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Modeling Longitudinal Trajectories of Word Production with the CDI

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Title: Modeling Longitudinal Trajectories of Word Production with the CDI
Language: English
Authors: Trevor K. M. Day (ORCID 0000-0003-2911-8312), Arielle Borovsky, Donna Thal, Jed T. Elison
Source: Developmental Science. 2025 28(4).
Availability: Wiley. Available from: John Wiley & Sons, Inc. 111 River Street, Hoboken, NJ 07030. Tel: 800-835-6770; e-mail: cs-journals@wiley.com; Web site: https://www.wiley.com/en-us
Peer Reviewed: Y
Page Count: 18
Publication Date: 2025
Sponsoring Agency: National Institutes of Health (NIH) (DHHS)
National Institute on Deafness and Other Communication Disorders (NIDCD) (DHHS/NIH)
National Science Foundation (NSF), Graduate Research Fellowship Program (GRFP)
Contract Number: R01MH104324
U01MH110274
Document Type: Journal Articles
Reports - Research
Descriptors: Language Skills, Measures (Individuals), Children, Models, Data Analysis, Longitudinal Studies, Speech Impairments, Language Impairments, Learning Disabilities, Scores, Infants, Toddlers
Assessment and Survey Identifiers: MacArthur Bates Communicative Development Inventories
DOI: 10.1111/desc.70036
ISSN: 1363-755X
1467-7687
Abstract: The MacArthur-Bates Communicative Development Inventories (CDI) are widely used, parent-report instruments of language acquisition. Here, we focus on the word-inventory sections of the instruments, and show two different approaches to modeling CDI data, based on real-world needs. First, we show that Words & Gestures data collected out-of-age-normed-range can be robustly adjusted to Words & Sentences scores. Second, we demonstrate a novel application of Gompertz growth curves to longitudinal CDI data, especially when the same timepoints were not collected between individuals (i.e., an accelerated longitudinal design). Gompertz curves provide a "growth rate" or an "age at maximum growth" parameter that can be used to summarize vocabulary development. We compare these parameters between healthy developing children in two longitudinal cohorts, as well as a cohort of children with a diagnosis of speech disorder, language disorder, or learning or reading disability, who we show to have lower growth rates. We hope these analyses and results inform future work on longitudinal CDI analyses.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1475002
Database: ERIC
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  Value: <anid>AN0186138041;5g501jul.25;2025Jun26.03:45;v2.2.500</anid> <title id="AN0186138041-1">Modeling Longitudinal Trajectories of Word Production With the CDI </title> <p>The MacArthur‐Bates Communicative Development Inventories (CDI) are widely used, parent‐report instruments of language acquisition. Here, we focus on the word‐inventory sections of the instruments, and show two different approaches to modeling CDI data, based on real‐world needs. First, we show that Words & Gestures data collected out‐of‐age‐normed‐range can be robustly adjusted to Words & Sentences scores. Second, we demonstrate a novel application of Gompertz growth curves to longitudinal CDI data, especially when the same timepoints were not collected between individuals (i.e., an accelerated longitudinal design). Gompertz curves provide a "growth rate" or an "age at maximum growth" parameter that can be used to summarize vocabulary development. We compare these parameters between healthy developing children in two longitudinal cohorts, as well as a cohort of children with a diagnosis of speech disorder, language disorder, or learning or reading disability, who we show to have lower growth rates. We hope these analyses and results inform future work on longitudinal CDI analyses.</p> <p>Keywords: CDI; growth curve; language acquisition; word learning</p> <hd id="AN0186138041-2">Introduction</hd> <p>One of the enduring foci of study in language acquisition is word learning. Beginning around 18 months and continuing into the second year of life, English‐learning children tend to undergo a rapid expansion in their productive vocabulary (Bloom [<reflink idref="bib3" id="ref1">3</reflink>]; Brown [<reflink idref="bib5" id="ref2">5</reflink>]; Fenson et al. [<reflink idref="bib14" id="ref3">14</reflink>], [<reflink idref="bib15" id="ref4">15</reflink>]; Fernald et al. [<reflink idref="bib16" id="ref5">16</reflink>]; Frank et al. [<reflink idref="bib18" id="ref6">18</reflink>]; Thal et al. [<reflink idref="bib42" id="ref7">42</reflink>]). This phenomenon is followed by an increase in phrase and sentence construction skills (Brown [<reflink idref="bib5" id="ref8">5</reflink>]; Fenson et al. [<reflink idref="bib14" id="ref9">14</reflink>]; Fernald et al. [<reflink idref="bib16" id="ref10">16</reflink>]; Thal et al. [<reflink idref="bib42" id="ref11">42</reflink>]) until an age of about 7–10 years (Hyams and Orfitelli [<reflink idref="bib23" id="ref12">23</reflink>]; Johnson and Newport [<reflink idref="bib24" id="ref13">24</reflink>]).</p> <hd id="AN0186138041-3">Summary</hd> <p></p> <ulist> <item> The MacArthur‐Bates Communicative Development Inventories are widely used language‐development instruments. We provide two novel analyses of the CDIs.</item> <p></p> <item> We show how to project scores from the infant CDI to the toddler CDI, for instances where the infant CDI was used for toddlers.</item> <p></p> <item> We provide a novel longitudinal analysis that creates a summary "growth score" for CDIs that accounts for ceiling and floor effects using Gompertz curves.</item> <p></p> <item> The rate of growth is found to be the highest at 24 months; suggesting researchers interested in trajectories of growth collect data past this age.</item> </ulist> <p>Word learning represents a conceptual middle ground between phoneme acquisition and the understanding and use of complex syntax. During toddlerhood, knowledge of words is conceptually easy to measure—what linguistic event is better‐known than a baby's first word?—and has been shown to predict a variety of outcomes at different ages (Fisher [<reflink idref="bib17" id="ref14">17</reflink>]; Marchman and Fernald [<reflink idref="bib28" id="ref15">28</reflink>]; Patrucco‐Nanchen et al. [<reflink idref="bib35" id="ref16">35</reflink>]). In this paper, we provide two analyses of word production data from parent‐report instruments. The first is a procedure for better estimating vocabulary size when an age‐inappropriate instrument was used. The second demonstrates a novel modeling technique for said vocabulary data.</p> <p>The emergence of word use is often explained by the realization that words function as symbols that refer to objects in the environment (Dore [<reflink idref="bib12" id="ref17">12</reflink>]; Dore et al. [<reflink idref="bib13" id="ref18">13</reflink>]; Reznick and Goldfield [<reflink idref="bib40" id="ref19">40</reflink>] and many others), further supported by growth in the ability to categorize objects (Gopnik and Meltzoff [<reflink idref="bib21" id="ref20">21</reflink>]), and improvement in word‐segmentation skills (Plunkett [<reflink idref="bib36" id="ref21">36</reflink>]; Walley [<reflink idref="bib45" id="ref22">45</reflink>]). Code for these analyses is available at https://github.com/TrevorKMDay/MCDI-analysis.</p> <hd id="AN0186138041-4">Vocabulary Spurts</hd> <p>In the study of word learning, the idea of a "vocabulary spurt" or "naming explosion" emerged in the 1980s. These terms refer to the same concept: A transition between an earlier period of slower, but non‐zero lexical development and a period of rapid word learning to reach an average of 300 words by 24 months (Fenson et al. [<reflink idref="bib14" id="ref23">14</reflink>]; Ganger and Brent [<reflink idref="bib19" id="ref24">19</reflink>]). The threshold for spurt‐like growth compared to slower development has been defined in multiple ways in the literature, but in general refers to growth between about 13 and 17 new words in a month, occurring roughly between 17 and 22 months (Choi and Gopnik [<reflink idref="bib7" id="ref25">7</reflink>]; Goldfield and Reznick [<reflink idref="bib20" id="ref26">20</reflink>]; Gopnik and Meltzoff [<reflink idref="bib21" id="ref27">21</reflink>]; Lifter and Bloom [<reflink idref="bib25" id="ref28">25</reflink>]; Mervis and Bertrand [[<reflink idref="bib31" id="ref29">31</reflink>]]; Poulin‐Dubois et al. [<reflink idref="bib37" id="ref30">37</reflink>]), see Table 1. See also Figures S1.1 and S1.2 for a visual depiction of the table below. Ninio ([<reflink idref="bib33" id="ref31">33</reflink>]) took a different approach, and defined the spurt as the 2‐month period with the longest rate of growth in an analysis of Hebrew data.</p> <p>1 TABLE Growth spurts as reported in the early English‐learning literature, in published parameters and normalized to words per month. Age at spurt given as reported in original paper, in a range or M (SD). Mean age at spurt is estimate for Mervis and Bertrand (1994) based on information reported.</p> <p> <ephtml> <table><thead><tr><th>Reference</th><th>Reported threshold</th><th>Words per month</th><th>Age at spurt (mo)</th></tr></thead><tbody><tr><td>Choi and Gopnik (<xref ref-type="bibr" rid="bibr7">1995</xref>)</td><td>10 object words in 3–4 weeks</td><td>10.0–14.3</td><td>16.7–21.4</td></tr><tr><td /><td>10 verbs in 3–4 weeks</td><td>10.0–14.3</td><td /></tr><tr><td>Lifter and Bloom (<xref ref-type="bibr" rid="bibr25">1989</xref>)</td><td>At least 20 words plus 12 words in 1 month</td><td>12.0</td><td>13–25</td></tr><tr><td>Gopnik and Meltzoff (<xref ref-type="bibr" rid="bibr21">1987</xref>)</td><td>10 object words in 3 weeks</td><td>14.3</td><td>15.7–21.7</td></tr><tr><td>Poulin‐Dubois et al. (<xref ref-type="bibr" rid="bibr37">1995</xref>)</td><td>15 object words in 1 month</td><td>15.0</td><td>20.53 (3.29)</td></tr><tr><td>Goldfield and Reznick (<xref ref-type="bibr" rid="bibr20">1990</xref>)</td><td>10 words in 2.5 weeks</td><td>17.1</td><td>15–22</td></tr><tr><td>Mervis and Bertrand ([<xref ref-type="bibr" rid="bibr31">31-32</xref>])</td><td>10 words in 14 days</td><td>21.4</td><td>∼21.4 (2.4)</td></tr><tr><td>20.0–20.7</td></tr><tr><td /><td /><td>10.0–21.4</td><td>13–21.4</td></tr></tbody></table> </ephtml> </p> <p>The idea of a "spurt" or "explosion" is defined by its nonlinearity. That is, children who undergo a spurt enter a period of rapid growth, after which the rate of word learning declines. Many authors have attempted to test for the existence of a "spurt." For example, Ganger and Brent ([<reflink idref="bib19" id="ref32">19</reflink>]) provide an analysis that compares a quadratic fit to a logistic fit of the <emph>rate of learning</emph> compared to <emph>total inventory size</emph>. In order to calculate vocabulary changes, parents were asked to report all the words their child used each day (beginning around 15.3 months and continuing for about a year), which was used by the authors to identify new words added to the child's inventory. If a spurt exists (i.e., there is an <emph>inflection point</emph> in the rate over time), then the logistic fit will show better model fit compared to the quadratic fit (which does not have an inflection point). Individuals for whom a logistic form fits their data better are concluded to have undergone a spurt, and vice versa. This reflects non‐monotonicity in the growth rate, a temporary change being considered a "spurt" and a child with an ever‐increasing rate being a "non‐spurter." The authors concluded that about 1 in 5 children demonstrated a "spurt" under these modeling conditions. That said, in a reanalysis of some of Ganger and Brent's diary data, Dandurand and Shultz ([<reflink idref="bib10" id="ref33">10</reflink>]) identified a mean of 6.5 spurts per individual using automatic maxima detection, suggesting that the identification of spurts is highly dependent on both conceptualization and implementation. Parladé and Iverson ([<reflink idref="bib34" id="ref34">34</reflink>]) also performed a longitudinal analysis of parent‐reported vocabulary sizes at 2‐week intervals between 8 and 19 months. They applied Ganger and Brent's method of comparing logistic to quadratic fits, concluding that half of the 18 participants exhibited a spurt.