A New Growth Rate Measure in Identifying Extended Gompertz Growth Curve and Development of Goodness-of-fit Test

Abstract

Growth is a fundamental aspect of a living organism. Growth curves play an important role in explaining the complex dynamics of growth trajectories. The development of a large class of growth models provides more choices to explain complex growth dynamics. However, identifying a suitable growth curve from a broad class of growth models becomes a challenging task. Relative Growth Rate (RGR) is the most popular measure in the growth-related study. It serves many purposes in growth curve literature, including constructing any goodness-of-fit index of some growth dynamics. However, the goodness-of-fit test based on RGR is restricted to only simple growth models. This study aims to develop a new growth rate function, instantaneous maturity rate (IMR), which can play an important role in identifying growth models. We have explored that the measure has synergy in mathematical form with IMR. However, unlike the hazard rate, IMR is a random variable when the size/RGR variable is stochastic. We have derived the exact and asymptotic distribution of this measure under the Gaussian setup of both the size and RGR variables. We have constructed a goodness-of-fit test for the extended Gompertz growth model based on the instantaneous maturity rate. We have checked the performance of the test through simulation studies as well as real data.

AMS 2010 subject classifications: 62Mxx, 92Bxx, 62P10

Keywords

Relative growth rate Instantaneous maturity rate goodness-of-fit test extended Gompertz growth

1 Introduction

The generalization of growth curves is an important aspect of growth curve literature and provides plenty of choices when growth curves are applied to real data.^[1-6] Identification of a suitable growth curve is a challenging task in growth curve analysis since growth curves are generally sigmoidal type. Again if we plot the empirical relative growth rate (RGR) over size or time, we can identify the growth curve models suitable for the given data.^[7] However, identifying growth curves using the shapes of RGR is a difficult task since RGR takes only two shapes, decreasing and bell shapes. So it is essential to develop a growth rate measure that can identify growth curves more accurately. A novel and unified growth rate called the Instantaneous Maturity rate (IMR) can address this issue. IMR at the time interval $(t, t + Δ t)$ is defined as the ratio of the RGR at the same time interval and the cumulative RGR yet to achieve beyond that interval.

In growth curve analysis, the goodness-of-fit index calculates the proximity between theoretical and empirical growth curves. A series of research works are carried out to propose more sophisticated goodness-of-fit tests for growth laws that can measure the proximity of the data to a given model Basu and Bhattacharjee^[8] has offered a goodness-of-fit test explicitly for the exponential growth law extending the finite difference approach of Hill^[9] through directly modelling the RGR variable. Ratios of logarithms of size data were used to offer interval-specific goodness-of-fit for the exponential polynomial growth curve model^[10] Chakraborty and Basu^[11] devolved the interval-specific goodness of fit test for the logistic growth curve model.

Chakraborty et al.^[7] considered finite differences of the ratios of logarithms of the empirical RGR for testing goodness of fit for the Gompertz growth curve model. RGR plays a fundamental role in developing these goodness-of-fit tests. It is a better tool compared to the size variable. Note that the growth curves Gompertz, logistic take only decreasing shapes of RGR. We have also seen the extensions of these growth curves, which take non-monotone shapes of RGR.^{[4, 6]} However, developing a goodness-of-fit test based on RGR is complicated. On the other hand, IMR is more beneficial to identify a growth model in comparison with RGR. In this study, we develop a goodness-of-fit test procedure for the extended Gompertz growth curve model (henceforth, Chakraborty-Bhowmick-Chattopadhyay-Bhattacharya model or CBCB model) proposed by Chakraborty et al.^[6] We have checked the performance of this test through simulation studies as well as real data.

We organize the rest of the article as follows. We define a new growth rate measure and discuss some important aspects of this measure in Section 2. We derive the new measure's exact and asymptotic distributions in Section 3. We develop the goodness-of-fit test procedure for the CBCB model in Section 4. We present a simulation study and a real-data example in Section 5. Concluding remarks are given in Section 6.

2 Growth Rate and Its Proposed Extension

The RGR is the widely used measure to quantify the population growth.^[12] The mathematical definition of the RGR function was proposed by Fisher^[13] as the per-unit rate of change in the value of the size variable. Let $X (t)$ be the magnitude of a suitably measured size variable at time point $t$ . The term “size” may refer to the height, weight, length, or abundance of a species. The RGR at time $t$ ; $t \geq 0$ is defined as the relative rate of change per–unit size over the time interval $(t, t + Δ t)$ , which is mathematically expressed as

\frac{1}{X (t)} \frac{d X (t)}{d t} = \frac{d ln X (t)}{d t} .

