MODELLING DIFFUSION: EXAMINING CABLE AS AN INNOVATION. By Benjamin J. Bates, and David R. Brimm

Transcription

1 MODELLING DIFFUSION: EXAMINING CABLE AS AN INNOVATION By Benjamin J. Bates, and David R. Brimm (At the time we were both Ph.D. students at the University of Michigan). Abstract: Presented at the 36th Annual Conference of the International Communication Association C h i c a g o, I L, May 1986 Contact: Benjamin J. Bates School of Journalism & Electronic Media University of Tennessee bjbates@utk.edu This research examined the statistical models which had been used to model the diffusion of cable television, and compared their performance to a variety of alternative statistical models, using data from a census of U.S. cable system penetration levels. The study found that the models only did a fair job of explaining differences in penetration rates (R-squares consistently below 0.30), and that the most widely used models were not the most effective for this set of data. Rather, this study found that a model that allowed for the presence of disadopters tended to be more effective.

2 Modeling Diffusion: Examining Cable as an Innovation The research fields of cable television and the diffusion of innovations both have a considerable research tradition. These traditions share one common focus: a deep concern with the concept of adoption, its process and effects. A further link between the two traditions can be seen if one considers cable television as an innovation, and the adoption of cable television in communities as an example of the diffusion of innovations. In any case, research in both fields has led to the development of a number of mathematical models of the process of diffusion; models which have focused on the determination of the rate and extent of adoption. Within the cable research tradition, this concern with the rate and extent of the adoption of this particular innovation manifested in a concern with the level of penetration for cable services. This concern with penetration was a central issue in both the research done on the likely impact of cable television upon over-the-air broadcasters (c.f. Fisher and Ferrall, 1966; Park, 1971a; Webster, 1983, 1984) and the work done upon the likely impact of regulation upon the development of cable, both in the early 1970s (c.f. Comanor and Mitchell, 1971; Crandall and Fray, 1974; Mitchell and Smiley, 1974; Sloan Commission, 1971) and later (Webb, 1983). The concern with the level of cable penetration in communities led to a number of early research attempts to predict either current or eventual ("ultimate") cable penetration levels (Comanor and Mitchell, 1971; Crandall and Fray, 1974; McGowan, Noll and Peck, 1971; Park, 1971b, 1972). More recently, other researchers (Bloch and Wirth, 1984; Ellickson, 1979; Webb, 1983) have examined and attempted to model the demand for cable television in communities, a concept which is inextricably linked with the issue of penetration. 1 An attempt was made in 1985 (Bates, 1985) to replicate and update the 1971(b) Park study in the wake of significant changes in the cable policy and programming environment. In that effort to deal with Park's model and procedure, which had been introduced as a typical model of the diffusion process, several potentially problematic aspects of adoption modeling emerged. This attempt at replication brought into focus the basic uncertainty over the appropriateness of any of the various theoretical models which have been used to illustrate cable penetration, or, for that matter, the diffusion process as manifest in the adoption of cable television. Accepting cable television as an example of an innovation, this paper will examine the various approaches being used to model the diffusion/adoption of innovations. We will address theoretical issues involved in the inherent assumptions of various models, and then apply basic models to a cross-section of American cable systems collected for the earlier replication (Bates, 1985) in order to consider the relative benefits and deficiencies of the models in explaining the diffusion process in cable. Such a consideration will be focused on the usefulness of the models in explaining differential diffusion, or penetration, but will not be limited to such factors. Review In much of the cable research noted above, the adoption of cable television has been viewed from a diffusion perspective. A great deal of emphasis in the diffusion literature has been devoted to the specification of mathematical models of adoption behavior. Gabriel Tarde (1903) was the first scholar to posit that cumulative adoption within a social system could be generally 1 In fact, most such models were themselves based upon considerations of penetration levels.

