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"COMBINING RELATED AND SPARSE DATA INLINEAR REGRESSION MODELS"
byWilfried VANHONACKER*
Donald LEHMANN**and
Fareena SULTAN***
N° 89 / 16
Associate Professor of Marketing, INSEAD, Boulevard de Constance,77305 Fontainebleau, France
** George E. Warren Professor of Business, Graduate School of Business,Columbia University, New York, NY10027
*** Assistant Professor, Graduate School of Business Admininstration,Harvard University, Cambridge, MA02163
Director of Publication:
Charles WYPLOSZ, Associate Deanfor Research and Development
Printed at INSEAD,Fontainebleau, France
COMBINING RELATED AND SPARSE DATA IN LINEARREGRESSION MODELS
Wilfried R. Vanhonacker*Donald R. Lehmann**
andFareena Sultan***
Revised February 1989
* Associate Professor of Marketing, INSEAD, Boulevard de Constance,77305 Fontainebleau, France
** George E. Warren Professor of Business, Graduate School of Business,Columbia University, New York, NY10027
*** Assistant Professor, Graduate School of Business Administration, HarvardUniversity, Cambridge, MA02163
We would like to thank Gerry Tellis who was kind enough to share hisextensive data base on price elasticity estimates.
-2-
COMBINING RELATED AND SPARSE DATA IN LINEAR
REGRESSION MODELS
ABSTRACT
Meta-analysis has become a popular approach for studying
systematic variation in parameter estimates across studies. This
paper discusses the use of meta-analysis results as prior
information in a new study. In contrast to hierarchical prior
distributions in a traditional Bayesian framework which are
characterized by complete exchangeability, meta-analysis priors
explicitly incorporate heterogeneity in prior vectors. This paper
discusses the nature of the meta-analysis priors, their properties,
and shows how they can be integrated into a familiar recursive
estimation framework as to enhance the efficiency of parameter
estimates in linear regression models. This approach has the added
advantage that it can provide such estimates when (a) the design or
data matrix is not of full rank, or (b) when observations are too
few to allow independent estimation. The methodology is
illustrated using published and new meta-analysis results in market
response and diffusion of innovation models.
Keywords: META-ANALYSIS; PRIOR INFORMATION; RECURSIVE ESTIMATION;
RANDOM COEFFICIENT REGRESSION; EXCHANGEABILITY.
- 3
1. INTRODUCTION
Meta-analysis has been promoted in a variety of fields as a way to
integrate research findings across various studies (Glass, McGaw, and Smith
1981, Hunter, Schmidt, and Jackson 1982, Peterson, Albaum, and Beltramini 1985,
Farley and Lehmann 1986, Hite and Fraser 1988, Sultan, Farley, and Lehmann
1988). Within an analysis-of-variance framework, the parameter estimates of
the various studies are viewed as imperfect replications of one overall but
unplanned experiment. This replication framework generalizes and enhances the
understanding of systematic variations in research findings based on design
characteristics. Some applications have appeared in the literature in recent
years and a number of interesting generalizations have been established and
discussed (see, e.g., Assmus, Farley, and Lehmann 1984, Farley, Lehmann, and
Ryan 1981, 1982, Houston, Peter, and Sawyer 1983, Reilly and Conover 1983).
Meta-analysis findings, however, have never been incorporated as a source
of prior knowledge or information in subsequent studies. Given the design
characteristics of a new study, it should be rather simple to derive some prior
parameter estimates using meta-analysis results. When prior knowledge is
updated with sample information on the new situation, resulting parameter
estimates tend to be more stable and efficient (see, e.g., Lilien, Rao, and
Kalish 1981); hence, an attempt should be made to include prior information in
the estimation procedure.
Gain in efficiency from the use of prior information has been the reason
for extensive use and development of Bayesian statistical methods. The work by
Lindley and Smith (1972) which advocates the use of hierarchical prior
distributions has become the foundation for numerous applications. Early
research on integrating independently generated research findings into priors
adopted this approach (DuMouchel and Harris 1983, Kuczera 1983). However, the
strong prior assumption of exchangeability underlying hierarchical prior
distributions has been criticized as an oversimplification (Kass 1983). The
assumption, usually captured in a normal prior (Learner 1978, p.49), states that
the research findings are related and that the data carry information about
that relationship. Because of incompatibility of research studies,
heterogeneity is almost certain to enter that relationship. Because prior
information on heterogeneity is often sparce, neither grouping nor the use of
conditional priors has been practical, and the traditional approach with its
strict assumption has survived valid criticisms.
- 4 -
In attempting to uncover systematic variations, meta-analysis results
explicitly recognize parameter heterogeneity across independently executed
studies. Hence, using the results as prior information circumvents the strict
exchangeability assumption underlying hierarchical Bayes priors and addresses
the valid concern often raised in empirical research relying on such priors.
Using a known recursive estimator in a linear regression framework, this paper
discusses how these priors can be integrated with sample information. Besides
the established gain in parameter efficiency, the approach has the added
advantage that it can provide updated parameter estimates in instances where
sample information is missing or insufficient to allow for independent
estimation. The reported empirical illustrations are very encouraging in that
respect.
2. DERIVATION OF META-ANALYSIS PRIORS IN A MULTIPLE REGRESSION FRAMEWORK
Published meta-analysis focus on individual parameters analyzed
independently using ANOVA. As the parameters in each of these studies are not
estimated independently, this approach to recovering systematic variance in
them has ignored their covariance structures. Casting meta-analysis within a
multivariate regression framework allows the incorporation of those covariances
which, in a familiar Generalized Least Squares (GLS) framework, enhances the
efficiency of the results and sharpens the statistical power of the
significance tests. Although conceptually not different from an ANOVA
approach, the regression approach is adopted here as efficient priors are
naturally preferred over inefficient ones (i.e., inefficient priors are
essentially "non-informative"). Furthermore, this approach simplifies
comparison to the traditional hierarchical Bayes method which it improves upon
given partial exchangeability. To set the tone for subsequent discussion and
pinpoint inherent limitations of the suggested approach, the meta-analysis
priors are derived here within a rigorous multiple regression framework.
Meta-analysis attempts to capture and explain systematic variations in
parameter estimates across various studies. Hence, a set of original studies
are available for which linear regression models can be specified as
y. X. 0. + u ifor i 1, 2, ... m (1)
5 -
where y i is a (n x 1) vector containing observations on the dependent variable
in study i, X i is a (n x p) matrix containing observations on p non-stochastic
predictors in study i, 01.is a (p x 1) vector containing the parameters, u i is
a (n x 1) disturbance vector, and m denotes the number of previous studies.