(2.1)

We observe that the RGR measure provides the information of growth rate for a specific time interval with respect to the size of the cohort. It indicates that RGR offers information regarding the earlier period and the current status but does not provide any information about the upcoming growth process. The experimenter needs to know that for any instant, how much RGR is necessary for a species to achieve to reach the usual size. We define IMR for a specific time interval $(t, t + Δ t)$ as the ratio of the RGR within that interval and the cumulative RGR, which is yet to achieve to reach maturity. So mathematically, Instantaneous Maturity Rate at time point t is defined as

m (t) = \frac{R (t)}{\int_{t}^{\infty} R (u) d u},

(2.2)

where $m (t)$ and $R (t)$ are IMR and RGR at time $t$ , respectively. Again, the denominator can be written as $\int_{t}^{\infty} R (u) d u = \int_{t}^{\infty} \frac{1}{X (u)} \frac{d X (u)}{d u} d u = \int_{t}^{\infty} d ln (X (u)) = ln K - ln X (t)$ , where $K$ is the asymptotic size. Hence, the IMR at time $t$ can also be written as $m (t) = \frac{R (t)}{ln K - ln X (t)}$ . If the asymptotic size is infinite, for example exponential growth, then $m (t) = 0$ for all time point $t$ . For the exponential growth, RGR at any time is constant and the RGR required to reach the asymptotic size is infinite. Hence the ratio of the two quantities in Equation (2.2) will be zero. We derive the IMR of Gompertz and extended Gompertz model in the following.

2.1 Gompertz Model

Gompertz model is one of the most frequently used curves, and it is well known for modelling tumour growth. The Gompertz growth law is governed by the equation

R (t) = b e^{- at} .

(2.3)

The size at time $t$ is given by $X (t) = X (0) e^{\frac{b}{a} (1 - e^{- at})}$ . So the asymptotic size is $K = X (0) e^{\frac{b}{a}}$ . The IMR at time $t$ is obtained by $m (t) = a$ . Hence the IMR of Gompertz growth at any time point is constant. Again, if the IMR of any growth law is constant, it can be shown that Gompertz's growth law. Hence constant IMR is an important characterization of the Gompertz growth, which can be used to identify the Gompertz growth. Note that using RGR, we can not distinguish Gompertz growth from the other growth curves.

2.2 Extended Gompertz Model

The Gompertz model can only decrease if we plot RGR over time. However, there is also a bell-shaped RGR which motivated to extend the Gompertz growth curve. We want to mention two important extensions of the Gompertz growth in this context. The first one was proposed by Bhowmick and Bhattacharya.^[4] and the second one was the CBCB model proposed by Chakraborty et al.^[6] The CBCB model enjoys all important properties, including capturing monotone and non-monotone shapes of RGR like the model proposed by Bhowmick and Bhattacharya.^[4] However, the CBCB model proposed by Chattopadhyay et al.^[6] has an extra advantage because we can get the simple analytical form of size. We have considered the CBCB model in our study. So it is essential to study the IMR of the model since the IMR will be used to develop the goodness-of-fit test for this growth law.

The RGR of the CBCB model is given by

R (t) = {be}^{- {at}^{c}} t^{c - 1},

(2.4)

where $a > 0$ , $b > 0$ , and $c > 0$ . Integrating the above equation, the size at time $t$ is given by

X (t) = X (0) exp [\frac{b}{ac} (1 - e^{- {at}^{c}})] .

(2.5)

This is the Gompertz growth model when $c = 1$ . The IMR of the extended Gompertz model is obtained as

m (t) = \frac{R (t)}{\int_{t}^{\infty} R (u) d u} = {act}^{c - 1} .

The IMR of the CBCB model takes three different shapes; decreasing, constant and increasing. If $c < 1$ IMR is a decreasing function of time. If c=1, then IMR is constant, which is the Gompertz growth model. In these two cases, RGR is a decreasing function of time. If $c > 1$ , then IMR is an increasing function of time, and RGR is bell-shaped.

Remark 1. We need to find the RGR to find the IMR of a growth curve. We want to mention the article Chakraborty et al(2019) where the authors used RGR to unify a broad class of growth curves, and the forms of RGR of many growth curves can be integrated with a simple function. For the other growth curves such as Tabatabai et al.^[2] RGR can be obtained as follows. The hyperbolistic model proposed by Tabatabai et al.^[2] is

\frac{d X (t)}{d t} = \frac{1}{K} X (t) (K - X (t)) (K β + \frac{θ}{\sqrt{1 + t^{2}}}),

where $β, θ$ , and $K$ are the parameters of the model. $β$ represents intrinsic growth rate and K represents carrying capacity. The RGR of the model is obtained as

R (t) = (K - X (t)) (β + \frac{θ}{K \sqrt{1 + t^{2}}}),

(2.7)

or equivalently

R (t) = \frac{α exp (- β K t - θ arcsinh (t))}{1 + α exp (- β K t - θ arcsinh (t))} (K β + \frac{θ}{\sqrt{1 + t^{2}}}) .

(2.8)

We can find the IMR using the RGR and the size.

