The Constant Rate Hypothesis
What is the CRH?
With lexical change as probably the sole exception, all language change takes place over
the set of relevant linguistic environments in which the linguistic variable is embedded.
Thus, a sound change occurs in contexts defined by such factors as preceding/following
environments, prosodic features, etc., each of which affects the change to a different
degree. A following nasal, for example, may be a more favorable environment for some
phonological changes than other environments, so that a greater number of innovative
variants is observed in this environment than elsewhere.
A question that arises here is: Does language change proceed faster in
favorable environments than less favorable ones? The Constant Rate Hypothesis (CRH,
also known as the Constant Rate Effect) proposed in Kroch (1989), says ‘No’ to this
question. The CRH predicts that the rate of change is uniform across the linguistic
environments in which the change occurs. To cite the original definition in Kroch (1989:
200),
… when one grammatical option replaces another with which it is in
competition across a set of linguistic contexts, the rate of replacement,
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properly measured, is the same in all of them. The contexts generally differ
from one another at each period in the degree to which they favor the
spreading form, but they do not differ in the rate at which the form spreads.
One alternative to CRH would be a model in which an innovative form spreads
at different rates in different contexts, so that, for example, in some “accommodating”
contexts where innovative forms appear more frequently the change proceeds at a faster
rate, whereas in other contexts, such forms appear only rarely, and the change chugs
along slowly. Basically, this was the idea of the Wave Model (WM) by Bailey (1973). The
WM holds that language change proceeds in an S-curve fashion, a point that Kroch
(1989) agrees with as well; but the WM also equated the favorableness of each context
with the rate of change and the order of appearance of change in each context, so the
rate of change is not equal across environments. If the innovative form appears first in a
given context, the rate of change is higher there than elsewhere. The situation is best
summarized in Bailey’s own words, “[W]hat is quantitatively less is slower and later;
what is more is earlier and faster” (Bailey 1973: 82). This is exactly opposite to what
the CRH predicts.
Originally, Kroch based his hypothesis on four historical syntactic changes
(Kroch 1989): the replacement of have by have got in British English (Noble 1985), the
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rise of the definite article in the Portuguese possessive NP (Oliveira e Silva 1982), the
loss of V2 order in Middle French (Adams 1987) and the rise of periphrastic do in
Modern English. Subsequent research added examples from Old English (Pintzuk 1995)
and Yiddish (Santorini 1993) as well. Most of these syntactic changes involve a change
in the abstract level of grammar, and not a mere alternation of linguistic elements at
the surface level, and this is a rather important point in the research program of the
CRH. This is because the constant rate effect is seen as a direct result of the fact that, as
far as a single syntactic change occurring in related contexts is concerned, the change
occurring in those contexts is due to a single change in parameter settings, just like a
reflex (or reaction) to a single stimulus being realized in different muscles or organs
simultaneously. Kroch further extends this view to claim that a speech community
where the change is in progress is in a state of diglossia with respect to a given
parameter setting, with two different grammars in competition; which setting is used in
a given moment of production is now seen as stylistic preferences or psycholinguistic
processing effects, a clear move away from the classical variationist paradigm that
would incorporate such frequencies into the grammar (Kroch 2001).
The CRH in Statistical Terms
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The meaning of the CRH can be easily captured in statistical terms. When there are two
variables X and Y, and one wants to predict the values of Y based on those of X, the
most popular statistical analysis to use is a linear regression of the form shown in (1):
(1)
Y= aX +b
where a is a slope and b is an intercept. Linear regression, however, is possible only for
variables of ratio or interval scale (e.g., sound frequencies, test scores, etc.) and
linguistic variation data is most commonly nominal (the assignment of items into
categories – male/female, following consonants, etc.). The only statistical operation
possible for nominal data is taking the frequency of its variant and computing its ratio
to the total, varying from 0 to 1. But this would mean that the Y value in (1) cannot
exceed that range. Also, the graph of the change is not linear (a straight line) as (1)
implies, but S-curved.
Alternatively, one can transform Y into a logit (a natural log of Y over 1-Y).
Because the logit term varies from -∞ to +∞ as the Y value changes from 0 to 1, the
upper limit issue no longer poses a problem.
(2)
ln(Y / (1 - Y)) = aX + b
The graph of (2) would still be a straight line, but of course the vertical axis now has a
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different scale. The equation (2) is solved with respect to Y as in (3):
(3)
Y = eax+b / (1 + eax+b)
This is the logistic equation. The graph of this equation forms an S-shape and so is an
ideal statistical tool for analysis of change phenomenon. Indeed, since its first use for
the analysis of coronary heart disease in the 1960s (Truett et al. 1967), the logistic
regression has been used extensively in such diverse fields as medical science, political
science, biology and a number of other scientific fields. Thus, its extension to the study
of language change was a very natural move.
