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A Simple Selection Test Between the Gompertz and Logistic
Growth Models∗
Pierre Nguimkeu†
Georgia State University
Abstract
This paper proposes a simple model selection test between the Gompertz and the Logistic growth
models based on parameter significance testing in a comprehensive linear regression. Simulations studies
are provided to show the accuracy of the method. Two real-data examples are also provided to illustrate
the implementation of the proposed method in practice.
Keywords: Gompertz function, Logistic function, Model selection, t-test;
1 Introduction
Let Yt be a time series taking nonnegative values. The Gompertz trend curve for Yt is given by
Yt = α1 exp(−β1e−γ1t), (1)
and the Logistic trend curve for Yt is given by
Yt = α2(1 + β2e−γ2t)−1, (2)
where t represents time and αi, βi, γi, i = 1, 2, are positive parameters. Model (1) and Model (2), together
with their multi-response and multivariate generalizations, are now widely used in applied research work
for modelling and forecasting the behaviour of many diffusion processes like the adoption rate of technology
based products (Chu et al. 2009, Gamboa and Otero 2009), population growth (Nguimkeu and Rekkas 2011,
Meade 1988), and marketing development (Mahajan et al. 1990, Meade 1984). In fact, the Gompertz and
Logistic curves both share the interesting property that their “S-shaped” feature are suitable to describe
processes that consist of a slow early adoption stage, followed by a phase of rapid adoption which then tails∗I would like to thank two anonymous referees, Associate Editor Yuya Kajikawa, and Daisuke Satoh, for useful comments.†Department of Economics, Andrew Young School of Policy Studies, Georgia State University; 14 Marietta Street NW, Suite
524, Atlanta, GA 30303, USA; Tel. (1)404.413.0162; Email: [email protected].
1
off as the adopting population becomes saturated. However, despite these visual and numerical similarities
there are fundamental differences between the two curves and one of the most important is that the Gompertz
function is symmetric whereas the Logistic function is asymmetric. Failing to account for these differences
and choosing an inappropriate growth curve for inference can lead to seriously misleading forecasts ( see Chu
et al. 2009 and Yamakawa 2013 for some empirical illustrations). The need to develop a reliable selection
procedure to discriminate between the two models in practice is therefore salient.
Unfortunately, in spite of the important request to selection between these models in practice, there rarely
exists a framework for statistical test between the two. The selection is usually made in an ad hoc basis
using criteria based on forecasting errors, on the plausibility of the estimated saturation levels, or on visual
evidence obtained from plotting the data in a special way, see for example, Gregg et al. (1964). A notable
exception is the approach of Franses (1994) who proposed a selection based on statistical significance testing
in an auxiliary regression which we briefly discuss in Section 2. Other approaches used are based on criteria
of fitness that require to actually estimate the two models and then compare their fits with historical data
through measures like R2, root mean squared errors (RMSE), mean absolute percentage error (MAPE),
root mean squared prediction errors (RMSPE) (see Chu et al. 2009, Yakamawa 2013). Such a procedure
is however not attractive as it requires to estimate both models by nonlinear regression methods involving
numerical optimization which is usually computer expensive and time consuming. There is thus a clear need
for selection methods between Gompertz and Logistic models which are easy to understand and inexpensive
to compute. In this context, it seems natural to investigate the use of statistical tests that require simple
estimation and easy computation.
This paper proposes a model selection test based on one linear regression and the significance test of
one parameter. Our approach is therefore similar in spirit to the one proposed by Franses (1994) who also
based their method to a single parameter significance testing. However, whereas the Franses (1994) method
requires to primarily impute the original data in order to get only strictly positive increments of Yt, our
approach is based on the original responses themselves regardless of their values. Thus, there is no loss or
distortion of information that could possibly undermine the result of our test which at the same time is more
straightforward to compute. We examine the empirical size and power performance of the proposed test
through Monte Carlo simulations and also provide real data examples to illustrate its usefulness in practice.
The results show that the proposed test performs reasonably well in finite samples and could be a better
alternative to the Franses’ test.
In Section 2 we discuss the transformations of the Gompertz and Logistic curves leading to our selection
procedure as well as the difference between our test and the Franses (1994) method. Section 3 provides
numerical studies including Monte Carlo simulations and two real-data examples. Some concluding remarks
2
are given in Section 4.
2 The Selection Procedure
Recall that Yt is our variable of interest and denote by yt = (Yt − Yt−1)/Yt−1 the relative increase in Yt.1
Let the Gompertz response function in (1) be denoted by g(t) = α1 exp(−β1e−γ1t).
