On comparison of relative growth rates under different environmental conditions with application to biological data

Abstract

In this paper, we discuss a few empirical estimates of relative growth rate and their distributions based on two well-established parametric structures of the size variable- normal and log-normal with different growth curve models as the mean process. We discuss some exact and asymptotic testing procedures applicable to test the equality of expected relative growth rates over two or more different environmental situations, e.g. distinct geographical locations, several biological species, etc. Along with the tests under the parametric structures for the size variable, we consider some nonparametric tests as well. We perform tests for two other important hypotheses—whether the times taken to reach the maximum values of the relative growth rates are equal for all environments and whether those maximum values are equal. We illustrate our methods with two real life examples. The performances of the various testing procedures are studied by means of their power functions. Our methods have immense practical value in growth rate modeling in the biological context.

Appendix

1.1 Proof of Theorem 2.1

The RGR estimate for the i-th individual from the j-th population at time t, i.e. $R_{ji}(t)$ can be written as

$$\begin{aligned} R_{ji}(t)= & {} \ln \left( \frac{X_{ji}(t+1)}{X_{ji}(t)}\right) = \ln \left[ \frac{\mu _j(t+1) \left( 1 + \frac{X_{ji}(t+1)-\mu _j(t+1)}{\mu _j(t+1)} \right) }{\mu _j(t) \left( 1 + \frac{X_{ji}(t)-\mu _j(t)}{\mu _j(t)} \right) }\right] \\= & {} \ln \left( \frac{\mu _j(t+1)}{\mu _j(t)}\right) +\ln \left[ 1+\left( \frac{X_{ji}(t+1) -\mu _j(t+1)}{\mu _j(t+1)}\right) \right] \\&-\ln \left[ 1+\left( \frac{X_{ji} (t)-\mu _j(t)}{\mu _j(t)}\right) \right] \\= & {} \ln \left( \frac{\mu _j(t+1)}{\mu _j(t)}\right) +\sum _{k=1}^{\infty } {(-1)}^{k-1}\frac{{(X_{ji}(t+1)-\mu _j(t+1))}^k}{k\mu _j^k(t+1)}\\&-\sum _{k=1}^{\infty }{(-1)}^{k-1}\frac{{(X_{ji}(t)-\mu _j(t))}^k}{k\mu _j^k(t)}\\\approx & {} \ln \left( \frac{\mu _j(t+1)}{\mu _j(t)}\right) +\frac{{X_{ji}(t+1)-\mu _j(t+1)}}{\mu _j(t+1)}-\frac{{(X_{ji}(t+1)-\mu _j(t+1))}^2}{2\mu _j^2(t+1)}\\&-\frac{{X_{ji}(t)-\mu _j(t)}}{\mu _j(t)} +\frac{{(X_{ji}(t)-\mu _j(t))}^2}{2\mu _j^2(t)}. \end{aligned}$$

Here we are retaining up to the 2nd order terms. Rao [37], Chakraborty et al. [13], Bhattacharya et al. [4] have made sincere efforts for establishing the convergence of the series with higher order terms. We have provided the justification for the above approximation in our main article.

Now, the approximate expression for the mean of $R_{ji}(t)$, i.e. $\mu ^*_j(t)$ in Eq. 3, follows from the expectation terms– $E[X_{ji}(t+1)] = \mu _j(t+1)$, $E[X_{ji}(t)] = \mu _j(t)$, $E[{(X_{ji}(t+1)-\mu _j(t+1))}^2] = \sigma _j(t+1, t+1)$ and $E[{(X_{ji}(t)-\mu _j(t))}^2] = \sigma _j(t, t)$ with the notations $V_{j,t+1} = \frac{\sqrt{\sigma _j(t+1,t+1)}}{\mu _j(t+1)}$, $V_{j,t} = \frac{\sqrt{\sigma _j(t,t)}}{\mu _j(t)}$.

