schmittlein, d. c., & mahajan, v. (1982). maximum likelihood estimation for an innovation diffusion...

Maximum Likelihood Estimation for an Innovation Diffusion Model of New ProductAcceptanceAuthor(s): David C. Schmittlein and Vijay MahajanSource: Marketing Science, Vol. 1, No. 1 (Winter, 1982), pp. 57-78Published by: INFORMSStable URL: http://www.jstor.org/stable/184074 .Accessed: 01/10/2013 06:47

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MAXIMUM LIKELIHOOD ESTIMATION FOR AN INNOVATION DIFFUSION MODEL OF NEW

PRODUCT ACCEPTANCE*

DAVID C. SCHMITTLEINt AND VIJAY MAHAJANt

A maximum likelihood approach is proposed for estimating an innovation diffusion model of new product acceptance originally considered by Bass (1969). The suggested approach allows: (1) computation of approximate standard errors for the diffusion model parameters, and (2) determination of the required sample size for forecasting the

adoption level to any desired degree of accuracy. Using histograms from eight different product innovations, the maximum likelihood

estimates are shown to outperform estimates from a model calibrated using ordinary least squares, in terms of both goodness of fit measures and one-step ahead forecasts.

However, these advantages are not obtained without cost. The coefficients of innovation and imitation are easily interpreted in terms of the expected adoption pattern, but individual adoption times must be assumed to represent independent draws from this distribution. In addition, instead of using standard linear regression, another (simple) program must be employed to estimate the model. Thus, tradeoffs between the maximum likelihood and least squares approaches are also discussed.

1. Introduction

In recent years, a number of models have been developed to represent the

spread of a new product in the marketplace (Wind, Mahajan and Cardozo

1981). In particular, diffusion theory has often been used to model the

first-purchase sales growth of a new product over time and space (Mahajan and Muller 1979). Diffusion theory suggests that there is a time lag in the

*Received March 1981. This paper has been with the authors for 1 revision.

Key words. Diffusion of innovations, new product forecasting, maximum likelihood. tDepartment of Marketing, The Wharton School, University of Pennsylvania, Philadelphia,

Pennsylvania 19104. The authors would like to thank Christopher Easingwood for supplying the data on medical

innovations, Abba Krieger and Yoram Wind for their helpful comments and suggestions. This research was supported by the Center for Marketing Strategy Research, The Wharton School, University of Pennsylvania.

57 MARKETING SCIENCE Vol. 1, No. 1, Winter 1982 0732-2399/82/0101/0057$01.25

Printed in U.S.A. Copyright ? 1982, The Institute of Management Sciences


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DAVID C. SCHMITTLEIN AND VIJAY MAHAJAN

adoption of products by different members of a social system. The product is first adopted by a select few innovators who, in turn, influence others to adopt it. Thus it is the "interaction" or interpersonal communication (word-of- mouth) between adopters and non-adopters that is posited to account for the rapid growth stage in the diffusion process (Rogers and Shoemaker 1971).

The best known first-purchase diffusion models of new product acceptance in marketing are those of Bass (1969), Fourt and Woodlock (1960), and Mansfield (1961). In particular, the Bass model has been successfully demon- strated in retail service, industrial technology, agriculture, and consumer durable sectors (Bass 1969; Dodds 1973; Nevers 1972). Some attempts have also been made to incorporate strategic marketing variables such as price (Robinson and Lakhani 1975; Bass 1980), advertising (Horsky and Simon 1978; Dodson and Muller 1978), promotion (Lilien, Rao and Kalish 1981), product interrelationships (Peterson and Mahajan 1978) and market size (Mahajan and Peterson 1978) into this model. Dodson and Muller (1978) and Lilien, Rao and Kalish (1981) have extended these modelling efforts to include repeat purchase. Furthermore, Lilien, Rao and Kalish (1981) and Bret- schneider and Mahajan (1980) have suggested Bayesian and feedback estimation procedures, respectively, to update the model parameters as additional sales data become available. A limited extension incorporating the spatial dimension into the diffusion model has been reported by Mahajan and Peterson (1979).