Differentiating g(t) and rearranging terms yields
g′(t)g(t)
= γ1[lnα1 − ln g(t)].
This suggests setting up a simple linear regression for the Gompertz model given in (1) with the form
yt = δ1 + ρ1 lnYt−1 + u1t. (3)
Likewise, if we denote by h(t) = α2(1 + β2e−γ2t)−1 the Logistic response function in (2), a similar manipu-
lation leads to the differential equation
h′(t)h(t)
= γ2[α2 − h(t)].
Hence, a linear regression model of the form
yt = δ2 + ρ2Yt−1 + u2t (4)
can be set up for the Logistic model given in (2). Testing (1) against (2) is therefore equivalent to testing
Model (3) against (4). Models (1) and (2) as well as Models (3) and (4) are clearly nonnested in the sense of
Cox (1961). For the latter models, it is desirable to use higher frequency data, if available, so that the first-
derivative approximation by the difference score is more precise. However, regardless of the time frequency,
the decision-rule provided by the test discussed below should not change, so long as one has enough data
and one uses a definition of first-derivative that is consistent with the frequency of the data and is applied
alike to both competitive models.
Following Davidson and MacKinnon (1981), an artificial comprehensive model can therefore be formulated
as follows:
yt = δ + γ lnYt−1 + θYt−1 + ut, (5)
where ut is an error term. It can be seen that when θ = 0, Model (5) reduces to (3). Thus, it might seem
that to test (3) against (4) we could simply estimate this model and test whether θ = 0.2 However, for the
1 One may instead consider yt = (Yt+1 − Yt)/Yt or the approximation yt = log Yt − log Yt−1and the results discussed wouldbe similar.
2This idea is similar to the J test that was first suggested by Davidson and MacKinnon (1981) for nonnnested regressions.
3
types of applications we consider here (i.e. growth curves), the series of interest, {Yt}, will usually display an
upward trend with no tendency of mean reversion, thus implying that they are non-stationary (Franses 1998,
pp. 67-68). Estimation of Model (5) using ordinary least squares might then lead to a spurious regression
with an inconsistent estimate of θ (see, e.g., Hamilton 1994, pp. 557-562, for a thorough discussion). In
order to avoid spurious regressions, the simplest and most recommended way to base a test on Model (5) is
to estimate a differenced version of it given by 3
∆yt = µ+ γ∆ lnYt−1 + θ∆Yt−1 + εt, (6)
where εt is the error term which can be assumed to be NID(0, σ2). We can estimate Model (6) by ordinary
least squares and test the null hypothesis that θ = 0 using an ordinary t-test for a desired significance level.
This provides an easy and reliable way to test for (3). Alternatively, we could test for γ = 0, which would
correspond to the logistic model given by (4). Since this can be done by simply interchanging the roles of the
two models in all the following discussions, we focus on the former case in the rest of the paper, for brevity.4
Note that the inclusion of the constant term µ is not strictly needed for the comprehensive specification of
the differentiated model in theory, but is useful in practice, for example to control for a possible nonzero
mean in the error term, and does not create any bias in the coefficients. Also, assuming normality of the
error terms is not a straightforward assumption, and is not a strict requirement for the test either. But it
is a common practice in time series analysis for estimating parameters and the resulting estimates have a
number of desirable properties even if the errors are non-normal. This assumption can be easily dropped
if the sample size is large enough since the normal distribution would validly approximate the asymptotic
distribution of the t-test. The selection method between a Gompertz and a Logistic curve based on (6)
uses all the in-sample observations. Hence, no observations are lost because of out-of-sample forecasting
performance evaluation. This is important in practice where only small samples are usually available. The
graphical illustration given in Figure 1 is also instructive. It shows a relationship between yt and Yt−1 that
is logarithmic for the Gompertz process and linear for the logistic process. This may be helpful to guide
the data analysis in practice, although a selection based only on visual evidence could be misleading or
imprecise. Other graphical methods based on different types of transformations on the variable of interest
are also available in Harvey (1984), Franses (1994).
3Although many researchers recommend routinely differencing non-stationary variables before estimating regressions, differ-encing may not be needed in the exceptional circumstance where the variables are cointegrated. In this case it is preferable toperform our selection test over Model (5), since differencing may cause a reduction of power (thanks to an anonymous referee forpointing this out). In our numerical applications, preliminary analysis have shown that they are not cointegrated. Cointegrationtests are easy to perform and are available in most standard statistical softwares.