With respect to the disturbance terms, the assumptions are made that
E(u i ) = 0
2
and E(u. 1.C.) = a. I1 j 1
= o
for all i
for i = j
otherwise.
(2)
In the current development of the methodology described hereafter, the
assumption of linearity in (1) is fundamental and represents a limitation.1
Moreover, to the extent that prior studies cannot be cast in models which are
linear in parameters, the approach cannot be adopted. However, the methodology
developed subsequently is not confined to cases where disturbances are
characterized by scalar variance-covariance matrices as defined in (2). We
confined ourselves to this case here for simplicity.
The fundamental assumption of meta-analysis is that systematic variation
in the parameter vectors O i can be related to study characteristics. For
example, systematic differences might be attributable to differences in product
category, industry, sample size, estimation method, etc. These study
characteristics define a multidimensional classification within which similar
studies can be grouped together. Within each cell, 1, of the classification,
the random coefficients assumption
0. =+ e.1
for all j cl5 ki (3)
holds with first and second moments
E(e.). 0 for all jEki , 1 1, 2, ... k,
and E(e. = E11 3 if i j and i, j Ek1
= 0 otherwise.
- 6 -
In these expressions, k 1 denotes the set of studies in group 1 of k groups. In
hierarchical Bayes priors (see, e.g., Kuczera 1983), the parameter vectors.01
for all i would be assumed to be drawn from a prior distribution (most likely
normal) with a mean vector and covariance matrix. Accordingly, the O i 's are
exchangeable and this prior distribution captures the "connectedness" between
all studies (DuMouchel and Harris 1983). In other words, each parameter vector
of a previous study contributes information about any single parameter in the
basic linear model. This strict assumption can be tested (see, e.g., Kuczera
1983), and lead to grouping of studies which could be analyzed separately.2
Capturing and describing the heterogeneity explicitly as achieved in meta-
analysis enables simultaneous analysis which enhances efficiency and
understanding.
The parameter vectors of the various studies can be combined as
a=s + e
where 0' . (0', 0'2 ... R' 0' (0; ... -01'( ) and e' = (el e' )
assuming studies similar in terms of design characteristics (i.e., partially
exchangeable) are grouped together. Note that vector has m vector entries,
one for each original study. The vector entries corresponding to similar
studies are identical 0i vectors. In other words, if the first three studies
belong to group one, then S = (01 01 •••)•
Of specific interest in meta-analysis are the variations in the mean
vectors 01 for 1 = 1, 2, ...k. Systematic variations arising from design
characteristics imply that 0 = Zc and, hence,
0 Zc e (4)
That is, the mean parameter vectors can be expressed as a linear combination,
c, of specific design vectors which constitute the columns of matrix Z. Within
the multidimensional classification scheme adopted above, the design vectors in
- 7 -
matrix Z describe or explain group membership of the m original studies in the
k groups. These vectors could contain observations on continuous, categorical,
or dummy variables. Note that expression (4) not only describes the heteroge-
neity among the previous studies but at the same time captures the systematic
link among them.
In practice, the parameter vectors are not observed. Traditional meta-
analysis relies on parameter estimates in attempting to estimate vector c.
Hence, the basic model postulated is
0 = Zc + E (5)
. - .whereW-(00',...0')with.containing unbiased least squares estimates
mS.
of the parameters in study i, and c = e + u with u = 0 - 0 or the difference
between the estimated parameters and their true values (i.e., ui =
i(X!X.) X'.ui
i i
given (2)).
Given the assumptions about the random components in (2), (4), and (5), it
is evident that
E(c) = 0
2 _1E, + a, (V,X,) 0
and
E (cc') = 2 = . (6)
20 Z
k + a (X'X )-
m m m
Invoking Generalized Least Squares (GLS) principles given the non-scalar
variance-covariance matrix in (6), BLUE estimates of vector c in (4) can be
obtained which equal 3
c = (Z'S2Z)Z'S2-i^0
(7)
-8 -
Stated simply,
_1 1c = DO with D = (Z'S2 Z) Z'S2
(8)
In a new study, a regression model similar to the ones specified in (1)
can be postulated. Specifically, we have
yo X 0 + u
oo o
with E(uo
) 0 and E(u u') a2
0 I. 4
o oGiven the meta-analysis results in (7),
a prior estimate of 00 can be obtained from
bo Zc
0
where Z defines the design of the current study in terms of theo
characteristics incorporated in the design matrix Z of the meta-analysis.
As shown in Appendix 1, the prior estimate bo can be expressed as
b = Z DMO + Z Da Dy0 0 0 0 0
(9)
_ -where y e + u with D, M, and d as defined in Appendix 1. This is the meta-
analysis prior which will be updated with sample information in the new study
using a familiar recursive estimator. The use of this estimator fits nicely
into the advocated regression approach and, as will be shown shortly, makes the
approach applicable when data in the new study are either incomplete or
insufficient for independent estimation.
3. RECURSIVE ESTIMATION
Updating the prior on 00 in (9) with sample information contained in y0
and Xo can be accomplished using the augmented system
uo
(10)
ZoDy
- 9 -
YoXo
ZDM0
b- Z Dd0 0■■•■••■• .■
When the prior information is with respect to a subset of the parameters in 00
(e.g., 001 ), the augmented system equals
r . ....111.
yo
bol
- Zol
D1d1
Xol
Zol
D1M
Sol +
Xo2
0
°o2
uo
Zol
D1y1
where bol
denotes the meta-analysis prior on Sol'
and all other elements are
similar to the corresponding ones in (10) with 6:3 = [04; 1 04; 2 1 and X0=[X01 X021
(where X01 contains the columns in Xo corresponding to the parameters in 13°1).
The system could be expanded further if independent prior information (e.g.,
separate meta-analysis results) was available for different subsets of 00.
This approach is similar to the Goldberger-Theil approach (Theil 1971)
with the basic difference that (a) the prior values for 00 , bo , are adjusted
downward by Z oDd and (b) the matrix associated with vector 0 0 in the prior
equation (i.e. ZoDM in (10)) is different from an identity matrix. It is worth
noting that
bo - va = Zo ic - Dal
and, hence,
b .:, - zopa - zoc _ Zo (Z'2 1 Z) 1 Z'S2 I a.