2.3 Synergy Between IMR and Hazard Rate

In actuarial sciences and demography, the hazard rate provides the most intuitive way to introduce analysts’ assumptions and knowledge in mortality laws.^[15] Hazard rate is the probability that an item will fail within the infinitesimal interval $(t, t + Δ t]$ given that it survives up to the time point t, which can be mathematically written as $h (t) = \frac{f (t)}{1 - F (t)}$ , where $f (t)$ and $F (t)$ are the failure density and distribution functions respectively. We find a synergy between the IMR and the hazard rate function in survival analysis but in the opposite sense. We consider the following definitions to establish the synergy of IMR with hazard rate.

Definition 1: The cumulative relative growth rate at time $t$ is defined by $\int_{0}^{t} \frac{d ln X (u)}{d u} d u = ln X (t) - ln X (0)$ .

Note that the minimum and maximum value cumulative relative growth rate are 0 and $ln K - ln X (0)$ respectively. To relate this with cumulative distribution we standardize the cumulative relative growth rate by dividing $(ln K - ln X (0))$ .

Definition 2: Standardized cumulative relative growth rate (SCRGR) is defined by $S (t) = \frac{ln X (t) - ln X (0)}{ln K - ln X (0)}$ . So $S (t)$ lies between $0$ and $1$ and $S (t)$ is an increasing function of time $t$ as $X (t)$ is an increasing function.

The SCRGR can be interpreted as the proportion of CRGR that the species have achieved up to $t$ . Now the proportion of RGR yet to reach by an organism can be expressed by the quantity $\overset{̅}{S (t)} = 1 - S (t)$ . The rate of change of SCRGR, $S^{'} (t)$ , can be viewed as the proportion of RGR achieved by the species for a specific interval $(t, t + dt]$ . The proportion is calculated with respect to the total RGR, i.e., the CRGR at $\infty$ . Note that the IMR, defined in Equation (2.2), can be written as the ratio of $S^{'} (t)$ and $\overset{̅}{S (t)}$ . The IMR has a strong synergy with the hazard rate, usually used in survival analysis. The hazard is the instantaneous rate of failure at time interval $(t, t + dt]$ , given that the individual survives up to time $t$ . This rate of failure within the prescribed interval $(t, t + dt]$ and the probability of failure beyond the time $t$ are comparable with the instantaneous RGR within the interval $(t, t + dt]$ and the RGR yet to achieve by the species to reach at maturity respectively. So the hazard rate defined in reliability analysis is analogous with the IMR in the case of growth curve analysis.

Now we will focus on the properties of IMR, which are also common attributes of the hazard function.

Note that $\overset{̅}{S (t)}$ behaves like $\overset{̅}{F (t)}$ . $\overset{̅}{S (0)}$ and $\overset{̅}{S (\infty)}$ are 1 and 0 respectively.

$S^{'} (t)$ has a close synergy with $f (t)$ .

Most importantly, IMR lies between 0 and $\infty$ , which is also the possible range of hazard rate.

3 Distribution of IMR

In growth curve analysis we have the size/RGR at several time points which is not necessarily deterministic. Usually we assume the size vector $X = (X (1), X (2), . . ., X (q))^{T}$ follows multivariate normal distribution with mean from the size of some specific growth curve and dispersion structure $Cov (X (s), X (t)) = ρ^{| s - t |} σ_{s} σ_{t}, for s, t = 1, 2, . . . ., q$ or the RGR vector $R = (R (1), R (2), . . ., R (q - 1))^{T}$ follows multivariate normal distribution with mean from the RGR of some specific growth curve and dispersion structure $Cov (R (s), R (t)) = ρ^{| s - t |} σ_{s} σ_{t}, for s, t = 1, 2, . . . ., q - 1 .$ Here AR(1) structure is used in the $Σ$ matrix as it is usually observed that observations at closer time points are highly correlated and as the difference increases, correlations between them diminishes.^{[16, 17]}

The IMR can be estimated empirically using the discrete analogue of the hazard rate function. Hence the IMR at time $t$ is given by

m (t) \approx \frac{R (t)}{\sum_{i = t}^{\infty} R (i)} \approx \frac{R (t)}{\sum_{i = t}^{q} R (i)} .

(3.1)

Therefore IMR is the ratio of two random variables, which is not the case for hazard function. Again, Hazard functions track how the failure rate changes with time. Similarly, for a given empirical estimate of RGR, the IMR can state the species’ actual maturity status. Although both IMR and the hazard have some conceptual similarity, they diverge in terms of randomness. Actually, in growth curve analysis, the IMR can serve two purposes. Its empirical estimate based on the given data can characterize a broad family of growth curves. Finally, we can compare the growth of the same species’ precise maturity status in different locations or different species in the same locations by the test of equality of expected values of IMR. The exact and asymptotic distributions are the primary tool to construct the statistical test of equality. We have derived the exact distribution of IMR under normality assumption as a natural interest of theoretical statistics. However, it does not have a direct connection with the development of the goodness-of-fit test. We have also developed the asymptotic sampling distributions of IMR. We have identified a transformed variable of IMR so that the new variable's mean process is constant. The asymptotic distributions of the transformed variable of IMR are also derived which is a useful tool for the goodness-of-fit test for the CBCB model.