One statement of the Bass model' is that

dN(t) _ _ ______

(P+^N^Ym ) (1) dt

( m ))(m - N(t))

where N(t) is the cumulative number of adopters at time t, m is the ceiling, p is the coefficient of innovation and q is the coefficient of imitation. In considering the timing of initial purchase of a new consumer product, Bass uses a discrete analog of equation (1). That is, P(t), the probability that an initial purchase will be made in the interval [t - 1, t] given that no purchase has yet been made, is a linear function of the number of previous adopters. Thus

P(t) =p + ( )N(t- 1).

Since N(0) = 0 the constant p is the probability of an initial purchase in the first time interval.

Given that the remaining number of nonadopters at time t- 1 is m- N(t - 1), the expected number of incremental adopters, X(t), in interval

'This model was developed for the diffusion of news in a social group, by Taga and Isii (1959).

58



INNOVATION DIFFUSION MODEL OF NEW PRODUCT ACCEPTANCE

[t- 1,t]

E(X(t)) = (P + N(t - 1))(m- N(t- 1)). (2)

The parameters p, q and m in equation (2) are generally estimated via

ordinary least squares (Bass 1969) or nonlinear optimization techniques (Dodds 1973) since we can write

X(t) =pm + (q -p)N(t - 1) - q N2(t 1) + (t) m

= a, + a2N(t- 1) + a3N2(t- 1) + E(t) (3)

where a, = pm, a2 = (q -

p), a3 = -

q/m, E[e(t)] = 0, Var[E(t)] = o2 and E(ti) is independent of E(tj) for i =j. Given regression coefficients al, a2 and a3, the estimates of the parameters p, q and m can be easily obtained. That is,

p= a/m (4)

q= -ma (5)

and

(-a - 2 - 46163

m=~~ A *(6) 2aj

As is clear from equations (4)-(6), in the presence of few time-series data

points and multicollinearity between variables, one may obtain parameter estimates which are unstable or possess wrong signs. Commenting on the inappropriateness of deriving the diffusion model parameters from regression equation (3) and equations (4)-(6), Heeler and Hustad (1980, p. 1020) write:

Clearly, an infinity of alternative reformulations of al, a2, and a3 are possible. All will yield the same predictions when substituted back. The good fit of the model to the data supports the utility of a quadratic power series for adoption-type time series. The result is not too surprising as unimodal data usually can be fitted closely with a quadratic Taylor series. The good fit does not in itself support the particular reformula- tion of a1, a2 or a3 into p, q, and m. Ceteris paribus p, q, and m have intuitive appeal because of their deduction from the processes behavioral scientists have observed in diffusion. But it is these social science findings rather than the model fit which justify p, q, and m.

59




To follow up on this point, note that the left side of equation (2) should theoretically be the derivative of N(t) and not the difference represented by X(t). This substitution causes a problem in that, as defined, X(t) will underes- timate dN(t)/dt for time intervals before the maximum adoption rate is reached and will overestimate after that point.

The purpose of this paper is to propose and develop a maximum likelihood approach to estimating the parameters of innovation diffusion models of new product acceptance; more specifically, a new product growth model proposed by Bass. It will be shown that the approach allows:

(i) computation of approximate standard errors for the diffusion model parameters-p, q, and m, and

(ii) determination of the required sample size for forecasting the adoption level to any desired degree of accuracy. That sample size is shown to depend both on the parameters determining the adoption process and the method used to group the time series data.

The next two sections outline the maximum likelihood estimates (MLE) and their sampling properties. These results are then used in ?4 to compare the maximum likelihood formulation with OLS estimates above for data on eight different product innovations. Finally, ?5 deals with the issue of required sample sizes for parameter estimation-particularly in forecasting the ultimate adoption level.

2. A Maximum Likelihood Approach

In his 1969 paper, Bass also states that, for eventual adopters, the likelihood of purchase at time t given that no purchase has yet been made can be written

f(t) f-(t + qF(t) (7) 1- F(t)

where f(t) is the likelihood of purchase at t and

F(t) = f(t)dt.

From equation (7) the probability density function for adoption at time t can be expressed as:

f(t) = (p + qF(t))(1 - F(t)).

