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2/20/2015
1
Statistics: Unlocking the Power of Data Lock5
STAT 250 Dr. Kari Lock Morgan
Confidence Intervals: Bootstrap Distribution
SECTIONS 3.3, 3.4
• Bootstrap distribution (3.3)
• 95% CI using standard error (3.3)
• Percentile method (3.4)
Statistics: Unlocking the Power of Data Lock5
Confidence Intervals
Population Sample
Sample
Sample
Sample Sample Sample
. . .
Calculate statistic for each sample
Sampling Distribution
Standard Error (SE): standard deviation of sampling distribution
Margin of Error (ME) (95% CI: ME = 2×SE)
statistic ± ME
Statistics: Unlocking the Power of Data Lock5
Reality
One small problem…
… WE ONLY HAVE ONE SAMPLE!!!!
• How do we know how much sample statistics vary, if we only have one sample?!?
BOOTSTRAP! Statistics: Unlocking the Power of Data Lock5
Your best guess for the population
is lots of copies of the sample!
Sample repeatedly from this “population”
Statistics: Unlocking the Power of Data Lock5
• It’s impossible to sample repeatedly from the population…
• But we can sample repeatedly from the sample!
• To get statistics that vary, sample with replacement (each unit can be selected more than once)
Sampling with Replacement
Statistics: Unlocking the Power of Data Lock5
Suppose we have a random sample of 6 people:
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Statistics: Unlocking the Power of Data Lock5
Original Sample
A simulated “population” to sample from
Statistics: Unlocking the Power of Data Lock5
Bootstrap Sample: Sample with
replacement from the original sample, using the same sample size.
Original Sample
Bootstrap Sample
Remember: sample
size matters!
Statistics: Unlocking the Power of Data Lock5
• How would you take a bootstrap sample from your sample of Reese’s Pieces?
Reese’s Pieces
Statistics: Unlocking the Power of Data Lock5
Your original sample has data values
18, 19, 19, 20, 21 Is the following a possible bootstrap sample?
18, 19, 20, 21, 22
a) Yes b) No
Bootstrap Sample
Statistics: Unlocking the Power of Data Lock5
Your original sample has data values
18, 19, 19, 20, 21 Is the following a possible bootstrap sample?
18, 19, 20, 21
a) Yes b) No
Bootstrap Sample
Statistics: Unlocking the Power of Data Lock5
Your original sample has data values
18, 19, 19, 20, 21 Is the following a possible bootstrap sample?
18, 18, 19, 20, 21
a) Yes b) No
Bootstrap Sample
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Statistics: Unlocking the Power of Data Lock5
Original Sample
BootstrapSample
BootstrapSample
BootstrapSample
.
.
.
Bootstrap Statistic
Sample Statistic
Bootstrap Statistic
Bootstrap Statistic
.
.
.
Bootstrap Distribution
Statistics: Unlocking the Power of Data Lock5
• What is the average mercury level of fish (large mouth bass) in Florida lakes?
• Sample of size n = 53, with 𝑥 = 0.527 ppm. (FDA action level in the USA is 1 ppm, in Canada the limit is 0.5 ppm)
•Key Question: How much can statistics vary from sample to sample?
Mercury Levels in Fish
Lange, T., Royals, H. and Connor, L. (2004). Mercury accumulation in largemouth bass (Micropterus salmoides) in a Florida Lake. Archives of Environmental Contamination and Toxicology, 27(4), 466-471.
Statistics: Unlocking the Power of Data Lock5
Mercury Levels in Fish
Statistics: Unlocking the Power of Data Lock5
You have a sample of size n = 53. You sample with replacement 1000 times to get 1000 bootstrap samples. What is the sample size of each bootstrap sample? (a) 53 (b) 1000
Bootstrap Sample
Statistics: Unlocking the Power of Data Lock5
You have a sample of size n = 53. You sample with replacement 1000 times to get 1000 bootstrap samples. How many bootstrap statistics will you have? (a) 53 (b) 1000
Bootstrap Distribution
Statistics: Unlocking the Power of Data Lock5
“Pull yourself up by your bootstraps”
Why “bootstrap”?
