Transcript Chapter 7 Slides

Paired Data: One
Quantitative Variable
Chapter 7
Introduction
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The paired datasets in this chapter have one pair
of quantitative response values for each
observational unit.
This allows for a built-in comparison.
Studies with paired data remove individual
variability by looking at the difference score for
each individual.
Reducing variability in data improves inferences:
 Narrower confidence intervals
 Smaller p-values when the null hypothesis is
false
Introduction
Our data that we will analyze will just be a
single quantitative variable.
 So things like mean and standard
deviation are important to look at, but
really nothing new for descriptive
statistics.

Section 7.1: Simulation-based method
 Section 7.2: Theory-based method

Section 7.1: Simulation-Based
Approach for Analyzing Paired Data
Example 7.1:
Rounding First Base
First Base
Imagine you’ve hit a
line drive and are
trying to reach second
base.
 Does the path that
you take to “round”
first base make much
a difference?
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Narrow angle
Wide angle
First Base
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Hollander and Wolfe (1999) report on a
Master’s Thesis by Woodward (1970) that
investigates base running strategies.
Woodward timed 22 different runners from a
spot 35 feet past home to a spot 15 feet
before second.
Each runner used each strategy (paired
design), with a rest between.
This paired design controls for the runner-torunner variability.
He used random assignment to decide which
path each runner should do first.
First Base
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Times for the first 10 runners
Subject
narrow
angle
wide
angle
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1
2
3
4
5
6
7
8
5.50 5.70 5.60
5.50
5.85
5.55 5.40
5.50 5.15
5.80
…
5.55 5.75 5.50
5.40
5.70
5.60 5.35
5.35 5.00
5.70
…
Dotplots of times for all 22 runners
9
10
First Base
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There is a lot of overlap in the
distributions and a fair bit of variability
Narrow
Mean
5.534
SD
0.260
Wide
5.459
0.273
Difficult to detect a difference between the
methods when there’s a lot of variation
First Base
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What are the observational units in this
study?
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What variables are recorded? What are their
types and roles?
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The runners (22 total)
Explanatory variable: base running method: wide
or narrow angle (categorical)
Response variable: time for middle of the route
from home plate to second base (quantitative)
Is this an observational study or an
experiment?
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Randomized experiment since the explanatory
variable was randomly applied to determined which
method each runner used first
First Base
These data are clearly paired.
 The paired response variable is time
difference in running between the two
methods (narrow angle – wide angle).
 Could we do wide angle – narrow angle?
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First Base
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Differences for the first 10 runners
Subject
narrow
angle
wide
angle
diff
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1
5.50
2
5.70
3
5.6
4
5.50
5
5.85
6
5.55
7
5.40
8
5.50
9
5.15
10
5.80
…
5.55
5.75
5.5
5.40
5.70
5.60
5.35
5.35
5.00
5.70
…
0.1
0.15
-0.05 0.05
0.15
0.15
0.10
…
-0.05 -0.05 0.1
A dotplot of the differences for all 22
runners.
First Base
The distribution is a bit skewed left, but
not too bad.
 Mean difference is 𝑥 d = 0.075 seconds
 Standard deviation is SDd = 0.0883 sec
 Standard deviation (0.0883) is smaller
than the original standard deviations of
the running times (0.260 and 0.273).
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First Base
The original dotplots with each
observation paired between the base
running strategies.
 What do you notice?
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First Base
Is the average difference of 𝑥 d = 0.075
seconds significantly different from 0?
 The parameter of interest is, µd, is the
population average difference in running
times by some population of runners when
using the narrow angle and the wide
angle. (narrow – wide)
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First Base
The hypotheses:
 H0: µd = 0
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Ha: µd ≠ 0
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On average, the mean of the differences between
the running times (narrow – wide) is 0.
On average, the mean of the differences in running
times (narrow – wide) is not 0.
If the parameter of interest is the population
average difference, then the corresponding
statistic is the sample average difference.
First Base
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How can simulation-based methods find
an approximate p-value?
