1

SupplementaryInformation:SupplementaryMethod.

Earlynon-immersionlearners.Subjectsreportinglearninginanon-immersion

environmentbeginningat1,2,or3yearsofageexhibitedstrangeresults(Fig.S1).Asnoted

inthemaintext,thesewereexcluded.

FigureS1.Performancecurvesfornon-immersionlearnerswithagesoffirstexposureofone,two,orthreeyears(indicatedbynumbersoverlaidonthelines).

Modelsofchangesinthelearningrate

Datapreparationwasidenticaltothatofthepermutationanalyses.Inordertofind

parametervaluesthatminimizedR2,weemployedDifferentialEvolutionfollowingalocal-

to-beststrategy,with500iterationsandapopulationsizeof10xthenumberofparameters

(Mullenetal.,2011).

1

11

1 1 1

1

1

1

11 1

2 2 2 2 2 22

2

22

2

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3 33 3

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years of experience

2

Fivedifferentmodelswereconsidered.TheExponentialLearningwithSigmoidal

Decay(ELSD)modelispresentedinthemaintext.Performancecurvesarederivedby

combiningthetwoequationsinthemaintextandintegrating:

𝑔 𝑡

=

1 − 𝑒()*+ ,(,- 𝑎 + 𝑏, 𝑡2 ≤ 𝑡4,𝑡 ≤ 𝑡4

(1 − 𝑒()*+ ,6(,- − 1)𝑒()*+ ,(,6 8

9: ;<

982=>?

982> @=@6=? 𝑎 + 𝑏, 𝑡 > 𝑡4, 𝑡2 ≤ 𝑡4

1 − 𝑒()*+ ,(,- 8

9: ;<

982> @-=@6=?

982> @=@6=? 𝑎 + 𝑏, 𝑡 ≥ 𝑡2 > 𝑡4

withtheadditionalofscaleparametersaandb,whichweresetto2.0and1.5,respectively,

inorderfortheresultstospantheempiricalrangeinlog-odds.TheELSDmodeliscapable

ofcapturingawiderangeofpossibilitiesintermsofhowlearningabilitychangesoverthe

lifespan(Figs.1,4E,S2).

FigureS2.TheELSDmodelcanconsiderlearningratedeclinesthatareslow(A),rapid(B),

0.000.250.500.751.00

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A B

C D

E F

Figure S2

current age current age

current age current age

current age current age

3

ordiscontinuous(F),andwhichoccuratanyagebetween1and40(compareBwithCwithE).Intheseexamples,theinitiallearningrateissetto1.0inordertobetterhighlightthedifferentshapepossibilities;however,initiallearningrateisalsoaparameterthatmustbefit.A:tc=1,r0=1,α=0.1,δ=34.B:tc=1,r0=1,α=0.5,δ=10.C:tc=1,r0=1,α=1,δ=35.D:tc=40,r0=1,α=1,δ=3.E:tc=40,r0=1,α=1,δ=-20.F:tc=20,r0=1,α=0.1,δ=0.

WealsoconsideredmodifiedversionsofELSD:adiscontinuousratechangemodel,

namelyasimplestepfunctioninwhichthelearningratechangedfromr0tor1atagetc,and

aflatratemodel,wherethelearningrateremainedconstant.Wealsoincludedvariantsof

theELSDandthediscontinuousratechangemodelinwhichthelearningrateschangedasa

functionofyearsofexperienceratherthanage.Inallcases,weusedthesamevaluesforthe

scaleparametersaandb(2.0and1.5,respectively).Thebestfittingparametersforthese

modelsaregiveninFigs.S4-S8.Foreaseofcomparison,theempiricaldatafromFigs.4and

7havebeencombinedinFig.S3.

NotethatDifferentialEvolutionrequiresdefiningarangeofpossiblevaluesforeach

parameter.Forallmodelsthelearningratewasconstrainedtobebetween0and1.Theage

atwhichthelearningratebegantochangeintheELDSanddiscontinuousmodelwas

constrainedtobebetween1and40yearsofageorexperience,asappropriate.IntheELSD

models,αcouldrangefrom0to1andδcouldrangefrom-50to+50.Theexperience

discountfactorEwassetto1formonolingualsandcouldrangebetween0and1for

simultaneousbilinguals,laterimmersionlearners,andnon-immersionlearners.

4

FigureS3.Empiricallymeasuredlearning-curves.A-B:Performanceasafunctionofyearsofexperienceformonolingualsandimmersionlearners(A)andnon-immersionlearners(B).C-D:Performanceasafunctionofageformonolingualsandimmersionlearners(C)andnon-immersionlearners(D).Ageofexposureisindicatedoneachcurve.

Inordertocomparethefitsofdifferentmodels,weconductedtenrunsofMonte

Carlosplit-halfcross-validation,splittingtherawdata,nottheaverageddata.Inpairwiset-

tests,theresultingR2sweresignificantlyhigherfortheELDSmodelthanforanyother

model.Thereasonisvisibleinthegraphs:theflat-ratemodelandthemodelswithrate

m0

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Figure S3

current age current age

5

changesbasedonexperienceratherthanagecouldnotcapturethedifferencesinultimate

attainmentacrossexposureages.Thediscontinuousrate-changemodelfitsnearlyaswell

asELSD,butcannotdistinguishtheslopesoftheperformancecurvesamongparticipants

whobeganlearningatvaryingamountsoftimeafterthelearningratechanged(thatis,in

adulthood).

Notethatbecauseourinterestwasinchangesinlearningasafunctionofageoffirst

exposureandexperience,themodelwasfittoperformancecurves,nottotheaggregate

dataasawhole:fittingtotherawdatadirectlywouldhaveoverweightedthemonolinguals,

whocontributednearlyhalfofthedata,andunderweightedthelaterlearners,vitiatingthe

goalsoftheanalysis.Thus,forinstance,wecalculatedthesquareddifferencebetweenthe

predictedvalueforimmersionlearnerswhobeganat5yearsandhad10yearsof

experienceagainstthemeanempiricalperformanceatthatpoint.

Notethatwesmoothedtheperformancecurveswithfive-yearfloatingwindowsin

ordertodampennoise.Weconfirmedthatsmoothingthedatadidnotaffectthepatternof

results:R2sbasedonnon-smootheddatawerelower,asexpected,buttheELSDmodelstill

fitsignificantlybetterthanallothermodels(R2=.66),andprovidedasimilarestimatefor

whentheunderlyinglearningratebegantochange(18.2yearsold).

Wealsoranthemodelsusingpercentcorrectasthemeasureofaccuracyrather

thanlog-odds,bothwithandwithoutsmoothedperformancecurves.Inbothcases,the

ELSDmodelfitsignificantlybetterthantheothersandproducedsimilarestimatesforthe

ageatwhichlearningratebeginstodecline(17.9yearsand18.1years,respectively).

6

FigureS4.Thebest-fittingflat-ratemodel(a,withr=.15;E=1.00,.72,.55,.13formonolinguals,simultaneousbilinguals,laterimmersionlearners,andnon-immersionlearners,respectively),R2=.66.A-B:Predictedperformanceasafunctionofyearsofexperienceformonolingualandimmersionlearners(A)andnon-immersionlearners(B).C-D:Predictedperformanceasafunctionofageformonolingualsandimmersionlearners(C)andnon-immersionlearners(D).E:Estimatedlearningrateasafunctionofage.

-0.20.00.20.40.6

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m0 2,5,8,

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Figure S4

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FigureS5.Thebest-fittingdiscontinuousratechangemodel(tc=20.7,r0=.20,r1=.04;E=1.00,.61,1.00,.27formonolinguals,simultaneousbilinguals,laterimmersionlearners,andnon-immersionlearners,respectively).R2=.86.A-B:Predictedperformanceasafunctionofyearsofexperienceformonolingualandimmersionlearners(A)andnon-immersionlearners(B).C-D:Predictedperformanceasafunctionofageformonolingualsandimmersionlearners(C)andnon-immersionlearners(D).E:Estimatedlearningrateasafunctionofage.