</p> <p>However, McMurray ([<reflink idref="bib30" id="ref35">30</reflink>]) argues that a vocabulary explosion is a natural consequence of word learning. Under this model, each word is learned at the same rate, but has a different time‐to‐acquisition, reflecting various levels of difficulty. Therefore, he argues that an "explosion" does not reflect specialized processes, but merely the accumulation of words as a child progresses from 0 to 60,000 words.</p> <hd id="AN0186138041-5">Measuring Vocabulary</hd> <p>Together, these analyses suggest that a (single) "vocabulary spurt" is not a universal feature of first‐language (L1) acquisition. However, the vocabulary‐spurt line of research neatly illustrates the problem of measuring and modeling vocabulary growth, which is constrained by a few factors.</p> <p>One constraint is the amount of sampling some researchers have considered necessary to measure vocabulary growth. For example, Ganger and Brent ([<reflink idref="bib19" id="ref36">19</reflink>]) and Dandurand and Shultz ([<reflink idref="bib10" id="ref37">10</reflink>]) used daily sampled vocabulary counts in three studies (with sample sizes of 38, 18, and 1). Due to the demands on the raters, sampling at such high frequency is not always possible, and does not scale well to larger samples. Furthermore, in studies where language is a covariate or secondary outcome, daily diary reports on vocabulary development would be excluded as too burdensome. Finally, these studies focused on early ages, and did not have to contend with an upper asymptote on words reported.</p> <p>The MacArthur‐Bates Communicative Development Inventories (CDI; Fenson et al. [<reflink idref="bib14" id="ref38">14</reflink>]) are widely used language development assessments, with core lists of words typically learned and spoken in early development. While the forms were carefully normed to capture common early words, the form for older children lists 680 items, a vocabulary size that children quickly grow past. As such, beyond the first few years of life, it is impossible for a rater to list all of the words a child knows. This means word lists like those on the CDI may capture only a subset of the words a child produces, which is in turn, a subset of the words a child knows. Secondly, word learning is clearly nonlinear in its trajectory, requiring analytical techniques to capture trends beyond univariate linear regression.</p> <p>In this paper, we estimate word‐learning trajectories based on CDI data. These instruments are widely used parent‐report questionnaires that have been shown to have high test‐retest reliability (<emph>r</emph>s = 0.87–0.95), high internal consistency (<emph>r</emph>s = 0.95–0.96), and high reliability compared to laboratory measures (<emph>r</emph>s = 0.33–0.85) (Fenson et al. [<reflink idref="bib14" id="ref39">14</reflink>]). Furthermore, the CDI has been adapted into at least 42 languages and dialects (stanford.wordbank.edu; January 2025).</p> <p>The English version of the CDI is divided into two assessments tailored to the communicative skills of two age groups: Words & Gestures (WG; normed for 8–18 months) and Words and Sentences (WS; normed for 16–30 months), each comprising a central word list (WG: 396 items; WS: 680). WG includes some additional items regarding early gestures and behaviors, and WS omits gestures, but includes morphological categories such as correct irregular nouns and verbs (e.g., <emph>children</emph> and <emph>ate</emph>), as well as incorrect overgeneralizations (e.g., *<emph>childrens</emph> and *<emph>ated</emph>). Additionally, WG asks parents to choose whether their child "understands" or "says and understands" an item, capturing both receptive and expressive language. WS, on the other hand, only asks if their child "says" a word, as parents were found to not be reliable reports of receptive language past 16 months (REF).</p> <p>The WS form also includes a 32‐item sentence complexity section that asks parents whether their child utters sentences like "That my truck" versus "That's my truck." It also collects non‐contemporaneous "three of the longest sentences you have heard your child say recently" for calculating mean length of utterance (MLU; Brown [<reflink idref="bib5" id="ref40">5</reflink>]). The forms have been carefully designed so that the items are age‐appropriate and representative of the child's larger linguistic development.</p> <p>Although longitudinal studies of the CDI have been reported in multiple languages (American Sign Language: Anderson and Reilly [<reflink idref="bib1" id="ref41">1</reflink>]; English: Bauer et al. [<reflink idref="bib2" id="ref42">2</reflink>]; Italian: Longobardi et al. [<reflink idref="bib26" id="ref43">26</reflink>]; Finnish: Stolt et al. [<reflink idref="bib41" id="ref44">41</reflink>]; Danish: Wehberg et al. [<reflink idref="bib46" id="ref45">46</reflink>]), the literature on the trajectory of growth on the CDI in English is small in comparison to the vast literature citing the CDI (as of January 24, 2025, the original Fenson and colleagues monograph has received more than 4500 citations on Google Scholar).</p> <p>One of those few longitudinal studies of the English CDI is Bauer et al. ([<reflink idref="bib2" id="ref46">2</reflink>]), who examined WG between 8 and 14 months of age (<emph>n</emph>  =  26), distinguishing two clusters of "rapid" and "slow" learners, which were validated against WS scores at 21 months. Over this age range, lexical development appears quadratic, and the authors reported that (for comprehension, but not production) the linear and quadratic effects were highly collinear and thus fixed the linear effect at 0.</p> <p>However, children eventually surpass the number of items available on the CDI. This means scores reach an upper asymptote, making a quadratic model unsuitable. Therefore, for studies that include children reaching ceiling on WS, a different functional form becomes necessary. Because scores reach the upper asymptote, a model must reflect the actual rate of growth, accounting for the limited number of items on the CDI. The longitudinal studies so far have used designs with the same sampling ages between participants. One of the studies analyzed in this paper was an accelerated longitudinal design, where participants were not studied at the same timepoints (e.g., always 12, 14, 16 months, etc.). Thus, we develop a more flexible modeling technique to compare language acquisition between participants.</p> <p>In this paper, we present two novel analyses of the CDI motivated bythe CDI data collected as part of the Baby Connectome Project (BCP; Howell et al. [<reflink idref="bib22" id="ref47">22</reflink>]), which collected WG and WS data from toddlers at 3‐month intervals, but also collected WG and WS forms above the normed range. Because BCP was an accelerated longitudinal study, visit ages were interposed, which can be challenging to analyze using conventional analyses. This paper describes two methods developed to analyze data from this study.</p> <p>First, we developed a method for simulating WG scores from individuals too old for the instrument norms (older than 18 months) to their would‐be score on WS. Forty‐seven percent of the BCP WG forms were collected above the normed range, up to a chronological age of 27.5 months. Collecting WG beyond the normed age range was done to serve as a reference for studies of young children with intellectual and developmental disabilities (IDD). Out‐of‐normed‐age‐range is common in studies of children with or at high‐risk for intellectual and developmental delays (e.g., Charman et al. [<reflink idref="bib6" id="ref48">6</reflink>]; Coonrod and Stone [<reflink idref="bib9" id="ref49">9</reflink>]; Luyster et al. [<reflink idref="bib27" id="ref50">27</reflink>]; Rague et al. [<reflink idref="bib39" id="ref51">39</reflink>]; Wu et al. [<reflink idref="bib47" id="ref52">47</reflink>]; Yoder et al. [<reflink idref="bib48" id="ref53">48</reflink>]; Zampini et al. [<reflink idref="bib49" id="ref54">49</reflink>]).</p> <p>When it came time to analyze the BCP CDI data longitudinally, this raised two questions. First, whether out‐of‐normed‐age‐range WG forms correctly estimated vocabulary size for typically developing children. And second, whether out‐of‐normed‐age‐range WG scores can be compared to and modeled alongside WS scores collected at the appropriate ages. This latter question applies to comparing within participants; who may have changed from WG to WS within the study, and also comparing between participants, who may have been studied using out‐of‐normed‐age‐range WG and WS at the same ages.</p> <p>Second, we used growth curves—specifically, Gompertz curves—to fit longitudinal data and calculate a summary growth value that can be used when sampling is not aligned between participants. None of the longitudinal techniques described above were sufficient for the BCP use‐case for two reasons. The forms were not collected densely enough for Ganger and Brent's ([<reflink idref="bib19" id="ref55">19</reflink>]) analysis, and because of the wider age range in BCP, individuals' trajectories are constrained by the lower and upper asymptotes at 0 and 680 words, making a linear or polynomial model (as suggested by Bauer and colleagues) unsuitable. A Gompertz curve fitted using nonlinear least squares is adaptable to the data at hand, and additionally models upper and lower asymptotes. However, Gompertz curves diverge from logistic models that also have upper and lower asymptotes by being asymmetrical and not assuming the rate approaching the upper asymptote is the exact inverse of the early growth.