A Simple Selection Test Between the Gompertz and Logistic ...econpn/Pierre_Nguimkeu/RESEARCH_files/gselectrev02.pdf · A Simple Selection Test Between the Gompertz and Logistic Growth

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].

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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

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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.

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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.

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Figure 1: The Gompertz and Logistic Curves

0 5 10 15 20 25 30 35 405

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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

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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

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Figure 2: Size function for n = 25

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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.

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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

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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

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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).

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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

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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

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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.

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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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