Accordingly, Dd can be interpreted as a least squares estimate of the design
matrix Z on the differences in group means. If no groups were identified
- 10 -
(i.e., complete exchangeability), the prior would equal Z oc. Moreover, ZoDd is
subtracted from this prior to adjust for the partial exchangeability captured
in the profiles contained in design matrix Z.
Expressing (10) compactly, we have:
where
and
y - X0o + v
y' = [y' (b - Z Dd)']o o o
X' = [X,') (Z0DM)']
v' = [ u ' (ZoDy)']•
With the variance-covariance matrix of v, Q, as derived in Appendix 2 thev
recursive estimator of 00 equals
1 _1 _I
0 = (X'Q X) X'S-2 y
v v
which has BLUE properties.
A couple of interesting observations can be made with respect to the
estimator shown in (12). The development thus far assumed that all information
for the new study was available. This is not always the case in practice. For
example, data on some predictors contained in X could be missing. Hence, the0
data matrix Xo would have columns containing zeros and a direct estimate of 0 o
could not be obtained.5 Note, however, that the augmented matrix X in (11) is
still of full rank; hence, a recursive estimate of all parameters in 00 is
obtainable despite the missing data. This is essentially an approach to data
transferability (Aigner and Leamer 1984) which, using completely exchangeable
priors, has been studied in Vanhonacker and Price (1988).
(12)
Recursive estimates can also be obtained when the number of observations
in yo and X0 are smaller than the number of predictors specified in the basic
model (i.e., the number of columns in X 0 ). Again, direct least squares
estimates of 0o could not be obtained since X' X would not be of full rank.
o o
The augmented system in (10), however, does provide estimates for 0o in this
case with degrees of freedom equal to the number of observations actually
available on the dependent and independent variables.
4. SOME ESTIMATION ISSUES
2
The above discussion assumed that E. (for i = 1, 2, ..., k), a. (for i =
0, 1, 2, ..., m), and d are known. In practice, these terms will have to be
2
estimated. a can be estimated as the average of the variances for theo
corresponding studies in the meta-analysis,
2 m 2
a o = (1/ml ) .2 a.
1 1=1 1
2
wherea.---[(y.-X..)'(y.-X4.)(n-Mand.contains the parameter131
estimates of the iILI
study, and m1 denotes the number of previous studies
belonging to the first set (to which the new study belongs).
The matrices E. for i = 1, 2, ..., k can be estimated as the average of
the covariance matrices for the corresponding studies in the meta-analysis,
m.
E i.--(1/m) .E1 i(0. - f3)(0. - 13.)'3 i
m.Si iwhe re0---.3
E;(0./m)withm.denoting the number of studies contained in thei
.th1-- set. Note that assuming normality and invoking the Lindley and Smith
(13)
(14)
- 12 -
(1972) results, the estimates in (13) and (14) are related to the empirical
Bayes estimates using non-informative priors (see Aigner and Learner 1984).
2Estimatesfora.(i = 1, 2, ...m) can be obtained from the estimation
results of the previous studies. Specifically, the variance-covariance
matrices of the parameter estimates from the previous studies are estimates for
2 -1 (ia. (V.X.) (1 = 1, 2, ...m) as incorporated in matrix 4, (see Appendix 2). The
difference between the mean parameter vectors contained in a can be derived
mistraightforwardly from the averaged estimates 6 i = J E, (13i /mi ) for i=1,2,...k.
5. EMPIRICAL ILLUSTRATIONS
Market Response Models
Two different empirical illustrations are discussed. The first example
relies on two widely cited meta-analysis in the market response literature.
Both focus on a single but different parameter and were executed independently.
The second example relies on the meta-analysis of both parameters of the
linear, discrete analog of a popular model in the diffusion literature.
The market response literature contains two meta-analyses: one on
advertising elasticities (Assmus, Farley, and Lehmann 1984) and one on price
elasticities (Tellis 1988). Accurate estimation of these parameters is
mandatory for optimal (e.g., profit maximizing) advertising spending and
pricing decisions. Elasticities are most often estimated using multiplicative
model specifications. Given that these can be linearized in parameters, the
procedure developed above can be implemented to enhance parameter efficiency
and stability in case of limited samples.
In this illustration, time-series observations for a major brand in a low-
priced consumer product category were used together with the published meta-
analyses. Bimonthly observations were available on market share (the response
variable), advertising share, and relative price. Relative price was defined
as the brands price over the mean price of the three major competitors in the
category. These three competitors together account for over two-thirds of the
volume sold in each time period.
- 13 -
A multiplicative response model was specified as
In MSt = a + 0 In AS
t + y In RP
t + S In MS
t-1 + u
t
where MSt denotes market share at time t, AS
t denotes advertising share at time
t, and RPt denotes relative price at time t. The parameters 0 and y denote,
respectively, the short-term advertising and price elasticities. Using the
meta-analysis prior on 0 from Assmus, Farley and Lehmann (1984) and the meta-
analysis prior on y from Tellis (1988), these parameters were estimated
recursively using five, six, seven, etc. sample observations. Before reporting
and discussing the results, a number of practical observations on the
implementation of the recursive estimator are in order.
Empirical studies seldom report standard errors of estimated parameters
(or exact t-values which would enable derivation of standard errors). The
data-base underlying Assmus, Farley, and Lehmann (1984) does not contain any
standard errors for the reported advertising elasticities. Tellis (1988)'s
data base contains some t-values but not for all price elasticities. Given
that published results are generally statistically significant, the standard
errors were derived assuming the corresponding t-values equaled three. With
this assumption, the entries for the matrices Q in (6) and T in (2-1) could be
derived. This assumption affected the weighting of the prior studies
incorporated in the meta-analysis data bases as well as the efficiency of the
derived parameter estimates.
Although the original data were used (with the original coding schemes),
the designs of both meta-analyses were reduced. This was done mainly to ensure
that the new study had a design profile which was matched in the data by at
least one study. Note that if that study belonged to an empty set, d in (9)
cannot be computed and, hence, no adjustment can be carried out as shown in
(10). The design reduction was done based on statistical significance. The
Assmus, Farley, and Lehmann (1984) design matrix Z was reduced from 26
variables to the 11 most significant variables (including the intercept). The
Tellis (1988) design matrix Z was reduced from 21 variables to 10 variables
(including the intercept). The derived Po - Zo Dd ] estimates equaled 0.0325
for the advertising elasticity and -2.3360 for the price elasticity.
