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M. Ramscar et al. / Topics in Cognitive Science 6 (2014)

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measures, which indicate that vocabulary growth in adulthood is marginal, such that
increases are only reliably detected in meta-analyses (Verhaeghen, 2003).
Unfortunately, psychometric vocabulary measures are virtually guaranteed to fail to
detect vocabulary growth in adults because they attempt to extrapolate vocabulary sizes
from sets of test words that are biased toward frequent types (Heim, 1970; Raven, 1965;
Wechsler, 1997). However, the distribution of word-types in language ensures both that
adult vocabularies overwhelmingly (and increasingly) comprise low-frequency types, and
that an individual’s knowledge of one randomly sampled low-frequency type is not predictive of his or her knowledge of any other randomly sampled low-frequency type. This
makes the reliable estimation of vocabulary sizes from small samples mathematically
impossible (Baayen, 2001).
2.2. Simulation Study 1: Why linguistic distributions confound vocabulary estimates
To illustrate these points, we analyzed the statistical properties of a state-of-the-art test
designed to measure the vocabularies of advanced learners of English (Lemhoefer & Broersma, 2012). The test samples 40 items (more than most standard vocabulary measures,
Bowles & Salthouse, 2008), and its design explicitly seeks to control for the way that the
shape of linguistic distributions makes vocabulary measurement a problem (unlike most
psychometric vocabulary measures). The upper left panel of Fig. 2 plots the words
employed in the test by their rank-frequency in the distribution of English lemmata in the
CELEX lexical database (Baayen, Piepenbrock, & Gulikers, 1995; in linguistics a lemma
is defined as a “canonical form,” such that the frequency of the lemma walk comprises
the individual frequencies of walk, walks, walked, etc.; basing this analysis on lemmata
frequencies ensured for more conservative estimates than counting inflected word forms
as separate items). As can be clearly seen, all of the types in the Lemhoefer and
Broersma test clearly belong to the higher frequency part of the English lexicon: Over
half of the lemmas in the CELEX database are lower in frequency than the test items.
To illustrate the way that the distribution of word types affects the measurement of
vocabulary growth over time, we constructed a word frequency distribution using the
lognormal-poisson model (Baayen, 2001), with parameters estimated from the distribution
of English lemmata. We then simulated 20 speakers incrementally sampling from this
distribution. (A simplifying assumption made here was that speakers sample the language
at the same rate.) The black circles in the upper right panel of Fig. 2 plot increase in
vocabulary size with “age” for one simulated learner (because the amount of language
individuals are exposed to varies dramatically [Hart & Risley, 1995], for the purpose of
these simulations, “age” is defined in terms of the number of lemma tokens a learner has
experienced, rather than time). As can be seen, although vocabulary size continually
increased in the simulations, the rate at which new lemmata were encountered in the
simulations decreased as learners’ experience grew. The gray circles then show the
vocabulary common to all 20 simulated speakers. This shared vocabulary is typically less
than half a speaker’s own vocabulary, and further, the rate at which new common
lemmata are encountered (i.e., learned) as compared to non-common lemmata decreased