The Lexicon-Grammar Relationship: Revisiting the

Critical-Mass Hypothesis

James A. DixonJames A. DixonUniversity of ConnecticutUniversity of Connecticut

Thanks to Virginia MarchmanThanks to Virginia Marchman

Emergence of Language

Language structures emerge from interactions Language structures emerge from interactions among various levels of the systemamong various levels of the system (Elman, 2004; (Elman, 2004; MacWhinney, 2004; Zevin & Seidenberg, 2004)MacWhinney, 2004; Zevin & Seidenberg, 2004) Lexical, semantic, syntactic, pragmatic, etc.,Lexical, semantic, syntactic, pragmatic, etc., Phonological representations emerge from:Phonological representations emerge from:

articulatory, acoustic, semantic interactions articulatory, acoustic, semantic interactions (Plaut & Kello, 1999)(Plaut & Kello, 1999)

Structures formed through repeated, real-time Structures formed through repeated, real-time interactionsinteractions Language acquisition within a unified systemLanguage acquisition within a unified system

Developmental Relations Language emerges from a unified system…Language emerges from a unified system…

Need to understand the developmental relations Need to understand the developmental relations within that systemwithin that system

How is lexical development related to grammatical How is lexical development related to grammatical development?development?

Developmental orderingDevelopmental ordering Does one item “lead” another in the system?Does one item “lead” another in the system? Do two items emerge synchronously?Do two items emerge synchronously?

Ordering imposes important constraints on theoryOrdering imposes important constraints on theory Primary type of evidence for modelingPrimary type of evidence for modeling

Models must demonstrate the same developmental Models must demonstrate the same developmental orderingsorderings

Two Related Goals

Present a new approach for testing hypotheses Present a new approach for testing hypotheses about orderingabout ordering Applied to the most recent MacArthur-Bates CDI dataApplied to the most recent MacArthur-Bates CDI data

Provide new information about the relationship Provide new information about the relationship between lexical and grammatical developmentbetween lexical and grammatical development

Grammar and the Lexicon Development of the lexicon and grammar strongly relatedDevelopment of the lexicon and grammar strongly related

Relationship is curvilinearRelationship is curvilinear Lexical development precedes grammatical developmentLexical development precedes grammatical development

Lexicon

Gra

mm

ar

Lexicon precedes Grammar

Observed in both cross-sectional and longitudinal Observed in both cross-sectional and longitudinal samples samples (Bates et al. 1994; Fenson et al., 1994; Dale et al., (Bates et al. 1994; Fenson et al., 1994; Dale et al., 2000)2000)

Similar relation observed for: Similar relation observed for: Italian Italian (Caselli, Casadio, & Bates, 1999)(Caselli, Casadio, & Bates, 1999) Hebrew Hebrew (Maital et al., 2000)(Maital et al., 2000) Icelandic Icelandic (Thordardottir, Weismer, & Evans, 2002)(Thordardottir, Weismer, & Evans, 2002) Spanish Spanish (Jackson-Maldonado et al., 2003)(Jackson-Maldonado et al., 2003) Spanish-English bilingual acquisition Spanish-English bilingual acquisition (Marchman et al., (Marchman et al.,

2004)2004)

Lexicon precedes Grammar

Relation holds when words related to grammatical Relation holds when words related to grammatical complexity are removed complexity are removed Grammatical function words Grammatical function words

PrepositionsPrepositions ConjunctionsConjunctions

Grammar and the Lexicon

Curvilinear relationship has important implicationsCurvilinear relationship has important implications Evidence of developmental ordering Evidence of developmental ordering Grammar emerges from the lexiconGrammar emerges from the lexicon ““Critical mass” hypothesis (Bates et al.)Critical mass” hypothesis (Bates et al.)

Grammar and the Lexicon

However, problems with this direct interpretationHowever, problems with this direct interpretation Measures of underlying variablesMeasures of underlying variables

LexiconLexicon GrammarGrammar

Interpreting the form of the function requires very Interpreting the form of the function requires very stringent assumptions about measuresstringent assumptions about measures Relationship between the measure and the underlying Relationship between the measure and the underlying

variablevariable Measure must be equally “responsive” to changes in the Measure must be equally “responsive” to changes in the

underlying dimension across the entire developmental underlying dimension across the entire developmental rangerange Interval ScalesInterval Scales

Underlying and Measured Levels

0102030405060708090100M

easu

re o

f L

exic

on

Und

erly

ing

Gra

mm

ar

0102030405060708090100

Mea

sure

of

Gra

mm

ar

Und

erly

ing

Lex

icon

Dev

elop

men

t

An Alternative Hypothesis

0102030405060708090100

Mea

sure

of

Lex

icon

Und

erly

ing

Gra

mm

ar0102030405060708090100

Mea

sure

of

Gra

mm

ar

Und

erly

ing

Lex

icon

Dev

elop

men

t

10090807060504030201000102030405060708090100

Measure of Lexicon

Mea

sure

of

Gra

mm

ar

Very serious problemVery serious problemObserved relationship not Observed relationship not

evidence of underlying formevidence of underlying form

Nonlinear mappings can create Nonlinear mappings can create curvilinear relationshipcurvilinear relationship

