Regression

Asastatisticalprocedure,simplelinearregressionhasalotoflimitations.Therealpowerofregressioncomeswithmultipleregressionandwhatwemeanbymultipleregressioniswhenwestartaddingindependentvariablestothemodel.So whenwethinkabouttheneedtoaddmorevariablestothemodel,I'dliketobeginwithsomethingofanexample.

Let'ssupposethatamarketinganalystwantstotesttherelationshipbetweenincomeandinternetusageandbelievesthatit'spositiveandprettystrong.Itmakessensetothinkthatpeoplewithmoreincomewouldhavebetterinternetaccess.Professionally,theymayneedmoreinternetaccess.Andsotheideabehinditis,thisanalystbelieves,thatasyourincomegoesup,weshouldalsoexpectyourinternetusagetogoup.Orsaidconversely,themoreinternetusageyouhave,chancesareyouhaveahigherincome.Andmoreover,thisparticularanalystbelievesthat'saprettystrongrelationship,feelsprettygoodabouthisorherconceptualhypothesisabouttherelationshipbetweenincomeandinternetusageandso,throughwhatevermeansoranother,acquiresdatathatteststherelationshipwithcorrelation.

Sotheycollectdatafromavarietyofdifferentrespondentsandmakesuretheyasktwothings--what's

Your

income,andhowmuchdoyouusetheinternet.Andlet'ssaythatthatcorrelationcomesback,well,lookingprettyweak.Ifwelookatascatterplotofthishypotheticaldata,perhapsweputinternetusageontheverticalaxisandincomeonthehorizontalaxis.Andlet'ssaythatitproducesadataplot,ascatterplot,thatlookslikethat--notmuchtoit.Andlet'ssaythatthecalculatedcorrelationcoefficientisonly0.04--nexttozero,basically,whichmeansalmostnocorrelationseemstoexistbetweeninternetusageandincome.

What'sgoingonhere?Isthemarketinganalystjustwrong?Arehisorherreallystrongintuitionabouttherelationshipbetweenthesevariablesjustthatfarofftarget?Oristheresomethingelseatplaythatwemightuncoverwiththedatabyaddingmorevariablestothissimplemodelthatsimplyincludesinternetusageand

income,butonlyproducesacorrelationof0.04amidstastrongbeliefthatthere'smoretoitgoingon?Well,wecantakealookatthemoretoitcomingup.

Suppose the analyst feels very confident in the relationship between internet usage and income and is quite surprised that the data do not support the hypothesis. What do you think might be the cause of the result? What might the analyst do to pursue this hunch further?