Using the initial value condition that F(0) = 0, integration of the above equation yields the cumulative distribution function (c.d.f.) for eventual

60




adopters:

(1 - e-bt) F(t) ae-b

(1+ ae-bt)

where a = q/p and b = (p + q). Since this c.d.f. is clearly only appropriate for eventual adopters the probabilities associated with it are conditional probabilities. Thus in a system where the probability of eventually adopting is c, the unconditional probabilities for adoption times are given by:

c(l - e-bt) F(t) = -b . (8)

(1 + ae-b)

In a sample of size M taken from the process the expected number of eventual adopters is cM. Again letting N(t) be the cumulative number of adopters by time t, we have

E(N(t)) = cMF(t).

Differentiating,

dE(N(t)) dEt [ +

q E(N(t)) [cM - E(N(t))].

The fact that this is not equivalent to equation (1) illustrates an important distinction between a model beginning with a distribution for adoption times (equations (7)-(8)) and one using the differential equation formulation (equation (1)-(2)).2 Thus, while parameters p and q may still be interpreted as coefficients of innovation and imitation (based on their impact on the c.d.f.- and hence on expected behavior) they are not directly comparable to p and q obtained by ordinary least squares.

Equation (8) represents the (c.d.f.) of adoption time for an individual chosen at random from the population. In the applications of the model, however, the individual adoption times are not known and instead a histogram with the number of individuals falling in each time interval is used to fit the model. That is, let T equal the number of time intervals for which the data are available and xi be the number of individuals who adopt the innovation in time interval (ti_ , ti), i = 1,2, . . , T. Then typically to -0 and t - oo, with Type I censoring occurring at time tT 1. In other words, one knows that XT individuals did not adopt by time tT_ , but has no other information about their adoption times, where by definition xT = M - Ci=l xi.

2We are grateful to Abba Krieger for pointing out this fact.

61




It has been well established (see, for example, Rao 1965) that under very general regularity conditions, maximum likelihood estimates are best asymptotically normal. That is, they are consistent, asymptotically normal and asymptotically efficient. In order to determine the maximum likelihood estimates of p, q, and m, we first generate the maximum likelihood estimates of a, b, and c. Note that, due to the 1-1 correspondence between a, b, and c and p, q, and m the maximum likelihood estimators for p, q, and m are easily obtained from the maximum likelihood estimators for a, b, and c by the following expressions:

p (Ab C)(9)

A ab (10) (a + 1)()

and

mi = cM (11)

Using equation (8), the likelihood function for the observed histogram of a particular innovation can be written as:

T-I

L(a,b,c, xi) =[ - F(t-Irl) ]T J [F(ti)- F(ti-1)]xi (12) i=1

and the logarithm of the likelihood function is given by:

T-1 - - Il-e -bt, -

l(a,b,c,xi)= xi [lnc + In- bI -bt ] i=l 1 +I ae - otI 1 + ae - bt~_ ,

+xln[--c --e-bt (13) 1 + ae-btT - (13)

Note that in this model the impact of the diffusion process is felt through the model's parameters. The coefficients of innovation and imitation determine the shape of the c.d.f., with such characteristics as a small value of q implying slow expected growth, etc. Given these parameters, the likelihood function (12) takes all individuals as independent draws from that particular distribution (with given expected diffusion characteristics). Although more direct word of mouth effects might be proposed, a tractable method for incorporating them in this framework has yet to be developed.

62




Explicit formulas for parameters a, b, and c which maximize l(a, b, c, xi) do not exist; hence the MLE's are obtained by using the Hooke-Jeeves acceler- ated pattern search (Himmelblau 1972). This procedure has previously been shown to be effective in fitting duration time distributions (Morrison and Schmittlein 1980) and repeat purchase data (Kalwani and Silk 1980). The

procedure restricts the values of the estimated parameters a, b and c to reflect the logical restrictions (a > 0, b > 0, 0 < c < 1). Copies of the two BASIC

programs which estimate the parameters and calculate their sampling properties (see below) are available from the first author.