• Lift yourself in the air simply by pulling up on the laces of your boots
• Metaphor for accomplishing an “impossible” task without any outside help
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Statistics: Unlocking the Power of Data Lock5
Sampling Distribution
Population
µ
BUT, in practice we don’t see the “tree” or all of the “seeds” – we only have ONE seed
Statistics: Unlocking the Power of Data Lock5
Bootstrap Distribution
Bootstrap “Population”
What can we do with just one seed?
𝑥
Estimate the distribution and variability (SE) of 𝑥 ’s from the bootstraps
µ
Statistics: Unlocking the Power of Data Lock5
Bootstrap statistics are to the original sample statistic
as
the original sample statistic is to the population parameter
Golden Rule of Bootstrapping
Statistics: Unlocking the Power of Data Lock5
Center
•The sampling distribution is centered around the population parameter
• The bootstrap distribution is centered around the
•Luckily, we don’t care about the center… we care about the variability!
a) population parameter b) sample statistic c) bootstrap statistic d) bootstrap parameter
Statistics: Unlocking the Power of Data Lock5
Standard Error
• The variability of the bootstrap statistics is similar to the variability of the sample statistics
• The standard error of a statistic can be estimated using the standard deviation of the bootstrap distribution!
Statistics: Unlocking the Power of Data Lock5
Confidence Intervals
Sample Bootstrap
Sample
. . .
Calculate statistic for each bootstrap sample
Bootstrap Distribution
Standard Error (SE): standard deviation of bootstrap distribution
Margin of Error (ME) (95% CI: ME = 2×SE)
statistic ± ME
Bootstrap Sample
Bootstrap Sample
Bootstrap Sample
Bootstrap Sample
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Statistics: Unlocking the Power of Data Lock5
• What is the average mercury level of fish (large mouth bass) in Florida lakes?
• Sample of size n = 53, with 𝑥 = 0.527 ppm. (FDA action level in the USA is 1 ppm, in Canada the limit is 0.5 ppm)
•Give a confidence interval for the true average.
Mercury Levels in Fish
Lange, T., Royals, H. and Connor, L. (2004). Mercury accumulation in largemouth bass (Micropterus salmoides) in a Florida Lake. Archives of Environmental Contamination and Toxicology, 27(4), 466-471.
Statistics: Unlocking the Power of Data Lock5
SE = 0.047
0.527 2 0.047 (0.433, 0.621)
Mercury Levels in Fish
We are 95% confident that average mercury level in fish in Florida lakes is between 0.433 and 0.621 ppm.
Statistics: Unlocking the Power of Data Lock5
Same process for every parameter! Estimate the standard error and/or a confidence interval for...
• proportion (𝑝)
• difference in means (µ1 − µ2 )
• difference in proportions (𝑝1 − 𝑝2 )
• standard deviation (𝜎)
• correlation (𝜌)
• ... Generate samples with replacement Calculate sample statistic Repeat...
Statistics: Unlocking the Power of Data Lock5
Hitchhiker Snails
A type of small snail is very widespread in Japan, and colonies of the snails that are genetically very similar have been found very far apart.
How could the snails travel such long distances?
Biologist Shinichiro Wada fed 174 live snails to birds, and found that 26 were excreted live out the other end. (The snails are apparently able to seal their shells shut to keep the digestive fluids from getting in).
What proportion of these snails ingested by birds survive?
Yong, E. “The Scatological Hitchhiker Snail,” Discover, October 2011, 13.
Statistics: Unlocking the Power of Data Lock5
Hitchhiker Snails Give a 95% confidence interval for the proportion of snails ingested by birds that survive.
a) (0.1, 0.2) b) (0.05, 0.25) c) (0.12, 0.18) d) (0.07, 0.18)
Statistics: Unlocking the Power of Data Lock5
Body Temperature What is the average body temperature of humans?
www.lock5stat.com/statkey
Shoemaker, What's Normal: Temperature, Gender and Heartrate, Journal of Statistics Education, Vol. 4, No. 2 (1996)
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Statistics: Unlocking the Power of Data Lock5
Other Levels of Confidence
• What if we want to be more than 95% confident?
