Outline Analytic Inference for a Correlation

STAT 113Analytic Inference for Correlation

Colin Reimer Dawson

Oberlin College

19 April, 2016

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Outline Analytic Inference for a Correlation

Outline

Analytic Inference for a Correlation

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Outline Analytic Inference for a Correlation

Testing a Correlation

Are tip percentages at First Crush Bistro correlated with the size ofthe bill?

H0 : ρ = 0

H1 : ρ 6= 0

Sample: 157 bills, measuring X = total bill and Y =tip %Sample stat: r = 0.135 between X and Y

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Outline Analytic Inference for a Correlation

Randomization TestAssume H0: no relationship.Fix the data, but randomize the pairings.

Randomization distribution centered at ρ0 = 0. Pretty symmetricand bell-shaped (despite outliers).

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Outline Analytic Inference for a Correlation

Testing a Correlation

Param. Stat. Randomization Theory SE Test Dist.

p p̂ Simulate from p0

√p0(1−p0)

nNormal

µ x̄ Bootstrap + shift s√n

tn−1

pA − pB p̂A − p̂B Scramble groups√

p̂(1−p̂)nA

+ p̂(1−p̂)nB

Normal

µA − µB x̄A − x̄B Scramble groups√

s2A

nA+

s2B

nBtmin(nA−1,nB−1)

µD x̄D Flip pairs∗ sD√nD

tnD−1

ρ r Scramble pairings√

1−r2

n−2∗∗ tn−2

∗∗

* Not in the textbook, and we didn’t actually do randomization for this case** Provided the relationship is reasonably linear

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Outline Analytic Inference for a Correlation

Doing the t-test

Demo

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