Using Randomization Methods to Build Conceptual Understanding in

Statistical Inference:Day 1

Lock, Lock, Lock, Lock, and LockMinicourse – Joint Mathematics Meetings

Boston, MAJanuary 2012

WiFi: marriotconference, password: 1134ams

Introductions:

NameInstitution

Schedule: Day 1Wednesday, 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 2Friday, 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. RegularlyB. OccasionallyC. Rarely/Never

QuestionHow 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• Data production (samples/experiments)

• Sampling distributions (mean/proportion)

• Confidence intervals (means/proportions)

• Hypothesis tests (means/proportions)

• ANOVA for several means, Inference for regression, Chi-square tests

• Data production (samples/experiments)• Bootstrap confidence intervals• Randomization-based hypothesis tests• Normal/sampling distributions

• Bootstrap confidence intervals• Randomization-based hypothesis 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

BootstrapSample

BootstrapSample

BootstrapSample

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.

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Bootstrap Statistic

Sample Statistic

Bootstrap Statistic

Bootstrap Statistic

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Bootstrap Distribution

We need technology!

Introducing

StatKey.

Example: Atlanta Commutes

Data: The American Housing Survey (AHS) collected data from Atlanta in 2004.

What’s the mean commute time for workers in metropolitan Atlanta?

Sample of n=500 Atlanta Commutes

Where might the “true” μ be?Time

20 40 60 80 100 120 140 160 180

CommuteAtlanta Dot Plot

n = 50029.11 minutess = 20.72 minutes

StatKeycan 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 :

𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐±2 ∙𝑆𝐸For the mean Atlanta commute time:

29.11±2 ∙0.92=29.11 ±1.84=(27.27 ,30.95)

Using the Bootstrap Distribution to Get a Confidence Interval – Version #2

Keep 95% in middle

Chop 2.5% in each tail

Chop 2.5% in each tail

For a 95% CI, find the 2.5%-tile and 97.5%-tile in the bootstrap distribution

95% CI=(27.35,30.96)

90% CI for Mean Atlanta Commute

Keep 90% in middle

Chop 5% in each tail

Chop 5% in each tail

For a 90% CI, find the 5%-tile and 95%-tile in the bootstrap distribution

90% CI=(27.64,30.65)

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?

B. How many years have you been teaching Intro Stat?C. What was the travel time, in hours, for your trip to

Boston for JMM?D. Including this one, how many times have you attended

the January JMM?E. ???

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?C. Have you ever attended a JSM meeting?D. ???E. ???

How do we assess student understanding of these methods(even on in-class exams without computers)?

See the green pages in the folder.

http://www.youtube.com/watch?v=3ESGpRUMj9E

Paul the Octopus

http://www.cnn.com/2010/SPORT/football/07/08/germany.octopus.explainer/index.html

• 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?

Paul the Octopus

• 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

Simulate!

Let p denote the proportion of games that Paul guesses correctly (of all games he may have predicted)

H0 : p = 1/2Ha : p > 1/2

Hypotheses

• 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

Randomization Distribution

• 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!

p-value

StatKey

81 0.00392

• 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

Randomization Test

• 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

Coming Attractions - Friday

Using Randomization Methods to Build Conceptual Understanding

of Statistical Inference:Day 2

Lock, Lock, Lock, Lock, and LockMinicourse- Joint Mathematics Meetings

Boston, MAJanuary 2012

• 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?

Cocaine Addiction

R R R R R R

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Desipramine Lithium

1. Randomly assign units to treatment groups

R R R R

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RRR R R R

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R = RelapseN = No Relapse

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2. Conduct experiment

3. Observe relapse counts in each group

LithiumDesipramine

10 relapse, 14 no relapse 18 relapse, 6 no relapse

1. Randomly assign units to treatment groups

10 1824

ˆ ˆ

24.333

new oldp p

• 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

Randomization Test

R R R R

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RRR R R R

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10 relapse, 14 no relapse 18 relapse, 6 no relapse

R R R R R R

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Simulate another randomization

Desipramine Lithium

16 relapse, 8 no relapse 12 relapse, 12 no relapse

ˆ ˆ16 1224 240.167

N Op p

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Simulate another randomization

Desipramine Lithium

17 relapse, 7 no relapse 11 relapse, 13 no relapse

ˆ ˆ17 1124 240.250

N Op p

• 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

Simulate!

StatKey

The probability of getting results as extreme or more extreme than those observed if the null hypothesis is true, is about .02. p-value

Proportion as extreme as observed statistic

observed statistic

• Why did you re-deal your cards?

• Why did you leave the outcomes (relapse or no relapse) unchanged on each card?