4∆ lnYt−1 and ∆Yt−1 cannot be perfectly correlated (since one cannot be obtained as an affine transformation of the other).However, it is good practice to examine the magnitude of the squared correlation of the two regressors and verify that collinearityis not present in the model, otherwise simply testing individual coefficients might not be sufficient. In our empirical examples,no evidence of collinearity was found.
4
Figure 1: The Gompertz and Logistic Curves
0 5 10 15 20 25 30 35 405
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
t
Y
GompertzLogistic
5.5 6 6.5 7 7.5 8 8.5 9 9.5 100
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Y
!Y
Y
Gompertz
Logistic
Franses (1994) showed that the Gompertz growth model given by (1) could be rewritten in the form
log(∆ log Yt) = a+ bt, (7)
and put forward a testing procedure that involves estimating by ordinary least squares the auxiliary regression
log(∆ log Yt) = a+ bt+ ct2 + νt (8)
and testing the null hypothesis that the estimated coefficient c is statistically different from zero. If this
coefficient turns out to be statistically different from zero then a Logistic specification should be estimated;
otherwise, a specification based on the Gompertz curve should be preferred. One major drawback of the
Franses (1994) procedure, however, is that, in practice, the values of ∆ log Yt may be negative, so that it
would not be possible to apply the second logarithmic transformation in the left-hand side of Equations (7)
and (8). Franses (1994) suggested that such observations be replaced by interpolated values or be treated as
missing, a solution that may well distort the original information, undermine the quality of the estimates, or
at least require the researcher to spend an extra time imputing the data. In contrast, the testing procedure
proposed in this paper uses the original data available and is straightforward and readily applicable without
requiring any further data imputation.
3 Numerical Studies
In this section, we provide both a Monte Carlo simulation study to gain a practical understanding of the
performance of our testing procedure as well as an application to real data examples to show how the test
could be used in practice. The focus of the simulation is to examine the size of the test, i.e. the frequency
of type I error, and the power of the test, i.e., the ability of rejecting the wrong model. The results from the
5
proposed test (denoted Proposed) is also compared with the Franses (1994) method (denoted Franses).
3.1 Monte Carlo Simulation
This section reports the results of a Monte Carlo study conducted to assess the small-sample performance
of the proposed test and also compare it to the Franses (1994) approach. Two data generating processes are
considered:
DGP1 : Yt = 20 exp(−β1e−γ1t) + u1t, u1t ∼ N(0, 0.1)
DGP2 : Yt = 20(1 + β2e−γ2t)−1 + u2t, u2t ∼ N(0, 0.01)
Table 1: Estimated size function for testing DGP1 against DGP2, at 5% significance level
n=20 n=30Parameters Proposed Franses Parameters Proposed Franses
β1 = 1 γ1 = 0.05 0.001 0.000 β1 = 1 γ1 = 0.05 0.000 0.0000.07 0.018 0.000 0.07 0.000 0.0000.10 0.012 0.000 0.10 0.003 0.0000.12 0.028 0.002 0.12 0.034 0.0010.15 0.031 0.000 0.15 0.072 0.003
β1 = 2 γ1 = 0.05 0.011 0.000 β1 = 2 γ1 = 0.05 0.000 0.0000.07 0.002 0.002 0.07 0.000 0.0010.10 0.011 0.000 0.10 0.001 0.0010.12 0.024 0.001 0.12 0.042 0.0060.15 0.000 0.010 0.15 0.003 0.000
β1 = 3 γ1 = 0.05 0.000 0.001 β1 = 3 γ1 = 0.05 0.000 0.0000.07 0.003 0.009 0.07 0.000 0.0000.10 0.065 0.000 0.10 0.003 0.0000.12 0.002 0.010 0.12 0.000 0.0080.15 0.002 0.014 0.15 0.007 0.008
β1 = 4 γ1 = 0.05 0.003 0.005 β1 = 4 γ1 = 0.05 0.000 0.0000.07 0.001 0.000 0.07 0.001 0.0010.10 0.003 0.000 0.10 0.000 0.0000.12 0.004 0.229 0.12 0.005 0.0020.15 0.003 0.017 0.15 0.000 0.001
The first part of the experiment involves estimating probabilities of a Type I error under DGP1 at
β1 ∈ {1, 2, 3} and γ1 ∈ {0.05, 0.07, 0.10, 0.12, 0.15} at 5% nominal level. The second part involves calculating
the power of the tests by estimating the rejection probabilities of the tests under the DGP2 for β2 ∈ {5, 10, 15}
and γ2 ∈ {0.3, 0.5, 0.6, 0.7, 0.8, 0.9} at the 5% level. We consider sample sizes of n = 20, n = 25 and n = 30
with 1000 replications each.