Remark 1. The discrete analogue of the cumulative RGR can be interpreted as the cumulative sum of change in log sizes for all the subintervals between $(0, t)$ , i.e. the cumulative RGR based on all subintervals between $(0, t)$ . The numerator and denominator of IMR can be written as $R (t)$ and $\sum_{t}^{\infty} R (t)$ respectively. Note that we should take the maximum value of t to be $\infty$ which is virtually impossible. However, in real life, the maximum value of t can be taken as the last time point where the experiment is done since in the case of sigmoidal growth curves $lim_{t \to \infty} R (t) = 0$ . So, in reality, the denominator can be taken as a finite sum.

3.1 The Exact Distribution of IMR

Theorem 1. Let $R = (R (1), R (2), . . ., R (q - 1))^{T}$ be the RGR variable at $q - 1$ time points which follow multivariate normal distribution with mean from the RGR of CBCB model and the dispersion matrix $Σ = (σ^{2} ρ^{| i - j |}), i, j = 1, 2, . . ., q - 1$ . For the given RGR vector, the exact distribution of IMR at time $t$ , $m (t)$ has the following probability density function

f (v) = ζ e^{- d} [\frac{1}{k} e^{- {kc}^{2}} + c \sqrt{\frac{π}{k}} \{1 - 2 Φ (- \sqrt{2 k} c)\}],

(3.2)

where

\begin{matrix} ζ & = & \frac{1}{2 π σ^{2} v^{2} \sqrt{p_{t}^{2} - {\tilde{ρ}}^{2}}} \\ k & = & \frac{p_{t}^{2}}{2 σ^{2} (p_{t}^{2} - {\tilde{ρ}}^{2})} [1 - 2 \frac{\tilde{ρ}}{{vp}_{t}^{2}} + \frac{1}{v^{2} p_{t}^{2}}] \\ c & = & \frac{μ (t) + \frac{\tilde{μ (t)}}{{vp}_{t}^{2}} - \frac{\tilde{ρ}}{{vp}_{t}^{2}} (μ (t) + v \tilde{μ (t)})}{1 - 2 \frac{\tilde{ρ}}{{vp}_{t}^{2}} + \frac{1}{v^{2} p_{t}^{2}}} \\ d & = & \frac{p_{t}^{2}}{2 σ^{2} (p_{t}^{2} - {\tilde{ρ}}^{2})} [(μ (t)^{2} + \frac{{\tilde{μ (t)}}^{2}}{p_{t}^{2}} - 2 \tilde{ρ} \frac{μ (t) \tilde{μ (t)}}{p_{t}^{2}}) - \frac{{[μ (t) + \frac{\tilde{μ (t)}}{{vp}_{t}^{2}} - \frac{\tilde{ρ}}{{vp}_{t}^{2}} (μ (t) + v \tilde{μ (t)})]}^{2}}{1 - 2 \frac{\tilde{ρ}}{{vp}_{t}^{2}} + \frac{1}{v^{2} p_{t}^{2}}}] \end{matrix}

Proof The proof of this theorem is discussed in detail in Appendix A. 1.

We depict the plot of the probability density functions of IMR for the CBCB model with parameters $a = 0.01, b = 1, c = 2$ . We have generated the RGR from the multivariate normal distribution with the specified parameter values over 20 time-points using different $σ$ and $ρ$ . We have calculated the density of IMR using these RGR. We plot the density functions of IMR for different parameter values of the $σ$ and $ρ$ over various time points in Figures 1 and 2. It is clear from Figures 1 and 2 IMR is positively skewed for all the cases. We can see from Figure 2 that IMR is more peaked if we increase the value of $ρ$ for all the time points.

We can see that the form of the density function of IMR (Equation 3.2) is complicated. The moments of this distribution are difficult to calculate. We have calculated the distribution of IMR for theoretical interest. We derive the asymptotic sampling distribution of IMR for the goodness-of-fit test, shown in the next subsection.

3.2 The Asymptotic Distributions of IMR

The asymptotic distributions play a key role in finding solutions when the exact distributions are complicated and challenging to work. We derive the asymptotic distributions (taking the no. of individuals large) of IMR assuming normality on both the RGR variable and the size variable.

Assumption 1. Let us assume that the RGR vector of $q - 1$ time points $R = (R (1), R (2), . . ., R (q - 1))^{T}$ , follow multivariate normal distribution with mean as the RGR from the CBCB model (Equation 2.4) and the dispersion matrix $Σ = (σ^{2} ρ^{| i - j |})$ , $i, j = 1, 2, . . ., q - 1$ . Let $R_{i} = (R_{i} (1), R_{i} (2), . . ., R_{i} (q - 1))^{T}, i = 1, 2, 3, . . ., n$ are n i.i.d. multivariate normal samples, where $R_{i} (t)$ be the RGR of $i^{th}$ individual at $t^{th}$ time point.