Synchrony with nonlinear Synchrony with nonlinear mapping is viable alternative mapping is viable alternative hypothesishypothesis

u Underlying Levelt GGuu = = αα + + β∗β∗LLuu + + εεuu

Measured Level GGmm = (G = (Guu))2 2 + + εεmgmg

LLmm = M(L = M(Luu)) + + εεmlml

Representing Developmental Relations

0102030405060708090100

Mea

sure

of

Lex

icon

Und

erly

ing

Gra

mm

ar0102030405060708090100

Mea

sure

of

Gra

mm

ar

Und

erly

ing

Lex

icon

Dev

elop

men

t

Another way to represent synchrony:Another way to represent synchrony:

Homoscedasticity, Heteroscedasticity, & Devlopmental Order

Assume that our error terms are largely Assume that our error terms are largely homoscedastichomoscedastic

Magnitude of variance constant across the entire developmental range: εu, εmg, εml

Usual OLS regression assumption

However, nonlinear mappings create However, nonlinear mappings create heteroscedasticityheteroscedasticity

Create systematic relationships between εu and levels of predictor

Gm = (Gu)2 + εmg

Gm = (α + β∗Lu + εu)2 + εmg

Gm = α2 + (β∗Lu)2 + εu2 + 2α∗(β∗Lu)+ 2α∗εu + 2(β∗Lu)∗εu

+ εmg

2(2(β∗β∗LLuu))∗ε∗εuu

Predicted LexiconPredicted Lexicon

Predicted GrammarPredicted Grammar

Gu = α + β∗Lu + εuU

nd

erl

yin

g G

ram

mar

Underlying Lexicon

Gm = (Gu)2 + εm

Measured Lexicon

Measu

red

Gra

mm

ar

LexiconGrammar01020304050607080901000102030405060708090100 LexiconGrammar01020304050607080901000102030405060708090100 LexiconGrammar01020304050607080901000102030405060708090100

Predicted Value of Grammar Predicted Value of Grammar Predicted Value of Grammar

Predicted Value of Lexicon Predicted Value of Lexicon Predicted Value of Lexicon

Predicted Systematic Heteroscedasticity

Specific nonlinear mappings predict specific Specific nonlinear mappings predict specific patterns of residualspatterns of residuals If the measure of grammar is an accelerating functionIf the measure of grammar is an accelerating function

Positive relationship between absolute value of residuals and Positive relationship between absolute value of residuals and predicted values of grammarpredicted values of grammar

If the measure of lexicon is an decelerating functionIf the measure of lexicon is an decelerating function Negative relationship between absolute value of residuals and Negative relationship between absolute value of residuals and

predicted values of lexiconpredicted values of lexicon Curvilinear pattern of residuals as secondary evidenceCurvilinear pattern of residuals as secondary evidence

MacArthur Communicative Development Inventories (CDI) Norming StudyNorming Study Participants from three sites: New Haven, Seattle, and San DiegoParticipants from three sites: New Haven, Seattle, and San Diego ““Toddler” sample: Toddler” sample: NN = 1128 = 1128 Ages 16 - 30 monthsAges 16 - 30 months Children with serious medical problems, hearing problems excludedChildren with serious medical problems, hearing problems excluded Reasonably representative demographicsReasonably representative demographics

SES diverse, but well above average SES diverse, but well above average ~ 50% female~ 50% female

Two measures of central interestTwo measures of central interest Lexicon: Vocabulary production checklistLexicon: Vocabulary production checklist

680 words, wide variety of categories680 words, wide variety of categories Grammar: Assessment of syntactic developmentGrammar: Assessment of syntactic development

37 sentence pairs37 sentence pairs Endorse version of sentence that is most like what child says Endorse version of sentence that is most like what child says

MacArthur Communicative Development Inventories (CDI)

LexiconLexicon: Words child uses: Words child uses BeeBee BugBug FrogFrog TigerTiger CookieCookie EggEgg WaterWater YogurtYogurt SoupSoup BibBib BootsBoots ZipperZipper

GrammarGrammar: Most like the way your child : Most like the way your child talks right nowtalks right now

““More cookie” vs “More cookies”More cookie” vs “More cookies”

““That my truck” vs “That’s my truck”That my truck” vs “That’s my truck”

““I make tower” vs “I making tower”I make tower” vs “I making tower”

““You fix it?” vs “Can you fix it?”You fix it?” vs “Can you fix it?”