It is important to note that the design profile of the new study matched
two studies out of 128 in the advertising-elasticity data base but only one out
of 421 in the price-elasticity data base. This limited representation has a
dual effect: first, the information contained in that subgroup will have
Table 1
ELASTICITY ESTIMATES FOR MARKET SHARE RESPONSE MODEL a
Number of Parameter Estimates (Estimated Variance)b
Based on
Observations Sample observations Meta—Analysis Prior on Single Meta — Analysis Prior on BothParameter Parameters
Advertising Price Advertising Price Advertising Price
5 0.2310(0.00) 1.7210(7.48) 0.0727(0.01) -2.3355(0.00) 0.0707(0.01) —2.3356(0.01)6 0.1530(0.01) -3.4815(12.8) 0.0633(0.00) -2.3363(0.00) 0.0639(0.00) —2.3359(0.01)7 0.1521(0.01) -3.4374(5.39) 0.0677(0.00) -2.3365(0.00) 0.0660(0.00) —2.3362(0.00)8 0.1544(0.00) -3.4914(4.67) 0.0684(0.00) —2.3365(0.00) 0.0672(0.00) —2.3363(0.00)9 0.1549(0.00) -4.0184(4.70) 0.0689(0.00) -2.3367(0.00) 0.0662(0.00) —2.3365(0.00)10 0.1520(0.00) —4.1684(4.05) 0.0679(0.00) -2.3368(0.00) 0.0650(0.00) —2.3366(0.00)11 0.0950(0.01) —3.5544(6.43) 0.0531(0.00) -2.3366(0.00) 0.0511(0.00) -2.3364(0.00)12 0.0462(0.01) —1.6317(10.7) 0.0372(0.00) -2.3357(0.01) 0.0383(0.00) -2.3356(0.01)13 0.0457(0.00) —3.5097(11.0) 0.0370(0.00) -2.3367(0.01) 0.0351(0.00) -2.3367(0.01)14 0.0554(0.01) —3.5970(9.87) 0.0408(0.00) -2.3367(0.01) 0.0390(0.00) -2.3367(0.01)15 0.0635(0.01) -4.2443(7.81) 0.0439(0.00) -2.3373(0.01) 0.0422(0.00) -2.3373(0.01)16 0.0582(0.01) -1.3461)5.05) 0.0420(0.00) -2.3348(0.01) 0.0431(0.00) -2.3348(0.01)17 0.0645(0.01) —1.0044(4.21) 0.0445(0.00) -2.3342(0.01) 0.0475(0.00) -2.3341(0.01)18 0.0548(6.00) —1.0553(3.78) 0.0452)0.00) -2.3342(0.01) 0.0558(0.00) -2.3340(0.01)19 0.0566(0.00) —1.0573(3.53) 0.0475(0.00) -2.3342(0.01) 0.0592(0.00) -2.3339(0.01)20 0.0610(0.00) -1.2296(3.26) 0.0504(0.00) -2.3344(0.00) 0.0605(0.00) -2.3341(0.00)
30 0.0739(0.00) -0.9645(1.45) 0.0640(0.00) -2.3320(0.00) 0.0642(0.00) -2.3320(0.00)
40 0.0937(0.00) -1.0894(0.99) 0.0806(0.00) -2.3294(0.01) 0.0832(0.00) -2.3293(0.01)
53 0.0493(0.00) 0.2452(0.55) 0.0473(0.00) -2.3083(0.01) 0.0563(0.00) -2.3112(0.01)
aMultiplicative response model with predictors advertising share, relative price, and lagged market share
bMete—analysis priors derived from Assmus, Farley, and Lehmann (1984) and/or Tellis (1988)
01.9 estimates
- 15 -
limited weight in the meta-analysis; second, as the adjustment of b o towards
the subgroup estimate is entirely a function of its members, if these are not
representative an inadequate adjustment can arise. These issues will have to
be kept in mind when evaluating the results, particularly for the price
response parameter.
The results are summarised in Table 1. Adding a single observation at a
time, the table shows the sample OLS estimates (no prior information), the
estimates using prior information on one of the two parameters, and the
estimates incorporating prior information on both parameters simultaneously.
The advertising parameter is very significant in the model and its estimate
gets adjusted very well in the direction of the "long-run" estimate (i.e., the
highly significant estimate over the entire sample observation period). For
example, the sample estimate with only five observations equals 0.2310;
incorporating prior information the estimate becomes, respectively, 0.0727 and
0.0707 which is much closer to the "long-run" estimate of 0.0493 and the
apparent equilibrium level between 0.05 and 0.06. For price, the sample
estimate is insignificant. Incorporating the corrected prior value of -2.336
draws the recursive estimate close to that value and makes it insensitive to
sample information. This is statistically reasonable; however, as the prior is
adjusted towards a single study with a highly significant price elasticity
(i.e., a reported t-value of -8.33), the significance of the recursive estimate
is intuitively unreasonable given limited representation. This raises an issue
worth further investigation given the otherwise very encouraging results.
New Product Diffusion: The Bass Model
In the modeling literature on the diffusion of innovations over time, the
Bass model has received considerable attention (Bass 1969).6 The basic premise
of the model can be stated as
P(t) = p + (q/m)Y(t) (15)
where P(t) denotes the probability of adoption at time t, Y(t) denotes the
cumulative number of adopters by time t, and p, q, and m are the three model
parameters. Accordingly, the probability of adoption at one point in time is
specified as a linear function in the number of adopters by that time. Hence,
the model describes the adoption process of an innovation as a contagion-type
process.
- 16 -
Following the algebra in the original paper by Bass (1969), it can be
shown that2
S(t) = a + bY(t) + c[Y(t)] (16)
where S(t) denotes the number of adopters at time t, and the parameters a, b,
and c are functions of the three original parameters as follows: a = pm, b = q-
p, and c = (-q/m). In discrete terms, model (16) is expressed as
St = a + bY
t-1 + cY
t-1 (17)
where t refers to a discrete time period (e.g., a month, quarter, year, etc.).
This quadratic equation was suggested by Bass (1969) as the basic model for
empirical analysis.