Theorem 2. Let us consider the Assumption 3.2 and define the RGR of the mean process as $\overset{̅}{R (t)} = \frac{1}{n} \sum_{i = 1}^{n} R_{i} (t), t = 1, 2, . . ., q - 1$ . If the IMR is estimated by $\hat{m (t)} = \frac{\overset{̅}{R (t)}}{\sum_{t}^{q - 1} \overset{̅}{R (t)}}$ then the asymptotic distribution of $\hat{m} = (\hat{m (1)}, \hat{m (2)}, \dots, \hat{m (q - 2)})^{T}$ is given by

\sqrt{n} (\hat{m} - μ_{m 1}) thicksim N_{q - 2} (0, Σ_{m 1})

μ_{m 1} = {(μ_{m 1} (1), \dots, μ_{m 1} (q - 2))}^{T} where μ_{m 1} (t) = \frac{μ (t)}{\sum_{k = t}^{q - 2} μ (k)}; t = 1, \dots, q - 2 .

and

Σ_{m 1} = D_{m 1}^{T} Σ D_{m 1} a n d D_{m 1} = (d_{i j}),

where

d_{ij} = \{\begin{matrix} \frac{\sum_{k = i}^{q - 1} μ (k) - μ (i)}{{(\sum_{k = i}^{q - 1} μ (k))}^{2}} & if j = i \\ - \frac{μ (j)}{{(\sum_{k = i}^{q - 1} μ (k))}^{2}} & if j = (i + 1), \dots, (q - 2) \\ 0 & otherwise . \end{matrix}

(3.3)

Proof Proof of theorem is given in Appendix A.2.

Remark 1. We consider the IMR vector $\hat{m} = (\hat{m (1)}, \hat{m (2)}, \dots, \hat{m (q - 2)})^{T}$ to calculate the distribution of IMR. We exclude the IMR of $(q - 1)^{th}$ time point as its empirical estimate is always 1.

Assumption 2. Normality assumption on size variable: Let us define $X = (X (1), X (2), . . ., X (q))^{T}$ be the size vector and $X_{i} = (X_{i} (1), X_{i} (2), . . ., X_{i} (q))^{T}, i = 1, 2, 3, . . ., n$ , are n i.i.d. multivariate normal samples having mean $μ^{*} (t)$ from the size of the CBCB model (Equation 2.5) and $Cov (X (s), X (t)) = σ^{2} ρ^{| s - t |}$ for $s, t = 1, 2, . . ., q$ . $X_{i} (t), i = 1, 2, . . ., n; t = 1, 2, . . ., q$ be the size of $i^{th}$ individual at $t^{th}$ time point.

Note that for a given $t$ , $X_{i} (t), i = 1, 2, . . ., n$ are the n i.i.d. samples from univariate normal with mean $μ^{*} (t)$ and variance $σ^{2}$ . We define $\overset{̅}{X (t)} = \frac{1}{n} \sum_{i = 1}^{n} X_{i} (t), t = 1, 2, . . ., q$ which can be thought as the size of the mean process. We can estimate the relative growth rate $R (t)$ at time point $t = 1, 2, . . ., q - 1$ by the following estimators:

$\hat{R (t)} = ln (\frac{\overset{̅}{X (t + 1)}}{\overset{̅}{X (t)}})$

$\hat{R (t)} = \frac{1}{n} \sum_{i = 1}^{n} ln (\frac{X_{i} (t + 1)}{X_{i} (t)})$

In the next two theorems we provide the asymptotic distribution of IMR vector $\hat{m} = (\hat{m (1)}, \hat{m (2)}, \dots, \hat{m (q - 2)})^{T}$ when RGR is estimated by the above two estimators respectively.

Theorem 3. Let us consider size vector $X$ follows a normal distribution with the mean and covariance structures as given in Assumption 3.2, and RGR is estimated by $\hat{R (t)} = ln (\frac{\overset{̅}{X (t + 1)}}{\overset{̅}{X (t)}})$ . If IMR is estimated using the Eq. (3.1) then

\sqrt{n} (\hat{m} - μ_{m 2}) thicksim N_{q - 2} (0, Σ_{m 2})

where,

μ_{m 2} = {(μ_{m 2} (1), \dots, μ_{m 2} (q - 2))}^{T} where μ_{m 2} (t) = \frac{ln (\frac{μ^{*} (t + 1)}{μ^{*} (t)})}{\sum_{k = t}^{q - 2} ln (\frac{μ^{*} (k + 1)}{μ^{*} (k)})}; t = 1, \dots, q - 2 .

and $Σ_{m 2} = D_{m 2}^{T} Σ^{*} D_{m 2}$ where, $D_{m 2} = (g_{ij})$ and

g_{ij} = \{\begin{matrix} \frac{\frac{- d (i)}{μ^{*} (i)} + \frac{1}{μ^{*} (i)} ln (\frac{μ^{*} (i + 1)}{μ^{*} (i)})}{d^{2} (t)}, & if j = i \\ \frac{1}{d^{2} (i) μ^{*} (i + 1)} & if j = i + 1 and d (i) = \sum_{k = i}^{q - 1} ln \frac{μ^{*} (k + 1)}{μ^{*} (k)} \\ 0 & otherwise . \end{matrix}

(3.4)

Proof

The proof of the theorem has been given in Appendix A.3.

