Transcript Introduction to Bootstrapping

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:
Bootstrap Sample: Sample with
replacement from the original sample, using
the same sample size.
Original Sample
Bootstrap Sample
Original Sample
Create a “sampling distribution” using this as our
simulated 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
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Bootstrap
Sample
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Bootstrap
Statistic
Bootstrap
Distribution
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
CommuteAtlanta
Dot Plot
n = 500
𝑥 =29.11 minutes
s = 20.72 minutes
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Time
Where might the “true” μ be?
180
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 ∙ 𝑆𝐸
For 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
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
That’s all folks
StatKey
can be found at
www.lock5stat.com
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.
B.
C.
D.
E.
Did you fly or drive to these meetings?
Have you attended any previous JMM meetings?
Have you ever attended a JSM meeting?
???
???
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
 12 
8
 0.0039
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
Desipramine
Lithium
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pˆ new  pˆ old
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 .333
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10 relapse, 14 no relapse
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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Simulate another
randomization
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16 relapse, 8 no relapse
pˆ N  pˆ O
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 0.167
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12 relapse, 12 no relapse
Simulate another
randomization
Desipramine
Lithium
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pˆ N  pˆ O
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 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
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
n = 50
𝑥 =98.26
s = 0.765
96
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98
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BodyTemp
100
Data from Allen Shoemaker, 1996 JSE data set article
101
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?
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 𝑥 =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 = 0.430
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 :  = 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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15 mean=15.25
B=Caffeine
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10 mean=12.25
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
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). 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
Measures from Scrambled Collection 1
Measures from Scrambled RestaurantTips
Slope :Restaurant tips
-60
-40
-20
0
20
slope (thousandths)
40
Dot Plot
Correlation: Malevolent uniforms
Dot Plot
60
-0.6
-0.4
-0.2
0.0
r
0.2
0.4
All
bell-shaped
What
do you
Mean :Body Temperatures
Diff means: Finger taps
distributions!
notice?
Measures from Sample of BodyTemp50
98.2
98.3
98.4
Dot Plot
Measures from Scrambled CaffeineTaps
98.5
98.6
Nullxbar
98.7
98.8
Proportion : Owners/dogs
0.4
0.5
phat
0.6
98.9
Dot Plot
99.0
-4
Measures from Sample of Collection 1
0.3
0.6
-3
-2
-1
0
Diff
1
2
3
4
Dot Plot
Mean : Atlanta commutes
Measures from Sample of CommuteAtlanta
0.7
0.8
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27
28
29
xbar
Dot Plot
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32
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 Interval:
𝑆𝑎𝑚𝑝𝑙𝑒 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 ± 𝑧 ∗ ∙ 𝑆𝐸
Hypothesis Test:
𝑆𝑎𝑚𝑝𝑙𝑒 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 − 𝑁𝑢𝑙𝑙 𝑃𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟
𝑆𝐸
Confidence Intervals
95%
-z*
z*
Hypothesis Tests
95%
Area is
p-value
Test
statistic
Yes! Students see the general
pattern and not just individual
formulas!
Confidence Interval:
𝑆𝑎𝑚𝑝𝑙𝑒 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 ± 𝑧 ∗ ∙ 𝑆𝐸
Hypothesis Test:
𝑆𝑎𝑚𝑝𝑙𝑒 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 − 𝑁𝑢𝑙𝑙 𝑃𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟
𝑆𝐸
Connecting CI’s and Tests
Measures from Sample of BodyTemp50
Dot Plot
Randomization
body temp means
when μ=98.6
98.2
98.3
98.4
98.5
Measures from Sample of BodyTemp50
98.6
xbar
98.7
98.8
98.9
99.0
Dot Plot
Bootstrap body
temp means from
the original sample
97.9
98.0
98.1
98.2
98.3
98.4
bootxbar
98.5
98.6
98.7
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
Simulation
13
Traditional
15
No AP Stat
26
4
DO inference
Simulation
Traditional
AP Stat
No AP Stat
18
24
10
6
UNDERSTAND
Simulation
Traditional
AP Stat
No AP Stat
17
25
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.