Since the original paper by Bass (1969), substantial research on the
specification and estimation of the model has been done. The specification has
been extended to incorporate explanatory variables such as price, advertising,
etc. (see, e.g., Mahajan and Peterson 1985). OLS estimation of the parameters
of model (17) has been shown to have a time aggregation bias (Srinivasan and
Mason 1986). Maximum likelihood estimation (Schmittlein and Mahajan 1982) as
well as Nonlinear Least Squares (NLS) of the continuous time Bass model
specification have been shown to be preferable over OLS with NLS being the
preferred estimator in terms of fit and predictive validity (Mahajan, Mason and
Srinivasan 1986, Srinivasan and Mason 1986). The linearity-in-parameters
constraint of the methodology suggested above confines us to the OLS estimation
of a, b, and c in the discrete analog model (17). Despite the shortcomings of
OLS, the estimates are easier to obtain and provide good starting values for
NLS. Within these constraints, a number of product/process innovations were
analyzed empirically.
A meta-analysis of Bass diffusion model estimates is contained in Sultan,
Farley, and Lehmann (1988). However, that study is very much in the tradition
of Assmus, Farley, and Lehmann (1984) and Tellis (1988) and is confined to an
independent analysis of individual parameters. In order to illustrate the full
power of the regression framework adopted above (which allows simultaneous
analysis of all parameters), a small meta-analysis was performed on the Bass
model results in Table 2.8 For simplicity of the illustration and in order to
preserve enough observations for the estimation of the variance-covariance
matrices in (14), only a single dummy predictor was specified in (5). In other
- 17 -
words, the various innovations listed in Table 2 were broken down into 2
groups. The variable was whether or not the innovation was a consumer durable
product innovation. A distinction was made between consumer and non-consumer
(or industrial) durables because clear differences exist in their diffusion
process characteristics. The absolute magnitudes of the potential adopter
populations are very different with consumer durables generally having much
larger population figures than non-consumer durables. This results in striking
differences in the values of, for example, parameter "a" in model (17) which
captures the number of adopters in the initial period of market introduction.
As indicated in the derivation of expression (7), the covariances of the
estimated parameters should be incorporated in order to secure efficient
estimation of the meta-analysis parameters (i.e., vector c in (5)). However,
these covariances are not reported in the studies whose results are shown in
Table 2. Only the variances or standard deviations of the parameter estimates
are given. Accordingly, vector c in expression (5) was estimated with
variance-covariance matrices in which all off-diagonal elements were zero.
The intercept estimates for parameters a, b, and c (representing the
coefficients for an industrial product) were equal to, respectively, 8.52905,
0.56810, and -0.0049512; the additive effect for a consumer durable innovation
equaled, respectively, 641,196.0, -0.25537, and 0.0049511. These additive
effects essentially indicate that (as discussed above) the potential adopter
populations for consumer durables are much larger than those for industrial
innovations, and consumer durable innovations diffuse slower on their own than
2industrial innovations. The R equaled 0.36. Given cross-sectional observa-
tions and a single predictor, this fit is impressive.
The methodology incorporating the meta-analysis results discussed above
was applied to color televisions. The sales data were obtained from Wind and
Mahajan (1985, p.215) and consist of annual observations from 1963 until 1970.
Some observations are in order with respect to the product and the data. The
Bass model has been fitted quite successfully to the color television data
(Bass 1969, Nevers 1972). Predictions of peak sales (which occur within the
period of observation) and the magnitude of the peak have been quite accurate.
Comparison of the results derived here with the published independent Bass
model predictions will constitute a relatively strong test of the power of the
recursive estimator.
Table 2
Bass Model Regression Results for Consumer/Industrial Innovation
PRODUCT/TECHNOLOGY Time
Period
4 of ob-
servations
b2
3 -7Electric Refrigerators '20-'40 21 104.67(10 ) 0.21305 -0.053913)10 ) 0.903
3 -7Home Freezers '46-'61 16 308.12(10 ) 0.15298 -0.077868(10 ) 0.742
3 -7Black end White TV's '46-'61 16 2,696.20(10 ) 0.23317 -0.025957(10 ) 0.576
3 -7Room Aircondition•rs '46-'61 16 175.69(10 ) 0.40820 -0.247770(10 ) 0.911
3 -7Clothers Dryers '48-'61 14 259.67(10 ) 0.33968 -0.236470(10 ) 0.896
3 -7Power Lawnmowers '48-'61 14 410.98(10 ) 0.32871 -0.075506(10 ) 0.932
3 -7Electr. Bed Coverings '49-'61 13 450.04(10 ) 0.23800 -0.031842(10 ) 0.976
3 -7
Automatic Coffee Makers '48-'61 14 1,008.20(10 ) 0.28435 -0.051242)10 ) 0.883
3 -7Steam Irons '49-'60 12 1,594.70(10 ) 0.29928 -0.058875(10 ) 0.828
3 -7Record Players '52-'61 10 543.94(10 ) 0.62931 -0.298170(10 ) 0.899
Holiday Inn Motels '54-'68 15 9.27866 0.35521 -0.000258 0.918
Howard Johnson '56-'69 11 10.64810 0.23687 -0.000502 0.759
Motor Lodges
Ramada Inns '62-'69 a 14.29150 0.14904 0.004850 0.746
Howard Johnson, '56-'69 11 24.58870 0.32699 -0.000150 0.930
Holiday Inn,
Ramada Inns
McDonald's Restaurants '55-'65 11 15.00440 0.52016 -0.000646 0.964
Rapid Bleach Proc. '59-'68 10 1.86071 0.54948 -0.014050 0.919
Conversion to 70% H 0 '627'68 7 5.26971 1.02542 -0.006090 0.9742 2
Deliv. Syst.
Continuous Bleach '44-'51 8 2.63906 0.90471 -0.014600 0.827
Ranges
Hybrid Corn '33-"41 9 3.51145 0.96654 -0.011107 0.926
4 Published results from Bess (1969) and Revers (1972)
CO
- 19 -
The recursive estimation results and their implied estimates for the
parameters of model (15) are shown in Table 3.9 Three sets of results are
shown: one set of prior estimates derived from the meta-analysis estimates, one
set derived from the priors and the first annual sales observations for color
televisions, and one set derived from the priors and the first two annual sales
observations. Estimation results with more data points converged with least
squares estimates based exclusively on color television sales observations
published in Bass (1969) and Nevers (1972). This rapid convergence indicates
that the sample observations contain much more information than the prior
information. This is intuitively reasonable since color televisions, relative
to the innovations listed in Table 2, are more an improvement of an existing
product (i.e., black/white television) than a discontinous innovation.