Theorem 4. Under the normality assumption (Assumption 3.1) on size variable if the relative growth rate $R (t)$ at time point $t = 1, 2, 3, . . ., q - 1,$ are estimated by

\hat{R (t)} = \frac{1}{n} \sum_{i = 1}^{n} ln \frac{X_{i} (t + 1)}{X_{i} (t)}

(3.5)

and the IMR at time point $t$ is estimated using Equation (3.1) then the asymptotic distribution of IMR is,

\sqrt{n} (\hat{m} - μ_{m 3}) thicksim N_{q - 2} (0, Σ_{m 3})

where

\begin{matrix} μ_{m 3} & = & {(μ_{m 3} (1), \dots, μ_{m 3} (q - 2))}^{T} where μ_{m 3} (t) \\ = & \frac{ln \frac{μ^{*} (t + 1)}{μ^{*} (t)} - \frac{1}{2} (V (t + 1)^{2} - V (t)^{2})}{\sum_{k = t}^{q - 2} (ln (\frac{μ^{*} (k + 1)}{μ^{*} (k)}) - \frac{1}{2} (V (t + 1)^{2} - V (t)^{2}))}; t = 1, \dots, q - 2 . \end{matrix}

and $Σ_{m 3}$ = $D_{g} {^{T}}_{(μ^{*})} Σ^{*} D_{g (μ^{*})}$

Proof

The proof of the above theorem can be done using the multivariate delta method.

4 Goodness-of-fit Test Using IMR

We need to find a function of the size or the RGR which is constant over the size or the time to develop a goodness-of-fit testing method for a growth model. For the CBCB model (Equation 2.4) we have $L (t) = \frac{d ln (\frac{d^{2} ln R (t)}{{dt}^{2}})}{d ln t} = constant .$ However, the expression of $L (t)$ is complex and involves multiple uses of the delta function. It may cause problems while considering the Gaussian approximation of $L (t)$ .

We propose an appropriate function of IMR, which is a constant and provides an essential characterization of the CBCB model using IMR. We have already seen that IMR of CBCB model is $m (t) = {act}^{c - 1}$ (see Equation 2.6). Taking the logarithm of both sides and using the differential operator, we get

\frac{Δ ln m (t)}{Δ ln t} = c - 1 .

We call the function $\frac{Δ ln m (t)}{Δ ln t}$ as $H (t)$ and we shall find the distribution of $H (t)$ which depends on the distribution of IMR.

Remark 1. The concept of Weibull probability plot can provide an alternate function of size variable which is constant. It can be shown that the standardized cumulative growth rate ( $S (t)$ ) for the CBCB model is $\frac{ln X (t) - ln X (0)}{ln K - ln X (0)} = 1 - e^{- {at}^{c}} .$ Therefore, we can have $\frac{Δ ln (- ln (1 - S (t)))}{Δ ln t} = c$ .

4.1 Distribution of $H (t)$

Let us write the empirical estimator of $H (t)$ as follows:

\hat{H (t)} = \frac{Δ ln \hat{m (t)}}{Δ ln t}

Let us denote $\hat{H} = {(\hat{H (1)}, \hat{H (2)}, \dots, \hat{H (q - 4)})}^{T}$ . We derive here the asymptotic distribution of $\hat{H}$ under the normality assumption of RGR and size variable respectively.

Theorem 5. If we consider the distributional assumption on the RGR variable then

\sqrt{n} (\hat{H} - μ_{H}^{(1)}) \sim N_{q - 4} (0, Σ_{H}^{(1)})

where

μ_{H}^{(1)} = {(μ_{H}^{(1)} (1), \dots, μ_{H}^{(1)} (q - 4))}^{T} w i t h μ_{H}^{(1)} (t) = \frac{Δ ln E (m (t))}{Δ ln t} = \frac{Δ ln μ_{m 1} (t)}{Δ ln t} \forall t

and

Σ_{H}^{(1)} = Ω Σ_{m 1} Ω^{T} where Ω = (ω_{ij})

and

ω_{ij} = \{\begin{matrix} \frac{1}{Δ ln i} & if j = i, i + 2 \\ \frac{- 2}{Δ ln i} & if j = i + 1 \\ 0 & otherwise . \end{matrix}

(4.1)

Proof.

The proof of the above theorem can be done using the multivariate delta method.

Theorem 6. Under the distributional Assumption 3.2 on size and if RGR is estimated by $\hat{R (t)} = ln (\frac{\overset{̅}{X (t + 1)}}{\overset{̅}{X (t)}})$ we have the following distribution

\sqrt{n} (\hat{H} - μ_{H}^{(2)}) \sim N_{q - 4} (0, Σ_{H}^{(2)}),

where

μ_{H}^{(2)} = {(μ_{H}^{(2)} (1), \dots, μ_{H}^{(2)} (q - 4))}^{T} w i t h μ_{H}^{(2)} (t) = \frac{Δ ln E (m (t))}{Δ ln t} = \frac{Δ ln μ_{m 2} (t)}{Δ ln t} f o r t = 1, 2, ..., q - 4

and

Σ_{H}^{(2)} = Ω Σ_{m 1} Ω^{T},

where $Ω$ is as given in Equation (4.4).