• How might you produce a 99% confidence interval for the average body temperature?
Statistics: Unlocking the Power of Data Lock5
Percentile Method
• For a P% confidence interval, keep the middle P% of bootstrap statistics
• For a 99% confidence interval, keep the middle 99%, leaving 0.5% in each tail.
• The 99% confidence interval would be
(0.5th percentile, 99.5th percentile)
where the percentiles refer to the bootstrap distribution.
Statistics: Unlocking the Power of Data Lock5
Bootstrap Distribution Best Guess at Sampling Distribution
Statistic
2 3 4 5 6 7 8
Best Guess at Sampling Distribution
Statistic
2 3 4 5 6 7 8
Observed
Statistic
Best Guess at Sampling Distribution
Statistic
2 3 4 5 6 7 8
Observed
Statistic
P%
Best Guess at Sampling Distribution
Statistic
2 3 4 5 6 7 8
Observed
Statistic
P%P%P%
Upper
Bound
Upper
Bound
Lower
Bound
• For a P% confidence interval:
Statistics: Unlocking the Power of Data Lock5
www.lock5stat.com/statkey
Body Temperature
We are 99% sure that the average body temperature is between
98.00 and 98.58
Middle 99% of bootstrap statistics
Statistics: Unlocking the Power of Data Lock5
Level of Confidence
Which is wider, a 90% confidence interval or a 95% confidence interval? (a) 90% CI (b) 95% CI
Statistics: Unlocking the Power of Data Lock5
Mercury and pH in Lakes
Lange, Royals, and Connor, Transactions of the American Fisheries Society (1993)
• For Florida lakes, what is the correlation between average mercury level (ppm) in fish taken from a lake and acidity (pH) of the lake?
𝑟 = −0.575
Give a 90% CI for
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Statistics: Unlocking the Power of Data Lock5
www.lock5stat.com/statkey
Mercury and pH in Lakes
We are 90% confident that the true correlation between average mercury level and pH of Florida lakes is between -0.702 and -0.433.
Statistics: Unlocking the Power of Data Lock5
Bootstrap CI Option 1: Estimate the standard error of the statistic by computing the standard deviation of the bootstrap distribution, and then generate a 95% confidence interval by Option 2: Generate a P% confidence interval as the range for the middle P% of bootstrap statistics
2statisti Sc E
Statistics: Unlocking the Power of Data Lock5
SE = 0.047
0.527 2 0.047 (0.433, 0.621)
Middle 95% of bootstrap statistics
Mercury Levels in Fish
Statistics: Unlocking the Power of Data Lock5
Bootstrap Cautions
• These methods for creating a confidence interval only work if the bootstrap distribution is smooth and symmetric
• ALWAYS look at a plot of the bootstrap distribution!
• If the bootstrap distribution is skewed or looks “spiky” with gaps, you will need to go beyond intro stat to create a confidence interval
Statistics: Unlocking the Power of Data Lock5
Bootstrap Cautions
Statistics: Unlocking the Power of Data Lock5
Bootstrap Cautions
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Statistics: Unlocking the Power of Data Lock5
Number of Bootstrap Samples
• When using bootstrapping, you may get a slightly different confidence interval each time. This is fine!
• The more bootstrap samples you use, the more precise your answer will be.
• For the purposes of this class, 1000 bootstrap samples is fine. In real life, you probably want to take 10,000 or even 100,000 bootstrap samples
Statistics: Unlocking the Power of Data Lock5
Summary The standard error of a statistic is the standard
deviation of the sample statistic, which can be estimated from a bootstrap distribution
Confidence intervals can be created using the standard error or the percentiles of a bootstrap distribution
Increasing the number of bootstrap samples will not change the SE or interval (except for random fluctuation)
Confidence intervals can be created this way for any parameter, as long as the bootstrap distribution is approximately symmetric and continuous
Statistics: Unlocking the Power of Data Lock5
To Do
Read Sections 3.3, 3.4
Do HW 3.3, 3.4 (due Friday, 2/7)
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