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, 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 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

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 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

(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

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 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).

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 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).

Bibliography

Adams, M. P. (1987). “From Old French to the theory of Pro-drop” Natural Language and Linguistic Theory 5, 1-32.

Bailey, C-J. (1973). Variation and linguistic theory. Washington, D.C.: Center for Applied Linguistics.

Guy, G. R. & Boyd, S. (1990). “The development of a morphological class” Language

Variation and Change 2, 1-18.

Kroch, A. S. (1989). “Reflexes of grammar in patterns of language change” Language Variation and Change 1, 199-244.

Kroch, A. S. (2001). “Syntactic change” In Baltin, M. R. & Collins, C. (eds.) The handbook of contemporary syntactic theory. Oxford: Basil Blackwell. 699-729.

Labov, W. (1963). “The social motivation of a sound change” Word 19, 273-309.

Labov, W. (1994). Principles of linguistic change. Volume 1: Internal factors. Cambridge:

Basil Blackwell.

Labov, W. (2001). Principles of linguistic change. Volume 2: Social factors. Cambridge: Basil

Blackwell.

Matsuda, K. (2003). “Constant rate hypothesis, age-grading and apparent time construct”

Penn Working Papers in Linguistics 9, 123-134.

Noble, S. (1985). “To have and have got” Paper presented at NWAVE 14, Georgetown University.

Oliveira e Silva, G. (1982). Estudo da regularidade na variação dos possessives no português do Rio de Janeiro. Ph.D. dissertation, Universidade Federal do Rio de Janeiro.

Paolillo, J. C. (2002). Analyzing linguistic variation: statistical models and methods. Stanford: CSLI Publications.

Pintzuk, S. (1995). “Variation and change in Old English clause structure” Language Variation and

Change 7, 229-260.

Rand, D. & Sankoff, D. (1990). GoldVarb (version 2.0). Montréal: Centre de recherches

mathématiques.

Rietveld, T. & van Hout, R. (1993). Statistical techniques for the study of language and language

behaviour. Berlin: Mouton de Gruyter.

Rousseau, P. & Sankoff, D. (1978). “Advances in variable rule methodology” In Sankoff, D

(ed.), Linguistic variation: models and methods. New York: Academic Press. 57-69.

Santorini, B. (1993). “The rate of phrase structure change in the history of Yiddish”

Language Variation and Change 5, 257-283.

Sigley, R. (2003). “The importance of interaction effects” Language Variation and Change.

15:227-253.

Truett, J., Cornfield J. & Kannel W. B. (1967). “A multivariate analysis of the risk of coronary heart

disease in Framingham” Journal of Chronic Disease 20 , 511– 524.

Yoneda, M. (1997). “Survey of standardisation in Tsuruoka, Japan: comparison of results from three

surveys conducted at 20-year intervals” Nihongo Kagaku 2, 24-39.

- 1 -