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Figure S5

current age current age

8

FigureS6.Thebest-fittingELSDmodel(tc=17.4,r=.20,α=.09,δ=.18;E=1.00,.63,1.00,.29formonolinguals,simultaneousbilinguals,laterimmersionlearners,andnon-immersionlearners,respectively).R2=.89.A-B:Predictedperformanceasafunctionofyearsofexperienceformonolingualandimmersionlearners(A)andnon-immersionlearners(B).C-D:Predictedperformanceasafunctionofageformonolingualsandimmersionlearners(C)andnon-immersionlearners(D).E:Estimatedlearningrateasafunctionofage.

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Figure S6

current age current age

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FigureS7.Thebest-fittingdiscontinuousratechangemodelwherethediscontinuityhappensafterasetnumberofyearsofexperienceratherthanatasetage(tc=11.2,r0=.18,r1=.06;E=1.00,.76,.57,.18formonolinguals,simultaneousbilinguals,laterimmersionlearners,andnon-immersionlearners,respectively).R2=.71.A-B:Predictedperformanceasafunctionofyearsofexperienceformonolingualandimmersionlearners(A)andnon-immersionlearners(B).C-D:Predictedperformanceasafunctionofageformonolingualsandimmersionlearners(C)andnon-immersionlearners(D).E:Estimatedlearningrateasafunctionofage.

0.100.15

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m0

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Figure S7

current age current age

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FigureS8.Thebest-fittingELSDvariantwherethediscontinuityhappensafterasetnumberofyearsofexperienceratherthanatasetage(tc=6.0,r=.22,α=.05,δ=7.8;E=1.00,.72,.53,.17formonolinguals,simultaneousbilinguals,laterimmersionlearners,andnon-immersionlearners,respectively).R2=.70.A-B:Predictedperformanceasafunctionofyearsofexperienceformonolingualandimmersionlearners(A)andnon-immersionlearners(B).C-D:Predictedperformanceasafunctionofageformonolingualsandimmersionlearners(C)andnon-immersionlearners(D).E:Estimatedlearningrateasafunctionofage.

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Figure S8

current age current age

11

Ultimateattainment.

InordertoimprovereadabilityofFigure6,meansandstandarderrorswere

calculatedusingathree-yearfloatingwindow,andcurveswererestrictedtoconsecutive

windowswithmorethantensubjects.However,datafromallsubjectswithatleast30

yearsofexperienceandnomorethan70yearsofagewereincludedinanalyses.This

resultedin107,125monolingualsavailableforanalysis.Numbersforimmersionlearners

andnon-immersionlearnersaregiveninthemaintext.

Weidentifiedsignificantchangesintheslopeoftheultimateattainmentcurveusing

multivariateadaptiveregressionsplines(MARS)(Friedman,1991)asimplementedinthe

earthpackageforR(Milborow,2014).MARSsuccessivelybreakslinearregressionlines

intomultiplesegments,eachwithitsownslope.Itthenprunesbreakpointsthatdolittleto

improvefit.Tofurtheravoidoverfitting,weused50-foldcross-validation.

Inordertoensurethattheresultswererobusttothemethodofbreakpoint

estimation,wealsoconsideredtwoothermethodsthatidentifybreakpoints.Thesecond

method(segmented)isaniterativesearchalgorithmthatfindsoptimalplacementfora

specifiednumberofbreakpoints(Muggeo,2014).Thelocationsofbreakpointschosenby

thisalgorithmoftendependontheinitialfirstguessastothelocationofthebreakpoints

(the“seed”),whichissetbytheresearcher.Weusedseveralprocedurestominimizeeffects

oftheseedandincreasethechancesoffindingtheoptimalplacementofthebreakpoints.

Foreachnumberofbreakpoints,weranthealgorithmwiththreedifferentsetsofstarting

seeds,choosingthebest-fittingresult.Moreover,weemployedabootstraprestarting

procedurewith25randomlyjitteredsamples.Wefitsegmentedmodelswith0,1,2,3,and

12

4breakpointsandchosethebest-fittingmodelbasedontheBayesianInformation

Criterion.

Thethirdmethod(optimalbreakpointplacement)wasaprocedurerecommended

byVanhove(2013),generalizedtomultiplebreakpoints(indevelopingthisgeneralization,

weareindebtedtocodewrittenbyDavidHitchcockoftheUniversityofSouthCarolina,

postedathttp://www.stat.sc.edu/~hitchcock/raw_piecewise_Rexample705.txt).We

consideredeverypossiblecombinationof0,1,2,or3breakpoints,withtherestrictionthat

thebreakpointmustbeplacedonawholenumberofyears.Foraspecificnumberof

breakpoints(e.g.,3),wechosethemodelwiththesmallestdeviation.Wethenchosefrom

amongtheresultingmodelsusingtheBayesianInformationCriterion.Resultsforallthree

setsofanalysesareshowninTableS1.Ultimateattainmentbegantodeclinerapidlyfor

immersionlearnersatageofexposureofabout12inallthreeanalysesandfornon-

immersionlearnersataround9,similartotheestimatesobtainedbyMARS.Ofthesix

analyses,fiveshowednoevidenceofaslowingofthedecline;thesixth(theoptimal

analysisappliedtothenon-immersionlearners)showedevidenceofaslowing—though

stillongoing—declineafteranexposureageof19.

TableS1.Estimatedbreakpoints.

Learners Method Breakpoints SegmentSlopesImmersion MARS 12 -.009,-.06 Segmented 11.4(2.4) -.007,-.04 Optimal 12 -.007,-.04 Non-Immersion MARS 9 +.01,-.06 Segmented 10.5(0.2) -.005,-.07 Optimal 9,12,19 +.01,-.05,-.10,-.02

13

Notethatinallcasestheresultingbreakpointsarestatisticallysignificantinthe

sensethattheyresultinoptimalfitsasjudgedbycross-validationortheBayesian

InformationCriterion.However,placingconfidenceintervalsontheplacementofthe

breakpointsortheslopesoftheresultingsegmentsisnon-trivialandremainsanareaof

activeresearch(segmentedprovidesconfidenceintervals,butthesearelikelyoverfitted,

sincetheyassumethenumberofbreakpointsisknown).Forthisreason,inthemaintext

wefocusoneffectsizes,asmeasuredintermsofthestandarddeviationofscoresby

simultaneousbilinguals(immersionlearnerswithanagefirstexposureof0).

Permutationanalyses.Performancecurves(proficiencyasafunctionofyearsof

experience)wereplottedfornon-immersionlearnersateachageofexposurefrom4to30,

andalsoformonolinguals(FigureS3A-B).Eachperformancecurvewasrestrictedto

consecutiveagesforwhichtherewereatleasttenparticipantsinthefive-yearwindow,

leaving244,840monolinguals,44,412immersionlearners,and257,998non-immersion

learners.Theotherdetailsofthisanalysisareinthemaintext.

SimulationsofPriorUltimateAttainmentStudies

Ineachsimulation,wesampledmonolingualsandimmersionbilingualswith

replacementfromourowndata.Allsubjectswererequiredtohaveatleast30yearsof

experiencewithEnglish,andimmigrantswererequiredtohaveminimalexposureto

Englishpriortoimmigration(followingthesamedefinitionfor“immersionlearners”used

elsewhereinthispaper).Forsimulatingmid-sizedstudies(N=275),therewerenoother

restrictions.Forsimulatinglargestudies(N=11,371),wematchedthenumberofboth

monolingualsandimmersionbilingualsasinourdataset(giventhejust-mentioned

constraints).Forsimulatingsmallstudies(N=69),wematchedthedemographicsof

14

JohnsonandNewport’s69subjectsascloselyaspossibleexceptwhereitconflictedwith

thejust-mentionedconstraints(30yearsofexperience,limitednon-immersionexposure).

Thatis,wesampled46non-nativespeakersand23nativespeakersfromourdatawith

replacement;thenon-nativeEnglishspeakersspokeaChineselanguageorKoreanandhad

immigratedtotheUnitedStates,and;subjectsmatchedtheageofimmigrationasreported

intheirpaper(seetheirTable1).JohnsonandNewportprovidenodemographic

informationaboutthenativespeakingcontrols.Thus,weselectedthe23nativespeakers

whowerebetween17and22yearsold,inclusive,ontheassumptionthatJohnsonand

Newport’snativespeakerswereundergraduates.

EffectofAnalysisDecisions.