The approximate expression for the variance of $R_{ji}(t)$, i.e. $\sigma ^*_j(t,t)$ in Eq. 4, follows from the variance and covariance terms– $Var(X_{ji}(t+1)-\mu _j(t+1))=\sigma _j(t+1,t+1)$, $Var(X_{ji}(t)-\mu _j(t))=\sigma _j(t,t)$, $Var({(X_{ji}(t+1)-\mu _j(t+1))}^2)=2\sigma _j^2(t+1, t+1)$, $Var({(X_{ji}(t)-\mu _j(t))}^2)=2\sigma _j^2(t, t)$, $Cov(X_{ji}(t+1)-\mu _j(t+1), X_{ji}(t)-\mu _j(t))=\rho _{j} \sqrt{\sigma _j(t+1,t+1)} \sqrt{\sigma _j(t,t)}$, $Cov({(X_{ji}(t+1)-\mu _j(t+1))}^2,{(X_{ji}(t)-\mu _j(t))}^2) = \rho _{j}^2 \sigma _j(t+1,t+1) \sigma _j(t,t)$ and the rest of the covariance terms are zeros. $\rho _j$ is the lag 1 autocorrelation of the size variable.

Similarly, the approximate expression for the covariance of $R_{ji}(s)$ and $R_{ji}(t)$, i.e. $\sigma ^*_j(s,t)$ follows from the Taylor series expansion of both of them with retaining up to second order terms and the covariance terms of the pairs– $Cov(X_{ji}(s+1)-\mu _j(s+1),X_{ji}(t+1)-\mu _{j}(t+1))=\rho _{j}^{|s-t|} \sqrt{\sigma _{j}(s+1,s+1)} \sqrt{\sigma _{j}(t+1,t+1)}$, $Cov(X_{ji}(s+1)-\mu _{j}(s+1), X_{ji}(t)-\mu _j(t))=\rho _{j}^{|s+1-t|}\sqrt{\sigma _{j}(s+1,s+1)}\sqrt{\sigma _{j}(t,t)}$, $Cov(X_{ji}(s)-\mu _j(s),X_{ji}(t+1)-\mu _{j}(t+1))=\rho _{j}^{|s-t-1|} \sqrt{\sigma _{j}(s,s)} \sqrt{\sigma _{j}(t+1,t+1)}$, $Cov(X_{ji}(s)-\mu _{j}(s), X_{ji}(t)-\mu _j(t))=\rho _{j}^{|s-t|}\sqrt{\sigma _{j}(s,s)}\sqrt{\sigma _{j}(t,t)}$, $Cov[(X_{ji}(s+1)-\mu _j(s+1))^2, (X_{ji}(t+1)-\mu _{j}(t+1))^2] =\rho _{j}^{2|s-t|} {\sigma _{j}(s+1,s+1)} {\sigma _{j}(t+1,t+1)}$, $Cov[(X_{ji}(s+1)-\mu _{j}(s+1))^2, (X_{ji}(t)-\mu _j(t))^2] = \rho _{j}^{2|s+1-t|}{\sigma _{j}(s+1,s+1)}{\sigma _{j}(t,t)}$, $Cov[(X_{ji}(s)-\mu _j(s))^2, (X_{ji}(t+1)-\mu _{j}(t+1))^2]=\rho _{j}^{2|s-t-1|} {\sigma _{j}(s,s)} {\sigma _{j}(t+1,t+1)}$, $Cov[(X_{ji}(s)-\mu _{j}(s))^2, (X_{ji}(t)-\mu _j(t))^2]=\rho _{j}^{2|s-t|} {\sigma _{j}(s,s)} {\sigma _{j}(t,t)}$ and the rest of the covariance terms are zeros.

1.2 Derivation of exact distribution of $\frac{R}{S}$ in Sect. 3.1

We consider the transformation $Z=\frac{R}{S}, W=S^2$ i.e. $S=\pm {\sqrt{W}}, R=\pm {Z\sqrt{W}}$. Here the supports of R and S are given by $-\infty<R<\infty , -\infty<S<\infty$ and hence, $0<W<\infty , -\infty<Z<\infty$. The transformation from (R, S) to (Z, W) is a two-to-one transformation.

