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
2
3
33 3
3 33 3
3
3
3
3
3
1
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4
0 20 40 60experience
accu
racy
(log
-odd
s)
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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rate
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racy
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-odd
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0.000.250.500.751.00
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rate
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-odd
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rate
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-odd
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rate
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racy
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-odd
s)
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accu
racy
(log
-odd
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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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89 101112 1314
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2930
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(log
-odd
s)C
A
D
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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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monolingualsage of exposure: 0-9 y.o.age of exposure: 10-19 y.o.age of exposure: 20-30 y.o.
m0 2,5,8,
11,14,17
20,23,26,29 4-30
m
258 11
14 172023
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294-30 (from top to bottom)
C
A
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Figure S4
current age current age
7
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.
0.050.100.150.20
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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.
m
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2629
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4-30 (from top to bottom)
4-30 (from top to bottom)
C
A
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E
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.
0.000.050.100.150.20
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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.
m
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232629
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Figure S6
current age current age
9
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
10 20 30 40 50 60 70yrs experience
rate
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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.
m0
2,5,8
11,14,17,20,23,26,29 4-30
m0 2
5 811
141720
2326
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C
A
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Figure S7
current age current age
10
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.
0.050.100.150.20
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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.
4-30
m0
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8 11
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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
0.01
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dens
ity
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ity
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ity
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ity0.00
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ity
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ity
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ity
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dens
ity
0.0
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1.2
−1.0 −0.5 0.0 0.5 1.0diff
density
0.0
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0
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density
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
2.0
2.5
3.0
3.5
4.0
0 10 20 30age of first exposure
accu
racy
(log
odd
s)
1.5
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accu
racy
(log
odd
s)
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racy
(log
odd
s)
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accu
racy
(log
odd
s)
A B
immersion learners non-immersion learners
99%
98%
96%
93%
89%
82%
accu
racy
(per
cent
)
99%
98%
96%
93%
89%
82%
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
25
811
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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
−60
−50
−40
−30
0 2 5 15 20effect size in years
log(
Baye
s Fa
ctor
)
N=300
−70
−60
−50
−40
−30
0 2 5 15 20effect size in years
log(
Baye
s Fa
ctor
)
N=450
−80
−70
−60
−50
−40
−30
−20
0 2 5 15 20effect size in years
log(
Baye
s Fa
ctor
)
N=600
−80
−60
−40
−20
0 2 5 15 20effect size in years
log(
Baye
s Fa
ctor
)
N=750
−90
−60
−30
0
0 2 5 15 20effect size in years
log(
Baye
s Fa
ctor
)
N=1500
−50
0
50
0 2 5 15 20effect size in years
log(
Baye
s Fa
ctor
)
N=2250
−100
−50
0
50
100
0 2 5 15 20effect size in years
log(
Baye
s Fa
ctor
)
N=3000
−100
0
100
0 2 5 15 20effect size in years
log(
Baye
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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Romance (N=4,220)
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cor
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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
3.0
3.5
4.0
0 10 20 30age of first exposure
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)
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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AoFE = 4 AoFE = 5 AoFE = 8
AoFE = 11 AoFE = 14 AoFE = 17
43
Fig.S23.Accuracybyageforeachofthe95criticalitems.Notethatbecausewehavedifferentnumbersofsubjectsateachage,thisispresentedintermsofpercentcorrect,
0.5
0.6
0.7
0.8
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.