Proof.

The proof of the above theorem can be done using the multivariate delta method.

4.2 The Null Hypothesis

We have the following distributional assumptions when the data are generated from the CBCB model

[H (t)] = θ_{H} (t) = constant \forall t

(4.2)

Alternatively, we can write the null hypothesis as

H_{0} = (H) = a constant vector against H_{1} = not H_{0}

(4.3)

The null hypothesis $H_{0}$ of a CBCB model is rejected unless the proposed test leads to insignificant results.

4.2.1 The Proposed Testing Methods

We follow the following approach to test the null hypothesis (4.8). Let $P$ be a $(q - 5) \times (q - 4)$ full rank matrix such that $P 1 = 0$ . We have transformed the random vector $H$ to a new random vector $U$ where $U = P^{T} H$ . Here, $U$ follows a $(q - 5)$ -variate normal distribution with mean $θ_{u} = P^{T} H$ and the dispersion matrix $Σ_{u} = P Σ_{H} P^{T}$ . It can be observed that under the null hypothesis in Equation (4.3), $θ_{u}$ becomes a null vector. Based on Theorem 5 and Theorem 6, we propose two test statistics to test the null hypothesis of the goodness-of-fit of the CBCB model:

Test statistic I:

T_{1} = n {\hat{U}}^{T} {\hat{Σ_{H}^{(1)}}}^{- 1} \hat{U}

(4.4)

Test statistic II:

T_{2} = n {\hat{U}}^{T} {\hat{Σ_{H}^{(2)}}}^{- 1} \hat{U}

(4.5)

Here $\hat{Σ_{H}^{(1)}}$ and $\hat{Σ_{H}^{(2)}}$ are the consistent estimators of $Σ_{H}^{(1)}$ and $Σ_{H}^{(2)}$ . Under the null hypothesis in Equation (4.3), both the test statistic follows chi-square distribution with $(q - 5)$ degrees of freedom.

5 Numerical Studies

5.1 Simulation Study

We observe the power of the test with simulated data to study the performance of the proposed test. First, we assume normality on the RGR variable of the growth model and generate a data set. We have simulated the data from the growth curve model with the RGR

R^{'} (t) = γ {[1 - e^{- a t^{c}}]}^{γ - 1} R (t); a > 0, c > 0, γ > 0.

(5.1)

The model reduces to the CBCB model when $γ = 1$ . We generate a sample of 15 individuals at the time points $t = 1, 2, . . ., 10$ having a joint multivariate normal distribution with mean structure given in Equation (5.1) with the parameter $a = 0.87, b = 0.1, c = 0.299$ . The covariance matrix of the distribution of the RGR variable follows AR(1) structure under homoscedasticity with the parameters $σ = 0.011$ and $ρ = 0.75$ . We calculate the test statistic given in Equation (4.9) using the simulated data set. We repeat the process 1000 times and note the number of rejections of the null hypothesis as the power of the test. Again we calculate the power of the test for different values of $γ$ . The plot of the power curve has been depicted in Figure 3a.

Now, we assume the multivariate normality on the size variable. We need the explicit form of the size variable (Equation 5.1) to simulate data. However, the closed form of size is not available. Hence we have approximated the RGR variable $R (t)$ by $\frac{Δ ln X (t)}{Δ t}$ using recursive formula. We generate an observation vector of the size variable for $t = 1, 2, . . . ., 10$ using the form of the RGR variable given in Equation (5.1) with the parameters $a = 0.32, b = 0.5, c = 1.05$ . We generate a multivariate normal sample of 15 individuals with the RGR vector as mean process and the specified dispersion matrix with $σ = 0.015$ , and $ρ = 0.7$ and calculate the value of test statistic given in Equation (4.10). We replicate this process 1000 times and take the proportion of samples leading to the rejection of the null as the test's observed power. Again we calculate the power for different values of $γ$ and represent the power curve in Figure 3b. We observe from Figure 3 that as the value of $γ$ shifts from 1, i.e., the true nature of the curves move away from the CBCB model, both the tests are more likely to reject.

5.2 Real-Data Example

We study the performance of the proposed test based on a real-life data set. The data set consists of measurements on the body weights of 48 different pigs during nine successive weeks of follow-up. The entire data is available at the website http://www.biostat.jhsph.edu/ fdominic/teaching/bio655/data/text/pigs.data. We have the observations for each individual pig on its RGR at eight different time points, and for a given time point, we have 48 independent and identically distributed observations. The objective is to check whether the growth of pigs follows the CBCB model.