So, how does the CRH translate into a logistic equation? Before answering this
question, one should note that the very first model of linear regression had only one
independent variable, namely, X. But as case after case reported by sociolinguists
demonstrates, most language variation is multivariate, in the sense that there is more
than one factor controlling the variation. These can be internal factors (grammatical/
phonological) or external ones (stylistic or sociological). If we add multiple independent
variables (X, Z, ...) and the intercept w, the simple equation (1) is extended to multiple
regression in (4), and correspondingly, to its logit counterpart in (5):
(4)
Y = aX + bZ + ... w
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(5)
ln (Y / (1 - Y)) = ln(X / (1 - X)) + ln(Z / (1 - Z)) + ... w
The equation in (5) is actually the model underlying the VARBRUL program, a
specialized multivariate logistic regression analysis freeware for variation analysis
(Rousseau and Sankoff 1978, Rand and Sankoff 1990). For example, suppose one is
looking at a variable use of the accusative case marker attached to an NP (say we code
“1” for the presence of the marker), and we picked out two factors, X and Z, as
independent variables, each denoting, respectively, i. the polarity of the sentence
(negative/affirmative) and ii. whether or not the NP is a pronoun. For negative
sentences with a pronoun, the model for predicting the case marker’s presence would be
as in (6):
(6)
ln (Yc / (1 - Yc)) = ln(Xn / (1 - Xn)) + ln(Zp / (1 - Zp)) + w
where Yc refers to the probability of the case marker’s presence, Xn and Zp are weights
(or parameter values) for the negative context and the pronoun context respectively, and
w is an intercept (or input probability in VARBRUL terminology).
Since logistic regression allows a continuous variable as one of its factors, we
could add another term for the time factor, f(t) to (6):
(7)
ln (Yc / (1 - Yc)) = f(t) + ln (Xn / (1 - Xn)) + ln(Zp /(1 - Zp)) + w
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Now, what the CRH predicts here is that the effects of X and Z are not affected by the
value of f(t), so that whatever point the time factor may indicate, all four parameter
values for X and Z (negative/affirmative and pronoun/non-pronoun) remain constant.
This is the meaning of independence in statistics; and the central tenet of the CRH is
simply that the contextual effect is independent of time. When the independence does
not hold, equation (7) requires an additional interaction term made up of the two or
more variables which interact with each other. Note here that the VARBRUL program,
as seen in the model in (5), assumes independence among the variables by default.
Previous research has shown that this assumption generally applies to internal factors
(but cf. Sigley 2003), so that linguistic factors rarely interact with each other, while
social variables such as sex, age or social class, often exhibit intricate interaction
patterns (Labov 1982).
It is important to note here that the “constant” part of the CRH lies in the
constancy of the rate of increase of the innovative form across contexts, which, in visual
form, would result in a series of parallel lines connecting the frequencies (in their logit
transforms) of the innovative form occurring in that environment. The rate of change in
a single context also remains the same throughout the course of the change (thus the
line becomes nearly straight), but this is simply because the raw frequencies are
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transformed into logits. That is, a logit transform is just a convenient way of
representing the slope of an S-curve (where the slope changes continually) in a single
value.
Further Issues
The CRH is an important hypothesis for the theory of language change, but its impact
seems to be spreading quickly in related problems in language variation and change in
the broad sense. Language acquisition is programmatically explored by Kroch himself
within the aforementioned idea of a parametrically diglossic speech community (Kroch
2001: 722). There has also been an attempt to relate the CRH to age-grading. Matsuda
(2003) showed that by replacing “time” with “age” and taking its contrapositive it is
possible to predict whether an observed synchronic age-correlation is a case of ongoing
change or an age-grading. The resulting proposition (the Extended Constant Rate
Hypothesis) reads as follows:
(8) If the linguistic contextual effect shows interaction with age, the observed
age-correlation cannot be a case of language change (and thus, it must be a
case of age-grading).
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On the other hand, if the contextual effect does not interact with age at all, (8)
does not say anything at all – meaning that the data is consistent with either a change
or an age-grading. It then follows that there are two different types of age-grading: one
that exhibits interaction between contextual effect and age, and another that does not.
Matsuda claimed the age-grading pattern of English t/d-deletion (Guy and Boyd 1990)
is the former type, while the classical cases of age-grading fall into the latter type.
As a final note on the CRH, it must be mentioned that the evidence for the
CRH so far comes from past syntactic changes, and this raises two interesting topics for
future research: firstly, whether the CRH applies to non-syntactic changes, especially
phonological changes (Kroch 1989: 240, fn.3). At this point some interesting
counterevidence emerges from the very first variationist study: the centralization of
diphthongs in Martha’s Vineyard involved an interaction between age and the weight of
the following segment (Labov 1963), a scenario that should not be possible under the
CRH. Since such a reorganization of environments does not seem so uncommon in
phonological changes, one may wonder whether the CRH is mainly valid for
morpho-syntactic changes.
The second topic for future research would be a demonstration of the CRH in a
real-time study of language change in progress. The Labovian methodology of observing
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language change in progress has revealed much about the nature of language change
(Labov 1994, 2001), and it is expected that the CRH should also benefit from real-time
study of speech communities, a practice that has grown rapidly enormously in recent
years in variationist sociolinguistics (Yoneda 1997).
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