</p> <hd id="AN0186138041-6">Part 1 Methods</hd> <p></p> <hd id="AN0186138041-7">Participants</hd> <p>The analyses that comprise Parts 1 and 2 of this report combine data from three sources: BCP, the Early Identification of Risk for Language Impairment (EIRLI) study, and the Stanford Wordbank (<ulink href="http://wordbank.stanford.edu/">http://wordbank.stanford.edu/</ulink>). The BCP represented a collaborative effort between the University of Minnesota (UMN) and the University of North Carolina (UNC; Howell et al. [<reflink idref="bib22" id="ref56">22</reflink>]). Participants were recruited from the respective metropolitan areas, Minneapolis/St. Paul, Minnesota, and Chapel Hill, North Carolina, USA. The primary objective of this project was to collect brain (structure and connectivity) and behavioral data from typically developing children. While 282 participants had at least one CDI form, only 150 had at least three measurements needed for longitudinal analysis. Data were collected between 2015 and 2020, when COVID‐19 interrupted the final stages of data collection</p> <p>Details on the recruitment strategy used in BCP can be found in Howell et al. ([<reflink idref="bib22" id="ref57">22</reflink>]), but in brief, most participants were assigned to cohorts with varying starting ages, but visits at 3‐month intervals. Approximately a third of participants had only one visit. Visits were denser at younger ages, in part due to the study design and in part due to the COVID pandemic, leading to fewer visits at ages older than 24 months. As mentioned above, 47% of WG forms were collected above the normed age, but additionally, 52% of BCP WS forms were collected above the normed range for that instrument, up to 38.3 months.</p> <p>The second project, EIRLI, consists of WS forms collected in the metropolitan Southern California, USA, area in the 1990s (see Phase IV of Thal et al. [<reflink idref="bib42" id="ref58">42</reflink>]). Unfortunately, three lexical categories went missing (<emph>Outside Things</emph> [31 tokens], <emph>Places to Go</emph> [<reflink idref="bib22" id="ref59">22</reflink>], <emph>Time Words</emph> [<reflink idref="bib12" id="ref60">12</reflink>]) for a total of 65 missing items, leaving 615 total words, as did the morphosyntactic categories (<emph>Word Forms</emph> and <emph>Endings</emph> for both nouns and verbs). Furthermore, it is only known for a subset of individuals whether they eventually received a diagnosis of a language disorder. Individuals without follow‐up are labeled "DX0," that is, "diagnostic status unknown." In order to determine diagnostic status, "parents were also asked if their children had received a diagnosis of speech disorder, language disorder, or learning or reading disability when they were between 4 and 7 years of age, and, if the answer was yes, they were asked to provide a copy of the clinical evaluation." (Thal et al. [<reflink idref="bib42" id="ref61">42</reflink>], 183). In this study, those who received a diagnosis are labeled "DX+," and those who were evaluated, but had no language disorder are labeled "DX−,"</p> <p>Thirdly, we compare these two datasets against English data obtained from Wordbank, a repository of cross‐sectional data from CDI forms globally (Frank et al. [<reflink idref="bib18" id="ref62">18</reflink>]). Data were obtained from Wordbank using the <emph>wordbankr</emph> package (ver. 1.0.0; Braginsky [<reflink idref="bib4" id="ref63">4</reflink>]). For analyses contrasting EIRLI with Wordbank data, individuals contributed by D. Thal to Wordbank were removed. See Table 2 for a summary of the basic demographic information common among the three datasets. For additional detail on the race/ethnicity of the samples (which were reported differently), see Table S2.1. We did not use race/ethnicity as a covariate, so for legibility, we show only the top three most common race/ethnicity categories here. Those categories were non‐Hispanic (or ethnicity not reported) White, Hispanic (all races), and Black (all ethnicity values, including missing). For more detail on reporting differences between the studies, see Supplementary Material 2. This section will not be recapitulated for Part 2.</p> <p>2 TABLE Demographic data for the three datasets.</p> <p> <ephtml> <table><thead><tr><th>Dataset</th><th>Individuals</th><th>Timepoints</th><th>Age (mo.)</th><th>Male (%)</th><th>Parent graduated college<ext-link /><sup>[a]</sup> (%)</th><th>Race/ethnicity (White, Black, Hispanic)</th></tr></thead><tbody><tr><td>EIRLI Dx−</td><td>582</td><td>1827</td><td>23.7 (6.57)</td><td>304 (52%)</td><td>443 (76%)</td><td>516 (89%)</td></tr><tr><td /><td /><td /><td /><td>381 (66%)</td><td>8 (1.4%)</td></tr><tr><td /><td /><td /><td /><td /><td>10 (1.7%)</td></tr><tr><td>EIRLI Dx+</td><td>102</td><td>357</td><td>24.3 (6.87)</td><td>65 (64%)</td><td>78 (77%)</td><td>88 (88%)</td></tr><tr><td /><td /><td /><td /><td>67 (66%)</td><td>4 (4%)</td></tr><tr><td /><td /><td /><td /><td /><td>2 (2%)</td></tr><tr><td>EIRLI Dx0</td><td>421</td><td>1,132</td><td>22.6 (6.36)</td><td>206 (49%)</td><td>263 (63%)</td><td>356 (85%)</td></tr><tr><td /><td /><td /><td /><td>258 (61%)</td><td>12 (2.9%)</td></tr><tr><td /><td /><td /><td /><td /><td>5 (1.2%)</td></tr><tr><td>EIRLI</td><td>1105</td><td>3,316</td><td>23.4 (6.56)</td><td>575 (52%)</td><td>784 (71%)</td><td>960 (87%)</td></tr><tr><td /><td /><td /><td /><td>706 (64%)</td><td>24 (2.3%)</td></tr><tr><td /><td /><td /><td /><td /><td>17 (1.5%)</td></tr><tr><td>BCP</td><td>150</td><td>547</td><td>21.2 (6.90)</td><td>73 (49%)</td><td>131 (89%)<ext-link /><sup>[b]</sup></td><td>122 (82%)</td></tr><tr><td /><td /><td /><td /><td>114 (77%)<ext-link /><sup>[c]</sup></td><td>2 (1.3%)</td></tr><tr><td /><td /><td /><td /><td /><td>10 (6.7%)</td></tr><tr><td>WB WG<ext-link /><sup>[e]</sup></td><td>3913</td><td>—</td><td>13.7 (2.97)</td><td>53%<ext-link /><sup>[d]</sup></td><td>68%<ext-link /><sup>[d]</sup></td><td>2238 (73%)[<sup>f</sup>]</td></tr><tr><td /><td /><td /><td /><td /><td>61 (2%)</td></tr><tr><td /><td /><td /><td /><td /><td>125 (4.1%)</td></tr><tr><td>WB WS<ext-link /><sup>[e]</sup></td><td>7905</td><td>—</td><td>23.7 (4.24)</td><td>52%<ext-link /><sup>[d]</sup></td><td>69%<ext-link /><sup>[d]</sup></td><td>3959 (74%)</td></tr><tr><td /><td /><td /><td /><td /><td>69 (1.3%)</td></tr><tr><td /><td /><td /><td /><td /><td>386 (7.3%)</td></tr><tr><td>Wordbank<ext-link /><sup>[e]</sup></td><td>11,818</td><td>—</td><td>20.4(6.07)</td><td>53%<ext-link /><sup>[d]</sup></td><td>69%<ext-link /><sup>[d]</sup></td><td>6197 (74%)</td></tr><tr><td /><td /><td /><td /><td /><td>482 (5.7%)</td></tr><tr><td /><td /><td /><td /><td /><td>511 (6.1%)</td></tr><tr><td>Total</td><td>15,284<ext-link /><sup>[e]</sup></td><td>10,519<ext-link /><sup>[e]</sup></td><td>21.0(6.3)</td><td>52%<ext-link /><sup>[d]</sup></td><td>54%<ext-link /><sup>[d]</sup></td><td>7279 (76%)</td></tr><tr><td /><td /><td /><td /><td /><td>508 (5.3%)</td></tr><tr><td /><td /><td /><td /><td /><td>538 (5.6%)</td></tr></tbody></table> </ephtml> </p> <p>1 <emph>Note</emph>: Wordbank (WB) includes all individuals (including those contributed by D. Thal). Percentages are given out of non‐missing values. These are descriptive statistics with no tests being performed. The bolded rows distinguish the complete samples from the subsamples.</p> <ulist> <item>2 <sups>[a]</sups>For the "parent graduated college" column, values are reported for both the mother (top) and father (bottom), except WB, which only stores the mother's education.</item> <item>3 <sups>[b]</sups>Excludes two missing values, includes two parents whose relationship was not specified.</item> <item>4 <sups>[c]</sups>Excludes two missing values, includes two nonbiological mothers and 31 relationships not specified.</item> <item>5 <sups>[d]</sups>Due to high rates of missingness (2%–51%), only percentages are reported for legibility.</item> <item>6 <sups>[e]</sups>Includes the individuals who contributed to WB by D. Thal.</item> </ulist> <hd id="AN0186138041-8">Analytic Strategy Part 1: Out‐of‐Normed‐Age‐Range WG to WS Conversion</hd> <p>We wanted to calculate within‐person trajectories of lexical development. Since some data in BCP came from using WG past the normed range, we used Wordbank data collected between 18 and 27 months to model what each individual's score would have been if WS had been used instead. This age range was selected because BCP collected WG data up to 27.5 months.</p> <p>We (<reflink idref="bib1" id="ref64">1</reflink>) simulated <bold>would‐be WS total score from WG scores and age</bold> using a model selection approach. The 680 words on WS are broken into 22 categories, like <emph>Animals</emph> (e.g. "alligator," "duck"), <emph>Descriptive Words</emph> (e.g. "full," "orange") or <emph>Action Words</emph> (e.g. "bite," "drive"). As individuals gain words in each of these categories at different times and potentially rates, we also (<reflink idref="bib2" id="ref65">2</reflink>) simulated would‐be <bold>scores for each WS category from WG category scores, age, and total WG score</bold>. After completing steps (<reflink idref="bib1" id="ref66">1</reflink>) and (<reflink idref="bib2" id="ref67">2</reflink>), an out‐of‐normed‐age‐range WG score can be compared to in‐range WS scores.</p> <p>For this analysis, CDI WS data were retrieved from Wordbank on February 15, 2023, including cross‐sectional data from 7905 unique participants with age data (16–30 months). Data were divided into a 75/25% training/test split at each age (Wordbank ages are floored to whole months), totaling 6883 training and 1970 test participants. Because many of the analyses in this paper rely on CDI category scores, we show the distribution of scores for all categories in Figure 1 to contextualize the numeric operations being performed. Rather than being normally distributed, most categories are dominated by low scores (e.g., Helping Verbs, Household Items, Pronouns), although others show a more even distribution (e.g., Animals, Toys, Vehicles). The bimodal distribution of the EIRLI data is evident in many frames, including <emph>Food and Drink</emph>, <emph>Body Parts</emph>, and <emph>Games and Routines</emph>.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/5G5/01jul25/desc70036-fig-0001.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="desc70036-fig-0001.jpg" title="1 Distribution of category scores within each project (BCP: red; [E]IRLI: green; Wordbank [WB]: blue). Note the per‐category y axes. Outside, Places, and Time Words were missing from the EIRLI data." /> </p> <p></p> <hd id="AN0186138041-10">Total Scores</hd> <p>The WG lexical inventory is a near‐perfect subset of the WS inventory, so we are able to simulate the scores a child would have received if the reporter filled out a WG form rather than a WS form. We created simulated WG scores by summing only the items present on WG. Next, we predicted the true WS score against the polynomial simulated WG score and age until the best fit was achieved.