3. Sampling Properties of the Parameter Estimates

As is the case with many common distributions, small sample properties of the MLEs for the model above are extremely difficult to obtain except through simulation. (Shenton and Bowman (1977) provide a comprehensive illustration of this problem). However, one can estimate those properties using the asymptotic normality of the MLEs. To this end, the asymptotic covariance matrix for the parameters a, b, and c is derived, and then used to generate the corresponding matrix for p, q, and m. The detailed formulas are included in the appendix and only the final equations are presented here.

It is well known that, under regularity conditions, joint maximum likelihood estimates (of, say, parameters 1,, ... , Ok) tend to a multivariate normal distribution as the sample size M becomes infinite, with covariance matrix

E= v-1

where the elements of the matrix V= [vij] are the expected values

v = -E aiasj (14)

and / is the log likelihood function. For the estimated parameters a, b, and c, the asymptotic covariance matrix

is

aaa Uab aac

= Obb bc = V- (15)

where the information matrix V is

Vaa Vab Vac

V = Vbb Vbc (16) L Vcc

63




Using equations (13) and (14), the elements of the information matrix V, equation (16), can be determined. The derivations of these elements are included in the appendix. Knowing V, 2 could clearly be written out explicitly; but since the determinant and cofactors of V do not reduce appreciably such a derivation did not seem worthwhile and was not undertaken. In an actual application, it is easier to calculate V and then invert the 3 x 3 matrix to find E. Note in the appendix that the covariance matrix, 2, depends on both the true values of parameters a, b, and c, and the method of grouping and censoring the adoption data (t,, . . , tT). (See equations A3-A16.) There- fore, in addition to obtaining confidence limits for the parameters, the efficiency of different data collection and report schemes can be evaluated.

Once the covariance matrix E for a, b, and c is obtained, the corresponding variances and covariances of the MLE's pf, q, and m can be written using the elements in equation (15) as: (see Kendall and Stuart, 1977, p. 247)

b2 Obb 2b a = ( a + 14 2 ab (17)

(a + 1a (a + 1)2 (a + 1)3

-bq 2 + a b(a - 1) (18) q = (a + 1) (a + i)2b a(ab' ( 18)

aq= (a+ (a+,)2 ()1)

b2 a2 2ab

c(a + a + 1) (a +

bca^ aac b(ac + auTc (21)

oa= M2bac (22)

4. Empirical Results

To illustrate the application of the maximum likelihood estimation procedure, time series data for four consumer durables and four medical technological innovations were examined. For the consumer durables, the time series were restricted to the early years of sales growth to help avoid replacement or repeat purchases, and the data were collected from the Statistical Abstracts of the United States.

64




All the published applications of the Bass model (including our application to four consumer durables) have assumed the availability of the adoption time series data on the entire population of the potential adopters. Since statistical inference from the estimation equation (3) is not possible, the utilization of such an application strategy is understandable. However, in some situations such a data base may be unavailable or prohibitively expensive to construct. As an alternative, one might estimate the life cycle of a new product from a survey of the target market; For example, a company contemplating a new product introduction may be interested in estimating the market potential (i.e., m) and industry growth rate (i.e., q). If the adoption census data are not available, it may be necessary to estimate these parameters from a survey of the target market consisting of both adopters and nonadopters.

With this in mind, survey data for four pieces of radiological equipment (Ultrasound, CT Head-Scanner, CT Body Scanner and Mammography) were analyzed. The data were collected from 209 hospitals throughout the U.S.A., i.e., M = 209. The hospitals were asked to identify themselves as adopters/ nonadopters and, if adopters, provide the date of adoption (see Robertson and Wind (1980) for details of this survey). The survey indicated that by 1978, of the 209 hospitals, 168 (80.38%) had adopted Ultrasound, 113 hospitals (54.07%) had adopted CT Head Scanner, 97 hospitals (46.41%) had adopted CT Body Scanner and 119 hospitals (56.94%) had adopted Mammography.

4.1. Consumer Durables

The four consumer durables examined were clothes dryers, room air conditioners, color televisions and dishwashers. In order to evaluate the effective- ness of maximum likelihood estimates, ordinary least squares estimates using the differential equation model (equation (3)) were also obtained.