3 mapped as an S-shaped curve. In the eight decades since, the prototypical view of accumulated adoption behavior in a system has been of a gradually increasing function with a single inflection point (Mahajan and Peterson, 1985). Pemberton (1936) found this model useful in case study analyses of adoption for postage stamps (the adopting unit was nations), constitutional and statute limitations of taxation (the adopting unit was American states), and compulsory school legislation (the adopting unit was, again, American states). In what is considered to be the classic synthesis on innovation diffusion, Rogers (1983) presented the "S-curve" as the basic form of the adoption process for innovations over time. In this and earlier editions, Rogers acknowledged the contribution of epidemiological research to the generation of this basic model (Rogers and Shoemaker, 1971; Rogers, 1983). An early example of this is the work of Bailey (1957) on the mathematics of epidemics, work which resulted in the development of an "Scurve" model, which is based upon a logistic model. In the field of communication, similar modeling has been done in the area of the diffusion of news. Building upon a rich tradition in this area (c.f. Larsen and Hill, 1954; Greenberg, 1964a, 1964b; Gantz, 1983), Dominick (1983) presented a number of summary news diffusion models or curves, noting that the importance of the news event influenced the basic shape of the curve. Dominick's various models, though, did tend towards the basic "S-curve," or simple transformations thereof. Such models, however, were basically designed to model diffusion solely in terms of exposure to the event in question. Adoption of an innovation, on the other hand, requires more than simple exposure, it requires a conscious effort to adopt the innovation after exposure. And as Rogers (1983, p.245) noted, the assumption of these models that there exists relatively free access of all members of a social system to one another is also not always applicable. This has resulted in the examination of a number of alternative models of the diffusion process, as evidenced in Dodd's (1955) and Olson's (1982) use of the logistic probability models and Mansfield's (1961) testing of deterministic and stochastic models of imitation of industrial innovation. In examinations of adoption, Hamblin, Miller and Saxton (1979) compared Gompertz curves to exponential epochs, while Gort and Konakayama (1982) developed a probabilistic model of adoption. Funkhouser and McCombs (1972) and Gray and von Broembsen (1974) developed stochastic models for the diffusion of information and Sharif and Ramanathan (1982) employed polynomial models for the diffusion of color television. There have been, thus, a number of potential mathematical models which have been used to examine the diffusion process. Almost uniformly, though, the venous models have followed the basic "S-curve" shape when considering adoption levels over time. As has been shown, however, there are a number of distinct mathematical models which reflect this basic form, which has led to a profusion of distinctive models which have been used in cable research efforts. Models This paper will examine a number of mathematical models of the diffusion process, from both a theoretical perspective and as applied to the adoption of cable television basic service. We will initially consider the various models, the assumptions which they make, and their relation with the diffusion process. Then, the various models will be applied to a collection of data on cable systems. In its simplest form, the diffusion process can be seen as a sequence of dichotomous variables based on the adoption, or non-adoption, of some innovation. In this case, where penetration is seen as a sequence of adopt/non-adopt states,