</p> <p>The exceptions to the perfect subset status are "in" and "inside", which are distinct items on WG, but counted as a single item on WS ("inside/in" in the <emph>Prepositions and Locations</emph> category). Therefore, when WG scores were simulated from WS to WG; "in/inside" endorsement was counted as two items (i.e., endorsement of both "in" and "inside").</p> <hd id="AN0186138041-11">Category Scores</hd> <p>Growth within the CDI lexical categories does not occur at the same time or (potentially) rate, so we performed a similar analysis on category level scores, predicting true WS category total score from simulated WG category total (i.e., subsetted), age, and total simulated WG total score. Fourteen items that are not in the same category between forms were appropriately reallocated to their WG lexical category when simulating. In doing so, WS scores can be estimated from category‐level totals rather than item‐level. The <emph>Sound Effect and Animal Sounds</emph> (<emph>SEAS</emph>) category is identical between instruments, and no <emph>Connecting Words</emph> items appear on WG. Thus, these categories are left out of the estimation procedure below (<emph>Connecting Words</emph> is explained later).</p> <p>We ran a set of nested models to identify the most parsimonious model across all categories in the training set. Prior to model evaluation, predicted values less than 0 were replaced with 0; and values greater than the category‐specific maximum were replaced with that value; and all values were rounded to the nearest whole word. Age was centered on 18 months, the minimum normed age range for WS. We run a series of models predicting category scores ("C"), rather than total scores.</p> <p></p> <ulist> <item> <bold> C Model 1 </bold> : Included the simple effect of simulated WG category (C) total and <emph>C</emph><sups>2</sups> on true WS core.</item> <p></p> <item> <bold> C Model 2 </bold> : Inspection of residuals from the previous step suggested a quadratic effect of age, so C Model 2 included age, <emph>C</emph>, age*<emph>C</emph>, and age<sups>2</sups>*<emph>C</emph><sups>2</sups>.</item> <p></p> <item> <bold> C Model 3 </bold> : Inspection of the residuals from C Model 2 showed a cubic dependency on total WG score (WG), so Model 3 included age, <emph>C</emph>, age*<emph>C</emph>, and age<sups>2</sups>*<emph>C</emph><sups>2</sups>, as well as <emph>WG</emph>, <emph>WG</emph><sups>2</sups>, and <emph>WG</emph><sups>3</sups>.</item> <p></p> <item> <bold> C Model 4 </bold> : Added the age/WG interactions in accordance with the best‐fitting total score model (T3).</item> </ulist> <p>Most categories are largely dependent on the <emph>C</emph> and <emph>WG</emph> terms, but some exceptions exist (<emph>Toys, Games and Routines)</emph>. Across categories, Models 1–2 showed a small, but significant improvement in AIC (mean ΔAIC = −178), and Model 2–3 a large improvement (mean Δ = −ΔAIC) effect. Trimming offered no improvement (mean ΔAIC = 61, but see Supplementary Material for discussion of an outlier), and Model 4 only a small improvement over the trimmed Model 3 (mean ΔAIC = −10). See Supplementary Material 3. Trimming nonsignificant terms offered only a reduction in accuracy, especially in <emph>Games and Routines</emph> (note that it is the only category with non‐significant effects of total WG score). Thus, we use the complete linear model to predict scores in the held‐out test sample.</p> <hd id="AN0186138041-12">Connecting Words</hd> <p>Because no <emph>Connecting Words</emph> items appear on WG, we performed a different type of model selection approach that iteratively regressed the true WS <emph>Connecting Words</emph> scores against simulated WG score, age, and all other lexical categories.</p> <hd id="AN0186138041-13">Part 1 Results</hd> <p></p> <hd id="AN0186138041-14">Total Score Prediction</hd> <p>The total‐only models predict total WS score remarkably well. Table 3 displays fit statistics for the training model (<emph>R</emph><sups>2</sups>) and root mean squared error (RMSE). The RMSE represents the average distance between true and predicted scores over the range of values (i.e., 0–680). A lower RMSE demonstrates better prediction, and can be interpreted in the context of the scale of values. An error of 10 words is much more meaningful for a smaller true value (Figure 2, bottom), however, prediction is better at lower values, in part because there is no simulation estimation to do: All of the items are represented already.</p> <p>3 TABLE Models estimating total WS score on total WG score (WG) and age. R2 calculated on the training sample (n = 6883) and RMSE on the held‐out test sample (n = 1970). The percent value is how large the error is relative to the maximum value of the instrument (<reflink idref="bib680" id="ref68">680</reflink>).</p> <p> <ephtml> <table><thead><tr><th>Model</th><th>Right‐hand terms</th><th>Training <italic>R</italic><sup>2</sup></th><th>Test RMSE</th></tr></thead><tbody><tr><td>T1</td><td>WG + WG<sup>2</sup></td><td>0.9981</td><td>14.7 (2.2%)</td></tr><tr><td>T2</td><td>WG + WG<sup>2</sup> + WG<sup>3</sup></td><td>0.9986</td><td>12.4 (1.8%)</td></tr><tr><td>T3</td><td>age*WG + age<sup>2</sup>*WG<sup>2</sup> + age<sup>3</sup>*WG<sup>3</sup></td><td>0.9987</td><td>11.9 (1.8%)</td></tr></tbody></table> </ephtml> </p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/5G5/01jul25/desc70036-fig-0002.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="desc70036-fig-0002.jpg" title="2 Error in words (left y axis) and as percentage of total WG inventory (i.e., 396; right y axis), by child sex (female: pink/solid; male: dashed/blue). Line shows mean error at age (top) or total WG score bin (n = 15; bottom). Ribbon shows ±  2σ. T1: quadratic model; T2: cubic model; T3: cubic model including cubic age." /> </p> <p></p> <p>Figure 2 shows the training residuals against age (binned; top) and total WG score (bottom), split into 15 bins (the same number of bins as age). The left <emph>y</emph> axis shows the average error and the 2 SD error range, showing that for T3, across all ages and the range of scores, 94% of the training population is predicted within 6% error (relative to the maximum total score, i.e., 680 words). Controlling for age and total WG score, there is no effect of sex on error: girls and boys are predicted equally well.</p> <hd id="AN0186138041-16">Category‐level Prediction</hd> <p>In order to evaluate the best way to predict category‐level scores, we perform a model‐selection approach based on summing the predicted category‐level scores and then summing them with the unmodified <emph>SEAS</emph> score (as it does not change between forms) and the predicted <emph>Connecting Words</emph> score. Because the <emph>Connecting Words</emph> score is calculated differently than the others, we report it first, then combine it with the other categories.</p> <hd id="AN0186138041-17">Connecting Words</hd> <p>First, we present the results of the <emph>Connecting Words</emph> analysis. None of the WS <emph>Connecting Words</emph> items appear on WG, so we regressed the true WS <emph>Connecting Words</emph> scores in an iterative approach. We started with the cubic effect of total simulated WG score (AIC: 17938, <emph>R<sups>2</sups> =</emph> 0.57), then the cubic age‐simulated WG interaction (as above; AIC: 17770, <emph>R<sups>2</sups></emph> = 0.58), which offered a significant improvement in model fit, <emph>F</emph>(<reflink idref="bib6" id="ref69">6</reflink>, 5925) = 40.9, <emph>p</emph> < 0.001. We fit a third model against the cubic age‐simulated WG interaction and all other lexical categories (AIC: 16011; <emph>R<sups>2</sups> =</emph> 0.69), which improved on the second model, <emph>F</emph>(<reflink idref="bib18" id="ref70">18</reflink>, 5907) = 116, <emph>p</emph> < 0.001.</p> <p>We trimmed nonsignificant terms from the third model (at <emph>p <</emph> 0.05/29 predictors), leaving the interaction between the cubic age and WG total terms, as well as three syntactic categories (<emph>Quantifiers, Question Words, Time Words</emph>) and one lexical category (<emph>SEAS</emph>), see the Supplementary Material 2. The AIC for this model was 16089, <emph>R</emph><sups>2</sups> = 0.69. The RMSE in the held‐out test sample (<emph>n =</emph> 1970) was 0.92 over possible category scores between 0 and 6 words, inclusive, that is, returned values below 0 or above 6 were replaced with 0 or 6. This value suggests the true <emph>Connecting Words</emph> can be predicted within roughly ±1 word. Relative to the small size of the <emph>Connecting Words</emph> category, this is a large error (±17%), but improves the total prediction. The intraclass correlation for this prediction was 0.819, with a 95% confidence interval of [0.804, 0.833], which is considered "excellent" (Cicchetti [<reflink idref="bib8" id="ref71">8</reflink>]), see also Table 4. This model is used in analyses, including the other categories going forward.</p> <p>4 TABLE The model selection approach taken to predicting category‐level scores. (Models starting with C.) Also included is the model predicting Connecting Words alone, trimmed (CW3t). The table shows the complete model, that is, C represents the simulated downward‐projected category score, S represents the SEAS total score (which is unchanged between forms), and CW the Connecting Words score (explained in text). Thus, the "Training R2" column shows the range of prediction R2 values (excluding SEAS and Connecting Words), and the second row, the R2 calculated on the sum scores compared to the true total score. The last three columns show the RMSE on the test dataset, including against the total, content, and function word scores (total items in each category in parentheses).</p> <p> <ephtml> <table><thead><tr><th /><th /><th /><th>Test RMSE</th></tr><tr><th>Model</th><th>Right‐hand terms</th><th>Training <italic>R</italic><sup>2</sup></th><th>Total</th><th>Content</th><th>Function</th></tr></thead><tbody><tr><td>CW3t</td><td>See text</td><td>0.6853</td><td>0.923</td><td /><td /></tr><tr><td>C1</td><td>∑(C ∼ C + C<sup>2</sup>) + S + CW</td><td>0.699–0.9950.988</td><td>17.9</td><td>14.0</td><td>6.48</td></tr><tr><td>C2</td><td>∑(C ∼ age*C + age<sup>2</sup>*C<sup>2</sup>) + S + CW</td><td>0.698–0.9950.988</td><td>17.1</td><td>13.7</td><td>6.09</td></tr><tr><td>C3</td><td>∑(C ∼ age*C + age<sup>2</sup>*C<sup>2</sup> + WG + WG<sup>2</sup> + WG<sup>3</sup>) + S + CW</td><td>0.837–0.9960.991</td><td>18.7</td><td>14.9</td><td>6.28</td></tr><tr><td>C4</td><td>∑(C ∼ age*C + age<sup>2</sup>*C<sup>2</sup> + age*WG + age<sup>2</sup>*WG<sup>2</sup> + age<sup>3</sup>*WG<sup>3</sup>) + S + CW</td><td>0.838–0.9960.996</td><td>18.8</td><td>15.0</td><td>6.28</td></tr><tr><td>C3t</td><td>C3, trimmed on category‐specific basis + S + CW</td><td /><td>18.6</td><td>14.8</td><td>6.31</td></tr></tbody></table> </ephtml> </p> <hd id="AN0186138041-18">Other Categories</hd> <p>As described in the Methods section, we evaluated iterative models to predict WS category scores from simulated WG scores, and evaluated the predicted category scores against the ground truth. As shown in Table 4, we also evaluated the total score by summing the category predictions (shown in the table with <emph>Σ</emph>, representing the sum of the individual predictions) along with the <emph>SEAS</emph> score and the predicted <emph>CW</emph> score (see above). For the rest of the categories, the model selection appears in Table 4, which shows that the cubic age‐WG interaction model (C4) does not offer an improvement over the noninteraction model, so we trim the nonsignificant terms from C3 to avoid over‐fitting.