Starting Inference with Bootstraps and Randomizations Robin H. Lock, Burry Professor of Statistics St. Lawrence University Stat Chat Macalester College,

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Presentation transcript:

Starting Inference with Bootstraps and Randomizations Robin H. Lock, Burry Professor of Statistics St. Lawrence University Stat Chat Macalester College, March 2011

The Lock 5 Team Robin & Patti St. Lawrence Dennis Iowa State Eric UNC- Chapel Hill Kari Harvard

Intro Stat at St. Lawrence Four statistics faculty (3 FTE) 5/6 sections per semester students per section Only 100-level (intro) stat course on campus Students from a wide variety of majors Meet full time in a computer classroom Software: Minitab and Fathom

Stat Traditional Topics Descriptive Statistics – one and two samples Normal distributions Data production (samples/experiments) Sampling distributions (mean/proportion) Confidence intervals (means/proportions) Hypothesis tests (means/proportions) ANOVA for several means, Inference for regression, Chi-square tests

QUIZ Choose an order to teach standard inference topics: _____ Test for difference in two means _____ CI for single mean _____ CI for difference in two proportions _____ CI for single proportion _____ Test for single mean _____ Test for single proportion _____ Test for difference in two proportions _____ CI for difference in two means

When do current texts first discuss confidence intervals and hypothesis tests? Confidence Interval Significance Test Moorepg. 359pg. 373 Agresti/Franklinpg. 329pg. 400 DeVeaux/Velleman/Bockpg. 486pg. 511 Devore/Peckpg. 319pg. 365

Stat Revised Topics Descriptive Statistics – one and two samples Normal distributions Data production (samples/experiments) Sampling distributions (mean/proportion) Confidence intervals (means/proportions) Hypothesis tests (means/proportions) ANOVA for several means, Inference for regression, Chi-square tests Data production (samples/experiments) Bootstrap confidence intervals Randomization-based hypothesis tests Normal distributions Bootstrap confidence intervals Randomization-based hypothesis tests

Toyota Prius – Hybrid Technology

Prerequisites for Bootstrap CI’s Students should know about: Parameters / sample statistics Random sampling Dotplot (or histogram) Standard deviation and/or percentiles

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 might the “true” μ be?

“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.

Atlanta Commutes – Original Sample

Atlanta Commutes: Simulated Population

Creating a Bootstrap Distribution 1. Compute a statistic of interest (original sample). 2. Create a new sample with replacement (same n). 3. Compute the same statistic for the new sample. 4. Repeat 2 & 3 many times, storing the results. 5. Analyze the distribution of collected statistics. Try a demo with Fathom

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:

Quick Assessment HW assignment (after one class on Sept. 29): Use data from a sample of NHL players to find a confidence interval for the standard deviation of number of penalty minutes.

Example: Find a confidence interval for the standard deviation, σ, of hockey penalty minutes. Original sample: s=49.1 Bootstrap distribution of sample std. dev’s SE=11.3

Quick Assessment HW assignment (after one class on Sept. 29): Use data from a sample of NHL players to find a confidence interval for the standard deviation of number of penalty minutes. Results: 9/26 did everything fine 6/26 got a reasonable bootstrap distribution, but messed up the interval, e.g. StdError( ) 5/26 had errors in the bootstraps, e.g. n=1000 6/26 had trouble getting started, e.g. defining s( )

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=(27.24,31.03)

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=(27.60,30.61)

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=(26.73,31.65)

What About Hypothesis Tests?

“Randomization” Samples Key idea: Generate samples that are (a)based on the original sample AND (a)consistent with some null hypothesis.

Example: Mean Body Temperature Data: A sample of n=50 body temperatures. Is the average body temperature really 98.6 o F? H 0 :μ=98.6 H a :μ≠98.6 Data from Allen Shoemaker, 1996 JSE data set article

Randomization Samples How to simulate samples of body temperatures to be consistent with H 0 : μ=98.6? Fathom Demo

Randomization Distribution Looks pretty unusual… p-value ≈ 1/1000 x 2 = 0.002

Choosing a Randomization Method A=Caffeine mean=248.3 B=No Caffeine mean=244.7 Example: Finger tap rates (Handbook of Small Datasets) Method #1: Randomly scramble the A and B labels and assign to the 20 tap rates. H 0 : μ A =μ B vs. H a : μ A >μ B Method #2: Add 1.8 to each B rate and subtract 1.8 from each A rate (to make both means equal to 246.5). Sample 10 values (with replacement) within each group.

Connecting CI’s and Tests Randomization body temp means when μ=98.6 Bootstrap body temp means from the original sample Fathom Demo

Fathom Demo: Test & CI

Intermediate Assessment Exam #2: (Oct. 26) Students were asked to find and interpret a 95% confidence interval for the correlation between water pH and mercury levels in fish for a sample of Florida lakes – using both SE and percentiles from a bootstrap distribution. Results: 17/26 did everything fine 4/26 had errors finding/using SE 2/26 had minor arithmetic errors 3/26 had errors in the bootstrap distribution

Transitioning to Traditional Inference AFTER students have seen lots of bootstrap and randomization distributions… Introduce the normal distribution (and later t) Introduce “shortcuts” for estimating SE for proportions, means, differences, slope…

Final Assessment Final exam: (Dec. 15) Find a 98% confidence interval using a bootstrap distribution for the mean amount of study time during final exams Results: 26/26 had a reasonable bootstrap distribution 24/26 had an appropriate interval 23/26 had a correct interpretation

What About Technology? Possible options? Fathom/Tinkerplots R Minitab (macro) JMP (script) Web apps Others? xbar=function(x,i) mean(x[i]) b=boot(Time,xbar,1000) Try a Hands-on Breakout Session at USCOTS! Applet Demo

Support Materials? We’re working on them… Interested in class testing?