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The null basically says the running path
doesn’t matter --- the times, on average, will
be the same for the two methods.
So we can use our same data set and
randomly decide which time goes with the
narrow and wide methods and compute a
mean difference. (Notice we don’t break our
pairs.)
We can repeat this process many times to
develop a null distribution.
First Base
The results of random swaps for the first
10 runners (done with coin flips)
 An average difference of 𝑥 d = -0.025
seconds
 Repeat many times to construct a null
distribution
Subject
1
2
3
4
5
6
7
8
9
10
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This
models
no
connection
between
times
narrow
5.55 5.70
5.50 5.50 5.70
5.60 5.40 5.50 5.15 5.70
angle
and the strategy used
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wide
angle
diff
…
5.50 5.75
5.60 5.40 5.85
5.55 5.35
5.35 5.00 5.80 …
0.05 -0.05
-0.1
0.05 0.05
0.15 0.15 -0.1
0.1
-0.15
…
First Base
Mean differences from 1000 repetitions
 Describe the shape of the distribution.
 The distribution appears to be centered at
about 0. Does that make sense?
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First Base
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Using the null distribution is the observed
average from the study of 0.075 out in the
tail?
First Base
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Only 2 of the 1000 repetitions of random
swappings gave a 𝑥𝑑 value at least as
extreme as 0.075
First Base
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We can also standardize 0.075 by dividing
by the applet’s estimate of the SD ≈ 0.024
0.075
to see we are
= 3.125 standard
0.024
deviations above zero.
First Base
Based on the p-value and standardized
statistic we have very strong evidence
against the null hypothesis.
 We can draw a cause-and-effect
conclusions since the researcher used
random assignment of the two base
running methods for each runner.
 There was not a lot of information about
how these 22 runners were selected to
decide if we can generalize to a larger
population.
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First Base
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Approximate a 95% confidence interval for
𝜇d:
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What does this mean?
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0.075 ± 2(0.024) seconds
(0.027, 0.124) seconds
We are 95% confident that, on average, the
narrow angle route takes 0.027 to 0.124
seconds longer than the wide angle route
Let’s try this out with the applet.
First Base
Alternative Analysis
 What do you think would happen if we
wrongly analyzed the data using a 2
independent samples procedure?
 I.e. the researcher selected 22 runners to use
the wide method and an independent sample
of 22 other runners to use the narrow
method, obtaining the same 44 times as in
the actual study.
 Would the p-value stay the same, increase,
or decrease?
First Base
Using the Two Means applet (which does an
independent test) we get a p-value of
0.1830
Does it make
sense that this
p-value is larger
than the one we
obtained
earlier?
Exercise and Heart Rate
Exploration 7.1
Section 7.2: Theory-based methods
for paired data.
H0: µd = 0
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Path doesn’t matter
Ha: µd ≠ 0
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Path does matter
narrow - wide
𝑥 d = 0.075 seconds
 SDd = 0.0883
seconds
 n = 22
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First Base
Our null distribution was centered at zero
and fairly bell-shaped.
 This can all be predicted (along with the
variability) using theory-based methods.
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To do this, our
sample size should
be at least 20.
Theory-based test
We can do theory-based methods with the
applet we used last time or the theorybased applet.
 With the applet we used last time, we
need to calculate the t-statistic:
𝑥𝑑
𝑡=
𝑠𝑑 𝑛
 With the theory-based applet, we just
need to enter the summary statistics and
use a test for a one mean.
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Theory-based results
First Base
The theory-based model gives slightly
different results, but we come to the same
conclusion. Which base running path used
does make a difference in the average times
(we can see that with our small p-value).
 We estimate the narrow angle path will take
between 0.036 to 0.114 seconds longer, on
average, to complete than the wide angle
path.
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Exploration 7.2
Comparing Auction Formats
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We will compare:
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Dutch auction the item for sale starts at a
very high price and is lowered gradually until
someone finds the price low enough to buy.
First-price sealed bid auction each bidder
summits a single sealed bid before a particular
deadline. After the deadline, the person with
the highest bid wins.