</p> <p>On the category level, the model did well overall. That said, some smaller syntactic categories showed larger errors and some residual non‐linearity (e.g., <emph>Helping Verbs</emph>, <emph>Prepositions and Locations</emph>, and <emph>Pronouns</emph>; Figure 3A). In order to simplify the presentation, the best category model is displayed in Figure 3, both on a category level, Figure 3A, and in sum (Figure 3B,C). Compared to the best total model (T3), the category‐level estimation performs slightly worse and with a slight bias toward overestimation. Based on the inflection point in Figure 3C it may be best to limit category‐level estimation to 250 words in the WG score or less.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/5G5/01jul25/desc70036-fig-0003.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="desc70036-fig-0003.jpg" title="3 Using the selected model (C3, trimmed), plot showing the error in category‐level predictions (A, note the variation in scale), as well as the error in words (left y axis) and by age (B) and by WG score (15 bins; C). The error from the best model (T3) predicting total score is overlaid in subplots (B) and (C) as the black/dashed line." /> </p> <p></p> <hd id="AN0186138041-20">Part 1 Discussion</hd> <p>We described three methods for projecting WG scores upward: total score, category scores (except Connecting Words), and a method for Connecting Words. Based on RMSE, the total score method provides better estimates of total score (11.9 compared to values > 17.0), which makes it the conceptually simpler and better statistical choice when only total inventory scores are needed. The category scores perform well, but only need to be estimated if subscores need to be calculated, for example, modeling verbs using Action Words, adjectives with Descriptive Words, or function words using the final seven[<reflink idref="bib1" id="ref72">1</reflink>] categories (Day and Elison [<reflink idref="bib11" id="ref73">11</reflink>]). Some residual nonlinearity and high error is evident in these categories, especially Helping Verbs and Pronouns. The Connecting Words projection is built on a different principle, since none of the six words appear on WG, and, with only six words, relative error is high. Due to this, it would be best to avoid modeling Connecting Words score alone, and only as part of a larger summed score (e.g., function words or total score).</p> <p>We demonstrated that, at least between 18 and 27 months, WG scores collected out of range on typically developing (TD) children can be used to simulate WS scores as needed. With the SD of error at 3% of 680 total items (i.e., 20 words), this approximates an ICC of greater than 0.99. In other words, if we were to compare, for example, reports from both parents, this agreement would represent a highly rigorous level of reliability. If one wanted to be even more conservative, this procedure could be used for only individuals with fewer than 250 words (see Figure 3B, where the error rate rises), which occurs at 23.3 months, based on a Gompertz model of the complete Wordbank data (<emph>k<subs>g</subs></emph> = 0.161). From the same plot (Figure 3C), it might also be appropriate to use this upward extension no older than 22–24 months, depending on the need of the individual study.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/5G5/01jul25/desc70036-fig-0004.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="desc70036-fig-0004.jpg" title="4 Shape characteristics of the Gompertz model (unbroken line). The inflection value is fixed at 36.79% of the upper asymptote. Here, the upper asymptote (A) is set at 10.0, maximum absolute growth rate (KU) to 1.5, time at inflection (Ti) to 2.0, and the starting point (W0) to 1.0. With a set asymptote and growth rate, time of inflection follows from a given starting point or vice versa. The maximum growth rate is represented by the tangent at inflection (dashed line). Duplicated from Tjørve and Tjørve ([43]), under the terms of the Creative Commons Attribution License." /> </p> <p></p> <p>It is important to note that the modeling here is based on TD children from Wordbank. This makes it suitable to project WG scores collected from TD children, but without a Wordbank‐like repository of instruments from children with developmental disorders, we are unable to create a similar model for that population. This model is best suited for studies like BCP, where forms collected to serve as a reference for an IDD sample are no longer being used in that context. We leave it to future researchers to decide whether our projection method is suitable for their data, but we note that for small vocabularies, little projection is typically implicated.</p> <p>We have provided an R package to apply these models to existing WG data (CRAN: https://cran.r‐project.org/package=cdiWG2WS or GitHub: https://github.com/TrevorKMDay/cdiWG2WS).</p> <hd id="AN0186138041-22">Part 2 Methods</hd> <p>CDI trajectories cannot be modeled with a linear trend, as CDI vocabulary growth is nonlinear, with asymptotes at 0 and 680 words. As children approach the upper age limits of the WS CDI, the apparent rate of growth slows. This, of course, does not reflect an actual decrease in the rate of word learning, but rather it becomes increasingly less likely that, for each given new word a child learns, it is represented on the CDI. Thus, we turn to Gompertz growth curves to model the specific trajectory of CDI and estimate a single, early‐childhood "growth rate," irrespective of the timing of assessments.</p> <hd id="AN0186138041-23">Analytic Strategy Part 2: Growth Curve Modeling</hd> <p>Gompertz curves are asymmetric sigmoid models that are frequently used to fit growth data such as plant, bird, tumor, and bacterial growth (Tjørve and Tjørve [<reflink idref="bib43" id="ref74">43</reflink>]). However, unlike a sigmoid growth curve, a Gompertz curve is not symmetric, which is advantageous because it does not assume early word learning has a perfectly reversed slope to later word learning. First, we will describe some mathematical properties of a Gompertz curve. These curves in this context are defined by a lower asymptote (i.e., 0 words) and an upper asymptote (here, 680 words), and an inflection point (<emph>T<subs>i</subs></emph>) the time at which the instantaneous rate of growth (i.e., the tangent line, in units of words per unit time, here, months) is the fastest rate of growth in the entire model. <emph>T<subs>i</subs></emph> is defined as occurring at <ephtml> <math display="inline" altimg="urn:x-wiley:1363755X:media:desc70036:desc70036-math-0001" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mfrac><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow><mi>e</mi></mfrac></mrow><annotation encoding="application/x-tex">$y = \frac{{max}}{e}$</annotation></semantics></math> </ephtml> . This means under these models, the fastest a child will be learning words occurs when they reach approximately 680/<emph>e</emph>  =  250 words on the CDI. This point is {<emph>T</emph><subs><emph>i</emph>,</subs><emph>W<subs>i</subs></emph>}. The previously mentioned fastest rate of growth at this point is <emph>k<subs>g</subs></emph>, which is dimensionless, but can be converted to words/month by multiplying it by <ephtml> <math display="inline" altimg="urn:x-wiley:1363755X:media:desc70036:desc70036-math-0002" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mfrac><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow><mi>e</mi></mfrac><annotation encoding="application/x-tex">$\frac{{max}}{e}$</annotation></semantics></math> </ephtml> , in which case it is known as the "universal growth rate," <emph>k<subs>U</subs></emph>. The convention for outcomes in Gompertz curves is to use <emph>W</emph>, which happens to align with <emph>words</emph>. See Figure 4 for characteristics of the Gompertz curve.</p> <p>There are multiple parameterizations of the Gompertz curve, which allow various features of the curve to vary. In this application, the lower (<emph>W</emph><subs>0</subs>) and upper asymptotes (<emph>A</emph>) do not vary, since every child begins with 0 words, and will eventually reach 680 words in typical development. Thus, we select a model that allows only time (<emph>t</emph>) and <emph>k<subs>g</subs></emph> to vary. In Tjørve and Tjørve ([<reflink idref="bib43" id="ref75">43</reflink>]), equation below is eq. 19. <ephtml> <math display="block" altimg="urn:x-wiley:1363755X:media:desc70036:desc70036-math-0003" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi><mfenced open="(" close=")"><mi>t</mi></mfenced><mo linebreak="badbreak">=</mo><mi>A</mi><msup><mfenced separators="" open="(" close=")"><mfrac><mi>A</mi><msub><mi>W</mi><mn>0</mn></msub></mfrac></mfenced><mrow><mi>e</mi><mi>x</mi><mi>p</mi><mfenced separators="" open="(" close=")"><mrow><mo>−</mo><mi>e</mi><mo>×</mo><msub><mi>k</mi><mi>U</mi></msub><mo>×</mo><mi>t</mi></mrow></mfenced></mrow></msup></mrow><annotation encoding="application/x-tex">$$\begin{equation*}W\left(t \right) = A{\left({\frac{A}{{{W_0}}}} \right)^{exp\left({ - e \times {k_U} \times t} \right)}}\end{equation*}$$</annotation></semantics></math> </ephtml></p> <p>We used the R (ver 4.2.2; R Core Team [<reflink idref="bib38" id="ref76">38</reflink>]) package stats function nls() to fit Gompertz curves using this formulation. For <emph>A</emph>, because 0 cannot be used in this model, we use the smallest value <emph>R</emph> can represent (i.e., .Machine$eps, or 2.2  ×  10<sups>−16</sups>). For <emph>W</emph><subs>0</subs>, we use max+1, so that an individual's curve will eventually reach max, rather than approach it asymptotically.</p> <p>Through applying this model, we achieve two readily interpretable values. They are nonlinearly dependent on one another, such that an earlier <emph>T<subs>i</subs></emph> always results in a higher <emph>k<subs>g</subs></emph> and <emph>k<subs>U</subs></emph>, which is linearly dependent on <emph>k<subs>g</subs></emph>. Each allows for a different approach to understanding the model.