1. The Lexicon-Grammar Relationship: Revisiting the Critical-Mass Hypothesis James A. Dixon University of Connecticut Thanks to Virginia Marchman

2. Emergence of Language Language structures emerge from interactions among various levels of the system (Elman, 2004; MacWhinney, 2004; Zevin & Seidenberg, 2004) Language structures emerge from interactions among various levels of the system (Elman, 2004; MacWhinney, 2004; Zevin & Seidenberg, 2004)  Lexical, semantic, syntactic, pragmatic, etc.,  Phonological representations emerge from:  articulatory, acoustic, semantic interactions (Plaut & Kello, 1999) Structures formed through repeated, real-time interactions Structures formed through repeated, real-time interactions  Language acquisition within a unified system

3. Developmental Relations Language emerges from a unified system… Language emerges from a unified system…  Need to understand the developmental relations within that system  How is lexical development related to grammatical development? Developmental ordering Developmental ordering  Does one item “lead” another in the system?  Do two items emerge synchronously? Ordering imposes important constraints on theory Ordering imposes important constraints on theory Primary type of evidence for modeling Primary type of evidence for modeling  Models must demonstrate the same developmental orderings

4. Two Related Goals Present a new approach for testing hypotheses about ordering Present a new approach for testing hypotheses about ordering  Applied to the most recent MacArthur-Bates CDI data Provide new information about the relationship between lexical and grammatical development Provide new information about the relationship between lexical and grammatical development

5. Grammar and the Lexicon Development of the lexicon and grammar strongly related Development of the lexicon and grammar strongly related  Relationship is curvilinear Lexical development precedes grammatical development Lexical development precedes grammatical development Lexicon Grammar

6. Lexicon precedes Grammar Observed in both cross-sectional and longitudinal samples (Bates et al. 1994; Fenson et al., 1994; Dale et al., 2000) Observed in both cross-sectional and longitudinal samples (Bates et al. 1994; Fenson et al., 1994; Dale et al., 2000) Similar relation observed for: Similar relation observed for:  Italian (Caselli, Casadio, & Bates, 1999)  Hebrew (Maital et al., 2000)  Icelandic (Thordardottir, Weismer, & Evans, 2002)  Spanish (Jackson-Maldonado et al., 2003)  Spanish-English bilingual acquisition (Marchman et al., 2004)

7. Lexicon precedes Grammar Relation holds when words related to grammatical complexity are removed Relation holds when words related to grammatical complexity are removed  Grammatical function words  Prepositions  Conjunctions

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

9. Grammar and the Lexicon However, problems with this direct interpretation However, problems with this direct interpretation  Measures of underlying variables  Lexicon  Grammar Interpreting the form of the function requires very stringent assumptions about measures Interpreting the form of the function requires very stringent assumptions about measures  Relationship between the measure and the underlying variable  Measure must be equally “responsive” to changes in the underlying dimension across the entire developmental range  Interval Scales

10. Underlying and Measured Levels Measure of Lexicon Underlying Grammar Measure of Grammar Underlying Lexicon Development

11. An Alternative Hypothesis Measure of Lexicon Underlying Grammar Measure of Grammar Underlying Lexicon Development Measure of Lexicon Measure of Grammar Very serious problem Very serious problem  Observed relationship not evidence of underlying form Nonlinear mappings can create curvilinear relationship Nonlinear mappings can create curvilinear relationship Synchrony with nonlinear mapping is viable alternative hypothesis Synchrony with nonlinear mapping is viable alternative hypothesis

12. u u Underlying Level t G u =  +  ∗ L u +  u   Measured Level  G m = (G u ) 2 +  mg  L m = M(L u ) +  ml Representing Developmental Relations Measure of Lexicon Underlying Grammar Measure of Grammar Underlying Lexicon Development Another way to represent synchrony: Another way to represent synchrony:

13. Homoscedasticity, Heteroscedasticity, & Devlopmental Order Assume that our error terms are largely homoscedastic Assume that our error terms are largely homoscedastic  Magnitude of variance constant across the entire developmental range:  u,  mg,  ml  Usual OLS regression assumption However, nonlinear mappings create heteroscedasticity However, nonlinear mappings create heteroscedasticity  Create systematic relationships between  u and levels of predictor  G m = (G u ) 2 +  mg  G m = (  +  ∗ L u +  u ) 2 +  mg  G m =  2 + (  ∗ L u ) 2 +  u 2 + 2  ∗ (  ∗ L u )+ 2  ∗  u + 2(  ∗ L u ) ∗  u +  mg 2(  ∗ L u ) ∗  u

14. Predicted Lexicon Predicted Grammar G u =  +  ∗ L u +  u Underlying Grammar Underlying Lexicon G m = (G u ) 2 +  m Measured Lexicon Measured Grammar

16. Predicted Value of Grammar Predicted Value of Lexicon

17. Predicted Systematic Heteroscedasticity Specific nonlinear mappings predict specific patterns of residuals Specific nonlinear mappings predict specific patterns of residuals  If the measure of grammar is an accelerating function  Positive relationship between absolute value of residuals and predicted values of grammar  If the measure of lexicon is an decelerating function  Negative relationship between absolute value of residuals and predicted values of lexicon  Curvilinear pattern of residuals as secondary evidence