Accordingly, one might expect a faster rate of diffusion which is the pattern
observed in the recursive estimates in Table 3 as sample observations are
added.
The corresponding unit sales forecasts are shown in Table 4. The pattern
of rapid diffusion is evident when comparing actual sales to the predictions
based on the prior estimates. Adding the first year sales figure adjusts the
forecasts in the right direction but there is still a significant
underprediction of actual sales. When the second year is added, the pattern is
adjusted further. A reasonable forecast is obtained with a slight
overprediction in the post sample years. This early 1965 forecast predicts
sales to peak in 1967 below the 7 million unit point. This forecast is more
acurate and much more reasonable than the industry expectations at that time
10(and even later; see Bass 1969, p.224). With a better-fitting meta-
analysis model than the one relied on in this illustration, the early forecast
is likely to improve further.
Although limited in scope and power, the empirical results are encouraging
and justify further rigorous testing of the suggested methodology.
- 20 -
Table 3
Estimation Results for Color Televisions
Parameter
Recursive Estimation
Meta-Analysis
Prior
Number of Observations Incorporated
1 2
a(103 )
b
7c(10
m(106
)
P
q
T*a
S(T*)(106
)b
)
641.205
0.31273
-0.10810
30.852
0.021
0.334
7.834
2.903
747.000
0.41389
-0.13918
31.445
0.024
0.438
6.315
3.824
747.000
1.01309
-0.42612
24.489
0.031
1.044
3.289
6.579
a Time of peak sales, or T* . (1/(p+q)7[1n(q/p)] (Bass 1969).
bMagnitude of peak sales, or S(T*) . [m(p+q)
21/4q (Bass 1969).
- 21 -
Table 4
Forecasts for Color Televisions
Forecast based on
Recursive Estimation
Actual
Number of Observations
Meta-Analysis Incorporated
Prior
1 2
Time 1968 1972 1970 1967
Peak Sales
Magnitude 5.982 2.903 3.824 6.579
(106
)
Sales (106
) 1963 0.747 0.641 0.747 0.747
1964 1.480 0.837 1.048 1.480
1965 2.646 1.077 1.445 2.792
1966 5.118 1.370 1.942 4.758
1967 5.777 1.702 2.518 6.579
1968 5.982 2.059 3.109 5.915
- 22 -
6. SUMMARY AND CONCLUSION
Meta-analysis has been postulated in a variety of fields as a way to
integrate research findings across studies. Within an analysis-of-variance
framework, the parameter estimates of the various studies are viewed as
imperfect replications of one overall but unplanned experiment. This
replication framework generalizes and enhances the understanding of systematic
variations in research findings.
Meta-analysis results have never been used, however, as a source of prior
information in subsequent studies. Within the design underlying the meta-
analysis, it is rather straightforward to obtain prior estimates for the
parameters of a new study. This paper described a methodology to update this
prior information with sample information for the new study. The methodology
relies on a random coefficients regression framework and accomplishes the
updating task using a recursive estimator. The procedure is versatile and
allows for efficient estimation even when independent estimation is infeasible
because the data are either incomplete or too sparse. Empirical examples in
the market response modeling literature and in the diffusion modeling
literature are encouraging.
- 23 -
FOOTNOTES
1. Linearity is an implicit assumption in ANOVA analyses.
2. Prior grouping is, of course, also possible but general lack of priorinformation on group membership makes this impractical.
3. The result shown in (7) suggests that the most efficient estimator of c inmodel (5) is the GLS estimator shown in (7).
4. Whether or not the similarity in model specification is justified dependson the problem studied and is largely determined by theoretical issues.Note, however, that if there is little similarity or analogy betweenprevious studies (incorporated in the meta-analysis) and the new study,the meta-analysis prior will be a non-informative prior.
5. Replacing missing values with zeros is the traditional approach in datatransferability (see e.g., Aigner and Learner 1984). Other approaches havebeen suggested in the literature on missing data (see, e.g., Dunsmuir andRobinson 1981, Gourieroux and Monfort 1981, Steward and Sorenson 1981).These approaches rely on likelihood functions and are often impractical.Working with zeros enables recursive estimation as a simple, practical,and conceptually appealing method to the problem investigated.
6. For a comprehensive review of this research area, see Mahajan and Peterson(1985).
7. At the aggregate level, parameter p in (15) denotes that fraction of thepopulation who adopt at a particular time whose decision to adopt was notinfluenced by the social environment (hence, the labeling of p as thecoefficient of "innovation"). Parameter q in (15) is labeled thecoefficient of "imitation" as it captures the effect of units sold on thelikelihood of adoption. Parameter m in (15) denotes the size of thepopulation which will ultimately adopt the innovation.
8. Water softeners and boat trailers from, respectively, Bass (1969) andNevers (1972) were not incorporated because of unrealistic (perhapsmisprinted) values.
'2 '2
9.Sincea.for i = 1,2,...,m were not available for previous studies, ao as1
2
shown in (13) could not be derived. The value for ao
incorporated in the
variance-covariance matrix 4 was set equal to the unit sales figure inv
year one.
10. With more than two sample observations, the results (not shown) converge tothe accurate forecasts shown in Bass (1969) and Nevers (1972).
- 24 -
Appendix 1: Derivation of Prior Estimate
Given expression (8), the prior estimate equals
b = Z DOo o
and, since 0 0 + u with terms as specified above,
b Z DO + ZoDu.
o o
Depending on the design of the new study, it will belong to one of the k
groups specified in terms of design characteristics in the meta-analysis. If
we arbitrarily assume that the study belongs to the first group (k), we can
write the parameter vector as
00 = 01 + eo .
Hence,
0. =+ e. for all jEk i(1-2)o jo
wheree.=e.-eo. For any other study 1 not belonging to set k„ we have
Jo
01 = 0o - + elo(1-3)
with j8i denoting the mean parameter vector for the set of studies containing
study 1.
Combining expressions (1-2) and (1-3) for all studies, we can specify the
augmented model
where e = [e' e'e' I, and M denotes a matrix of m identify matrices`o 2 0 mo
stacked on top of one another. Vector d contains zeros in the entries
corresponding to the parameter vectors for all studies belonging to the same
set as the new study, and the difference in mean parameter vectors (i.e.,
61 ) in the entries corresponding to parameters vectors for all studies
which do not belong to the same set as the new study. Accordingly, the prior
estimate b in (1-1) can be expressed as shown in (9).0
- 25 -
Appendix 2: Computation of Variance-Covariance Matrix Qv.