For the case $0<s<\infty ; -\infty<r<\infty$, the inverse mappings are given by $S=\sqrt{W} ; R=Z\sqrt{W}$. One can simply calculate the Jacobian of this transformation and is equal to $\frac{1}{2}$. In this case, the exponent in the joint pdf of R and S simplifies to $-\frac{1}{2(1-\rho ^2)}\left[ aw+2b\sqrt{w}+c\right]$ where $a=z^2-2\rho z+1, b=\rho k-kz+mz\rho -m, c=k^2-2\rho km+m^2$.

In case $-\infty<s<0 ; -\infty<r<\infty$, the inverse mappings are $S=-\sqrt{W}; R=-Z\sqrt{W}$. Again, one can simply calculate the Jacobian of this transformation and is equal to $\frac{1}{2}$. Here the exponent equals $-\frac{1}{2(1-\rho ^2)}\left[ aw-2b\sqrt{w}+c\right]$ where a, b and c are defined as earlier.

Combining these two, we have the joint pdf of (Z, W) as

$$\begin{aligned} f(z,w)&= \frac{1}{2}\frac{1}{2\pi \sqrt{(1-\rho ^2)}}{\exp \left[ - \frac{a}{2(1-\rho ^2)}\left( {\left( \sqrt{w}+\frac{b}{a}\right) }^2 +\left( \frac{c}{a}-\frac{b^2}{a^2}\right) \right) \right] }\\&\qquad + \frac{1}{2}\frac{1}{2\pi \sqrt{(1-\rho ^2)}}{\exp \left[ -\frac{a}{2(1-\rho ^2)} \left( {\left( \sqrt{w}-\frac{b}{a}\right) }^2+\left( \frac{c}{a}-\frac{b^2}{a^2}\right) \right) \right] } \end{aligned}$$

The marginal density of Z is obtained by integrating over W as follows

$$\begin{aligned} f_1(z)= & {} \frac{1}{2\sqrt{2\pi }}\exp {\left[ \frac{1}{2(1-\rho ^2)}\left( c-\frac{b^2}{a}\right) \right] }\left[ I_1+I_2\right] \end{aligned}$$

where one can easily evaluate $I_1$ and $I_2$ by integrating the corresponding terms and they are

$$\begin{aligned} I_1= & {} 2\left[ \frac{1}{a\sqrt{2\pi }}\exp {\left( -\frac{b^2}{2\left( 1-\rho ^2\right) } \right) }-\frac{b}{\root 2 \of {a^3}}\left( 1-\Phi \left( \frac{b}{\sqrt{a\left( 1-\rho ^2\right) }} \right) \right) \right] \\ I_2= & {} 2\left[ \frac{1}{a\sqrt{2\pi }}\exp {\left( -\frac{b^2}{2\left( 1-\rho ^2\right) } \right) }+\frac{b}{\root 2 \of {a^3}}\left( 1-\Phi \left( \frac{-b}{\sqrt{a\left( 1-\rho ^2\right) }}\right) \right) \right] \end{aligned}$$

Combining these two, our result follows.

1.3 Derivation of the asymptotic null distribution of $T_1$ and $T_2$

The null hypothesis, $H_0: \nu _1=\nu _2=\ldots \nu _k=\nu (say)$; after simple calculation $T_1$ can be written as

$$\begin{aligned} T_1= & {} {\sum _{j=1}^k}n_j(\overline{R_j}-\nu )'{\widehat{\Sigma _j}} ^{-1}(\overline{R_j}-\nu )-{\sum _{j=1}^k}n_j(\widehat{\nu }-\nu )' {\widehat{\Sigma _j}}^{-1}(\widehat{\nu }-\nu ). \end{aligned}$$