The empirical size performance of the test are presented in Table 1 for the sample sizes n = 20 and
n = 30, and in Figure 2 for n = 25, at a nominal significance level of 5%. The results indicates that
while the empirical size of both tests can be below the nominal level of 5% the proposed test clearly dom-
inates the Franses test whose rejection probabilities tends to be consistently close to zero. The size of the
tests does not seem to be sensitive with the different sample sizes considered. The results of the power
study are displayed in Table 2 for sample sizes n = 20 and n = 30, as well as in Figure 3 for the sample size
6
Figure 2: Size function for n = 25
0.05 0.1 0.15 0.2
0
0.01
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0.05
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0.07
0.08
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Pro
bab
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β=1
0.05 0.1 0.15 0.2
0
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0.04
0.05
0.06
0.07
!
Pro
bab
ilit
y
β=2
0.05 0.1 0.15 0.2
0
0.01
0.02
0.03
0.04
0.05
0.06
!
Pro
bab
ilit
y
β=3
0.05 0.1 0.15 0.2
0
0.02
0.04
0.06
0.08
0.1
!
Pro
bab
ilit
y
β=4
Proposed
Franses
Nomina lProp osed
Franses
Nominal
Proposed
Franses
Nomina l
Proposed
Franses
Nomina l
Student Version of MATLAB
n = 25. The powers of the proposed test are reasonably high in most cases and occasionally hit the limit of 1.
Compared to the Franses’ test, the proposed test performs remarkably better. In fact, although the
Franses test also exhibits high powers in many cases, there are several cases in which it completely lacks power.
This is not surprising, given the nature of this test which is partially based on a quadratic approximation of
the original responses (see Equation (8) above and the discussion in Franses 1994). The test may therefore
lack power in some instances, perhaps because the quadratic function has neither an inflexion point nor a
saturation level, two key features of the functional forms being tested. Note, however, that the type I error
of the proposed test is still not well controlled even though it performs better than the Franses (1994) test.
There are cases where the empirical size of the test exceeds the nominal level. These cases are, nonetheless,
very few and the corresponding values mostly remain within an acceptable range.
7
Table 2: Estimated power function for testing DGP1 against DGP2, at 5% significance level
n=20 n=30Parameters Proposed Franses Parameters Proposed Franses
β2 = 5 γ2 = 0.3 0.530 0.106 β2 = 5 γ2 = 0.3 0.445 0.0030.5 0.102 0.001 0.5 0.254 0.0590.6 0.574 0.051 0.6 0.166 0.8280.7 0.737 0.288 0.7 0.343 0.8270.8 0.481 0.096 0.8 0.595 0.5430.9 0.832 0.387 0.9 0.395 0.826
β2 = 10 γ2 = 0.3 0.221 0.971 β2 = 10 γ2 = 0.3 0.074 0.0640.5 0.839 0.004 0.5 0.281 0.3170.6 0.391 0.005 0.6 0.331 0.9610.7 0.418 0.000 0.7 0.607 0.7180.8 0.358 0.136 0.8 0.312 0.9790.9 0.674 0.063 0.9 0.671 0.528
β2 = 15 γ2 = 0.3 0.160 0.982 β2 = 15 γ2 = 0.3 0.047 0.0940.5 0.455 0.144 0.5 0.179 0.0320.6 0.930 0.012 0.6 0.181 0.3400.7 0.440 0.005 0.7 0.484 0.6260.8 0.692 0.015 0.8 0.594 0.6640.9 1.000 0.052 0.9 0.950 0.412
Figure 3: Power function for n = 25
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
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Pro pos ed
Fran s es
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Prop os ed
F ran s es
Pro p os ed
F ran s esProp os ed
F ran s es
Student Version of MATLAB
8
3.2 Empirical examples
An application of the proposed selection method is illustrated using two examples taken from Franses (1994)
and the results are compared to traditional criteria such as R2, RMSE, MAPE and out-of-sample predic-
tion errors. The first example consists of the official figures on tractor ownership in Spain over the period
1951-1976. The observations are plotted in Figure 4. The right panel of Figure 4 also depicts the graph of
the yt series with respect to the Yt−1 series.
