the lexicon-grammar relationship: revisiting the critical-mass hypothesis james a. dixon university...
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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
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An Alternative Hypothesis
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10090807060504030201000102030405060708090100
Measure of Lexicon
Mea
sure
of
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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
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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