Simple Linear Regression...Simple Linear Regression Statistics: Unlocking the Power of Data Lock5 To Do Read Section 9.1 Do HW 9.1 (due Friday, 12/4) Title Sec9-1 Author Kari Lock

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Statistics:UnlockingthePowerofData Lock5

STAT250Dr.KariLockMorgan

SimpleLinearRegression

SECTION9.1 • Inferenceforcorrelation• Inferenceforslope• Conditionsforinference


QuestionoftheDay

Isthesizeofcertainregionsofyourbrainassociatedwiththesizeofyoursocialnetwork?


SocialNetworksandtheBrain� Datafrom 40studentsatCityCollegeLondon

�Howtomeasurebrainsize?

�Howtomeasuresocial networksize?

Source:R.Kanai,B.Bahrami,R.Roylance andG.Ree(2011).Onlinesocialnetworksizeisreflectedinhumanbrainstructure,ProceedingsoftheRoyalSocietyB:BiologicalSciences.10/19/11.


MeasuringBrainSize� Structural Magnetic Resonance Imaging (MRI)

� Voxel-based morphometry (VBM) tocomputeregional greymatter volume based onT1-weightedanatomical MRIscans

� Brainregions foundsignificant ininitial study¡ Amygdala (emotion and emotionalmemory)¡Middle temporal gyrus (social perception)¡ Entorhinal cortex(memory and navigation)¡ Superior temporal sulcus (perception of others)

� Response: normalized z-score ofgreymatterdensity forthese brain regions


BrainRegions

ImagefromDoourBrainsDetermineourFacebookFriendCount? (www.nature.com)


SocialNetworksandtheBrain� Howtomeasure sizeofsocialnetwork?

¡ Howmanywerepresentatyour18thor21stbirthdayparty?¡ Ifyouweregoingtohaveapartynow,howmanypeoplewouldyouinvite?

¡ What isthetotalnumberoffriendsinyourphonebook?¡ Write downthenamesofthepeopletowhomyouwouldsendatextmessage markingacelebratoryevent. Howmanypeopleisthat?

¡ Write downthenamesofpeopleinyourphonebookyouwouldmeet forachatinasmallgroup(onetothreepeople).Howmanypeopleisthat?

¡ Howmanyfriendshaveyoukeptfromschoolanduniversitywhomyoucouldhaveafriendlyconversationwithnow?

¡ Howmanyfriendsdoyouhaveon‘Facebook’?¡ Howmanyfriendsdoyouhavefromoutsideschooloruniversity?¡ Write downthenamesofthepeopleofwhomyoufeelyoucouldaskafavorandexpecttohaveitgranted.Howmanypeopleisthat?

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SocialNetworksandtheBrain

r =0.436

Istheassociationsignificant?


Parameter Distribution Standard Error

Proportion Normal

Difference inProportions

Normal

Mean t,df =n – 1

Difference inMeans t,df = min(n1, n2) – 1

Correlation t,df =n– 2

Standard ErrorFormulas

(1 )p pn−

2

nσ

1 1

1

2 2

2

(1 ) (1 )p p p pn n− −+

2 21 2

1 2n nσ σ+

1− ρ2

n− 2


SocialNetworks andtheBrain• IsthegreymattervolumeoftheseregionsofthebrainsignificantlycorrelatedwithnumberofFacebookfriends?

• Fromn=40people,wefindr =.436.Isthissignificant?


SocialNetworks andtheBrain0 : 0: 0aH

H ρρ=≠

t = r − 0

SE= 0.436

0.156= 2.99

This provides strong evidence that the grey matter density of these regions of the brain and number of Facebook friends are positively correlated.

n = 40 ≥ 30

1.Statehypotheses:

2.Check conditions:

3.Calculate test statistic:

4.Compute p-value:

5. Interpret incontext:

r = 0.436

SE = 1− 0.4362

40− 2= 0.156

p-value=0.0048


SocialNetworksandtheBrain

ShouldyougooutandaddmoreFacebookfriendstoincreasethesizeofyourbrain?

a) Yesb)No


R2

R2 istheproportionofthevariabilityintheresponsevariable,Y,thatis

explainedbytheexplanatoryvariable,X

� Forsimplelinearregression, R2 =r2 (R2 isjustthesamplecorrelation squared)

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R22 0.67R = 2 0.09R =

HowmuchdoesthevariabilityinYdecreaseifyouknowX?


RegressioninMinitab� Stat->Regression->FittedLinePlot

0.4362 =0.19


Prediction

Shouldyouusethisequationtopredictthenormalizedsizeoftheseregionsofyourbrain?

a) Yesb)No


SampletoPopulation� Everythingwehavedonesofarwithregressionisbasedsolelyonsampledata

�Now,wewillextendfromthesampletothepopulation

� Statistical inference!