Table 1 summarizes the MLE and OLS estimates for the parameters p, q, and m and also provides the estimated time of peak sales, t*. (For the sake of brevity, regression coefficients a,, a2, and a3 in equation (3) are not reported. Only the estimates of p, q, and m generated via equations (4)-(6) are provided). We stress again that although MLE and OLS estimates of p and q have similar behavioral interpretations, they come from different models and are not strictly comparable. The mean absolute deviation and mean squared error are reported in Table 2, and the actual and fitted sales estimates for the four products are depicted in Figures 1-4. Some important comments on these results are warranted:

* For dishwashers, the OLS procedure yielded an inappropriate sign for the regression coefficient a3, i.e., a, = 166.1017, a2 =.0985, a3 =.6874 x 10-5 (with R2 =.90). As equation (5) indicates, a positive sign for regression coefficient (3 would give a negative sign for the parameter q. The MLE estimates, however, possessed correct signs for all products analyzed. Natu- rally, the high R 2-value supports the observation made by Heeler and Hustad (1980) that the quadratic sales response function generally fits the data well.

* For clothes dryers, room air conditioners and color T.V. Figures 1-3 and the statistics reported in Table 2 clearly indicate that the maximum likelihood

65



TABLE 1

Parameter Estimates for Consumer Durables*

Period OLS MLE Product Covered p q c m(103) t* p q c m(103) t*

Clothes 1949- .0237 .3248 .2937 15652 7.5114

Dryers 1961 .0120 .3512 .2995 15960 9.2972 (.000003) (.000087) (.00011) (5.64) (0.0023)

Room Air 1949- .0200 .3891 .3249 17314 7.2552 Conditioners 1961 .0072 .4228 .3299 17581 9.4741

(.0000013) (.000074) (.000095) (5.07) (0.0017)

Color T.V. 1963- .0443 .6295 .5519 36117 3.9388 1970 .0160 .6566 .5901 38619 5.5202

(.000002) (.000080) (.000097) (6.37) (0.00068)

Dish- 1949- washers 1961 - -- .0035 .1282 .7696 41013 27.3258

(.000005) (.00018) (.00204) (3.44) (0.0383)

*For MLEs, numbers in parentheses are standard errors. Sample size taken as number of households in the last data collection period:

# households, 1961: 53,291,000 # households, 1970: 65,444,000

-l z

z

z

-el On

S

3

L-




TABLE 2

Fit Statistics for Consumer Durables

Mean Absolute Deviation Mean Squared Error Product OLS MLE OLS MLE

Clothes Dryers 113.86 95.63 22286.14 16106.10 Room Air Conditioners 180.58 136.59 42354.81 28759.00 Color T.V. 444.06 232.00 317979.00 121496.00 Dishwashers - 33.15 - 1688.04

estimates consistently provide a better fit to the data. In addition the standard error generated by the maximum likelihood procedure indicate the general stability of the estimated parameters.

In order to obtain some sense of the predictive validity of the two estimation approaches, one-step-ahead sales forecasts were generated for clothes

dryers, room air conditioners and color televisions. The fit statistics reported in Table 3 again indicate the superiority of the maximum likelihood estimates over those generated using the differential equation model with ordinary least

squares.

1500

1400

1300

1200

1100

1000

900

~ ~800 t/ o?~-? Actual 8? // 0

_ __ ~C-----0 MLE Fitted

0f 700 ,- OLS Fitted

600

500

400 E /'

300

200

100 , Year

1949 51 53 55 57 59 61

FIGURE 1. Actual, MLE and OLS Fitted Number of Adopters for Clothes Dryers.

67




3 G_ Q Actual

---- - MtE FITTED

- OLS FITTED

Year 1949 51 53 55 57 59 61

FIGURE 2. Actual, MLE and OLS Fitted Number of Adopters for Room Air Conditioners.

7000

6500

ur

L'i w

ca n

LA. 0 Q: tY

z

LUi co

-O Actual

.---- MLE FITTED

j-__ _ OLS FITTED

Year

1963 64 65 66 67 68 69 70

FIGURE 3. Actual, MLE and OLS Fitted Number of Adopters for Color Televisions.