</p> <p></p> <ulist> <item> <emph>T<subs>i</subs></emph> : The time in months of maximum growth (i.e., the age at words = 250).</item> <p></p> <item> <emph>k<subs>U</subs></emph> : The fastest rate of growth, in words (on the CDI) per month.</item> </ulist> <p>We fit Gompertz curves to both the entire dataset, and to individuals, since one of the motivating factors for this analysis was the accelerated longitudinal design of BCP, which makes it difficult to compare individuals when they do not all have the same time points. Across the BCP (<emph>n</emph> = 206) and EIRLI datasets, 1197 individuals had two or more time points. Among the EIRLI participants, the sample sizes were: DX0 <emph>n</emph> = 321, DX− = 571, and DX+ <emph>n</emph> = 99. When appropriate, scores were adjusted upward following the results of Analysis 1.</p> <p>Participants had varying amounts of data (although no more than five assessments). Because EIRLI data were collected at regular intervals (<reflink idref="bib16" id="ref77">16</reflink>, 20, 24, 28, and 33 or 36 months), and not in an accelerated design like BCP, we used individuals from that dataset to evaluate the number of timepoints necessary to create a good‐fitting Gompertz curve on an individual basis. We did this with all five timepoints, iteratively removing a time point to estimate the minimum number of timepoints for this strategy. We removed the oldest timepoint(s) in reverse order to reflect the distribution of BCP data, which is sparser at older age ranges.</p> <p>Following the establishment of data inclusion criteria based on EIRLI data, we calculated individual‐level Gompertz curves on participants with sufficient data. Furthermore, we calculated <emph>T<subs>i</subs></emph> for the individual categories of the lexical inventories and among the content and function word categories together (Day and Elison [<reflink idref="bib11" id="ref78">11</reflink>]).</p> <p>We fit Gompertz curves to all three datasets using both the complete data (where available) and the EIRLI set of lexical categories. In doing so, BCP and Wordbank have a sample growth rate calculated on all data and "EIRLI‐like" data, excluding the categories missing from the EIRLI data. Estimating <emph>k<subs>g</subs></emph> using the full‐data and EIRLI‐like estimates produces values within 0.001 of one another, that is, BCP: 0.15 ± 0.001; Wordbank: 0.17 ± 0.001. The <emph>k<subs>g</subs></emph> value for the EIRLI data was 0.17. Converted to <emph>k<subs>U</subs></emph> with the appropriate maximum value, the difference in estimates was within four words/month and the difference in estimates of <emph>T<subs>i</subs></emph> was less than 0.2 months, or 6 days, see Table 5.</p> <p>5 TABLE Model values with 95% confidence intervals for the complete CDI (All) and the categories included in the EIRLI data. ntp: Number of timepoints. kU is the "universal growth rate," in units of the analysis.</p> <p> <ephtml> <table><thead><tr><th /><th /><th>Rate of growth <italic>k<sub>U</sub></italic> (words/month)</th><th>Age at maximum rate of growth <italic>T<sub>i</sub></italic> (months)</th></tr><tr><th /><th><italic>n</italic><sub>tp</sub></th><th>All</th><th>EIRLI</th><th>All</th><th>EIRLI</th></tr></thead><tbody><tr><td>BCP</td><td>732</td><td>37.7 [37.2, 38.1]</td><td>34.0 [33.6, 34.4]</td><td>24.9 [24.6, 25.2]</td><td>24.9 [24.6, 25.2]</td></tr><tr><td>Wordbank</td><td>7955</td><td>42.5 [42.4, 42.7]</td><td>38.7 [38.4, 38.9]</td><td>22.1 [22.0, 22.1]</td><td>21.9 [21.8, 22.0]</td></tr><tr><td>EIRLI</td><td>3316</td><td /><td>38.7 [38.4, 38.9]</td><td /><td>21.9 [21.8, 22.1]</td></tr></tbody></table> </ephtml> </p> <p>These data are also shown in Figure 5, divided by form (WG: red; WS: blue) and by included data (all: solid; EIRLI categories only: dashed). Within each panel, the red lines are the maximum available words and inflection point (i.e., <ephtml> <math display="inline" altimg="urn:x-wiley:1363755X:media:desc70036:desc70036-math-0004" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mfrac><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow><mi>e</mi></mfrac><annotation encoding="application/x-tex">$\frac{{max}}{e}$</annotation></semantics></math> </ephtml> ) for the complete form (solid) and EIRLI (dashed). We conclude that the maximum‐adjusted summary values are comparable across the complete and reduced‐maximum input data. In other words, in the EIRLI data, the maximum growth rate and its timing would be the same without the missing categories.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/5G5/01jul25/desc70036-fig-0005.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="desc70036-fig-0005.jpg" title="5 Gompertz curves fit to the complete data for all three datasets. All CDI categories in solid lines, the reduced set available in the EIRLI data in dashed lines. Red horizontal lines are the maximum available and inflection points with each set of categories." /> </p> <p></p> <hd id="AN0186138041-25">Participant‐Level Growth Curves</hd> <p>Within the EIRLI data (DX0/DX‐), we calculated <emph>k<subs>g</subs></emph> values using five timepoints, where available, and then removed the last data point to calculate a <emph>k<subs>g</subs></emph> value using only four timepoints, to evaluate the number of timepoints needed for a reliable curve and growth rate value. Finally, we repeated the process to calculate a three timepoint <emph>k<subs>g</subs></emph> value using only the first three time points. When we compared four‐ and three‐timepoint <emph>k<subs>g</subs></emph> values, individuals with only four timepoints were included. Results from these analyses suggest that it is necessary to have four timepoints and for the individual to reach 250 words ( <ephtml> <math display="inline" altimg="urn:x-wiley:1363755X:media:desc70036:desc70036-math-0005" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mfrac><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow><mi>e</mi></mfrac><annotation encoding="application/x-tex">$\frac{{max}}{e}$</annotation></semantics></math> </ephtml> ).</p> <p>Only 65 BCP participants (44%) and 405 EIRLI individuals (across all diagnoses; 82%) meet these criteria. A further 57 BCP participants (38%) and 74 EIRLI participants (15%) have at least three time points and reach one‐half the inflection point. A complete description of the data quality of the two longitudinal groups is given in Supplementary Table 3.1, and for diagnostic status, Supplementary Table 3.2. Both sets of BCP participants will be included under the "sufficient" and "insufficient"[<reflink idref="bib2" id="ref79">2</reflink>] data labels to maximize data inclusion.</p> <hd id="AN0186138041-26">Part 2 Results</hd> <p></p> <hd id="AN0186138041-27">Estimating the Minimum Number of Timepoints</hd> <p>The correlation between <emph>k<subs>g</subs></emph> estimates using 5/4 points was: <emph>r</emph>  =  0.99,   <emph>n</emph>  =  266; the correlation between 5/3 points was <emph>r</emph>  =  0.95,   <emph>n</emph>  =  251; and the correlation between 4/3 points was <emph>r</emph>  =  0.96,   <emph>n</emph>  =  332. All correlations were significant. As mentioned, not all participants had five data points, and some models failed to converge. Those participants are excluded. In total, five participants (<1%) failed to converge, all in the EIRLI dataset, all had three timepoints, but reached the suggested minimum score of ( <ephtml> <math display="inline" altimg="urn:x-wiley:1363755X:media:desc70036:desc70036-math-0006" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mfrac><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow><mi>e</mi></mfrac><annotation encoding="application/x-tex">$\frac{{max}}{e}$</annotation></semantics></math> </ephtml> = 215 for EIRLI), see section 7.1.</p> <p>However, the variance is higher among those with a lower rate of growth, for example, a median split (<emph>k<subs>g</subs></emph> median = 0.16) on the best‐estimate rate of growth (i.e., 5‐point) shows lower correlations among the low group compared to the high, see Table 6.</p> <p>6 TABLE Correlations between kg estimates given n points used in the calculation, on a median split of kg values (high kg indicates a higher rate of growth).</p> <p> <ephtml> <table><thead><tr><th>Low <italic>k<sub>g</sub></italic> group (<italic>n</italic> = 133)</th><th>High <italic>k<sub>g</sub></italic> group (<italic>n</italic> = 118–133)</th></tr></thead><tbody><tr><td /><td>5</td><td>4</td><td>3</td><td /><td>5</td><td>4</td><td>3</td></tr><tr><td>5</td><td /><td /><td /><td>5</td><td /><td /><td /></tr><tr><td>4</td><td>0.95</td><td /><td /><td>4</td><td>1.0</td><td /><td /></tr><tr><td>3</td><td>0.77</td><td>0.84</td><td /><td>3</td><td>0.96</td><td>0.94</td><td /></tr></tbody></table> </ephtml> </p> <p>Therefore, we contrasted estimates of <emph>k<subs>g</subs></emph> between the best estimate and the 4‐ and 3‐point solutions, shown in Figure 6, with vertical lines at the inflection point, <ephtml> <math display="inline" altimg="urn:x-wiley:1363755X:media:desc70036:desc70036-math-0007" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>615</mn><mi>e</mi></mfrac><mo>≈</mo><mn>227</mn></mrow><annotation encoding="application/x-tex">$\frac{{615}}{e} \approx 227$</annotation></semantics></math> </ephtml> , and one‐half that value, approximately 113.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/5G5/01jul25/desc70036-fig-0006.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="desc70036-fig-0006.jpg" title="6 The difference in estimates using 3, 4, and 5 points according to the participants inventory total at 28 months (the fourth timepoint). Values are calculated against the best estimate (all five timepoints). The difference between the 3‐ and 5‐point estimate in orange, and the difference between the 4‐ and 5‐point estimate in green. Both are scaled against the 5‐point estimate. Vertical red lines are placed at the upper asymptote (615) divided by e and 2e." /> </p> <p></p> <p>As can be seen in Figure 6, using four points creates <emph>k<subs>g</subs></emph> values very similar to the best estimate, whereas there is considerably more variability (predominantly underestimating) in maximum rate of growth, when using only three points. The deflection in the 4–5 line is driven by the leftmost outlier.</p> <p>In order to model CDI data with a Gompertz curve, we suggest (<reflink idref="bib1" id="ref80">1</reflink>) that it is necessary to have at least four time points; and (<reflink idref="bib2" id="ref81">2</reflink>) that the participant reaches the inflection point on the inventory.