We calculate the Chi-square test statistic $T_{1}$ and its value turned out to be $44.4317$ , and the $p$ -value is $1.63 \times 10^{- 5}$ . So the test suggests that the pig-growth data does not follow the CBCB model. Note that Chakraborty et al.^[7] developed the Goodness of test procedure for the Gompertz growth model, which can be obtained in our proposed approach taking $c = 1$ . We get the same conclusion that data does not originate from the Gompertz growth model. However, in this study, we also see that it is not even originated from the larger class of growth model, the CBCB model.

6 Concluding Remarks

Development of Goodness of fit test has achieved significant importance since last decades^{[7, 10, 11]} apart from the usual model selection criteria, namely R-square, adjusted R-square AIC, BIC, etc. In most of these studies, the authors developed the goodness-of-fit test by identifying transform functions of size and RGR variables. Note that RGR cannot capture the correct status of the species growth. Hence, a sophisticated and accurate measure of the growth of the species is demanding. Our study proposed a refined measure IMR for capturing the true status of growth and used this rate function for the goodness of fit test for a complex growth curve. The present study ensures that the proposed goodness-of-fit test based on IMR is powerful as the power increases with a slide departure from the underlying model. The goodness-of-fit test for some growth models, such as the extended Gompertz model proposed by Bhowmick and Bhattacharya.^[4] can be developed using the IMR.

Footnotes

Acknowledgements

We are thankful to the anonymous reviewers for their suggestions that greatly improved the revised version of the manuscript from its earlier versions. We sincerely thank the Associate Editor Professor Aditya Chattopadhyay and the Former Editor-in-Chief Professor Uttam Bandyopadhyay for their valuable suggestions.

Declaration of Conflicting Interests

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Appendix

References

Tsoularis

, Wallace

Analysis of logistic growth models. Math Biosci . 2002; 179(1): 21–55.

Tabatabai

, Williams

, Bursac

Hyperbolastic growth models. Theor Biol Med Model . 2005; 14.

Koya

, Goshu

Generalized mathematical model for biological growths. Open J of Model Simul . 2013; 1: 42–53.

Bhowmick

, Bhattacharya

A new growth curve model for biological growth: Some inferential studies on the growth curve of

Cirrhinus mrigala. Math Biosci . 2014; 254: 28–41.

Crescenzo

, Spina

Analysis of a growth model inspired by Gompertz and Korf laws, and an analogous birth-death process. Math Biosci . 2016; 282: 121–134.

Chakraborty

, Bhowmick

, Chattopadhyay

, Bhattacharya

Physiological responses of fish under environmental stress and extension of growth (curve) models. Ecol Model . 2017; 363: 172–186.

Chakraborty

, Bhattacharya

, Basu

, Bandyopadhyay

, Bhattacharjee

Goodness-of-fit testing for the Gompertz growth curve model. METRON . 2014; 72: 45–64.

Basu

, Bhattacharjee

A natural goodness-of-fit test for the exponential growth curve model based on the relative growth rate. Bull Calcutta Stat Assoc . 2006; 58: 231–232.

Hill

A short note on growth curve analysis. Biometrics . 1968; 4: 192–196.

10.

Bhattacharya

, Basu

, Bandyopadhyay

Goodness-of-fit testing for exponential polynomial growth curves. Commun Stat Theory Methods . 2008; 38: 340–363.

11.

Chakraborty

, Basu

A natural goodness-of-fit testing procedure for the logistic growth curve model. Calcutta Stat Assoc Bull . 2008; 60: 53–70.

12.

Hoffmann

, Poorter

Avoiding bias in calculations of relative growth rate. Ann Bot . 2002; 90: 37–42.

13.

Fisher

Some remarks on the methods formulated in a recent article on the quantitative analysis of plant growth. Ann Appl Biol . 1921; 7: 367–372.

14.

Chakraborty

, Bhowmick

, Chattopadhyay

, Bhattacharya

A novel unification method to characterize a broad class of growth curve models using relative growth rate. Bull Math Biol . 2019; 81: 2529–2552.

15.

Veres-Ferrer

, Pavia

On the relationship between the reversed hazard rate and elasticity. StatPapers . 2014; 55: 275–284.

16.

Mukhopadhyay

, Hazra

, Bhowmick

, Bhattacharya

On comparison of relative growth rates under different environmental conditions with application to biological data. METRON . 2016; 74: 311–337.

17.

Pal

, Bhowmick

, Yeasmin

, Bhattacharya

Evolution of model specific relative growth rate: Its genesis and performance over fisher's growth rates. J Theor Biol . 2018; 444: 11–27.

A New Growth Rate Measure in Identifying Extended Gompertz Growth Curve and Development of Goodness-of-fit Test

Abstract

Keywords

1 Introduction

2 Growth Rate and Its Proposed Extension

3 Distribution of IMR

4.1 Distribution of H ( t )

5.1 Simulation Study

6 Concluding Remarks

Footnotes

Acknowledgements

Declaration of Conflicting Interests

Funding

Appendix

References

4.1 Distribution of $H (t)$