68

2100

1900

1700

S0 ,_ 1500

< 1300 o

i 1100

900

700

500



INNOVATION DIFFUSION MODEL OF NEW PRODUCT ACCEPTANCE 69

600 - /,

--->O Actual

500 - O-- -_ MLE Fitted

/0

, 400 -

o / X ,

200

1 00 , , , ,, Year

1949 51 53 55 57 59 61

FIGURE 4. Actual and MLE Fitted Number of Adopters for Dishwashers.

4.2 Medical Innovations

The maximum likelihood and ordinary least squares parameter estimates for the four products analyzed are summarized in Table 4, with fit statistics given in Table 5. The actual and fitted adoption estimates are plotted in Figures 5-8. Some important comments on these results follow:

* For Ultrasound and Mammography, the OLS procedure yielded a negative sign for the regression coefficient a&l. As equation (4) indicates, a negative sign for regression a& would give a negative sign for either p or m.

TABLE 3

One-Step-Ahead Forecast Performance

Forecast Mean Absolute Deviation Mean Squared Error Product Period OLS MLE OLS MLE

Clothes Dryers 1955-1960 544.62* 284.97 464299.00* 135904.00 Room Air Conditioners 1956-1960 589.29 426.47 538381.00 245072.00 Color T.V. 1969-1972 3211.23 2559.93 11,664,233 8,529,629

*The OLS procedure yielded inappropriate signs for the regression coefficients for the period 1949-1955.




TABLE 4

Parameter Estimates for Medical Technological Innovations*

Maximum Likelihood OLS** Product p q c p q c

Ultrasound .00618 .4342 .9377 (.0025 ) (.0183) (.0196)

CT Head Scanner .01252 .9875 .5886 .06962 1.19165 .53110 (.00055) (.0428) (.0391)

CT Body Scanner .00988 .9901 .5894 .05719 1.78881 .42967 (.00069) (.0689) (.0606)

Mammography .00256 .6462 .5942 - -

(.00008) (.0192) (.0357)

* For MLEs, numbers in parentheses are approximate standard errors. **OLS procedure yielded wrong parameter signs for ultrasound and mammography.

TABLE 5

Fit Statistics for Medical Technological Innovations

Mean Absolute Deviation Mean Squared Error Product OLS MLE OLS MLE

Ultrasound 2.98 15.47 CT Head Scanner 4.43 3.81 32.16 21.42 CT Body Scanner 3.80 4.65 23.21 32.34 Mammography 1.91 - 7.30

* Although the OLS procedure yielded correct parameter signs for Head Scanner and Body Scanner, the estimates of the parameter c, the ultimate penetration level, are less than the penetration levels already achieved by these two products. As mentioned earlier, the survey identified 54.07% adopters of CT Head Scanner and 46.41% adopters of CT Body Scanner. Both of these adoption levels are higher than the ultimate penetration levels suggested by the OLS procedure.

* The maximum likelihood estimates for all the four products possess correct signs and their values are very plausible. The standard errors suggest the general stability of these estimates. Furthermore, Table 5 and Figures 5-8 suggest that these estimates provide a good fit to the adoption data.

5. Sample Size Determination

To further illustrate the application of the sampling properties in ?3, required sample sizes for forecasting ultimate market penetration are calculated for the four medical innovations. Here, the parameter of interest is c, the percent market penetration (i.e., m/M). Clearly, a similar analysis could be carried out separately for p (coefficient of innovation) or q (coefficient of imitation). In fact, using the estimated covariance matrix, equation (15), the sample size needed to generate a given joint confidence region (for all three

70



INNOVATION DIFFUSION MODEL OF NEW PRODUCT ACCEPTANCE 71

Actual 0--

Fitted 0- -- o

\\

Year

1965 67 69 71 73 75 77

FIGURE 5. Actual and MLE Fitted Number of Adopters for Ultrasound.

79

------ Actual

O ---0 Fitted

Year

1972 73 74 75 76 77 78

FIGURE 6. Actual and MLE Fitted Number of Adopters for CT Head Scanners.

28

25

20

15

10

w 0-

c--

z




O--o Actual 40 -

0---- Fitted

g 30- / cc

20

/ /

/e

10 -

I /

7 6 - i ' Year

1973 74 75 76 77 78

FIGURE 7. Actual and MLE Fitted Number of Adopters for CT Body Scanners.