Manypriorstudieshaveincludedimmigrantswhohadsignificantamountsof

educationinthetargetlanguagepriortoimmigration(Hakutaetal.,2003;Johnson&

Newport,1989).Thisraisesanissue:shouldresearchersdatetheonsetoflearningtothe

ageatfirstexposureortheageatimmigration(Johnson&Newport,1989)?Eitheroption

introducesimprecision:thefirsttreatsimmersionandnon-immersionlearningequally,

whereasthelatterassumesnon-immersionlearningiscompletelyineffective.Thus,either

optionintroducesbothnoise(sincepre-immigrationexposurevariesbetweensubjectsand

betweenstudies)andbias(sinceolderimmigrantstypicallyhavemorepre-immigration

exposure;seeJohnson&Newport,1989;Flegeetal.,1999).FollowingDeKeyseretal.

(2010),wesidesteppedthisproblembyexcludingimmigrantswhohadsubstantialpre-

immigrationexposuretoEnglish.

Similarly,ultimateattainmentanalysescomparelearnerswhohaveatleastXyears

ofexperience.Intheabsenceofsoliddata,priorresearchersemployedavarietyofcut-offs

15

(Fig.5A).Ourlargersampleallowedustoidentifytheappropriatecutoffdirectly:about30

years(atleastforourdata).Thissuggeststhatthesmallercut-offsinpreviousstudies

disadvantagedlater-learners,who(forobviousreasons)tendtohavelessexperienceand

thusarefartherfromasymptote.Moreover,differentstudiesuseddifferentcut-offs,which

canintroducevariabilityinultimateattainmentcurves(Fig.S9).

FigureS9.Ultimateattainmentcurvesrevealedbyourdata,usingdifferentcut-offsforminimumyearsofexperience.

However,whiletheseanalyticdecisionscanhaveasignificanteffectforstudieslike

ours,whichhavelargedatasets(Fig.S9),theyappeartohavelittleeffectontypically-sized

studies.WeconcludedthisbasedonsimulatingJohnson&Newport’s(1989)studywithor

withouttheanalyticdecisionsabove.Thatis,inthesimulationsdiscussedinthemaintext,

wesimulatedrunningJohnson&Newport’sstudywiththesamenumberofsubjectshailing

fromthesamecountriesandarrivingatthesameages,butotherwiseusingourown

analyticdecisions(limitedpre-immigrationexposure,andatleast30yearsofexperience

16

withEnglish).WeranasecondsetofsimulationsinwhichwematchedJohnson&

Newport’sanalyticdecisions,matchingtheirsubjects’numbersofyearsinAmericaand

numberofyearsofpre-immigrationexposure(asreportedintheirpaper).Ascanbeseen

inFig.S10,therangeofresultswassimilar,suggestingthattheseanalysesdecisionswerea

fairlyminorcontributortothedifferencesacrosspriorstudies.

FigureS10.Weconductedtwosetsof2,500simulatedexperiments.Inthefirstset(toprow),wedrew69subjectsfollowingthedemographicsofJohnson&Newport(1989)ascloselyaspossible(includingnativelanguage,pre-immigrationEnglishexposure,andageattest).Likewise,wefollowedtheminconductinganalysesintermsofageofarrivalratherthanageoffirstexposure.Wemodifiedthesemethodsinthesecondset(bottomrow)tomatchwhatweusedinourownanalysesbyrestrictingimmigrantstominimalpre-immigrationexposure,followingthedefinitionusedinthemaintext.NotethatthesesimulationsarealsoreportedinFig.8(toprow).AsinFig.8,threeanalyseswereconsidered.Fromlefttoright:correlationbetweenonsetageandultimateattainmentpriorto16yearsold.minusafter16yearsold;firstsubgroupofsubjectstobesignificantlyworsethanmonolingualsinat-test;onsetageatwhichperformancebeginstodeclinemorerapidly,ifany.Blue:estimatesfrompriorstudies.Red:estimatesfromcurrentstudy.Notethatthey-axisvariesacrosspanels.

ItemEffects.

Itisplausiblethatcriticalperiodeffectsmightdifferforthoseaspectsofgrammar

thataretypicallymasteredearlyinfirst-languageacquisitionasopposedtothosethatare

Figure SY

Analyses followingJohnson & Newport

(1989)

correlation:onset<16 - onset>16

end of optimal period:t-test

end of optimal period:segmented regression

0.00

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ity

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ity0.00

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Analyses followingcurrent study

17

masteredlate.Ourdata,however,providedlittlesupportforthishypothesis:Thebest-

fittingmodelsoflearningindicatedthatlearningratebegantoslowatapproximatelythe

sametimeforthe47itemsthataremasteredbytheyoungestmonolingualEnglish-

speakersinoursample(ages7-8)asforthe48itemsthataremasteredonlybytheolder

ones:17.3yearsoldand18.2yearsold,respectively.1

Likewise,ultimateattainmentanalysisandpermutationanalysisoftheperformance

curvesbothsupportedthisfinding,withonecomplication:Veryfewimmersionlearners

missedanyoftheearly-mastereditems.Thus,whilebreakpointanalysesofultimate

attainmentcurves(FigureS11)usingMARSfoundthatattainmentbegantodropsteeply

fornon-immersionlearnersatagesofexposureof12yearsforearly-mastereditems(B=-

.12)andat9yearsforlate-mastereditems(B=-.07),andforimmersionlearnersat9years

forlate-mastereditems(B=-.04),immersionlearnersremainedneartheceilingonearly-

mastereditemsregardlessoftheirageoffirstexposure:Evenimmersionlearnerswho

beganlearningEnglishat25yearsoldmissed,onaverage,fewerthanoneoftheearly-

mastereditems.Thus,althoughMARSwasabletoidentifyabreakpoint(3yearsold),

performancedeclinedonlynegligiblyafterthatage(B=-.01).(Thereisanapparentdecline

1Itemswereconsideredmasterediftheywereansweredcorrectlybyatleast22ofthe23monolingualEnglish-speakersages7-8(wecombinedthetwoyoungestagecategoriesinordertoachievesensibleN).Whilethiswasasomewhatarbitrarychoice,itwastheonlyoneweconsidered,mitigatingsomewhatconcernsaboutposthocanalyses.Asinthemainanalyses,thebest-fittingmodelsinvolvedsigmoidalratechange(R2=.85forearly-mastereditemsandR2=.87forlate-mastereditems).Early-mastereditems:tc=17.3,r=.27,α=.07,δ=-4.1;E=1.00,.42,1.00,.22.Late-mastereditems:tc=18.2,r=.17,α=.10,δ=2.3;E=1.00,.65,.95,.33.Recallalsothatthenaturalrangeofthemodelis(0,1),andthusweusedscaleparametersaandb(2and1.5,respectively)tomapthemodel’srangeontotheempiricallyobservedrangeofscores[inthecaseoftheprimaryanalyses,approx.(1.5,3.5);cf.FigureS3].Thus,parameterbhadtobeadjustedto2.25and1fortheearly-masteredandlate-mastereditems,respectively,eachofwhichhasadifferentempiricalrange[approx.(2.25,4.25)forearly-mastereditemsand(1,3)forlate-mastereditems].

18

startingataround19yearsold,butitisnotstatisticallysignificant.)

FigureS11.Ultimateattainmentforearly-mastereditems(A)andlate-mastereditems(B),smoothedforpresentationwithathree-yearfloatingwindow.Shadowsrepresent+/-1SE.AsinFigure2,dataweresmoothedbyathree-yearfloatingwindow,andonlyconsecutivewindowswithmorethan10subjectsshown.

Permutationanalysisofthelearningcurves(FigureS12)revealssimilarresults.

Performancecurvesarereliablyshallowerfornon-immersionlearnersbyanageoffirst

exposureof10yearsforearly-mastereditems(p=.009)andby12yearsforlate-mastered

items(p=.03).2Immersionlearnersshowsignificantlyshallowerperformancecurvesby

anexposureageof11yearsforlate-mastereditems(p=.002).3However,forearly-

2Onearly-mastereditems,thereisasignificanteffectat7yearsold(p=.03),butthisdisappearsfor8yearsold(p=.09)and9yearsold(p=.36)andsoislikelyduetonoise;incontrast,everyperformancecurvefrom10yearsoldonshowsasignificanteffect.Notethatthroughout,theseanalysesarenotcorrectedformultiplecomparisons.3Infact,thereisasignificantdifferenceat2yearsold(p=.04),butnotat5(p=.89)or8(p=.28),sothisisagainprobablynoise.Noteagainthattheseanalysesarenotcorrectedformultiplecomparisons.

