What Can We Do When Conditions Aren’t Met? Robin H. Lock, Burry Professor of Statistics St. Lawrence University BAPS at 2011 JSM Miami Beach, August 2011.

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Presentation transcript:

What Can We Do When Conditions Aren’t Met? Robin H. Lock, Burry Professor of Statistics St. Lawrence University BAPS at 2011 JSM Miami Beach, August 2011

Example #1: CI for a Mean To use t* the sample should be from a normal distribution. But what if the sample is clearly skewed, has outliers, …?

Example #2: CI for a Standard Deviation Example #3: CI for a Correlation What is the distribution?

Alternate Approach: Bootstrapping “Let your data be your guide.” Brad Efron – Stanford University

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

Example #1: 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. Important point: The basic process is the same for ANY parameter/statistic. Bootstrap sample Bootstrap statistic Bootstrap distribution

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:

Example #2 : Find a confidence interval for the standard deviation, σ, of prices (in $1,000’s) for Mustang(cars) for sale on an internet site. Original sample: n=25, s=11.11 Bootstrap distribution of sample std. dev’s SE=1.61

Using the Bootstrap Distribution to Get a Confidence Interval – Method # 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=(27.34,31.96)

90% CI for Mean Atlanta Commute For a 90% CI, find the 5%-tile and 95%-tile in the bootstrap distribution Keep 90% in middle Chop 5% in each tail 90% CI=(27.52,30.66)

99% CI for Mean Atlanta Commute For a 99% CI, find the 0.5%-tile and 99.5%-tile in the bootstrap distribution Keep 99% in middle Chop 0.5% in each tail 99% CI=(26.74,31.48)

What About Technology? Possible options? Fathom R Minitab (macro) JMP Web apps Others? xbar=function(x,i) mean(x[i]) x=boot(Margin,xbar,1000) x=do(1000)*sd(sample(Price,25,replace=TRUE))

(coming soon)

Example #3: Find a 95% confidence interval for the correlation between size of bill and tips at a restaurant. Data: n=157 bills at First Crush Bistro (Potsdam, NY) r=0.915

Bootstrap correlations 95% (percentile) interval for correlation is (0.860, 0.956) BUT, this is not symmetric…

Method #3: Reverse Percentiles Golden rule of bootstraps: Bootstrap statistics are to the original statistic as the original statistic is to the population parameter

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 #3: Pool the 20 values and select two samples of size 10 (with replacement) 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

Materials for Teaching Bootstrap/Randomization Methods?