Give your data the boot: What is bootstrapping? and Why does it matter? Patti Frazer Lock and Robin H. Lock St. Lawrence University MAA Seaway Section.

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Presentation transcript:

Give your data the boot: What is bootstrapping? and Why does it matter? Patti Frazer Lock and Robin H. Lock St. Lawrence University MAA Seaway Section Meeting Plattsburgh, October 2010

Bootstrap confidence intervals and randomization hypothesis tests provide an alternate way to DO and to TEACH statistical inference.

Why bootstrap intervals and randomization tests?

Top Ten Reasons for using simulation-based inference Five

5. Maintain student interest by foreshadowing inference from day 1 and getting to the key ideas of inference very early in the course. When do current texts first discuss intervals and tests? Confidence IntervalSignificance Test pg. 359pg. 373 pg. 329pg. 400 pg. 486pg. 511 pg. 319pg. 365

4. Develop students’ intuitive understanding of the key ideas of statistical inference. Descriptive stats Sampling and design Probability distributions Statistical inference formulas Current model in intro stats: The underlying concepts behind intervals and tests are hard. Is this the best way to build understanding?

3. Help students understand the global picture for intervals and tests, rather than memorize a list of formulas. We’d like students to see the general pattern rather than a string of (what can appear to them to be) unrelated formulas.

2. Flexibility!!!  Few underlying assumptions  Works for any parameter  Same methods apply to many situations

1. It’s the way of the past and the future. "Actually, the statistician does not carry out this very simple and very tedious process, 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

“... despite broad acceptance and rapid growth in enrollments, 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 … and the future.

Top Five Reasons to use simulation-based inference : 5. Maintain interest by getting to inference early. 4. Develop understanding of the key ideas. 3. Help students understand the global picture. 2. Flexibility. 1.It’s the way of the past and the future.

What is a bootstrap? and How does it give an interval?

Example: Atlanta Commutes Data: The American Housing Survey (AHS) collected data from Atlanta in What’s the mean commute time for workers in metropolitan Atlanta?

Sample of n=500 Atlanta Commutes Where is “true” μ?

“Bootstrap” Samples Key idea: Sample with replacement from the original sample using the same n. Assumes the “population” is many, many copies of the original sample.

Creating a Bootstrap Distribution (using Fathom) 1. Start with the sample in a collection. 2. Define the statistic of interest (as a measure). 3. Create a sample with replacement (same n). 4. Collect the measures for lots of samples. 5. Analyze the distribution of collected measures.

Bootstrap Distribution of 1000 Atlanta Commute Means

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 : For the mean Atlanta commute time:

Using the Bootstrap Distribution to Get a Confidence Interval – Version # Keep 95% in middle Chop 2.5% in each tail

Using the Bootstrap Distribution to Get a Confidence Interval – Version # 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 95% CI=(27.33,31.00)

90% CI for Mean Atlanta Commute Keep 90% in middle Chop 5% in each tail For a 90% CI, find the 5%-tile and 95%-tile in the bootstrap distribution 90% CI=(27.52,30.68)

99% CI for Mean Atlanta Commute Keep 99% in middle Chop 0.5% in each tail For a 99% CI, find the 0.5%-tile and 99.5%-tile in the bootstrap distribution 99% CI=(27.02,31.82)

Other Parameters? Find a 95% confidence interval for the standard deviation, σ, of Atlanta commute times. Original sample: s=20.72

Other Parameters? Find a 98% confidence interval for the correlation between time and distance of Atlanta commutes. Original sample: r =0.807 (0.71, 0.87)

Questions? For more info: Patti Frazer Robin