</p> <hd id="AN0186138041-29">Estimating Individual Growth Curves</hd> <p>Using the criteria explored above, the mean <emph>k<subs>U</subs></emph> for BCP‐sufficient was 39.1 words/month, and for the insufficient group, 35.6. This is reflected in statistically different (<emph>t</emph>(25.6) = 2.9; <emph>p</emph> = 0.008) <emph>T<subs>i</subs></emph> values of 24.3 and 26.7 months, respectively. Likewise, in the EIRLI data, the mean <emph>k<subs>U</subs></emph> for the DX‐group was 39.3, DX0 was 37.6, and DX+ was 34.3. These values are converted to mean Ti values of 22.1, 23.0, and 25.0 months, respectively. With the DX‐group as reference, there was no effect of the DX0 group, <emph>p</emph> = 0.067; but the DX+ group had significantly higher (i.e., delayed) <emph>T<subs>i</subs></emph> values, <emph>β</emph> = 3.7, <emph>p</emph> < 0.001. Across these three tests, <emph>p</emph> values were Bonferroni corrected for three comparisons. The distribution of <emph>T<subs>i</subs></emph> values is shown in Figure 7.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/5G5/01jul25/desc70036-fig-0007.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="desc70036-fig-0007.jpg" title="7 Distribution of Ti values across datasets and groups. BCP is divided into sufficient data (4–5 timepoints and values exceeding 251 words) and insufficient data (3 timepoints or values between 126 and 251 words). EIRLI is divided by diagnostic status. Y‐axis labels are for informational purposes, not an establishment of "precocious" or "delayed" status." /> </p> <p></p> <p>Between BCP and EIRLI (with sufficient data), we regressed a linear model on <emph>T<subs>i</subs></emph>, regressing time at maximum growth against dataset and diagnostic status (BCP was treated as DX0). There was no effect of the DX0 group, <emph>β</emph> = −0.44, <emph>p</emph> = 0.34, but the DX+ group had a delayed <emph>T<subs>i</subs></emph>, <emph>β</emph> = 3.0, <emph>p</emph> < 0.001. The difference between the BCP/EIRLI DX0 group and DX− was −0.70, <emph>p</emph> = 0.06.</p> <p>These curves can quite reasonably capture variations in trajectories. For example, Figure 8 shows four randomly chosen participants (two per dataset), with <emph>k<subs>U</subs></emph> values between 29.9 and 49.5 and <emph>T<subs>i</subs></emph> values between 28.4 and 19.6 months. We can also note that this method allows estimation of values at ages where data was not collected. For example, these four participants' estimated inventory sizes at 22 months (a timepoint in which no participant had data) were A: 67.2, B: 274, C: 286, and D: 120.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/5G5/01jul25/desc70036-fig-0008.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="desc70036-fig-0008.jpg" title="8 Example Gompertz curves from selected participants from BCP (top/green, "B" prefix) and EIRLI DX− (bottom/purple, "E" prefix). Fit values kU, Ti for each participant shown on their frame. The blue line shows the comparison at 22 months." /> </p> <p></p> <hd id="AN0186138041-32">Covariates</hd> <p>Because of the distinctly lower scores in BCP, we tested covariates on BCP and EIRLI separately. We regressed <emph>T<subs>i</subs></emph> on the demographic data available in EIRLI (<emph>n</emph> = 405). There was a significant effect of DX+ status, <emph>β</emph> = 3.29, <emph>p</emph> < 0.001, DX0 status, <emph>β</emph> = 1.02, <emph>p</emph> = 0.012, male sex, <emph>β</emph> = 1.12, <emph>p</emph> = 0.009, but no effect of mother or father's education level (college graduate vs. not), pmother = 0.086; pfather = 0.448, all <emph>p</emph> values Bonferroni corrected for multiple comparisons. In the BCP sample (<emph>n</emph> = 109), there were no significant effects of sex, mother's college graduate status, or income‐to‐needs ratio (defined as the median value of the family‐supplied income range divided by the number of family members).</p> <hd id="AN0186138041-33">Category Curves</hd> <p>The same method can be applied to individual lexical categories to establish when the rate of maximum growth is for SEAS compared to Action Connecting Words, for example. The results of this analysis are presented in Table ST3.3 and graphically in Figure 9. We observe the trend that BCP is persistently lower (i.e., later age of maximum growth) than Wordbank and the EIRLI DX0/DX− groups across all categories. A notably wide gap appears in the SEAS category.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/5G5/01jul25/desc70036-fig-0009.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="desc70036-fig-0009.jpg" title="9 Group age of fastest acquisition (Ti) across datasets (BCP: blue circles; WB: green squares) and EIRLI diagnostic groups (DX−: purple diamonds, DX+: red upward triangles, DX0: pink downward triangles). Higher values represent later age of acquisition. Category grouped Ti values highlighted with vertical lines (content vs. function words; all subordinate categories to the left)." /> </p> <p></p> <p>We fit the models against the total items across all categories within the content‐ and function‐word groups. This gives the age at which children were learning items in that category the fastest, see Figure 9, where older ages represent more delayed learning. For example, sound effects are being learned much earlier (around 14–16 months), compared to verbs (Action Words, around 22–25 months). Furthermore, we divided the inventory categories into content and function word groups, based on a previous factor analysis. The function‐word group included Words about Time, Pronouns, Question Words, Prepositions and Locations, Quantifiers and Articles, Helping Verbs, and Connecting Words, and the content‐word group the rest of the lexical categories (Day and Elison [<reflink idref="bib11" id="ref82">11</reflink>]).</p> <p>Of note are the delayed trajectories for the BCP sample, which are equivalent or worse than the EIRLI DX+ group in many categories. BCP, like the others, is a well‐educated, high‐SES sample. A true delay in learning trajectories between EIRLI (mid‐1990s) and BCP (mid‐2010s) would suggest children are learning words later. This would seem unlikely, contrasting with the Flynn effect of rising IQ scores over time, see Trahan et al. ([<reflink idref="bib44" id="ref83">44</reflink>]) for a recent meta‐analysis pegging the IQ inflation effect at 2.3 points per decade.</p> <p>Years of acquisition are available for some Wordbank data (nWG = 1794, nWS = 4785). We performed some simple linear models on the forms individually, which obviates the nonlinearity. Controlling for age, there was no effect of year for WG production (<emph>p</emph> = 0.152) or comprehension values (<emph>p</emph> = 0.375). Surprisingly, however, there was an effect of year in the WS production data, <emph>β</emph> = −0.91, <emph>p</emph> < 0.001 after Bonferonni correction for three multiple comparisons, which has the effect of a loss of 23.66 words (3% of 680) between 1988 and 2014, the earliest and latest years in the Wordbank database.</p> <p>One possibility is the method of responding to each item. BCP parents participated online, using radio buttons to mark each individual word, which is more time consuming than filling in a bubble sheet. This may have raised the barrier to endorsing an item; or parents fatigued sooner after filling out as many as 680 individual radio buttons. However, this phenomenon bears further investigation.</p> <hd id="AN0186138041-35">Part 2 Discussion</hd> <p>Because of the nonlinearity imposed by having a finite list of words, we show that a Gompertz curve is a suitable method for analyzing CDI inventory totals. Gompertz curves provide a readily interpretable growth rate (<emph>k<subs>U</subs></emph>) and age at maximum growth (<emph>T<subs>i</subs></emph>), and are robust to timepoints that differ between individuals. This method can be used on an individual level to provide summary values for individuals' growth rates, either overall or within lexical categories.</p> <p>One important comparison is the contrast between Table 1, reporting parameters from the literature regarding the "vocabulary spurt" and Table 5, reporting model parameters from our study. While the spurt literature suggests maximum spurt rates of somewhere between 10 and 17 words per month or so, we show that the fastest rate of growth in words per month (<emph>k<subs>g</subs></emph>) is likely much higher, between 37 and 42 words per month or so. While we suggest a very different typical maximum rate of growth, the timing (<emph>T<subs>i</subs></emph>) is at the higher end of the previously suggested range, at about 21–24 months.</p> <p>In the group analyses, we show that the timing of the maximum growth rate varies between categories of words; which may prove useful for researchers studying the acquisition of specific kinds of items. This also suggests researchers should prioritize including assessments between 20 and 36 months or so to capture the full range of vocabulary trajectories, rather than the earlier ages suggested by the spurt literature.</p> <p>In the individual analyses, it's important to consider some drawbacks. Like any nonlinear model, a Gompertz curve requires multiple timepoints. Our modeling finds at least four timepoints area required and that, on an individual level, at least one of those timepoints is greater than the inflection point (680/<emph>e</emph>  =  250), which tends to occur around 24 months. We also combine data across WG and WS, which increases the time frame over which development is sampled. Finally, in showing that the EIRLI DX+ individuals have a slower growth rate (and therefore delayed <emph>T<subs>i</subs></emph>), we demonstrate external validity for our measures. However, the difference is not robust enough to conclusively identify DX+ individuals based on growth rate alone.</p> <hd id="AN0186138041-36">General Discussion</hd> <p>The primary goal of the present analyses was to provide a basis for a longitudinal model of CDI inventory scores. Whereas previous work has used a quadratic form at younger ages, the data available here included older timepoints, meaning the upper asymptote has to be taken into account. We also diverge from previous work in suggesting an application of a Gompertz growth curve, rather than a logistic regression, as the former does not constrain the model to be symmetric. We show this is suitable, given the conditions that a child has at least four timepoints, and reaches 250 words in the input data.</p> <p>However, the BCP data was constrained by the fact that the project collected WG data at ages older than typical. Thus, we provide a supplementary analysis using Wordbank data that shows how to project out‐of‐normed‐age‐range WG scores onto WS to allow comparison between totals collected on two different forms to a high degree of accuracy.</p> <hd id="AN0186138041-37">Implications for the "Vocabulary Spurt"</hd> <p>We can also consider whether the Gompertz models provide evidence for or against a "spurt." First, the Gompertz model makes assumptions about the functional form of lexical development, necessarily constraining the model to a spurt‐like form, additionally taking into account the upper asymptote. Because of the accelerated design of BCP, we cannot take a model‐selection approach (e.g., data acquired in the middle of the age range may fit a linear model better than the Gompertz because there is no data at the extremes) to determine spurt status.