MacArthur Communicative Development Inventories (CDI) Word production and grammatical complexity increased with ageWord production and grammatical complexity increased with age

r’s (1127) = .68, .64, respectivelyr’s (1127) = .68, .64, respectively Measure of lexicon (word production) strongly related to measure of grammar Measure of lexicon (word production) strongly related to measure of grammar

(grammatical complexity)(grammatical complexity) Grammar predicted by Lexicon: RGrammar predicted by Lexicon: R22 = .72 = .72 Grammar predicted by Lexicon and LexiconGrammar predicted by Lexicon and Lexicon22: R: R22 = .78 = .78

Measured Lexicon

Measu

red

Gra

mm

ar

Predicted Lexicon

Predicted Grammar

Predicted Grammar and abs(Residual) positively related:r = .41

Predicted Lexicon and Signed Residual curvilinearrelationship:CubicNo negative relationship

Measured LexiconM

easu

red

Gra

mm

ar

Predicted Value of Grammar Predicted Value of Grammar Predicted Value of Grammar

Predicted Value of Lexicon Predicted Value of Lexicon Predicted Value of Lexicon

Predicted Lexicon

Predicted Grammar

Predicted Grammar and abs(Residual) positively related:r = .41

Predicted Lexicon and Signed Residual curvilinearrelationship:CubicNo negative relationship

Measured LexiconM

easu

red

Gra

mm

ar

Synchrony versus Priority

Synchrony is a viable alternative hereSynchrony is a viable alternative here Patterns of residuals exactly in line with Patterns of residuals exactly in line with

nonlinear mappingnonlinear mapping Had we found nice homoscedastic residualsHad we found nice homoscedastic residuals

Synchrony would be disconfirmedSynchrony would be disconfirmed• Unless error was related in strange waysUnless error was related in strange ways

Additional predictions from nonlinear mappingAdditional predictions from nonlinear mapping Provide converging evidence on the nature of the Provide converging evidence on the nature of the

nonlinear mappingnonlinear mapping

Multiply-determined Systems

Language development occurs in a unified systemLanguage development occurs in a unified systemGrammar and lexicon related, but not deterministicallyOther co-developing factors also have effectsWorking memory (Robinson et al., 2001)Social interaction (Tomasello et al., 2003)

These effects have been riding along in the error term: Gu = α + β∗Lu + εu

εu = (εu’ + [Wu, Su, etc.,])

Age as a proxy for other co-developing factorsAge as a proxy for other co-developing factorsεu = (εu’ + Au)

Gm = (α + β∗Lu + εu)2 + εmg

Gm = (α + β∗Lu + εu’ + Au)2 + εmg

Predicts that Age interacts with Lexicon (LPredicts that Age interacts with Lexicon (Lu u x Ax Auu))

Multiply-determined Systems

Predicts that Age interacts with Lexicon (Lu x Au)

Age and Lexicon x Age added to the modelLexicon x Age : B = .0015, t (1123) = 5.82Residuals remained heteroscedasticCorrelation between abs(Residual) and predicted values• r = .41Heteroscedasticity not caused by co-developing factors

Synchrony-Nonlinear Mapping Hypothesis

Age interacts with lexicon as predicted Age interacts with lexicon as predicted by the nonlinear mapping hypothesisby the nonlinear mapping hypothesis

Unexpected result

Pattern of residuals and the interaction Pattern of residuals and the interaction with age consistent with nonlinear with age consistent with nonlinear mappingmapping

Synchrony between lexical and syntactic Synchrony between lexical and syntactic development (and nonlinear mapping)development (and nonlinear mapping)

Hypothesis fits the data quite nicely

Grammar and the Lexicon Lexicon does not precede grammar, Lexicon does not precede grammar,

develop togetherdevelop togetherFits with idea of unified system, but has Fits with idea of unified system, but has

different implicationsdifferent implicationsModels should not demonstrate “lexicon-

precedes-grammar” orderingPossible reciprocal influencesLexicon --> GrammarGrammar --> Lexicon

Mutually driven by a third factor

Nonlinear Mapping as a Fundamental Issue

Presented nonlinear mapping as a Presented nonlinear mapping as a problem for interpreting developmental problem for interpreting developmental orderingordering

Issue is more general…Issue is more general…Every domain in psychology faces this problemEvery domain in psychology faces this problemFunctional form depends on the mapping Functional form depends on the mapping

between the underlying variable and the between the underlying variable and the measuremeasure

RT measure of activationRT measure of activationLikert scale measure of risk perceptionLikert scale measure of risk perceptionStroop interference measure of automatic Stroop interference measure of automatic

processingprocessing

A General Strategy for Evaluating Functional Form

Nonlinear mappings must create Nonlinear mappings must create systematic relationships among systematic relationships among underlying variablesunderlying variables

Error becomes correlated with Error becomes correlated with predictors and, therefore, predicted predictors and, therefore, predicted valuesvalues

First level of evidenceFirst level of evidence

Other “minor” contributing factors will Other “minor” contributing factors will also become correlatedalso become correlated

Second level of evidenceSecond level of evidence

Could also manipulate the underlying Could also manipulate the underlying variancevariance

Third level of evidenceThird level of evidence