Given expression (11)
2 1a 0
E(vv') = Q =v
0 Z DE(yy')D'Z'
2 1a 0
0 ZoDyD'Z'
o
where E( yY') = 4'. Accordingly, * = Efe + u)(e + u)']
Assuming that e and u are independent and knowing that the new study belongs to
the first set, it can be shown rather easily that
21-
2E, + a,(X1X1)
2
2E, + a2(V2X2) • • E
Et • El
11, (2-1)
Et Et
(El+Ek) + a 2, m(XrIX171)-1
E
or simply y = IQ + E® E, where E denotes an (m x]
and denotes a Kronecker product. Note that from
two diagonal elements in sy, it is evident that the
m) matrix containing all ones
the expression of the first
corresponding prior studies
Et
Et
belong to the set which also contains the new study.
- 26 -
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87/42 Damien NEVEN and
Jacques-P. THISSE
87/43 Jean CABSZEVICZ andJacques-F. THISSE
87/44 Jonathan HAMILTON,
Jacques-P. THISSEand Anita VESKAMP
87/45 Karel COOL,
David JEMISON andIngemar DIERICKX
07/46 Ingemar DIERICKXand Karel COOL
1988
88/01 Michael LAVRENCE andSpyros MAKRIDAKIS
88/02 Spyros MAKRIDAKIS
88/03 James TEBOUL
88/04 Susan SCHNEIDER
88/05 Charles VYPLOS2
88/06 Reinhard ANGELMAR
88/07 Ingemar DIERICKX
and Karel COOL
88/08 Reinhard ANGELMAR
and Susan SCHNEIDER
88/09 Bernard SINCLAIR-
DESCACNd
88/10 Bernard SINCLAIR-
DESCAGNe
88/11 Bernard SINCLAIR-
DESCAGNi
"Season/MY, size premium and the relationshipbetveen the risk and the return of Frenchcommon stocks", November 1987
"Combining horizontal and vertical
differentiation: the principle of max-min
differentiation", December 1987
"Location", December 1907
"Spatial discrimination: Bertrand vs. Cournot
in a model of location choice", December 1987
"Business strategy, market structure and risk-return relationshipsi a causal interpretation',December 1987.
'Asset stock accumulation and sustainabllityof competitive advantage", December 1987.
'Factors afacting judgemental forecasts andconfidence intervals', January 1988.
"Predicting recessions and other turning
points", January 1988.
'De-industrialize service for quality', January
1988.
'National vs. corporate culture: implicationsfor human resource management', January 1988.
'The svinging dollar: is Europe out of step?",
January 1988.
"Les conflits dans les canaux de distribution",
January 1988.
'Competitive advantage: a resource based
perspective", January 1988.
"Issues in the study of organizational
cognition", February 1988.
"Price formation and product design through
bidding', February 1988.
"The robustness of some standard auction game
forms", February 1988.
"Vhen stationary strategies are equilibriumbidding strategy: The single-crossing
property", February 1988.
"Business firms and managers in the 21stcentury', February 1988
'Alexithymia in organizational life: theorganization man revisited", February 1988.
'Ilse Interpretation of strategies: a study ofthe impact of CEOs on the corporation',March 1988.
' The production of and returns from industrialinnovation, an econometric analysis for adeveloping country', December 1987.
' Market efficiency and equity pricingsinternational evidence and Implications forglobal investing', March 1988.
'Monopolistic competition, costs of adjustmentand the behavior of European eaployaent',September 1987.
' Reflections on 'Veit Unemployment' inEurope*, November 1987, revised February 1988.
'Individual bias in judgements of confidence',March 1988.
"Portfolio selection by mutual funds, nnequilibrium model', March 1988.
'De-industrialize service for quality',March 1989 (88/01 Revised).
' Proper Ouadrattc Functions vith an Applicationto AT&T', May 1987 (Revised March 1988).
'Equilibres de Mash-Gournot dans le aarcheeuropeen du gaz: un ca.s 06 les solutions enboucle Gu y ette et en feedback coincident',Mars 1988
80/29 Naresh K. MALFMTRA,Christian PINSON andArun K. JAIN
88/11 Sumantra GHOSHAL andChristopher BARTLETT
88/38 Manfred KETS DE VRIES
88/39 Manfred KETS DE VRIES
88/40 Josef LAKONISROK andTheo VERMAELEN
88/41 Charles VYPLOSZ
88/42 Paul EVANS
'Consumer cognitive complexity and thedimensionality of multidimensional scalingconfigurations', May 1988.
' Creation, adoption, and diffusion ofinnovations by subsidiaries of multinationalcorporations', June 1988.
' The Motivating Role of Enry : A ForgottenFactor in Management, April 88.
'The Leader as Mirror : Clinical Reflections•,July 1988.
' Anomalous price behavior around repurchasetender offers', August 1988.
"Assysetry In the EMS, intentional orsystemic7', August 1988.
' Organizational development in thetransnational enterprise', June 1988,
88/12 Spyros MAKRIDAKIS
88/11 Manfred KETS DE VRIES
88/14 Alain NOEL
88/15 Anil DF.OLALIKAR andLars-Hendrik ROLLER
88/16 Gabriel HAVAVINI
83/17 Michael BURDA
08/18 Michael BURDA
88/19 M.J. LAWRENCE andSpyros HAKRIDAKIS
88/20 Jean DERMINE,Damien NEVEM andJ.E. TIIISSE
88/21 James TEBOUL
88/22 Lars-Hendrik ROLLER
88/23 Sjur Didrik FLAMand Georges ZACCOUR
813/30 Catherine C. ECKEL 'The financial fallout froa Chernobyl: riskand Theo VERMAELEN perceptions and regulatory response', May 1988.
88/32 Kasra FERDOVS and
'International manufacturing: positioningDavid SACKRIDER plants for success', June 1988.
88/33 Mihkel M. TOMBAK 'The importance of flexibility Inaanufacturing', June 1988.
88/34 Mihkel M. TOMBAK 'Flexibility: an Important dimension inmanufacturing', June 1988.
88/35 MIhkel M. TOMBAK
'A strategic analysis of investment In flexiblemanufacturing systems', July 1988.
88/36 Vikas TIEIREVALA and 'A Predictive Test of the NOD model thatBruce BUCHANAN Controls for Non-stationarity', June 1968.