1.5

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A B

immersion learners non-immersion learners

99%

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accu

racy

(per

cent

)

99%

98%

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93%

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accu

racy

(per

cent

)

19

mastereditems,immersionlearnersdonotshowsignificantlyshallowerperformance

curvesuntilanageoffirstexposureof23years.Again,thisfinalresultmaybeinfluenced

byceilingeffects:by10yearsofexperience,nearlyalltheseperformancecurvesareabove

3.4(equivalenttomakingasingleerror).

Wealsoconsideredwhethertherewasconsistencyinitemdifficultyfordifferent

typesoflearners.Inordertoavoidceilingeffects,weconsideredlearnerswith7-10years

ofexperience,thuscapturingtheearlieststageoflearningforwhichwehavedataforevery

learnertype.Withinthisgroup,wecomparedmonolinguals(N=82),simultaneous

bilinguals(N=35),andimmersionlearnerswithexposureages1-5(N=77),6-10(N=

314),11-15(N=287),and16-20(N=82).

Foreverybilingualgroup,by-itemperformancewashighlycorrelatedwiththatof

monolinguals:simultaneousbilinguals(r=.75,logBF=35.8,p<.0001),immersionlearners

withexposureages1-5(r=.81,logBF=47.4,p<.0001),immersionlearnerswithexposure

ages6-10(r=.81,logBF=46.1,p<.0001),immersionlearnerswithexposureages11-15(r

=.77,logBF=39.2,p<.0001),andimmersionlearnerswithexposureages16-20(r=.73,

logBF=37.4,p<.0001),wherelogBFisthelogoftheBayesFactorp(H1|d)/(p(H0|d)(see

Wagenmakers,2007).Importantly,thecorrelationforthelatestimmersionlearners(r=

.73)wasalmostidenticaltothatfortheearliestimmersionlearners(r=.75).

20

FigureS12.Performancecurvesforearly-learneditemsareshowforimmersionlearnersin(A)andnon-immersionlearnersin(B).Performancecurvesforlate-learneditemsareshownforimmersionlearnersin(C)andnon-immersionlearnersin(D).Notethatthey-axisscaleisdifferentforthetoptwopanelsvs.thebottomtwo.

L1Effects

Inthissection,weassessevidencethatcertainaspectsoflearningofEnglishare

significantlydifferentforoneofthelanguagegroups(Chinese,WesternGermanic,etc.;see

maintext)relativetotheothers.Wefocusedonimmersionlearners,wheretherangein

m0

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monolingualsage of exposure: 0-9 y.o.

age of exposure: 10-19 y.o.age of exposure: 20-30 y.o.

Figure S10

current age current age

current age current age

21

outcomesislarger.Weconsidereddifferencesinasymptoticperformance(ultimate

attainment),thelengthoftheoptimalperiod,andtheshapeoflearningcurves.

Foreachtypeofanalysis,wefirstconductedaseriesof“power”simulationsto

determineourabilitytodetecteffectsofvarioussizes.4Thesesimulationsservethree

purposes:1)determiningthelikelihoodwecoulddetectmeaningfuldifferencesinourdata,

giventhesamplesizesavailable;2)providingsomeguidanceonsamplesizefor

researcherswhoaredesigningfollow-upstudies,and;3)providingsomeintuitioninto

BayesFactorsforreadersunfamiliarwiththem.Forthepurposesof(1),itwouldbeideal

tousetheactualNsfromourdataandtakeintoaccounttheunevendistributionofsubjects

acrossconditions.However,wefoundthatthisresultsincomplex,confusinggraphs(cf.3),

andwasnotespeciallyhelpfulforprovidingguidanceonsamplesizes(cf.2),sinceit

merelyshowsthatinmostcaseswehavelimitedpower,ratherthanindicatinghowmany

subjectswouldbeneededformorepower.Thus,weelectedtousearangeofbalancedNs,

whichwebelievewillultimatelybethemostusefultothereader.

Tocompareperformanceacrosspopulations,weusedBayesFactorstocomparea

modelwherethatlanguagegroupistreatedseparatelyfromtheothers(M1)againstthe

nullmodelthattreatsalllanguagegroupsthesame(M0).BayesFactorsrepresenthow

muchmorelikelythedataareunderonemodelcomparedtotheother:

4Strictlyspeaking,“statisticalpower”referstoaconstructinnullhypothesissignificancetesting(theprobabilityofrejectingthenullhypothesisgivenaparticulareffectsizeandsamplesize).However,thereisafairlystraightforwardextensiontoBayesFactorAnalysis:theprobabilityofthedatafavoringthealternativehypothesis(logBF>0)givenaparticulareffectsizeandsamplesize.Wehopethatthereaderwillforgivethisabuseofterminology,sinceitresultsinmuchsimplerprose.

22

𝐵𝐹9E = 𝑃(𝐷|𝑀9)𝑃(𝐷|𝑀E)

wherethesubscriptsofBFdenotewhichmodelisthenumerator:

𝐵𝐹E9 = 1

𝐵𝐹9E=

𝑃(𝐷|𝑀E)𝑃(𝐷|𝑀9)

ThusBF10=3meansthatthedataarethreetimesmorelikelyunderM1relativetoM0.We

usethenaturallogarithmoftheBayesFactorbecauseitmakes0clearlyinterpretable:

log(BF10)<0moreevidenceforM0

log(BF10)=0equalevidenceforM1andM0

log(BF10)>0moreevidenceforM1

BayesFactorshavemanyadvantagesoverp-values,includingthefactthattheyquantify

evidenceforthenullhypothesis(Wagenmakers,2007).Mostimportantly,theyare

guaranteedtoselectthecorrectmodelasNincreasestoinfinity.P-values,incontrast,have

afixedTypeIerrorrateof0.05.However,BayesFactorscanbecomplextocalculate.

Throughout,weusetheBICapproximationtotheBayesFactor(Wagenmakers,2007).At

smallsamplesizes,thismethodwilltendtofavorthenull(M0)morethandoother,less

tractablealternatives(assamplesizesincrease,thedifferencedisappears).However,this

matchespsychologists’commonlystatedpreferenceforfavoringthenullhypothesis.Inany

case,thealternativesprovedintractablewithdatasetsaslargeasours.

Ideally,wewouldtreatbothsubjectsanditemsasrandomfactors(Baayen,

Davidson,&Bates,2008;Clark,1973).Unfortunately,itisunclearhowtocalculateBICfor

suchmodels.Whilesomeapproximationshavebeensuggested,wegenerallyfoundthat

thisgaveusunreliableresultsforunbalanceddesigns.Thus,weusefixedeffectsmodels

throughout.

23

AsymptoticPerformance.Weaskedwhetherasymptoticperformance(definedas

performancebyindividualswithatleast30yearsofexperienceandnomorethan70years

old)differsreliablydependingonfirstlanguage.Wefirstpresentpoweranalysesassessing

ourabilitytodetectmeaningfuldifferences,followedbyouractualresults.

Weconductedpoweranalysesthroughsimulation.Tosimulatethecasewherethere

isnodifferencebetweengroups,wedrewtwogroupsofNsubjectsfromtheasymptotic

simultaneousbilingualdata,samplingwithreplacement.Tosimulatedifferencesbetween

groupsofsizee(measuredasdifferenceinlog-odds),wefirstfitabinomialmixedeffects

modeltotheasymptoticsimultaneousbilingualdatausinglme4inR.Wethenused

predict.merModfromthelme4packagetogeneratedatafortwogroupsofNsubjects:one

withtheoriginalinterceptandonewithaninterceptthatdifferedbyefromtheoriginal.

Thismethodallowedustotakeintoaccountcorrelationsbetweenitemsandwithin

subjects.Wethenrecalculatedthelog-oddsofacorrectanswerforeachsubject(again,

usingtheempiricallogitfunction)andcomparedtwofixed-effectlinearregressionmodels:

onewithafixedeffectofsubjectgroup,andonewithout.