Figure 4: Plots of Yt and plots of yt with respect to Yt−1 for Tractors in Spain
1950 1955 1960 1965 1970 19750
5
1015
2025
3035
4045
t
Y t
0 10 20 30 400.05
0.1
0.15
0.2
0.25
Yt−1
y t
Tractors yt=f(Yt−1)
Student Version of MATLAB
A visual analysis of this graph shows a curvature in the relationship between yt and Yt−1 similar to
the stylized relationship depicted by the right panel of Figure 1. Although this empirical graph should be
interpreted with cautious, it seems to suggests that the series are closer to a Logistic process. This conjecture
is further confirmed by the t-statistic for the parameter θ in Model (6) which has a value of −2.658, thus
statistically different from zero at a 5% significance level.5 This result is consistent with several authors,
including Harvey (1984) and Mar-Molinero (1980) who also argued that the tractors data in Spain followed
a Logistic growth curve, as well as Franses (1994) who obtained a selection t-statistic of −3.7404. Moreover,
our conclusion is further supported by a comparison of the models based on R2, RMSE, MAPE and
prediction errors based on out-of-sample forecasts (RMSPE) (see Table 3.2 and Figure 3.2) .
The second example uses the annual stock of cars series in the Netherlands from 1965 to 1989. The
graph of the yt series with respect to the Yt series is depicted in Figure 6 and seems to visually suggest a
logarithmic relationship that is similar to the stylized one depicted in the right panel of Figure 1 so that
a Gompertz curve may indeed be appropriate. It is however obvious that the graphical visualization is
not very convincing in these examples which is why such an approach should be used with cautious. The
results obtained from comparing R2, RMSE, MAPE and out-of-sample prediction errors (RMSPE) further
support that the choice of the Gompertz model may indeed be appropriate (see Table 3.2 and Figure 3.2).
The value of the t-statistic for the parameter θ in Model (6) is 0.123, which is statistically not significant at
5In both examples, tests for residuals normality indicate no distributional misspecification. The squared correlation between∆ lnYt−1 and ∆Yt−1 is estimated at 0.103 in the Tractors data and at 0.639 in the Cars data, and for both exanples there areno symptoms of collinearity in the linear regression given by Model (6).
9
Table 3: Model Comparison Results
Tractors in Spain Cars in the Netherlands
Methods Gompertz Logistic Gompertz Logistic
R2 0.999 0.999 0.997 0.996
RMSE 1.775 0.607 316.1 363.5
MAPE 10.2% 4.64% 1.33% 2.11%
RMSPE 12.68 11.97 179.4 271.7
Franses -3.7404 1.031
Proposed -2.658 0.123
Figure 5: Forecast Performance of the Models for Tractors (left) and Cars (right) Data
1950 1955 1960 1965 1970 19750
5
10
15
20
25
30
35
40
45
Year
Trac
tor O
wne
rshi
p
Actual dataGompertzLogistic
Fitted region Forecast region
Student Version of MATLAB
1965 1970 1975 1980 1985 19901000
1500
2000
2500
3000
3500
4000
4500
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5500
Year
Car O
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Actual dataGompertzLogistic
Fitted region Forecast region
Student Version of MATLAB
the 10% level, therefore confirming that the Gompertz curve is more adequate. This result is also consistent
with that of Franses (1994), who obtained a selection t-stat of 1.031.
Figure 6: Plots of Yt and plots of yt with respect to Yt−1 for Cars in the Netherlands
1965 1970 1975 1980 1985 19901
2
3
4
5
6
t
Y t
1 2 3 4 5 60
0.05
0.1
0.15
0.2
Yt−1
y t
Cars yt=f(Yt−1)
Student Version of MATLAB
Although in this application all the methods lead to the same conclusion about the selection decision,
it is evident that the methods based on R2, and prediction errors are either less precise and/or are less
attractive because they require nonlinear numerical optimization which are computer intensive and suffer
from the well-known potential non-convergence problems in practice.
10
4 Conclusion
This paper has provided a model selection test between the Gompertz and the Logistic models. The idea
of the test exploits differential equations underlying both processes which can be estimated and tested in
the form of linear regressions. The test is more insightful and more accurate than alternative approaches
currently used in practice. The test is also easier to compute than the Franses (1994) selection test as it
uses readily available data and does not require any further data imputation as the latter does. Simulation
results show that the test has acceptable size although it can be conservative. Simulations also show that
in most cases, the power is very high and often hits the limit of one. To illustrate the practical use of the
proposed method two real-data examples are provided and the results are compared to traditional measures
such as R2, RMSE and prediction errors based on out-of-sample forecasts. Note that the method proposed
in this paper is based on the assumption that either the Gompertz or the Logistic model is in fact the correct
model. Although these two models are the most widely used to describe and forecast the trend of a wide
variety of such data other models might better fit the data in practice, in which case a method to evaluate
the adequacy of the selected model is still needed. The idea of the test developed in this paper can be
extended to model comparison between other types of growth curves. The author plans to investigate this
in a future research.
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