25

0--0 Actual

20 - - ---- Fitted

15

10

1965 67 69 71 73 75 77 79

FIGURE 8. Actual and MLE Fitted Number of Adopters for Mammography.

Year

72




parameters simultaneously) could also be determined. Nevertheless, ultimate market penetration is often a quantity of interest and may be the single most

important forecast obtained. It must be stressed that the needed sample size will depend on both the true

values of the parameters and on the time intervals chosen for data collection.

Yearly data were obtained for the medical innovations, and the periods covered were

Ultrasound: 1965-1978 CT Head Scanner: 1972-1978 CT Body Scanner: 1973-1978 Mammography: 1965-1978

With these time intervals and the parameter values in Table 4, sample sizes needed to forecast c to varying degrees of accuracy are given in Table 6. The first two columns refer to 95% confidence intervals which are within 5 and 1

percentage points of the true market penetration. The last two columns perform the same function for 99% confidence intervals.

Although no claims are made for generalizability of these numerical results, the patterns which emerge in Table 6 are still noteworthy. The nearly tenfold range in sample size needed across products indicates a great deal of sensitiv- i'y to the particular characteristics of the diffusion process being modeled. Thus, the fact that a sample size of 1200 was sufficient for estimating market penetration to within 5 percent must be taken with a note of caution.

Another interesting result concerns the difficulty of reducing the confidence interval below 5 percent. Since the standard error declines proportionately with vM, the number of observations must be increased by a factor of 25 to narrow the confidence interval from ?5% to ? 1%. For this reason, an alternative method of reducing the standard error may be considered in some studies. Generally, lengthening the period of data collection (e.g., from 1978 to 1980) or collecting more frequent observations will also increase the reliability of the parameter estimates. Repeating the calculations in ?3 for various data collection schemes (i.e., choices of tl, .. ., t) one can compare the benefits of each approach with an assessment of the expected costs.

TABLE 6

Required Sample Sizes for Estimating Market Potential

95% Confidence 99% Confidence Product ?.05 ? 01 ?.05 ?.01

Ultrasound 123 3084 213 5324 CT Head Scanner 491 12,275 847 21,186 CT Body Scanner 1179 29,485 2036 50,892 Mammography 409 10,233 706 17,662

73




Conclusions and Summary

This paper has proposed a maximum likelihood approach to estimating the parameters of an innovation diffusion model suggested by Bass. The currently utilized models leading to ordinary least squares or non-linear estimation procedures cause some difficulties in developing the diffusion model parameter estimates. As pointed out by Heeler and Hustad (1980), utilization of the regression analog of the Bass model, equation (3), provides a justification for fitting a quadratic time-series sales response function rather than "true" estimates of the model parameters p, q, and m. Furthermore, in the presence of few time-series data points and multicollinearity between the variables, these procedures may yield inappropriate signs for the diffusion parameters, and raise questions as to their usefulness for model estimation or forecasting.

In recent years, some attempts have been made to develop alternative approaches to estimate diffusion model parameters. The published examples include the Bayesian estimation approach suggested by Lilien, Rao and Kalish (1981) and the feedback approach proposed by Bretschneider and Mahajan (1980). However, no sampling distribution theory presently exists for the estimates provided by these models. In contrast, the maximum likelihood approach outlined in this paper is easy to use and can help one estimate the product life cycle of a new product from survey data. In addition to providing confidence regions for the parameters, it can also be used to assess the efficiency of different data collection and report schemes.

Another criterion for comparing the two approaches, which complements those presented in ?4, involves the pattern (over time) of deviations between the fitted and actual number of adopters. As one indicator of this pattern, the correlations between errors in successive periods were calculated and are given in Table 7. For the five products where the OLS procedure yielded appropriate parameter estimates, these autocorrelations were always higher (in absolute value) than the correlations using the MLE's. Thus the maximum likelihood fitted values are both closer (in terms of average deviations) to the actual values, and lead to errors which are, in one sense, less systematic than those from ordinary least squares.