where $\overline{R_j} = \frac{1}{n_j}{\sum _{i=1}^{n_j}}R_{ji}$ and $\widehat{\nu } ={\left( \sum _{j=1}^{k}n_j\widehat{\Sigma _j}^{-1}\right) }^{-1} \sum _{j=1}^{k}n_j \widehat{\Sigma _j}^{-1}\overline{R_j}$. Now, under $H_0$, $\sqrt{n_j}{\widehat{\Sigma _j}}^{-\frac{1}{2}}(\overline{R_j}-\nu ) \xrightarrow {d} N_{q-1}(\mathbf 0 ,I)$ since $\sqrt{n_j}{\Sigma _j}^{-\frac{1}{2}}(\overline{R_j}-\nu ) \xrightarrow {d} N_{q-1}(\mathbf 0 ,I)$ by multivariate CLT and $\widehat{\Sigma _1} \xrightarrow {d} \Sigma _1$ by consistency of the individual statistics. Hence, $n_j(\overline{R_j}-\nu )'{\widehat{\Sigma _j}}^{-1}(\overline{R_j}-\nu ) \xrightarrow {d} \chi ^2_{q-1}$ independently for all $j=1, \ldots , k$. Thus, ${\sum _{j=1}^k}n_j(\overline{R_j}-\nu )' {\widehat{\Sigma _1}}^{-1}(\overline{R_j}-\nu ) \xrightarrow {d} \chi ^2_{k(q-1)}$.

For the second term in the expression of $T_1$, $E(\widehat{\nu })=\nu$, $Var(\widehat{\nu })=\left( \sum _{j=1}^{k}n_j{\Sigma _j}^{-1}\right) ^{-1}$. Accordingly, we have ${\sum _{j=1}^k}n_j(\widehat{\nu }-\nu )'{\widehat{\Sigma _j}}^{-1} (\widehat{\nu }-\nu ) \xrightarrow {d} \chi ^2_{(q-1)}$

We are left with showing independence of $(\overline{R_j}-\widehat{\nu })$ and $\widehat{\nu }$. Due to their asymptotic normality, it suffices to show that they are uncorrelated. A straightforward calculation shows– $Cov(\overline{R_j},\widehat{\nu })= {\left( \sum _{j=1}^{k}n_j{\Sigma _j}^{-1}\right) }^{-1}$, $Cov(\widehat{\nu },\widehat{\nu })= {\left( \sum _{j=1}^{k}n_j{\Sigma _j}^{-1}\right) }^{-1}$ and $Cov (\overline{R_j}-\widehat{\nu }, \widehat{\nu }) = 0$. Hence, we can say that $T_1$ is asymptotic $\chi ^2$ with df $k(q-1)-(q-1)=(k-1)(q-1)$.

Now we will discuss about the asymptotic distribution of $T_2$. For n positive observations $X_1,\ldots ,X_n$, the approximation of geometric mean to arithmetic mean provided by [47] is

$$\begin{aligned} G= & {} \overline{X}\left( 1-\frac{M_2(\overline{X})}{2{\overline{X}}^2}+ \frac{M_3(\overline{X})}{3{\overline{X}}^3}-\frac{M_4(\overline{X})}{4{\overline{X}}^4}+\cdots \right) \end{aligned}$$

(14)

where G is the geometric mean, $\overline{X}$ is the arithmetic mean and $M_k(X)=\frac{1}{n}\sum _{i=1}^{n}{(X_i-\overline{X})}^k$. Now, if $\frac{M_k(\overline{X})}{k{\overline{X}}^k}$ is small for $k= 2,\ldots$, then we have $\overline{X} \approx G$. It is a very realistic assumption in our case and also follows from Remarks 1–3. Another reason is as follows. Suppose we write the equation as $G=M\overline{X}$. Then $\overline{X_j}(t+1)={\frac{G_j(t+1)}{M_j(t+1)}}$ and $\overline{X_j}(t)={\frac{G_j(t)}{M_j(t)}}$ and thus, ${R_j}^{*}(t) = \ln \left( {\frac{G_j(t+1)}{M_j(t+1)}} / {\frac{G_j(t)}{M_j(t)}}\right) =\ln \left( \frac{G_j(t+1)}{G_j(t)} \right) -\ln \left( \frac{M_j(t+1)}{M_j(t)} \right) \approx \ln \left( \frac{G_j(t+1)}{G_j(t)}\right)$ as $M_j(t+1) \approx M_j(t)$ which is even more realistic assumption given that the lag between two time points is small. Using the approximation, we replace $\overline{X_j}(t)$ by $\root n_j \of {\prod _{i=1}^{n_j} X_{ji}(t)}$ in the expression of ${R_j}^{*}(t)$ to obtain ${R_j}^{*} = \overline{R_j}$ and $\widehat{\nu ^{*}} = \widehat{\nu }$. Hence the expression of $T_2$ becomes approximately same as $T_1$, i.e., the asymptotic distribution of $T_2$ is similar to that of $T_1$.