• Thepopulation/true simple linearmodel is

𝑦 = 𝛽$ + 𝛽&𝑥 + 𝜀

• β0 and β1,areunknownparameters

• Can usefamiliar inference methods!

Intercept Slope

Simple Linear Model

Randomerror


InferencefortheSlope� Testforwhethertheslopeissignificantlydifferentfrom0(whetherthereisanylinearrelationshipbetweenxandy):

� Confidenceintervalforthetrueslope!!

H0 :β1 =0Ha :β1 ≠0

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• Confidenceintervalsandhypothesistests fortheslopecanbedoneusingthefamiliarformulas:

• PopulationParameter:β1,SampleStatistic:𝛽)&

• Uset-distribution withn – 2degreesoffreedom

Inference fortheSlope

sample statistic null valueSE

t −=

*sample statistic t SE± ×


RegressioninMinitabStat ->Regression->Regression->FitRegressionModel


Inference forSlope

Istheslopesignificantlydifferentfrom 0?(a)Yes(b)No

n =40

Givea95%confidenceintervalforthetrueslope.


Hypothesis Test


RegressioninMinitabStat ->Regression->Regression->FitRegressionModel


TwoQuantitative Variables• Thet-statistic (andp-value)foratest foranon-zeroslopeandatestforanon-zerocorrelation areidentical!

• Theyareequivalentwaysoftesting foralinearassociation betweentwoquantitativevariables.

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Confidence Interval


Multiple Testing?


FalsePositive (TypeIError) Protection

� Tofurtherprotect againstTypeIerrors, theyperformedtwoindependentanalysisontwoseparatesamples(n=125,thenn=40)


Real-WorldNetworkSize�Whataboutreal-worldnetwork size?


Inference based onthe simple linearmodel isonlyvalidifthe followingconditions hold:

1) Linearity2) Constant Variability ofResiduals3) Normality ofResiduals4) Independence

Conditions


• Therelationship between x and y islinear (itmakes sense todrawalinethrough thescatterplot)

Linearity

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DogYears

• 1dogyear=7humanyears• Linear:humanage=7×dogage

Charlie

• Fromwww.dogyears.com:“Theoldrule-of-thumbthatonedogyearequalssevenyearsofahumanlifeisnotaccurate.Theratioishigherwithyouthanddecreasesabitasthedogages.”

LINEAR

ACTUAL

A linearmodelcanstillbeuseful,evenifitdoesn’tperfectlyfitthedata.


“Allmodelsarewrong,butsomeareuseful”

-GeorgeBox


Residuals (errors)

( )~ 0,i N εε σTheerrorsarenormallydistributed

Theaverageoftheerrorsis0

Thestandarddeviationoftheerrorsisconstantforallcases

Conditions forresiduals:

Checkwithahistogram

(Alwaystrueforleastsquaresregression)

Constantspreadofpointsaround

thelineStatistics:UnlockingthePowerofData Lock5

RegressioninMinitabIstheassociationapproximatelylinear?a) Yesb) No

Isthespreadofthepointsaround thelineapproximatelyconstant?a) Yesb) No


HistogramofResidualsAretheresidualsapproximatelynormallydistributed?a) Yesb) No


Non-ConstantVariability

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Non-NormalResiduals


• Casesmustbeindependentofeachother(onecase’s valuesdoesnotaffect anothercase’svalues)

• Most common violationofthis:dataovertime

• Whatwouldmaketheindependenceconditionsatisfiedorviolatedinthesocial networkandbrainsizedata?

Independence


• Iftheassociation isn’tlinear:don’tusesimplelinearregression

• Ifvariabilityisnotconstant, residualsarenotnormal,orcasesnotindependent:Themodelitselfisstillvalid,butinference maynotbeaccurate

• Ifyouwanttodosomething morefancysotheconditionsaremet…takeSTAT 462!

Conditions notMet?


1) Plotyourdata!• Associationapproximatelylinear?• Outliers?• Constantvariability?

2) Fitthemodel(least squares)3) Usethemodel • Interpretcoefficients• Makepredictions

4) Lookathistogramofresiduals(normal?)5) Inference(extend topopulation)• Inferenceonslope (intervalandtest)

Simple Linear Regression


ToDo� ReadSection9.1

� DoHW9.1(dueFriday,12/4)

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Simple Linear Regression...Simple Linear Regression Statistics: Unlocking the Power of Data Lock5 To Do Read Section 9.1 Do HW 9.1 (due Friday, 12/4) Title Sec9-1 Author Kari Lock