18. MacArthur Communicative Development Inventories (CDI) Norming Study Norming Study Participants from three sites: New Haven, Seattle, and San Diego Participants from three sites: New Haven, Seattle, and San Diego “Toddler” sample: N = 1128 “Toddler” sample: N = 1128 Ages 16 - 30 months Ages 16 - 30 months Children with serious medical problems, hearing problems excluded Children with serious medical problems, hearing problems excluded Reasonably representative demographics Reasonably representative demographics  SES diverse, but well above average  ~ 50% female Two measures of central interest Two measures of central interest  Lexicon: Vocabulary production checklist  680 words, wide variety of categories  Grammar: Assessment of syntactic development  37 sentence pairs  Endorse version of sentence that is most like what child says

19. MacArthur Communicative Development Inventories (CDI) Lexicon: Words child uses Lexicon: Words child uses  Bee  Bug  Frog  Tiger  Cookie  Egg  Water  Yogurt  Soup  Bib  Boots  Zipper Grammar: Most like the way your child talks right now  “More cookie” vs “More cookies”  “That my truck” vs “That’s my truck”  “I make tower” vs “I making tower”  “You fix it?” vs “Can you fix it?”

20. MacArthur Communicative Development Inventories (CDI) Word production and grammatical complexity increased with age Word production and grammatical complexity increased with age  r’s (1127) =.68,.64, respectively Measure of lexicon (word production) strongly related to measure of grammar (grammatical complexity) Measure of lexicon (word production) strongly related to measure of grammar (grammatical complexity)  Grammar predicted by Lexicon: R 2 =.72  Grammar predicted by Lexicon and Lexicon 2 : R 2 =.78 Measured Lexicon Measured Grammar

21. Predicted Lexicon Predicted Grammar Predicted Grammar and abs(Residual) positively related: r =.41 Predicted Lexicon and Signed Residual curvilinear relationship: Cubic No negative relationship Measured Lexicon Measured Grammar

22. Predicted Value of Grammar Predicted Value of Lexicon

23. Predicted Lexicon Predicted Grammar Predicted Grammar and abs(Residual) positively related: r =.41 Predicted Lexicon and Signed Residual curvilinear relationship: Cubic No negative relationship Measured Lexicon Measured Grammar

24. Synchrony versus Priority Synchrony is a viable alternative here Synchrony is a viable alternative here  Patterns of residuals exactly in line with nonlinear mapping  Had we found nice homoscedastic residuals  Synchrony would be disconfirmed Unless error was related in strange waysUnless error was related in strange ways Additional predictions from nonlinear mapping Additional predictions from nonlinear mapping  Provide converging evidence on the nature of the nonlinear mapping

25. Multiply-determined Systems Language development occurs in a unified system Language development occurs in a unified system  Grammar and lexicon related, but not deterministically  Other co-developing factors also have effects  Working memory (Robinson et al., 2001)  Social interaction (Tomasello et al., 2003)  These effects have been riding along in the error term: G u =  +  ∗ L u +  u   u = (  u ’ + [W u, S u, etc.,]) Age as a proxy for other co-developing factors Age as a proxy for other co-developing factors   u = (  u ’ + A u )  G m = (  +  ∗ L u +  u ) 2 +  mg  G m = (  +  ∗ L u +  u ’ + A u ) 2 +  mg Predicts that Age interacts with Lexicon (L u x A u )  Predicts that Age interacts with Lexicon (L u x A u )

26. Multiply-determined Systems Predicts that Age interacts with Lexicon (L u x A u )  Age and Lexicon x Age added to the model  Lexicon x Age : B =.0015, t (1123) = 5.82  Residuals remained heteroscedastic  Correlation between abs(Residual) and predicted values r =.41  Heteroscedasticity not caused by co-developing factors

27. Synchrony-Nonlinear Mapping Hypothesis Age interacts with lexicon as predicted by the nonlinear mapping hypothesis Age interacts with lexicon as predicted by the nonlinear mapping hypothesis  Unexpected result Pattern of residuals and the interaction with age consistent with nonlinear mapping Pattern of residuals and the interaction with age consistent with nonlinear mapping Synchrony between lexical and syntactic development (and nonlinear mapping) Synchrony between lexical and syntactic development (and nonlinear mapping)  Hypothesis fits the data quite nicely

28. Grammar and the Lexicon Lexicon does not precede grammar, develop together Lexicon does not precede grammar, develop together Fits with idea of unified system, but has different implications Fits with idea of unified system, but has different implications  Models should not demonstrate “lexicon-precedes- grammar” ordering  Possible reciprocal influences  Lexicon --> Grammar  Grammar --> Lexicon  Mutually driven by a third factor

29. Nonlinear Mapping as a Fundamental Issue Presented nonlinear mapping as a problem for interpreting developmental ordering Presented nonlinear mapping as a problem for interpreting developmental ordering Issue is more general… Issue is more general…  Every domain in psychology faces this problem  Functional form depends on the mapping between the underlying variable and the measure  RT measure of activation  Likert scale measure of risk perception  Stroop interference measure of automatic processing

30. A General Strategy for Evaluating Functional Form Nonlinear mappings must create systematic relationships among underlying variables Nonlinear mappings must create systematic relationships among underlying variables Error becomes correlated with predictors and, therefore, predicted values Error becomes correlated with predictors and, therefore, predicted values  First level of evidence Other “minor” contributing factors will also become correlated Other “minor” contributing factors will also become correlated  Second level of evidence Could also manipulate the underlying variance Could also manipulate the underlying variance  Third level of evidence