Cocaine AddictionYou want to know what would happen

• by random chance (the random allocation to treatment groups)

• 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

Data: A random sample of n=50 body temperatures.

Is the average body temperature really 98.6oF?

BodyTemp96 97 98 99 100 101

BodyTemp50 Dot Plot

H0:μ=98.6 Ha:μ≠98.6

n = 5098.26s = 0.765

Data from Allen Shoemaker, 1996 JSE data set article

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 SamplesHow to simulate samples of body temperatures to be consistent with H0: μ=98.6?

1. Add 0.34 to each temperature in the sample (to get the mean up to 98.6).

2. Sample (with replacement) from the new data.3. Find the mean for each sample (H0 is true).

4. See how many of the sample means are as extreme as the observed 98.26.

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 = 0.430n = 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 : = 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 MethodA=Sleep 14 18 11 13 18 17 21 9 16 17 14 15 mean=15.25

B=Caffeine 12 12 14 13 6 18 14 16 10 7 15 10 mean=12.25

Example: Word recall

Option 1: Randomly scramble the A and B labels and assign to the 24 word recalls.

H0: μA=μB vs. Ha: μA≠μB

Option 2: Combine the 24 values, then sample (with replacement) 12 values for Group A and 12 values for Group B.

Reallocate

Resample

Question

In Intro Stat, how critical is it for the method of randomization to reflect the way data were collected? A. Essential B. Relatively important C. Desirable, but not imperative D. Minimal importance E. Ignore the issue completely

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). RockB). PaperC). 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

• Descriptive Statistics – one and two samples

• Confidence intervals (means/proportions)• Hypothesis tests (means/proportions)

• ANOVA for several means, Inference for regression, Chi-square tests

• Data production (samples/experiments)

• Normal/sampling distributions

• Bootstrap confidence intervals• Randomization-based hypothesis 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

slope (thousandths)-60 -40 -20 0 20 40 60

Measures from Scrambled RestaurantTips Dot Plot

r-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

Measures from Scrambled Collection 1 Dot Plot

Nullxbar98.2 98.3 98.4 98.5 98.6 98.7 98.8 98.9 99.0

Measures from Sample of BodyTemp50 Dot Plot

Diff-4 -3 -2 -1 0 1 2 3 4

Measures from Scrambled CaffeineTaps Dot Plot

xbar26 27 28 29 30 31 32

Measures from Sample of CommuteAtlanta Dot Plot

Slope :Restaurant tips Correlation: Malevolent uniforms

Mean :Body Temperatures Diff means: Finger taps

Mean : Atlanta commutes

phat0.3 0.4 0.5 0.6 0.7 0.8

Measures from Sample of Collection 1 Dot PlotProportion : Owners/dogs

What do you notice?

All bell-shaped distributions!

Bootstrap and Randomization Distributions

The students are primed and ready to learn about the normal distribution!

Transitioning to Traditional Inference

Confidence Interval:

Hypothesis Test:

• Introduce the normal distribution (and later t)

• Introduce “shortcuts” for estimating SE for proportions, means, differences, slope…

z*-z*

95%

Confidence Intervals

Test statistic

95%

Hypothesis Tests

Area is p-value

Confidence Interval:

Hypothesis Test:

Yes! Students see the general pattern and not just individual formulas!

Connecting CI’s and Tests

Randomization body temp means when μ=98.6

xbar98.2 98.3 98.4 98.5 98.6 98.7 98.8 98.9 99.0

Measures from Sample of BodyTemp50 Dot Plot

97.9 98.0 98.1 98.2 98.3 98.4 98.5 98.6 98.7bootxbar

Measures from Sample of BodyTemp50 Dot Plot

Bootstrap body temp means from the original sample

What’s the difference?

Fathom Demo: Test & CI

Sample mean is in the “rejection region”

Null mean is outside the confidence interval

Technology SessionsChoose 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

Formulas and Theoretical Distributions

39 19

67% 33%

Student Preferences

Which way do you prefer to do inference?

Bootstrapping and Randomization

Formulas and Theoretical Distributions

42 16

72% 28%

Student Preferences

Which way of doing inference gave you a better conceptual understanding of confidence intervals and hypothesis tests?

Bootstrapping and Randomization

Formulas and Theoretical Distributions

42 16

72% 27%

Student Preferences

DO inference Simulation Traditional

AP Stat 18 10

No AP Stat 24 6

LEARN inference Simulation Traditional

AP Stat 13 15

No AP Stat 26 4

UNDERSTAND Simulation Traditional

AP Stat 17 11

No AP Stat 25 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.