08/17 Murugappa KRISRNAN "Regulating Price-Liability Competition ToLars-Hendrik ROLLER Improve Welfare', July 1988.
88/24 E. Espen ECKBO andHervig LANGOIIR
' Information disclosure, means of payment, andtakeover premia. Public and Private tenderoffers in Prance', July 1985, Sixth revision,April 1968.
' The future of forecasting', April 1988.
"Semi-competitive Couruot equilibrium inmultistage oligopolies', April 1988.
' Entry glue vith resalable capacity',April 1988.
'The multinational corporation as a network:perspectives from lnterorganlzational theory',May 1988.
88/41 B. SINCLAIR-DESCACNE 'Group decision support systems implementBaycsian rationality', September 1988.
88/44 Essam MARMOU0 and
'The state of the art and future directionsSpyros MAKRIDAKIS, in combining forecasts', September 1988.
88/45 Robert KORAJCZYX
'An empirical investigation of internationaland Claude VIALLET asset pricing', November 1986, revised August
1988.
88/46 Yves DOZ and
'From intent to outcome: a process frameworkAmy SHUEN
for partnerships', August 1988.
80/25 Everette S. GARDNERand Spyros MAKRIDAKIS
88/26 Sjur Didrik FLAMand Georges ZACCOUR
88/27 Hurugappa KRISHNANLars-Hendrik ROLLER
88/28 Sumantra GROSRAL andC. A. BARTLETT
88/47 Alain BULTE2,Els GIJSBRECHTS,
Philippe NAERT andPiet vANDEN Matta
88/48 Michael BURDA
88/49 Nathalie DIERKENS
88/50 Rob WEITZ andArnoud DE MEYER
88/51 Rob VE1TZ
88A52 Susan SCHNEIDER andReinhard ANCELMAR
88/53 Manfred KETS DE VRIES
88/54 Lars-Hendrik ROLLER
and Mihkel M. TONBAK
88/55 Peter ROSSAERTSand Pierre MILLION
88/56 Pierre MILLION
88/59 Martin KILDUFF
88/60 Michael BURDA
89/61 Lart-Rendr ik ROLLER
88/62 Cynthia VAN NULLS,Theo YERMAELEN andPaul DE VOUTERS
'Asymmetric cannibalism between substitute!tees listed by retailers", September 1988.
'Reflections on 'Veit unemployment' in
Europe, II', April 1988 revised September 1988.
'Inforaatfon asymmetry and equity issues",
September 1988.
'Managing expert systems: from inceptionthrough updating', October 1987.
'Technology, pork, and t h e organisation, theImpact of expert systems', July 1988.
'Cornitfon and orgamleational analysis, who'•
minding the store', September 1988.
'Whatever happened to the philosopher-king: theleader's addiction to power, September 1988.
'Strategic choice of flexible productiontechnologies and velfare implications',
October 1988
'Method of moments tests of contingent claimsasset pricing models', October 1988.
"Size-sorted portfolios and the violation of
the random walk hypothesis! Additionalempirical evidence and implication for tests
of asset pricing models', June 1988.
'The Interpersonal structure of decision
asking: a social comparison approach toorganizational choice', November 1988.
'Is mismatch really the problem/ Some estimates
of the Chelvood Cate II model with US date',Septeaber 1988.
'Modelling cost structure! the Bell System
revisited', November 1988.
'Regulation, taxes and the market for corporatecontrol in Belgium', September 1988.
88/63 Fernando NASCIMENTOand Vilfried R.VANHONACKEA
88/64 Kasra FERDOVS
88/65 Arnoud DE MEYERand Kasra FERDOVS
88/66 Nathalie DIERKENS
88/67 Paul S. ADLER andKasra FERDOVS
1989
89/01 Joyce K. BYRER andTavfik JELASSI
89/02 Louis A. LE BLANCand Tavfik JELASSI
89/03 Beth H. JONES andTavfik JELASSI
89/04 Kasra FERDOVS andArnoud DE MEYER
89/05 Martin KILDUFF andReinhard ANCELMAR
89/06 Mihkel M. TOMBAK andB. SINCLAIR-DESGAGNE
89/07 Damien J. NEVEN
89/08 Arnoud DE MEYER andHellmut SCHOTTE
89/09 Damien NEVEN,Carmen MATUTES andMarcel CORSTJENS
89/10 Nathalie DIERKENS,Bruno GERARD andPierre BILLION
"Strategic pricing of differentiated consumerdurables in a dynamic duopoly: a numerical
analysis", October 1988.
"Charting strategic roles for internationalfactories", December 1988.
"Quality up, technology dovn", October 1988.
"A discussion of exact measures of informationassymetry: the example of Myers and Majlufmodel or the importance of the asset structureof the firm", December 1988.
'The chief technology officer", December 1988.
"The impact of language theories on DSSdialog", January 1989.
"DSS software selection: a multiple criteria
decision methodology", January 1989.
"Negotiation support: the effects of computerintervention and conflict level on bargainingoutcome", January 1989."Lasting improvement in manufacturingperformance: In search of a nev theory",January 1989.
"Shared history or shared culture? The effectsof time, culture, and performance oninstitutionalization in simulatedorganizations", January 1989.
"Coordinating manufacturing and businessstrategies: I", February 1989.
"Structural adjustment in European retailbanking. Some view from industrialorganisation", January 1989.
"Trends in the development of technology andtheir effects on the production structure inthe European Community", January 1989.
*Brand proliferation and entry deterrence',February 1989.
"A market based approach to the valuation ofthe assets in place and the growthopportunities of the firm", December 1988.
88/57 Vilfried VANHONACKX8 'Date transferability: estiaatin4 the response
and Lydia PRICE effect of future *vents based on historicalanalogy', October 1988.
88/58 8. SINCLAIR-DE5CAGNE 'Assessing econoalc inequality', November 1988.
and Mibkel M. TOM8AX
89/11 Manfred KETS DE VRIES
and Alain NOEL
"Understanding the leader-strategy interface:
application of the strategic relationship
interviev method", February 1989.
89/12 Vilfried VANHONACKER "Estimating dynamic response models vhen the
data are subject to different temporal
aggregation", January 1989.
89/13 Manfred KETS DE VRIES "The impostor syndrome: a disquieting
phenomenon in organizational life", February
1989.
89/14 Reinhard ANGELMAR "Product innovation: a tool for competitive
advantage", March 1989.
89/15 Reinhard ANGELMAR "Evaluating a firm's product innovation
performance", March 1989.