ForeverylevelofN,weconducted200simulationsfore=0and100simulationsfor

allotheres.Weincludedbothpositiveandnegativeesinordertoaccountforanyceiling

effects(largepositiveeswerenotpossiblebecauseofceiling).Theresultsofour

simulationsareshowninFig.S13.

24

FigS13.ExpectedBayesFactorsbasedonarangeofNspercondition(panels)andeffectsizes(x-axis).Forreference,e=0.145isthedifferencebetweenasymptoticsimultaneousbilingualsandmonolinguals.Notethatthey-axisscalevariesacrosspanels.

0

20

40

0 .1 −.1 .145−.145 .25 −.25 .5 −.5 −.75 −1effect size in log−odds

log(

Baye

s Fa

ctor

)N=50

0

25

50

75

0 .1 −.1 .145−.145 .25 −.25 .5 −.5 −.75 −1effect size in log−odds

log(

Baye

s Fa

ctor

)

N=100

0

50

100

150

200

0 .1 −.1 .145−.145 .25 −.25 .5 −.5 −.75 −1effect size in log−odds

log(

Baye

s Fa

ctor

)

N=250

0

100

200

300

0 .1 −.1 .145−.145 .25 −.25 .5 −.5 −.75 −1effect size in log−odds

log(

Baye

s Fa

ctor

)

N=500

0

200

400

600

0 .1 −.1 .145−.145 .25 −.25 .5 −.5 −.75 −1effect size in log−odds

log(

Baye

s Fa

ctor

)

N=1000

0

500

1000

1500

0 .1 −.1 .145−.145 .25 −.25 .5 −.5 −.75 −1effect size in log−odds

log(

Baye

s Fa

ctor

)

N=2500

0

1000

2000

3000

0 .1 −.1 .145−.145 .25 −.25 .5 −.5 −.75 −1effect size in log−odds

log(

Baye

s Fa

ctor

)

N=5000

25

Ascanbeseen,withsmallN,BayesFactorsfavorthenull.However,asNincreases,

theBayesFactorisextremelylikelytofavorthecorrectmodel.Withonly50subjectsper

condition,onlyfairlylargeeffectsontheorderof0.5canbereliablydetected.For

comparison,0.5isapproximatelyhalfthedifferencebetweenasymptoticmonolingualsand

ouryoungestmonolinguals(7yearsold).Reliablydetectingthedifferencebetween

asymptoticmonolingualsandbilinguals(e=0.145)requiresaround500subjectsper

condition.

Forcomparison,wehaveincludedstandardpoweranalysesbasedonp-valuesin

TableS2(weusedbinomialmixedeffectsregressionwithmaximalrandomslopes).The

comparisonofFig.S13andTableS2nicelydemonstratesthefactthatusingp-valuesmakes

onemorelikelytorejectthenullhypothesis—notonlywhenthenullisfalsebutalsowhen

itistrue(Wagenmakers,2007).Notethatwhene=0,theprobabilityoftheBayesFactor

erroneouslysupportingthealternativehypothesisisnegligible,evenforsmallN.In

contrast,whenusingp-values,TypeIerrorremainsaconstant0.05independentofN(as

bydefinition).

TableS2.Poweranalysesforstudyingasymptoticbehavior,usingp-values,forvariouseffectsizes.N/condition 0.1 -0.1 0.145 -0.1450.25 -0.25 0.5 -0.5 -0.75 -150 .22 .31 .24 .32 .44 .49 .78 .92 1.0 1.0100 .38 .33 .36 .53 .63 .73 .98 1.0 1.0 1.0250 .49 .45 .62 .67 .91 .96 1.0 1.0 1.0 1.0500 .68 .66 .86 .90 .99 .99 1.0 1.0 1.0 1.01000 .84 .84 .97 .97 1.0 1.0 1.0 1.0 1.0 1.02500 .99 .97 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.05000 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

26

Toanalyzeourdata,wedivviedupimmersionlearnersintogroupsbyageoffirst

exposure(AoFE):0,1-5,6-10,11-15,and16-20.Wethenanalyzedresultsforanylanguage

groupwithatleast50subjectsinaparticularAoFEbin.Asdescribedinthemaintext,for

eachlanguagegroup,weaskedwhetherthedatawouldbebetterfitbyassumingadistinct

meanforthatlanguagegroupasopposedtoallothersubjects.AscanbeseeninFig.S14,

resultswerehighlysimilaracrosslanguagegroupswithinaparticularAoFEbin.Giventhis,

itisnotsurprisingthatBayesFactoranalysestypicallysupportedthenullhypothesis

(logBF<0)(TableS3).TheexceptionsinvolvedsuperiorperformancebyRomance

speakersatAoFE=0,superiorperformancebyWesternGermanicspeakersatAoFE1-5,

andsuperiorperformancebyChinesespeakersatAoFE6-10.Giventhelackof

systematicityandtherelativelysmallBayesFactors,thesearemostlikelyspurious.For

comparison,wehaveincludedtraditionalp-valuesforthesameanalyses,basedon

binomialmixedeffectsregression.Asexpected,thesetendtomorestronglyfavorthe

alternativehypothesis(Wagenmakers,2007).Notewithcorrectionformultiple

comparisons,𝛼=.0037.

27

FigureS14.Boxplots(inred)overlaidonviolinplots(white)forasymptoticimmersionbilingualsoverall(left)andforfivelanguagefamilies(right).A:ageoffirstexposure=0(simultaneousbilinguals).B:ageoffirstexposure1-5.C:ageoffirstexposure6-10(notethatforthis,thereweretoofewUralicspeakerstoinclude).

●●● ●●

●

●

●

●●●●1

2

3

4

5

ChineseN=1728

RomanceN=742

SlavicN=164

UralicN=195

W GermN=1433

language group

●●●

1

2

3

4

5

allN=7895

log−

odds

cor

rect

●●●

●

●●

1

2

3

4

5

ChineseN=301

RomanceN=400

SlavicN=153

UralicN=66

W GermN=305

language group

●1

2

3

4

5

allN=2708

log−

odds

cor

rect

1

2

3

4

5

ChineseN=66

RomanceN=89

SlavicN=59

W GermN=74

language group

1

2

3

4

5

allN=497

log−

odds

cor

rect

A

B

C

28

TableS3.Evidencethatasymptoticperformancewassignificantlydifferentforalanguagegrouprelativetotherest,usingbothBayesFactorandp-value.AoFE Chinese Romance Slavic Uralic West.Germ.0

logBF -4.4 2.4 -4.5 -3.2 -2.0 p-val .295 .000 .254 .450 .0001-5

logBF -3.7 -3.9 -2.8 -3.6 2.6 p-val .243 .060 .785 .302 .0006-10

logBF 1.7 -3.0 0.3 NA -2.5 p-val .003 .473 .000 NA .019

OptimalPeriod.Wefirstpresentpoweranalysesassessingourabilitytodetect

meaningfuldifferences,followedbyouractualresults.

Forpoweranalysis,weconductedaseriesofsimulationstodetermineourabilityto

detectdifferencesofvarioussizes.Wefirstfitasegmentedregressionmodelwithasingle

breakpointtothefullultimateattainmentdatasetusingthesegmentedpackageinR,as

describedabove(“UltimateAttainment”).Foragivensamplesize(asexplainedbelow),we

thengeneratedtwodatasets:oneusingtheoriginalmodelandonewiththebreakpoint

shiftedbyeyears.5

Weconsideredninesamplesizes:Ns=155,310,775,1550,3100,7750,15500,

31000,and11371percondition.ThefirsteightsamplesizesinvolvedN=xsubjectsperage

offirstexposurefrom0to31,wherex=5,10,25,50,100,250,500,and1000,

5Theslopeofthefirstsegmentwasadjustedsothattheheightofthecurveatthebreakpointwaskeptthesameacrossthetwomodels(asdescribedabove,thatslopewasverysmall,sothisdecisionhadalimitedeffect).Theslopeafterthebreakpointwasleftunchanged.

29

respectively.Forthefinalsimulation,foreachconditionwedrewexactlythenumberof

subjectsateachageoffirstexposurethatwasfoundinouractualultimateattainment

dataset.Thus,thisfinalsimulationbetterrepresentstheunequalsamplesizesinouractual

data.Weconducted100simulationsforeachlevelofNande,exceptfore=0,wherewe

ran200.ResultsareshowninFig.S15.