4 statistical theory suggests the use of a logistic transformation of the penetration variable for penetration (Neter and Wasserman, 1974). This logistic response model also follows the basic shape of the "S-curve" generally acknowledged as the basic model for the diffusion process, and was incorporated in several cable penetration studies (Mitchell and Smiley, 1974; Park, 1972). One possible limitation of this model, noted by several researchers, is the fact that the pure logistic response model has an upper asymptote of one. That is, the model assumes that penetration will ultimately reach 100%, an assumption that most researchers familiar with the cable industry are unwilling to make. For that reason, some studies of cable penetration (Comanor and Mitchell, 1971; Park, 1971) used a modification of the logistic response model, incorporating a reciprocal transformation of the independent variable. 2 This particular model, based upon the logistic reciprocal model, also follows the basic shape of the "S-curve," although it permits the upper asymptote, or limit of penetration, to be less than 100 percent. In other words, it drops the assumption that, sooner or later, everyone will adopt the innovation, and allows for the presence of non-adopters of the innovation within the community, to some degree. However, as noted by Sharif and Ramanathan (1982), in many innovation adoption situations there are not only adopters and non-adopters of an innovation in a community, but also there may be disadopters, those who may initially adopt the innovation 2 Park's (1971) model used a reciprocal transformation on the system age variable, modified by a polynomial function of system size. Comanor and Mitchell (1971), to increase the impact of system age, used the reciprocal of age squared in their model. only to reject it at a later point. In developing modified stochastic models which allow for "disadoption" or "desertion" behavior, Gray and von Broembsen (1974, p. 238) have argued:... [M]odified models... seem to be worthwhile extensions of the basic models and can be applied to less restrictive situations.... Traditional approaches to diffusion (cf. Rogers and Shoemaker, 1971: Ch. 2) deal almost exclusively with diffusion curves which increase to a limit of unity... and ignore the possibility of other types of curves. It seems clearly possible that under certain types of circumstances the number of individuals... having adopted a particular practice can decline over time. One model which Sharif and Ramanathan (1982) found to fit the diffusion process in communities where disadopters, as well as nonadopters, may be present in the community was the polynomial model. This model initially follows the general form of the "S-curve," but permits the existence of declining penetration rates after a certain point in time. The polynomial model, like Park's (1971) version of the logistic model discussed earlier, allows for an asymptote, or highest level of penetration, of less than 100%. There have been other models which have been used to estimate cable penetration which do not follow the basic "Scurve" shape. For example, one model is known as the double logarithm model, as it incorporates logarithmic transformations of both dependent and independent variables. This model has been frequently used to model economic demand (Johnston, 1972), and has served as the basis for several estimates of the level of demand for cable in a community (c.f. Perrakis and Silva-Echenique, 1983), although the basic form is quite similar to that of a strictly linear equation. This is quite distinct from the basic "S-curve," although it does

5 correspond somewhat to the middle portion of that curve. There are other basic models which can be seen as corresponding somewhat to parts of the "S-curve." These include two other logarithmic transformations: the exponential, which uses a logarithmic transformation of the dependent variable, and follows the first part of the "Scurve;" and the logarithmic model, which is based upon the logarithmic transformation of the independent variable, and roughly corresponds to the second half of an "S-curve." Although neither of these models has been specifically used in a cable penetration study encountered by the authors, it could be argued that there is a theoretical foundation for the use of the logarithmic model in particular. According to Rogers (1983), the first part of the basic "S-curve" model of the diffusion of innovations corresponds to a period when members of the community first become aware of the innovation, and are exposed to it. Cable as an innovation is distinctive in two ways that act to minimize, or do away with, this initial period of exposure. First, most cable systems generate a great deal of publicity and thus exposure during the licensing process. Thus, by the time that the cable service is available for adoption, most members of the community are likely to already have had a strong awareness of cable as an innovation. Further, since the product of cable television is quite similar to that of broadcast television, most members of the community are also aware of the nature and benefits of this particular innovation prior to its actual availability. For these reasons, it is possible, if not likely, that the adoption process would follow the second half of the traditional "S-curve" when applied to the product of cable television. These two models are hampered, though in that neither has an asymptote, or limiting upper value. Thus, it would be possible, under these models, to have a penetration of greater than 100%. Two other basic models were considered. First, because of its use in more general diffusion studies, a model built upon the Gompertz curve was included for analysis. Also, as a standard for comparison, and also due to the similarity in form to the double log model, a straight linear model was included for consideration. It was also decided, due to the lack of asymptotes for some of the models and the likelihood of disadopters of the innovation of cable television, to integrate several of the transformations of the dependent variable with polynomial forms of the basic independent variable, system age. The various models discussed above all can be seen as somewhat appropriate for the modeling of the diffusion and adoption of innovations in a community, although some curves seem to be theoretically more appropriate to the study of cable adoption in communities. For example, it seems probable that both nonadopters and disadopters exist in all communities. That suggests, first, that models, or curves, should have an asymptote of less than 1 (100%). The presence of disadopters, on the other hand, suggests the appropriateness of polynomial-based models. It is important, however, to consider more than the theoretical appropriateness of any model. After all, the proof of any statistical model is its ability to accurately portray reality, to reflect what it models. Thus, it is necessary to consider how accurately the models can portray, or reflect, the penetration process. For this reason, we will fit the various models to a set of data on cable systems in the U.S., and consider how well the various models account for variations in levels of penetration within communities. Methods