1.4 Derivation of the asymptotic null distribution of $\tau$

Here our null hypothesis is given by $H_0 : m_1 = m_2 = \cdots = m_k$, $\widehat{m_j}$ is an unbiased estimator of $m_j$ with $Var(\widehat{m_j})= V_j$ and $\widehat{V_j}$ denotes a consistent estimator of $V_j$ for all $j=1,2, \ldots , k$. Under $H_0 : m_1 = m_2 = \cdots = m_k = m$(say), we consider an estimator $\widehat{m}$ of m and a pivotal quantity $\widetilde{m}$ given by $\widehat{m}={\left( \sum _{j=1}^{k}\frac{1}{\widehat{V_j}}\right) }^{-1}{\sum _{j=1}^{k}\frac{\widehat{m_j}}{\widehat{V_j}}}$ and $\widetilde{m}={\left( \sum _{j=1}^{k}\frac{1}{{V_j}}\right) }^{-1}{\sum _{j=1}^{k}\frac{\widehat{m_j}}{{V_j}}}$ respectively.

Also, we consider a proper test statistic $\tau$ in our set-up which is given by $\tau = \sum _{j=1}^{k}\frac{{(\widehat{m_j}-\widehat{m})}^2}{\widehat{V_j}}$.

As $\widehat{V_j}$ is a consistent estimator of $V_j$, the asymptotic null distribution of $\tau$ is same as that of the pivotal quantity given by $\tau ^{*} = {\sum _{j=1}^{k}}\frac{{(\widehat{m_j}-\widetilde{m})}^2}{V_j}$.

Thus, we derive the asymptotic distribution of $\tau ^{*}$ instead of $\tau$. After a simple calculation, we can write $\tau ^{*}$ in the form

$$\begin{aligned} \tau ^{*}= & {} {\sum _{j=1}^{k}}\frac{{(\widehat{m_j}-\mu )}^2}{V_j} - {{(\widetilde{m}-m)}^2}{\sum _{j=1}^{k}}\frac{1}{V_j} \end{aligned}$$

Now, under $H_0$, $E(\widehat{m_j}) = m_j = m$ and $Var(\widehat{m_j})=V_j$ and $\widehat{m_j}$ for $j=1,\ldots ,k$ are independent, $E(\widetilde{m}) = m$ and $Var(\widetilde{m}) = {\left( {\sum _{j=1}^{k}}\frac{1}{V_j}\right) }^{-1}$, $Cov(\widehat{m_j},\widetilde{m}) = {\left( {\sum _{j=1}^{k}}\frac{1}{V_j}\right) }^{-1}$. Thus, under $H_0$, $\sum _{j=1}^{k}\frac{{(\widehat{\mu _j}-\mu )}^2}{V_j} \xrightarrow {d} {\chi ^2}_{k}$, ${{(\widetilde{\mu }-\mu )}^2}\sum _{j=1}^{k}\frac{1}{V_j} \xrightarrow {d} {\chi ^2}_{1}$ and $Cov(\widehat{\mu _j}-\widetilde{\mu },\widetilde{\mu }) = 0 ~ \forall j=1, \ldots , k$. Thus, $\tau ^{*} \xrightarrow {d} {\chi ^2}_{k-1}$ and also, $\tau \xrightarrow {d} {\chi ^2}_{k-1}$.