30

Fig.S15.ViolinplotsofloggedBayesFactorsfor9differentsamplesizes(panels)and7differenteffectsizes(x-axis).Seetextforexplanationofhowthesesubjectsweredistributedacrossagesoffirstexposure.Notethatthey-axisscalevariesacrosspanels.

Thus,basedonthesesimulations,weareunlikelytodetectadifferenceinoptimal

−5

0

5

0 2.5 −2.5 5 −5 7.5 −7.5effect size in years

log(

Baye

s Fa

ctor

)N=155

−10

−5

0

5

10

0 2.5 −2.5 5 −5 7.5 −7.5effect size in years

log(

Baye

s Fa

ctor

)

N=310

−10

0

10

20

30

0 2.5 −2.5 5 −5 7.5 −7.5effect size in years

log(

Baye

s Fa

ctor

)

N=775

0

20

40

60

0 2.5 −2.5 5 −5 7.5 −7.5effect size in years

log(

Baye

s Fa

ctor

)

N=1550

0

40

80

0 2.5 −2.5 5 −5 7.5 −7.5effect size in years

log(

Baye

s Fa

ctor

)

N=3100

0

100

200

0 2.5 −2.5 5 −5 7.5 −7.5effect size in years

log(

Baye

s Fa

ctor

)

N=7750

0

200

400

600

0 2.5 −2.5 5 −5 7.5 −7.5effect size in years

log(

Baye

s Fa

ctor

)

N=15500

0

300

600

900

0 2.5 −2.5 5 −5 7.5 −7.5effect size in years

log(

Baye

s Fa

ctor

)

N=31000

−10

0

10

20

30

0 2.5 −2.5 5 −5 7.5 −7.5effect size in years

log(

Baye

s Fa

ctor

)

N=11371

31

periodofmuchlessthan7.5years,whichisaround60%ofthelengthoftheoptimalperiod

measuredoverallsubjects.However,iffutureresearchersfindamechanismforrecruiting

asubjectgroupmoreevenlydistributedacrossagesoffirstexposure,theycouldget

considerablybetterprecisionwithfewersubjects.

Wethenanalyzedourdata.Resultsdidnotfavorthehypothesisofaseparate

breakpointforChinese(logBF=-7.9),Romance(logBF=-12.6),Slavic(logBF=-9.1),or

Turkic(logBF=-9.6).Thesegmentedpackagewasunabletoidentifyanybreakpointfor

UralicorWesternGermanic.Thisseemstobeduetolargeamountsofnoiseratherthan

clearevidenceagainsttheexistenceofabreakpoint(seeFig.S16).

FigureS16.UltimateattainmentasafunctionofageoffirstexposureforUralic(left)andWesternGermanicspeakers(right).Inordertobettershowthevariability,nosmoothingwasused.Notethatthey-axisscalevariesacrosspanels.

LearningCurves.Wealsoconsideredwhetherfirstlanguageaffectedhowquickly

0 5 10 15 20 25

2.5

3.0

3.5

4.0

age of first exposure

log-odds

0 5 10 15 20 25 30

2.0

2.5

3.0

3.5

4.0

4.5

5.0

age of first exposure

log-odds

Western GermanicUralic

32

Englishwaslearned.Ideally,wewouldmeasurehowlongittakestoreachasymptoteasa

functionoffirstlanguage.However,mostparametriccurves—includingtheoneweusedin

ourmainmodel—requireaninfiniteamountoftimetoreachasymptote.Wecould

alternativelymeasurehowlongittakesforsubjectstogetwithinεofasymptoteforsomeε,

butthentheresultsmaydependontheεchosen,particularlygiventhatanyreasonableε

willbesmallrelativetotheamountofstatisticalnoiseinthedata(wehaverelativelyfew

subjectsattheagesthataremostrelevantforthisanalysis).

Moreinformationisavailableifwecomparelanguagegroupsintermsoftheshape

oftheirlearningcurves(thecurvesrelatingperformanceandyearsofexperience).We

choseamethodthatputasfewaprioriconstraintsontheshapeofthelearningcurveas

possible.Specifically,wefitalinearregressiononaccuracy(inlog-odds)withyearsof

experience,languagegroup,andtheirinteractionasfixedeffects.Thus,norelationshipis

assumedbetweenperformanceatageaandatagea+1.Weaskedwhetherthismodelfit

betterthanonewithoutlanguagegrouporitsinteractions.Becauseourinterestisinthe

learningcurve,werestrictedanalysestothefirstthirtyyearsofexperience(fromanalyses

above,weknowthatdataafterthistimeishighlysimilaracrossthelanguagegroups,so

includingtheseageswoulddiminishourabilitytodetectdifferencesinlearning).

Wefirstpresentpoweranalysesassessingourabilitytodetectmeaningful

differences,followedbyouractualresults.

Ourpoweranalysesshowthatthesensitivityandflexibilityofthismethodcomesat

acost:Itrequiresmanysubjects.Thiswasconfirmedinaseriesofsimulations(Fig.S15).In

eachsimulation,wegeneratedtwodatasets.Inthefirst,wesampledfromour

simultaneousbilingualswithreplacement.Fortheseconddataset,wesimulated

33

completionoflearningeyearsearlierbyfirstadjusting“yearsofexperience”intheoriginal

dataaccordingtothefollowingformula:

𝐴 = 7 + (30 − 𝑑) − 7 ∗ R(S

TE(S𝐴 ≤ (30 − 𝑑)

(30 − 𝑑) + 70 − (30 − 𝑑 ) ∗ R(TESE(TE

30 < 𝐴 ≤ 70

Thishastheeffectof“squeezing”thecurvepriorto30yearsofexperienceinto30-eyears

ofexperience(thedataafter30yearsofexperienceis“stretched”tocompensate).Wethen

sampledtheseconddatasetwithreplacement.

Weconsidered9samplesizeswith5,10,15,20,25,50,75,100,or150subjectsper

levelofyearsofexperience.AsFig.S17shows,ifonegrouplearned3xfasterthanthe

other,wewouldneedaround2,250subjectspergroup(75pergroupperyearof

experience).Todetect“only”adifferenceof2x,weneedaround4,500subjectspergroup

(150pergroupperyearofexperience).Thesevaluesareinfactoptimistic,inthatthey

assumeanevendistributionofsubjectsacrossyearsofexperience,andthusequal

precisionatallpointsinthecurve,whichisnotthecase.

34

FigureS17.ViolinplotsforloggedBayesFactorsforninedifferentsamplesizespercondition(panels)andfivedifferenteffectsizes(x-axis).Theeffectsizereflectsthenumberofyearsthelearningcurvewasshiftedleft(seetext).Notethatthey-axisscalevariesacrosspanels.

−60

−50

−40

−30

0 2 5 15 20effect size in years

log(

Baye

s Fa

ctor

)N=150

−70

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0 2 5 15 20effect size in years

log(

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s Fa

ctor

)

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log(

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ctor

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log(

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log(

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log(

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s Fa

ctor

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100

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log(

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s Fa

ctor

)

N=4500

35

Giventhesesimulations,werestrictedanalysesforsubgroupswherewehadatleast

1,500subjects(approx.)ormore.Withinsimultaneousbilinguals,thisincludedChinese,

Romance,andWesternGermanic.Bayesfactoranalysessuggestednodistinctionbetween

Romanceandtheremaininglanguages(logBF=-4.8)orbetweenWesternGermanicand

theremaininglanguages(logBF=-3.1).Therewasweakevidenceinfavoroftreating

Chineseseparately(logBF=2.0).However,theactualdifferencebetweenlearningcurves

appearstobeslight(Fig.S18,left)andmayonlyreflectnoise.

Lookingatlaterlearners(agesoffirstexposurefrom1-5yearsold),onlyChinese

hadsufficientlymanysubjects.Inthiscase,therewasstrongevidencefortreatingChinese

separatelyfromtheothergroups(logBF=4.4).FrominspectionofFig.S18(right),this

seemstoreflectsomewhatfasterinitiallearningandperhapsanearlierdecline.However,

asthefigureshows,thedataarefairlynoisy,reflectingtherelativelysmallnumberof

subjectsforthistypeofanalysis.