6 The data used for this study was taken from a census of cable systems taken from the edition of the Television and Cable Factbook (Factbook). The data were gathered from all cable systems within the continental U.S. listed as currently providing basic service to consumers. Exclusion of those systems with significant incomplete, missing, or clearly invalid basic information reduced the number of systems in the basic data set from over five thousand to Further instances of missing data reduced the applicable data set for specific analyses and procedures, with most fitted models dealing with samples of around 3700 systems. The measures for subscriber and household counts used to generate the penetration estimates, as well as broadcast signals received, utilized the definitions and measures of the Factbook. The age of the cable system was measured by the number of months from the date on which service was first provided to its community to the date for which the subscriber counts were given. The rate of market penetration was defined as the number of subscribers to the basic cable service divided by the number of households passed by the cable system. Penetration, that is, was defined as that portion of the households which could adopt cable which have adopted that innovation. Based upon the census of 3961 systems, cable systems averaged a penetration rate of 66.3%, and an average age of months. It should be noted that, as the data were originally collected for a replication of the 1971 Park study (Bates, 1985), they do not include measures of several factors identified by other cable penetration studies as influencing the decision to subscribe to cable television, and thus the final rate, and amount, of adoption. As this paper, however, was dealing with very basic models of penetration over time, the absence of these additional factors was not considered to invalidate the basic procedures of this study. Results and Analyses A number of basic models were then fitted to the data on penetration and system age through the use of basic least squares regression techniques. In all cases the regressions proved to be significant at the.005 level, as did the coefficients for the independent variables corresponding to the age of the system. The various regressions and their corresponding R 2 and F-statistic values are given in Table 1. The results of the fitting of the various models to this data on U.S. cable systems provide some interesting findings. Of the three basic models used in cable penetration research, one (the log-reciprocal) ranked last of the tested models, while the others proved to have less explanatory power than the pure linear model which fits none of the theoretical diffusion assumptions. Only two of the tested models did, in fact, yield more accurate portrayals of the adoption process: the logarithmic model, which reflected only the second half of the standard "Scurve;" and the polynomial model, which made provision for the presence of disadopters as well as nonadopters of the innovation in the community. The polynomial model proved to be somewhat superior to all other basic models considered in its ability to explain variations in cable penetration levels. None of the models, however, could explain more than 30 percent of the variation in penetration rates among cable systems; only five could explain more than 25%. This lack of explanatory power is likely to be at least partly attributable to the various factors which were left out of these models, such as the price of cable service, household income, and the precise services provided by the cable system. Most earlier studies had included at least some of these various other measures; in fact, they had also stratified their data on cable systems along various dimensions, and fitted separate models to each type of cable

