Bootstrap and Randomization Distributions Dot Plot
Measures from Scrambled Collection 1
Slope :Restaurant tips
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Correlation: Malevolent uniforms
Dot Plot
Measures from Scrambled RestaurantTips
-20 0 20 slope (thousandths)
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All bell-shaped What do you Mean :Body Temperatures Diff means: Finger taps distributions! notice? Dot Plot
Measures from Sample of BodyTemp50
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98.6 Nullxbar
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0.5 phat
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Proportion : Owners/dogs
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Measures from Scrambled CaffeineTaps
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Measures from Sample of Collection 1
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Mean : Atlanta commutes
Dot Plot
Measures from Sample of CommuteAtlanta
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The students are primed and ready to learn about the normal distribution!
Transitioning to Traditional Inference • Introduce the normal distribution (and later t) • Introduce “shortcuts” for estimating SE for proportions, means, differences, slope…
Confidence Intervals
95%
-z*
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Hypothesis Tests
95%
Area is p-value
Test statistic
Yes! Students see the general pattern and not just individual formulas!
Connecting CI’s and Tests Dot Plot
Measures from Sample of BodyTemp50
Randomization body temp means when μ=98.6 98.2
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98.6 xbar
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Dot Plot
Measures from Sample of BodyTemp50
Bootstrap body temp means from the original sample 97.9
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98.3 98.4 bootxbar
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What’s the difference?
Fathom Demo: Test & CI
Sample mean is in the “rejection region”
Null mean is outside the confidence interval
Technology Sessions Choose Two! (The folder includes information on using Minitab, R, Excel, Fathom, Matlab, and SAS.)
Student Preferences Which way did you prefer to learn inference (confidence intervals and hypothesis tests)? Bootstrapping and Randomization
39 67%
Formulas and Theoretical Distributions
19 33%
Student Preferences Which way do you prefer to do inference? Bootstrapping and Randomization
42 72%
Formulas and Theoretical Distributions
16 28%
Student Preferences Which way of doing inference gave you a better conceptual understanding of confidence intervals and hypothesis tests? Bootstrapping and Randomization
42 72%
Formulas and Theoretical Distributions
16 27%
Student Preferences LEARN inference AP Stat No AP Stat
Simulation 13 26
Traditional 15 4
DO inference AP Stat No AP Stat
Simulation 18 24
Traditional 10 6
UNDERSTAND AP Stat No AP Stat
Simulation 17 25
Traditional 11 5
Thank you for joining us! More information is available on www.lock5stat.com Feel free to contact any of us with any comments or questions.
