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Models and Modeling in Introductory Statistics
Robin H. LockBurry Professor of Statistics
St. Lawrence University
2012 Joint Statistics MeetingsSan Diego, August 2012
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What is a Model?
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What is a Model?
A simplified abstraction that approximates important features of a
more complicated system
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Traditional Statistical Models
PopulationYN(μ,σ)
Often depends on non-trivial mathematical ideas.
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Traditional Statistical Models
Relationship𝑌 𝛽0+𝛽1 𝑋+𝜀
Predictor (X)
Resp
onse
(Y)
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“Empirical” Statistical Models
A representative sample looks like a mini-version of the population.
Model a population with many copies of the sample.
BootstrapSample with replacement from an original sample to study the behavior of a statistic.
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“Empirical” Statistical ModelsHypothesis testing: Assess the behavior of a sample statistic, when the population meets a specific criterion.
Create a Null Model in order to sample from a population that satisfies H0
Randomization
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Traditional vs. Empirical
Both types of model are important, BUTEmpirical models (bootstrap/randomization) are• More accessible at early stages of a course• More closely tied to underlying statistical
concepts• Less dependent on abstract mathematics
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Example: Mustang Prices
Estimate the average price of used Mustangs and provide an interval to reflect the accuracy of the estimate.
Data: Sample prices for n=25 Mustangs
Price10 20 30 40 50
MustangPrice Dot Plot
𝑥=15.98 𝑠=11.11
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Original Sample Bootstrap Sample
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Original Sample
BootstrapSample
BootstrapSample
BootstrapSample
.
.
.
Bootstrap Statistic
Sample Statistic
Bootstrap Statistic
Bootstrap Statistic
.
.
.
Bootstrap Distribution
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Bootstrap Distribution: Mean Mustang Prices
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Background?
What do students need to know about before doing a bootstrap interval?
• Random sampling• Sample statistics (mean, std. dev., %-tile)• Display a distribution (dotplot)• Parameter vs. statistic
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Traditional Sampling Distribution
Population
µ
BUT, in practice we don’t see the “tree” or all of the “seeds” – we only have ONE seed
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Bootstrap Distribution
Bootstrap“Population”
What can we do with just one seed?
Grow a NEW tree!
𝑥
Estimate the distribution and variability (SE) of ’s from the bootstraps
µ
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Round 2
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Course Order• Data production• Data description (numeric/graphs)• Interval estimates (bootstrap model)• Randomization tests (null model)• Traditional inference for means and
proportions (normal/t model)• Higher order inference (chi-square,
ANOVA, linear regression model)
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Traditional models need mathematics,
Empirical models need technology!
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Some technology options:• R (especially with Mosaic)• Fathom/Tinkerplots• StatCrunch• JMP
StatKeywww.lock5stat.com
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Three Distributions
One to Many Samples
Built-in data Enter new data
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Interact with tails
Distribution Summary Stats
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Smiles and LeniencyDoes smiling affect leniency in a college disciplinary hearing?
Null Model: Expression has no affect on leniency
4.12
4.91
LeFrance, M., and Hecht, M. A., “Why Smiles Generate Leniency,” Personality and Social Psychology Bulletin, 1995; 21:
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Smiles and LeniencyNull Model: Expression has no affect on leniency
To generate samples under this null model:• Randomly re-assign the smile/neutral labels to
the 68 data leniency scores (34 each).• Compute the difference in mean leniency
between the two groups, • Repeat many times• See if the original difference, , is unusual in the
randomization distribution.
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StatKey
p-value = 0.023
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Traditional t-testH0:μs = μn H0:μs > μn
𝑡= 4.91−4.12
√ 1.52234+1.68
2
34
=0.790.39=2.03
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Round 3
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Assessment? Construct a bootstrap distribution of sample means for the SPChange variable. The result should be relatively bell-shaped as in the graph below. Put a scale (show at least five values) on the horizontal axis of this graph to roughly indicate the scale that you see for the bootstrap means.
Estimate SE? Find CI from SE? Find CI from percentiles?
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Assessment? From 2009 AP Stat: Given summary stats, test skewness
Find and interpret a p-value
ratio0.94 0.96 0.98 1.00 1.02 1.04 1.06
Measures from Collection 1 Dot Plot
𝑅𝑎𝑡𝑖𝑜=𝑥
𝑚𝑒𝑑𝑖𝑎𝑛 Given 100 such ratios for samples drawn from a symmetric distribution
Ratio=1.04 for the original sample
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Implementation Issues
• Good technology is critical
• Missed having “experienced” student support the first couple of semesters
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Round 4
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Why Did I Get Involved with Teaching Bootstrap/Randomization Models?
It’s all George’s fault...
"Introductory Statistics: A Saber Tooth Curriculum?"
Banquet address at the first (2005) USCOTS
George Cobb
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Introduce inference with “empirical models” based on simulations from the sample data (bootstraps/randomizations), then approximate with models based on traditional distributions.
Models in Introductory Statistics