However, the benefits of the MLE approach are not obtained without cost. As noted in ?2, once the parameters are chosen (and, hence, the expected shape of the adoption curve is set) individual adoption times are taken as independent draws from that distribution. This may be appropriate for in-

TABLE 7

Correlation Between Errors in Successive Periods

Product OLS MLE

Clothes Dryers 0.264 0.188 Room Air Conditioners 0.237 0.122 Color T.V. 0.220 - 0.078 CT Head Scanner 0.175 - 0.089 CT Body Scanner - 0.42 - 0.076

74




novators or imitators separately, but is surely only an approximation of behavior for the population as a whole. Thus, at present the choice is between a model (equation (1)) which more explicitly incorporates word of mouth effects, but whose (least squares) estimation leaves something to be desired, and a model (equation (8)) with simpler assumptions about individual behavior, but a more attractive estimation procedure. Though the data sets employed above have been limited in both number and scope, the findings support use of the latter option, especially for applications where forecasting is important, and/or when survey data are available.

Appendix

The asymptotic covariance matrix for MLE's (aC, b, c) is

Oaa ab Oac

= Obb bc = V- (A1) acc

where the information matrix V is

Vaa Vab Vac

V= Vbb Vbc (A2) Vcc

with elements

T 2 Vaa M[ E i 2Ci] (A3)

i=1 Ai Vab M 2(a + I)E - Di - 1) ], (A4)

Vac McBT[ 1+ , (A5)

N 2

Vbb = M(a + l) (a+ 1) A + FJ (A6)

c = -M(a )[ c D (A7) Vbc=-M(a+ 1) - -+ A (A7) C T\

75




and

Vcc = M + (A8)

In equations (A3)-(A8), Ai, Bi, Ci, Di, Ei, Fi, GT and HT are determined by using the following equations expressed in terms of the estimated parameters a, b, and c.

Ai = c 1- -bt c 1 - e-bti- + ae -bt 1 + ae -

(A9)

e-2bti( _ e-bj ) e -bt-i l e -b- _,

Bi = -c (AIO)

[1 + ae-b-t] [1+ ae-bt',]

e--2bti j - e -2bt,-( e- bt,-

EC=c 'e -c' (All) [1 + ae-bt"] [1 + ae-bt ']

D.=c C 'e~bt* ,-c -, i(A12) [l+ae'1 [ + ae-'1]

tie -

2bti ti- l e -

2bti_- Ei = c

- (A13) [ l+ae- bi]3 [ l+ae-bt"-1]3'

and

ti2e-t(l - ae-bt)

[1 +ae-bt]3

ti2_ le-bti-( - ae -bti-) -c . (A14)

[1+ ae-bt-']3

The above equations (A9)-(A14) can be used for i = 1,2, . .., T- 1. For i= T, however, the same equations can be used by setting the first term in

equation (A9) equal to one, and omitting the first term in equations (A10)-

76




(A14) for BT through FT. Finally,

tT le-bt-'(I - e- bt- I Gr= l-c - -T 3) (A15)

[1 + ae-b-']

and

1- ebtT-I T 1 -+ aebtT (A16)

References

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78 DAVID C. SCHMITTLEIN AND VIJAY MAHAJAN

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Article Contentsp. 57p. 58p. 59p. 60p. 61p. 62p. 63p. 64p. 65p. 66p. 67p. 68p. 69p. 70p. 71p. 72p. 73p. 74p. 75p. 76p. 77p. 78

Issue Table of ContentsMarketing Science, Vol. 1, No. 1 (Winter, 1982), pp. 1-122Front MatterLaunching Marketing ScienceNews: A Decision-Oriented Model for New Product Analysis and Forecasting [pp. 1 - 29]A Marketing Decision Support System for Retailers [pp. 31 - 56]Maximum Likelihood Estimation for an Innovation Diffusion Model of New Product Acceptance [pp. 57 - 78]The Marketing Mix Decision under Uncertainty [pp. 79 - 92]A Descriptive Model of Consumer Information Search Behavior [pp. 93 - 121]Back Matter [pp. 122 - 122]

schmittlein, d. c., & mahajan, v. (1982). maximum likelihood estimation for an innovation diffusion...

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