36

FigureS18.Learningcurves(withoutsmoothing)forimmersionlearnerswithageoffirstexposure0(left)and1-5(right).Althoughonlythefirst30yearsareusedforanalyses,weplotthrough80yearsoldforreference.

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all (N=30,417)

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rect

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all (N=8,595)

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cor

rect

Chinese (N=1485)

37

EducationDifferences.

Toinvestigatetheeffectsofeducationonultimateattainment,wecategorized

participantsaccordingtowhethertheirhighestlevelofeducationwassecondary(high

schooldiplomaorless:578immersionlearnersand4,359non-immersionlearners),

undergraduate(partialorcompleteundergraduatestudies:4,411immersionlearnersand

6,309non-immersionlearners),orgraduate(partialorcompletegraduatestudies:6,382

immersionlearnersand18,006non-immersionlearners).Asmallnumberofsubjectswere

excludedfornotreportingeducationlevel.ResultsareshowninFig.S19.Theoverall

shapesofthecurvesaresufficientlysimilartooneanotherthatnoformalanalysiswasrun.

FigureS19.Ultimateattainmentforimmersionlearnersandnon-immersionlearners,byeducationlevel,smoothedwithathree-yearfloatingwindow.Therewereinsufficientimmersionlearnerswithonlysecondaryeducationforanalysis.

1.5

2.0

2.5

3.0

3.5

4.0

0 10 20 30age of first exposure

accu

racy

(log

odd

s)

1.5

2.0

2.5

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4.0

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accu

racy

(log

odd

s)

1.5

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accu

racy

(log

odd

s)

1.5

2.0

2.5

3.0

3.5

4.0

0 10 20 30age of first exposure

accu

racy

(log

odd

s)

99%

98%

96%

93%

88%

82%

accu

racy

(per

cent

)

monolingualsimmersionlearners

non-immersionlearners

secondary

graduate

undergrad

undergradgraduate

99%

98%

96%

93%

88%

82%

accu

racy

(per

cent

)

Figure 6

38

GenderDifferences.

FigureS20separatestheultimateattainmentdataformaleandfemaleparticipants.

Thesameexclusionsusedelsewherewereapplied(e.g.,restrictinganalysestoconsecutive

windowswithatleast10subjects).Iftheoffsetofthecriticalperiodisdrivenbypuberty,

onemightexpectwomen’slearningratetobegintodeclineearlierthanmen’s.However,

thelearningmodelestimatedaslightlylateronsetforadeclineintheunderlyinglearning

rateinwomen(19.3yearsold)thaninmen(17.9yearsold).6Becausetheunderlying

learningrateisatheoreticalestimatewhichrequiresintensivecomputationalresourcesto

derive,itwasnotfeasibletousepermutationanalysistodeterminewhetherthegender

differenceinthisestimateisstatisticallysignificant.Butstatisticallyanalyzingthegender

differenceintheageatwhichtheultimatelevelofattainmentdeclinesismoretractable.

Permutationanalysisshowedthattheeffectgenderonultimateattainmentwasnot

significanteitherfortheimmersionlearners(12yearsoldforfemales,8yearsoldfor

males,p=.80)orforthenon-immersionlearners(11yearsoldforfemales,9yearsoldfor

males;p=.43).7(Analternativewouldbetousethemodel-comparisonapproachusedin

6Men:tc=17.9r=.16,α=.09,δ=.04;E=1.00,.66,.98,.31.Women:tc=19.3r=.21,α=.10,δ=1.3;E=1.00,.66,.95,.28.7Aswiththemainanalyses,ultimateattainmentanalysesfocusedonparticipantswithatleast30yearsofexperienceandwhowerenoolderthan70yearsofage.5,110immersionlearnersweremale,and6,207werefemale.14,043non-immersionlearnersweremale,and15,565werefemale.Thepermutationanalyseswereconductedbyshufflingparticipants’gendersseparatelyforeachlearnertype(immersion,non-immersion)andeachageoffirstexposure.Forimmersionandnon-immersionlearnersofeachgender,MARSanalyseswereapplied,andtheyoungestageforwhichtheslopewasmorenegativethan-.02wasrecorded(becauseoftenthefirstsegmentoftheMARSregressionisslightlynegative,thearbitraryandrelativelymodestthresholdof-.02wasusedtodefine“substantialdecline”).Thisprocesswasrepeated100times.Thetwo-tailedp-valueisthenumberofiterationsforwhichtheabsolutedifferenceintheageatwhichultimateattainmentbegansubstantialdeclineformenasopposedtowomenwasequaltoorgreater

39

“L1Effects”,above.Wedevelopedthatanalysisafterthepermutationoneusedhereand

didnotre-runanalyseswiththenewmethod.)

FigureS20.Ultimateattainmentcurvesformen(A)andwomen(B),smoothedforpresentationwithathree-yearfloatingwindow.Blue=immersionlearners,Red=non-immersionlearners.Shadedareasrepresent+/-1SE.AsinFigure6,dataweresmoothedbyathree-yearfloatingwindow,andonlyconsecutivewindowswithmorethan10subjectsshown. Itispossiblethatmoresensitiveanalyseswouldfindstatisticallysignificant

evidenceofasexdifferenceinthecriticalperiod(thoughitwouldbeintheopposite

directionofwhatispredictedbythesexdifferenceintheonsetofpuberty).Interestingly,

therewasastrikingsexdifferenceinoverallperformance:acrosstheentireagerange,

thanthedifferenceactuallyobserved,calculatedseparatelyforimmersionandnon-immersionlearners.

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 10 20 30age of first exposure

accu

racy

(log

odd

s)

femalemale

femalemale

40

womenoutperformedmen(p<.01),consistentwithaliteratureshowingafemale

advantageincertainverbalabilities(Geary,2010).Thedifferencewastrueofeachofthe

typesoflearners(monolinguals,immersionlearners,andnon-immersionlearners;ps<

.01),andacrosstherangeofageandexperience(seeFigureS20).Thecurrentdatadonot

speaktotheextenttowhichthedifferenceistheresultofbiologicalorculturalcauses.

AsymptoticPerformance

Figure5Bshowsthatmonolingualsandsimultaneousbilingualsreachasymptoteon

ourtestataround30yearsofage.Theresultsforlater-learnersalsosuggestprotracted

periodsofimprovement(Fig.4).BecauseFig.4iscompact,wehavere-plottedsomeofthe

curvesforimmersionlearnersandnon-immersionlearnersinFigs.S21andS22,

respectively.Notethegraphsforimmersionlearnersinvolvefarfewersubjectsandsoare

noisier,evenwiththesmoothing(seecaption).

41

FigureS21.Panelsshowlog-oddsaccuracyasafunctionofyearsofexperienceforimmersionlearners,byageoffirstexposure(AoFE).GraphsinvolvesthesamesmoothingandexclusionsasinFig.3.Onlythefirstsixcurvesareshownbecauselatercurvesareshorterandprovidenoevidenceaboutasymptote.

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AoFE = 1-3 AoFE = 4-6 AoFE = 7-9

AoFE = 10-12 AoFE = 13-15 AoFE = 16-18

42

FigureS22.Panelsshowlog-oddsaccuracyasafunctionofyearsofexperiencefornon-immersionlearners.GraphsinvolvesthesamesmoothingandexclusionsasinFig.3.SixcurvesareshowncorrespondinginexposureagetothesixinFig.S12.SeealsoFig.S1.

ResultsbyItem

Thelearningcurvesformonolingualsvariedacrossitems.Ratherthangrapheachof

these95learningcurvesseparately,wehaveprovidedtherawdata,whichthereadercan

usetogenerateanyvisualizationofinterest.Inthemeantime,weprovideasinglegraph

showingall95learningcurves.Whiletheyarenotindividuallydistinguishable,thisgraph

providessomeintuitionabouttheby-itemvariability.

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43

Fig.S23.Accuracybyageforeachofthe95criticalitems.Notethatbecausewehavedifferentnumbersofsubjectsateachage,thisispresentedintermsofpercentcorrect,

0.5

0.6

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0.9

1.0

10 20 30 40age

correct

44

ratherthanlog-odds.(Readerswhotrycreatingthisgraphinlog-oddswillunderstandtheissue.)