7 system. In this tradition, and since the various models evidenced similar overall goodness-of-fit, it seemed possible that different kinds of cable systems might be best explained by different specific models. Thus, it was decided to stratify the data on cable systems along two dimensions considered significant in the explanation of cable penetration, and fit a selection of the dozen basic models to the stratified samples. The two dimensions utilized to differentiate cable systems were system size, as measured by the number of households passed, and basic service type, as indicated by the number of commercial broadcast signals received and retransmitted by the cable system. The results of the application of these models to the stratified samples of cable systems are given in Table 2 for system size and Table 3 for service type. The results of the fitting of these models to the stratified samples tend to reinforce the general conclusion of the appropriateness of the polynomial and logarithmic models in explaining adoption, or penetration, levels. For both stratifications, the polynomial model provided the best "fit" of the data for the largest strata, and was on of the three best models for all strata, supporting the basic finding that the polynomial model seems to provide the best basis for the explanation of cable penetration. It should be noted, however, that contrary to the rather lackluster performance of the three basic types of models used in the bulk of cable penetration studies in this study, the logreciprocal model proved to be considerably more accurate models for cable systems passing between 10,000 and 50,000 households. The fact that this model explained from 10 to 20 percent more of the variation in penetration rates than the next best model for systems of this size suggests that there might be differences in the process of adoption in communities of that size than in either larger or smaller communities. The ability of this model to provide significantly more accurate model of cable penetration for a specific segment of cable systems suggests a need for further study of the disparate performance of this curve in modeling adoption. The most interesting result of this research, though, was the poor showing of the fundamental models widely used in cable research. And that two models which had not been utilized as the foundation for such attempts to model or predict cable penetration have proved to provide a more accurate reflection of the adoption process as applied to cable television. This suggests that most previous research in this field have not utilized the best, or most appropriate, models for their analysis. The results of this consideration of alternative models also illustrates the usefulness of further research into the nature of cable as an innovation, and into the adoption of cable television service as an example of the process of innovation adoption in communities. Finally, the results also support the recent call for multi-method analyses, or at least the incorporation of a greater level of concern for the appropriateness of models and their inherent assumptions, in any research dealing with the modeling of social processes.

8 Bibliography Bailey, N. T. J. (1957). The Mathematical Theory of Epidemics. New York: Haffner Bates, Benjamin J. (1985). "Future Growth of Cable Television: A Replication and Update." Unpublished paper presented at the 35th Annual International Communication Association conference, Honolulu, HI, May Bloch, H., and Wirth, M. 0. (1984). "The Demand for Pay Services on Cable Television." Information Economics and Policy, 1(4), Comanor, W. S., and Mitchell, B. M. (1971). "Cable television and the impact of regulation." Bell Journal of Economics and Management Science, 2(1), Crandall. R. W., and Fray, L. L. (1974). "A re-examination of the prophecy of doom for cable television." Bell Journal of Economics and Management Science, 5(1), Dodd, S. C. (1955). "Diffusion is predictable: Testing probability for laws of interaction." American Sociological Review, 20, Dominick, J. R. (1983). The Dynamics of Mass Communication. Reading, MA: Addison-Wesley Ellickson, B. (1979). "Hedonic Theory and the Demand for Cable Television." American Economic Review, 69(1), Fisher, F. M., and Ferrall, V. E. (1966). "Community Antenna Systems and Local Station Audience." Quarterly Journal of Economics, 80(2), Funkhauser, G. R. and McCombs, M. E., (1972). "Predicting the diffusion of information to mass audiences." The Journal of Mathematical Sociology,2, Gantz, W. (1983). "The diffusion of news about the attempted Reagan assassination." Journal of Communication, 33(1), Gort, M., and Konakayama, A. (1982). "A model of diffusion in the production of an innovation." American Economic Review, 72, Gray, L. N. and von Broembsen, M. H. (1974). "On simple stochastic diffusion models" Journal of Mathematical Sociology.3, Greenberg, B. S. (1964a). "Diffusion of news of the Kennedy assassination." Public Opinion Quarterly,28, Greenberg, B. S. (1964b). "Person-to-person communication in the diffusion of news events." Journalism Quarterly,41, Hamblin, R. L., Miller, J. L. L., and Saxton, D. E. (1979) "Modeling use diffusion." Social Forces, 53, Johnston, J. (1972). Econometric Methods. New York: McGraw-Hill Larsen, 0. N., and Hill, R. J. (1954). "Mass media and interpersonal communication in the diffusion of a news event." American Sociological Review, 19, Mahajan, V. and Peterson, R.A. (1985). Models for Innovation Diffusion. Beverly Hills: Sage Mansfield, E. (1961). "Technical change and the rate of imitation." Econometrica, 29, McGowan, J. J., Noll, R. B., and Peck, M. J. (1971). "Prospects and Policies for CATV." Appendix B to: Sloan Commission, On the Cable: The Television of Abundance. New York: McGraw-Hill Mitchell, B. M., and Smiley, R. H. (1974). "Cable, cities, and copyrights." Bell Journal of Economics and Management Science, 5(1), Neter, J., and Wasserman, W. (1974). Applied Linear Statistical Models. Homewood, IL: Richard D. Irwin. Olson, J. A. (1982). "Generalized Least Squares and Maximum Likelihood Estimation of the Logistic Function for