</p> <p>If there was a categorical difference between "spurters" and "non‐spurters," as suggested by previous authors, we might find a bimodal distribution of growth values, one peak clustered around quickly growing "spurters," and a wider peak of individuals that are growing at a non‐exponential rate. We do not find evidence for such a distribution. Although there is variance within groups, each dataset shows an approximately normal distribution of growth values. Due to the differences in instruments between the CDI and other studies, the rate of growth for our models easily exceeds the most conservative "spurt threshold" previously reported in the literature. That said, in roughly half the sample, the age of maximum growth (the "peak" of the spurt) occurred past the typical upper bound of the spurt (i.e., 24 months), BCP: 58%; EIRLI DX−: 47%, DX0: 59%, and DX+: 69%, which highlights the importance of considering a wider age rage when analyzing early lexical development.</p> <hd id="AN0186138041-38">Limitations and Future Directions</hd> <p>The primary limitation is the demographics of the samples, with the exception of the Wordbank WS parents, more than 68% of the sample had earned a bachelor's degree, more than twice the US rate of 32% (McElrath and Martin [<reflink idref="bib29" id="ref84">29</reflink>]). The BCP represents a convenience‐sampled cohort (the battery included nighttime magnetic resonance imaging scans, a heavy burden on families, especially longitudinally) that was majority White and wealthy.</p> <p>Future studies should apply the Gompertz framework to a wider range of children, especially children in lower‐income households, as well as in other languages: The CDI has been translated into more than 60 languages (Frank et al. [<reflink idref="bib18" id="ref85">18</reflink>]). Secondly, our modeling of the necessary timing of the input timepoints was limited by data availability, a more robust analysis of how many timepoints are sufficient and when would inform future application of Gompertz curves to CDI data.</p> <hd id="AN0186138041-39">Conclusions</hd> <p>In summary, we demonstrate two main findings: First, WG scores can be projected to WS scores with a high degree of reliability when originally collected outside of the normed age range. This provides a basis for comparison between samples with WG and WS instruments collected at overlapping age ranges.</p> <p>Second, we describe a framework for calculating longitudinal summary scores of growth rate in CDIs. We suggest that the fastest rate of word growth occurs around 24 months, reaching an average maximum rate of about 40 words per month, in contrast to previous research. However, our findings do not support the existence of two groups of "spurters" and "non‐spurters," instead finding a continuous distribution of peak growth rates. Future studies can adopt this modeling when designing their sampling strategy as a way of summarizing multiple CDIs—both WG and WS.</p> <hd id="AN0186138041-40">Author Contributions</hd> <p> <bold>Trevor K. M. Day</bold>: conceptualization, data curation, formal analysis, funding acquisition, investigation, methodology, software, writing – review and editing, writing – original draft, visualization. <bold>Arielle Borovsky</bold>: writing – review and editing, resources. <bold>Donna Thal</bold>: resources. <bold>Jed T. Elison</bold>: conceptualization, project administration, supervision, resources, writing – review and editing.</p> <hd id="AN0186138041-41">Acknowledgments</hd> <p>This work was supported by NIH awards R01 MH104324 and U01 MH110274 (JTE), and NIDCD DC018593 (AB). Collection of the EIRLI data was supported by NIDCD DC000482 (DT). TKMD was supported by the NSF Graduate Research Fellowship Program at Minnesota (#2020295366) and a T32 postdoctoral fellowship at Georgetown (#T32DC019481). The funders had no role in the study design, data collection, analysis, data interpretation, or the writing of this report. We thank the anonymous reviewers for their helpful feedback and Dr. Sarah Phillips for her feedback on the second draft of this manuscript.</p> <hd id="AN0186138041-42">Conflicts of Interest</hd> <p>The authors declare no conflicts of interest.</p> <hd id="AN0186138041-43">Data Availability Statement</hd> <p>Wordbank is available at wordbank.stanford.edu. Data from the Baby Connectome Project and the Early Identification for Risk of Language Impairment studies are available upon request from authors J. T. Elison and A. Borovsky, respectively.</p> <p>GRAPH</p> <ref id="AN0186138041-44"> <title> Footnotes </title> <blist> <bibl id="bib1" idref="ref41" type="bt">1</bibl> <bibtext> Time Words, Pronouns, Question Words, Prepositions and Locations, Quantifiers and Articles, Helping Verbs, and Connecting Words.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref42" type="bt">2</bibl> <bibtext> Does not reach four timepoints and 250 words; but has at least three timepoints and 125 words.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref1" type="bt">3</bibl> <bibtext> Funding: This work was supported by NIH awards R01 MH104324 and U01 MH110274 (J.T.E.), and NIDCD DC018593 (A.B.). Collection of the EIRLI data was supported by NIDCD DC000482 (D.T.). T.K.M.D. was supported by the NSF Graduate Research Fellowship Program at Minnesota (#2020295366) and a T32 postdoctoral fellowship at Georgetown (#T32DC019481).</bibtext> </blist> </ref> <ref id="AN0186138041-45"> <title> References </title> <blist> <bibtext> Anderson, D., and J. Reilly. 2002. " The MacArthur Communicative Development Inventory: Normative Data for American Sign Language." Journal of Deaf Studies and Deaf Education 7, no. 2 : 83 – 106. https://doi.org/10.1093/deafed/7.2.83.</bibtext> </blist> <blist> <bibtext> Bauer, D. J., B. A. Goldfield, and J. S. Reznick. 2002. " Alternative Approaches to Analyzing Individual Differences in the Rate of Early Vocabulary Development." Applied Psycholinguistics 23, no. 3 : 313 – 335. https://doi.org/10.1017/S0142716402003016.</bibtext> </blist> <blist> <bibtext> Bloom, L. 1976. 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  Data: Journal Articles<br />Reports - Research
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Language+Skills%22">Language Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Measures+%28Individuals%29%22">Measures (Individuals)</searchLink><br /><searchLink fieldCode="DE" term="%22Children%22">Children</searchLink><br /><searchLink fieldCode="DE" term="%22Models%22">Models</searchLink><br /><searchLink fieldCode="DE" term="%22Data+Analysis%22">Data Analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Longitudinal+Studies%22">Longitudinal Studies</searchLink><br /><searchLink fieldCode="DE" term="%22Speech+Impairments%22">Speech Impairments</searchLink><br /><searchLink fieldCode="DE" term="%22Language+Impairments%22">Language Impairments</searchLink><br /><searchLink fieldCode="DE" term="%22Learning+Disabilities%22">Learning Disabilities</searchLink><br /><searchLink fieldCode="DE" term="%22Scores%22">Scores</searchLink><br /><searchLink fieldCode="DE" term="%22Infants%22">Infants</searchLink><br /><searchLink fieldCode="DE" term="%22Toddlers%22">Toddlers</searchLink>
– Name: SubjectThesaurus
  Label: Assessment and Survey Identifiers
  Group: Su
  Data: <searchLink fieldCode="SU" term="%22MacArthur+Bates+Communicative+Development+Inventories%22">MacArthur Bates Communicative Development Inventories</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1111/desc.70036
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 1363-755X<br />1467-7687
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: The MacArthur-Bates Communicative Development Inventories (CDI) are widely used, parent-report instruments of language acquisition. Here, we focus on the word-inventory sections of the instruments, and show two different approaches to modeling CDI data, based on real-world needs. First, we show that Words & Gestures data collected out-of-age-normed-range can be robustly adjusted to Words & Sentences scores. Second, we demonstrate a novel application of Gompertz growth curves to longitudinal CDI data, especially when the same timepoints were not collected between individuals (i.e., an accelerated longitudinal design). Gompertz curves provide a "growth rate" or an "age at maximum growth" parameter that can be used to summarize vocabulary development. We compare these parameters between healthy developing children in two longitudinal cohorts, as well as a cohort of children with a diagnosis of speech disorder, language disorder, or learning or reading disability, who we show to have lower growth rates. We hope these analyses and results inform future work on longitudinal CDI analyses.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2025
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1475002
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1475002
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1111/desc.70036
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 18
    Subjects:
      – SubjectFull: Language Skills
        Type: general
      – SubjectFull: Measures (Individuals)
        Type: general
      – SubjectFull: Children
        Type: general
      – SubjectFull: Models
        Type: general
      – SubjectFull: Data Analysis
        Type: general
      – SubjectFull: Longitudinal Studies
        Type: general
      – SubjectFull: Speech Impairments
        Type: general
      – SubjectFull: Language Impairments
        Type: general
      – SubjectFull: Learning Disabilities
        Type: general
      – SubjectFull: Scores
        Type: general
      – SubjectFull: Infants
        Type: general
      – SubjectFull: Toddlers
        Type: general
      – SubjectFull: MacArthur Bates Communicative Development Inventories
        Type: general
    Titles:
      – TitleFull: Modeling Longitudinal Trajectories of Word Production with the CDI
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Trevor K. M. Day
      – PersonEntity:
          Name:
            NameFull: Arielle Borovsky
      – PersonEntity:
          Name:
            NameFull: Donna Thal
      – PersonEntity:
          Name:
            NameFull: Jed T. Elison
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 07
              Type: published
              Y: 2025
          Identifiers:
            – Type: issn-print
              Value: 1363-755X
            – Type: issn-electronic
              Value: 1467-7687
          Numbering:
            – Type: volume
              Value: 28
            – Type: issue
              Value: 4
          Titles:
            – TitleFull: Developmental Science
              Type: main
ResultId 1