45

Materials

Allitemsareincludedbelow.Asnotedinthemaintext,wherepossiblewegrouped

multiplegrammaticalityjudgmentsintoasinglemultiple-choicequestion.Thus,Questions

9-35areinfact124distinctquestions.

Becausethegrammaticalityjudgmenttaskistime-consumingandunsuitablefor

probingcertaingrammaticalphenomena,wealsoincludeditemsthatrequiredmatchinga

sentencetoapicture(e.g.,toprobetopicalizationandtheapplicationoflinkingrules).

Questions1-8areofthatformat.

Thecorrectanswersaregiveninthenextsection.

Clickonthepicturethatbestmatchesthesentence.

1.Thedogwaschasedbythecat.

2.Itwasthechickenthatscaredthelion.

46

3.Itwasthelionthattheelephantbit.

4.Everyhikerclimbedahill.

5.Itwasthetigerthatthemonkeyhugged.

6.Itwasthemonkeythatpushedthebear.

47

7.Thedogwaspushedbythecat.

8.Everychildrodeanelephant.

48

49

Four-AlternativeForcedChoice8

9.Whichofthefollowingsentencessoundsmostnatural?

a.Ishan'tbecomingtothepartyafterall.

b.Iwon'tbecomingtothepartyafterall.

c.Both

d.Neither

10.Whichofthefollowingsentencessoundsmostnatural?

a.Whatageareyou?

b.Howageareyou?

c.Howoldareyou?

c.Whatoldareyou?

8Includingtheabovetwoquestionsintheanalysesisnotstraightforward,bothbecausetheyaretheonlyquestionsthatarenotabinaryforcedchoiceandbecausesomeoftheoptionsareexcludedfromtheanalysis.Thereareseveralwaysofcodingtheresponses,thoughthechoiceisunlikelytohavemucheffect:Accuracywasextremelyhighforthesequestions,andsotheycontributeverylittletothevariance,andmoreovertheyrepresentonlyatinyfractionoftheincludeddata.Forsimplicity,weelectedtoanalyzetheitemsaboveasifeachoptionwasanindependentforcedchoice(thus,forexample,participantsarecreditedfortwocorrectanswersiftheydonotselect(b)or(c)onQuestion9).

50

Fillintheblank(Chooseallthatapply)

11.I_________for6hoursbydinnertime.

a.willhavestudied

b.willhavebeenstudying

c.willhadstudied

d.willbestudying

12.Thepeople___________angry.

a.is

b.be

c.were

d.are

13.Theman____________arrivedyesterdayneedsawakeupcallatnine.

a.that

b.whom

c.which

d.where

14.Wewonthegame,_______wedid!

a.so

b.yes

c.no

51

d.although

15.I________medicine.

a.studies

b.reads

c.study

d.read

16.Hebrokehisleg,soheis________.

a.inthehospital

b.inhospital

c.onhospital

d.onthehospital

17.ItoldSallyIwasworriedabouttheexam.Shesaid,"Don'tworry.____________"

a.He'llberight!

b.She'llberight!

c.Itbeokay!

d.It'llbeokay!

18.Ifhe_______,hewouldhavehelpedher.

a.knew

b.hadbeenknowing

52

c.hadknown

d.haveknown

19.Mybrotherandsister_________playingtennisat11pmlatertonight.

a.are

b.will

c.were

d.was

20.I_______thestory.

a.said

b.replied

c.declared

d.told

21.MygrandmotherreallylovedJohn.Sheleftallhermoneyto______.

a.he

b.him

c.her

d.it

22.They________betraveling,butI'mnotsure.

a.may

53

b.can

c.would

d.have

23.John____thelibrarythebook.

a.gave

b.donated

c.distributed

d.contributed

24.Sally_____Mary.

a.laughed

b.happied

c.giggled

d.tickled

25.________livesintheWhiteHouse.

a.APresidentObama

b.ThePresidentObama

c.ThesePresidentObama

d.PresidentObama

26.Thesunisin________.

54

a.thesky

b.asky

c.ansky

d.sky

27.Ibelievein_________.

a.thesejustice

b.justice

c.ajustice

d.thejustice

28.Sorrytodisturbyou_________.

a.withtheweekend

b.undertheweekend

c.attheweekend

d.ontheweekend

29.Iwould_________gohome.

a.like

b.prefer

c.rather

d.want

55

30.Bill_________thecupwithwine.

a.poured

b.filled

c.drained

d.dripped

31.Iplay___________thesoccerteam.

a.at

b.in

c.on

d.inside

56

Chooseallthataregrammatical

32.

a.Johnagreedthecontract.

b.Sallyappealedagainstthedecision.

c.I’llwritemybrother.

d.I’mjustaftertellingyou.

e.Thegovernmentwasunabletoagreeonthebudget.

f.Iafteratedinner.

g.WhodidSueaskwhySamwaswaiting?

h.Hethoughthecouldwinthegame.

33.

a.Thecommitteeweredividedonthequestion.

b.SheresignedThursday.

c.Hesaidthatsheistakingatrip.

d.Hesaidthatshewastakingatrip.

e.Sallyswamtwomiles.Woreapairof100goggles.

f.I'mgoingtoWisconsinnextweek.

g.Heencouragedhertotravelsaroundtheworld.

h.I’mwantingdessert.

34.

a.Iworkedforfiveyears.

57

b.WhodidBillaskwhyJanewastalkingto?

c.Whowhomkissed?

d.Johnwenttothestore.Boughticecream.

e.I’mfinishedmyhomework.

f.I'mfinishedwithmyhomework.

g.Wedidgothebeach.

h.HebeworkingTuesdays.

35.

a.Yesterday,Johnwantedtowontherace.

b.Uptheaudience'sexpectations,thecriticsbuilt.

c.I’mdonedinner.

d.Hewaspulledoverbythepolicefordriving120milesperhour.

e.Hestayworking.

f.Thedogthemanownsbarked.

g.Ieatsdinner.

h.WhereisthepenthatIgaveittoyouyesterday?

58

Scoring

Belowweprovidethecorrectanswerforthe95criticalitems.Fortheother37

items,the“correct”answervariedbydialect.

1.Bottom2.Bottom3.Top5.Bottom6.Bottom7.Top9a.Incorrect9d.Incorrect10b.Incorrect10c.Incorrect11c.Incorrect11d.Incorrect12a.Incorrect12b.Incorrect12d.Correct13c.Incorrect13d.Incorrect14c.Incorrect14d.Incorrect15a.Incorrect15b.Incorrect15c.Correct16c.Incorrect16d.Incorrect17a.Incorrect17c.Incorrect17d.Correct18b.Incorrect18c.Correct18d.Incorrect19a.Correct19b.Incorrect19c.Incorrect19d.Incorrect20a.Incorrect20b.Incorrect20c.Incorrect20d.Correct21a.Incorrect21b.Correct

59

21c.Incorrect21d.Incorrect22a.Correct22b.Incorrect22c.Incorrect22d.Incorrect23c.Incorrect23d.Incorrect24a.Incorrect24b.Incorrect24c.Incorrect24d.Correct25a.Incorrect25b.Incorrect25c.Incorrect25d.Correct26a.Correct26b.Incorrect26c.Incorrect26d.Incorrect27a.Incorrect27b.Correct27c.Incorrect27d.Incorrect28a.Incorrect28b.Incorrect29a.Incorrect29b.Incorrect29c.Correct29d.Incorrect30a.Incorrect30b.Correct30c.Incorrect30d.Incorrect31a.Incorrect31d.Incorrect32e.Correct32f.Incorrect32h.Correct33d.Correct33e.Incorrect33f.Correct33g.Incorrect34a.Correct34b.Incorrect34c.Incorrect

60

34d.Incorrect34f.Correct34h.Incorrect35a.Incorrect35b.Incorrect35d.Correct35e.Incorrect35g.Incorrect35h.Incorrect

61

SupplementaryReferences

Geary,DC(2010)Male,female:Theevolutionofhumansexdifferences.(American

PsychologicalAssociation,Washington,DC)2ndedition.

Muggeo,VMR(2014)Segmented:AnRpackagetofitregressionmodelswithbroken-line

relationships.Rversion0.4-0.0.http://CRAN.R-project.org/web/packages/segmented.