9 Technology Diffusion." Technological Forecasting and Social Change, 21, Park, R. E. (1971a). "The Growth of Cable Television and its Probable Impact on Over-The-Air Broadcasting." American Economic Review, 61(2), Park, R. E. (197 lb). "Future Growth of Cable Television." Journal of Broadcasting, 15(3), Park, R. E. (1972). "Prospects for cable in the 100 largest television markets." Bell Journal of Economics and Management Science, 3(1), Pemberton, H.E. (1936). "The curve of culture diffusion rate." American Sociological Review, 1, Perrakis, S., and Silva-Echenique, J. (1983). "The Profitability and Risk of CATV Operators in Canada." Applied Economics, 15, Rogers, E. M. (1983). Diffusion of Innovations. 3rd Edition. New York: The Free Press Rogers, E. M., and Shoemaker, F. F. (1971). Communication of Innovations: A Cross-cultural approach. New York: The Free Press Sharif, M. N., and Ramanathan, K. (1982). "Polynomial Innovation Diffusion Models." Technological Forecasting and Social Change, 21, Sloan Commission on Cable Communications (1971). On the Cable: The Television of Abundance. New York: McGraw- Hill Tarde, G. (1903). The Laws of Imitation. (Tr.) Elsie Clews Parsons. New York: Holt Television Digest, Inc. (Annual). Television and Cable Factbook. Washington, DC: Television Digest. Webb, G. K. (1983). The Economics of Cable Television. Lexington, MA: Lexington Webster, J. G. (1983). "The Impact of Cable and Pay Cable Television on Local Station Audiences." Journal of Broadcasting, 27(2), Webster, J. G. (1984). "Cable Television's Impact on Audience for Local News." Journalism Quarterly, 61(2),

10 Table 1. Basic Regression Models Model Formula R 2 F-Statistic Rank Logistic Logit(P) = b 0 + b 1 A Log-reciprocal ln(p) = b 0 b 1 /A Polynomial P = b 0 + b 1 A + b 2 A Double Log ln(p) = b 0 + b 1 ln(a) Logarithmic P = b 0 + b 1 ln(a) Exponential ln(p) = b 0 + b 1 A Gompertz ln(ln(100p)) = b 0 + b 1 A Linear P = b 0 + b 1 A Logistic-polynomial Logit(P) = b 0 + b 1 A + b 2 A Log-polynomial ln(p) = b 0 + b 1 A + b 2 A Gompertz-polynomial ln(ln(100p)) = b 0 + b 1 A + b 2 A Logistic-log Logit(P) = b 0 + b 1 ln(a) P = penetration, A = system age, b i = constant terms, Logit(P) = ln{p/(1 P} Note: All models, estimated coefficients (b i ), were statistically significant at p <.005

11 Table 2. Appropriateness of Selected Models, by System Size Model R 2 Estimates by System Size (in households) No. "Best" Top Three ,000-24,999 25,000-49,999 50,000+ Logistic Log-reciprocal Polynomial Double log Logarithmic Logistic-polynomial Linear N All regressions proved statistically significant at a level of p <.001

12 Table 3. Appropriateness of Selected Models, by Signal Carriage Model R 2 Estimates by Off-Air Signals Carried No. "Best" Top Three Logistic Log-reciprocal Polynomial Double Log Logarithmic Logistic-polynomial Linear N All regressions proved statistically significant at a level of p <.001