Suppose a group of n untagged juvenile individuals in the same tank so that, as in many experiments with fish, n is not very high and body weights can be easily recorded for all animals at the first and the last samplings. In addition, suppose that there is no mortality. In such a simple experimental situation the arithmetic mean of individual specific growth rates, G, will be

$$G=\frac{1}{n}{\displaystyle \sum _{\text{i}}{G}_{\text{i}}}=\frac{1}{n}{\displaystyle \sum _{\text{i}}\frac{1}{t}\left(\ln W_{\text{i},t}-\ln W_{\text{i},0}\right)}=\frac{1}{t}\left(\frac{1}{n}{\displaystyle \sum _{\text{i}}\ln W_{\text{i},t}}-\frac{1}{n}{\displaystyle \sum _{\text{i}}\ln W_{\text{i},0}}\right)$$

Since the summation of the logarithms of i numbers equals the logarithm of the product of those i numbers and, in addition, the product of a constant by a logarithm can be written as the logarithm of a power expression, it is now possible to write,

$$G=\frac{1}{t}\left[\frac{1}{n}\ln \left({\displaystyle \prod _{\text{i}}{W}_{\text{i},t}}\right)-\frac{1}{n}\ln \left({\displaystyle \prod _{\text{i}}{W}_{\text{i},0}}\right)\right]=\frac{1}{t}\left[\ln \left(\sqrt[\text{n}]{{\displaystyle \prod _{\text{i}}{W}_{\text{i},t}}}\right)-\ln \left(\sqrt[\text{n}]{{\displaystyle \prod _{\text{i}}{W}_{\text{i},0}}}\right)\right]$$

By definition, the n-root of the product of i numbers is the geometric mean of those i numbers. Thus if µ_t and µ₀ denote the geometric means of the final and initial individual weights, the final expression of G becomes,

$$G=\frac{\ln {\mu }_{\text{t}}-\ln {\mu }_0}{t}=\frac{\ln \left({\mu }_{\text{t}}/{\mu }_0\right)}{t}$$

The above expression is very similar to that for the specific growth rate for one individual, except that individual weights have been replaced by geometric means. If initial and final arithmetic means of body weights are used instead of geometric means, a bias is produced in the calculation. The quantity of this bias will depend on the change in the squared coefficient of variation for body weights, as explained below. The geometric mean, µ, can be approximated by developing the terms lnW_i, where W_i represents each individual weight, as a Taylor series centred at the value M, where M is the arithmetic mean of individual weights; then taking expectations, E[ ],

$$\ln \mu =\ln M+{\displaystyle \sum _{\text{i}=\text{1}}^{\infty }\frac{{\left(-1\right)}^{\text{i}+\text{1}}}{\text{i}{M}^{\text{i}}}E\left[{\left(W_i-M\right)}^{\text{i}}\right]}$$

The Taylor series will be developed till the third term to obtain an approximation based on the arithmetic mean and variance of body weights; keeping in mind that the second term is zero, the sought expression is (Jean and Helms 1983)

$$\ln \mu \approx \ln M-0.5\frac{V_{\text{W}}}{M^2}=\ln M-0.5C^2$$

In the above approximation, V_w denotes the variance in fish weights and C sets for the coefficient of variation of fish weights. Now, substituting the above approximation into the expression of the true mean G,

$$G=\frac{\ln {\mu }_{\text{t}}-\ln {\mu }_0}{t}\approx \frac{\ln M_{\text{t}}-\ln M_0-0.5\left(C_t^2-C_0^2\right)}{t}$$

$$G_M\approx G+0.5\frac{\Delta C^2}{t}$$

where G_M is the growth rate as affected by the bias due to the use of arithmetic means of fish weights. The following results are now apparent:

∆C = 0 ⇒ G_M = G

∆C > 0 ⇒ G_M > G

∆C < 0 ⇒ G_M < G

The relative bias, 100×(G_M – G)/G, can be calculated as

$$100\frac{G_M-G}{G}\approx 100\frac{\Delta C^2}{2\ln \left(M_{\text{t}}/M_0\right)-\Delta C^2}$$

For example, Petursdottir (2002) performed a series of experiments on the growth of tagged individuals of the arctic charr Salvelinus alpinus (Linnaeus, 1758); the M_t/M₀ ratio was in the range 1.8-2.1, and the initial and final coefficients of variation of fish weights were approximately 6.6% and 33%; if the experiments had been conducted with untagged fish, the relative bias due to the arithmetic mean would have been in the range 7%-10%.

Let us now think of a more complex but also more frequent experimental situation with juvenile animals: a group of untagged fish in the same container, whose body weights are recorded from time 0 to time t and with mortality in the same time interval; in addition, suppose that geometric means are used instead of arithmetic ones, so the bias due to the arithmetic mean can be ruled out. In this case, there is still a potential bias in the calculation of G, because the first sampling includes both dead fish and fish surviving to time t. This fact was early recognized in the field of fisheries research (Ricker 1975) and later on, and perhaps inconspicuously, in the field of fish culture (Otterå 1992). Moreover, some authors have followed the “mortality of the smaller” principle (Folkvord and Otterå 1993) to correct this potential bias in G.

The size of the selective mortality bias can be expressed as a function of fish size in the surviving and dead populations. This goal can be attained by splitting the factors within ln µ₀ (i.e. within the geometric mean of fish weights at time 0) into two groups according to the survival or non-survival of each individual and then rewriting the expression of G,

$$\ln {\mu }_0=\ln \sqrt[n]{{\displaystyle \prod _{\text{i}}{W}_{\text{i},0}}}=\frac{1}{n}\ln \left({\displaystyle \prod _{j\in surviving}{W}_{\text{j},0}}\right)+\frac{1}{n}\ln \left({\displaystyle \prod _{k\in dead}{W}_{\text{k},0}}\right)$$

Let s and d be the number of surviving and dead animals, respectively, µ_s and µ_d the geometric mean weights at time 0 of the surviving and dead individuals in the interval [0, t], respectively, and m=d/n the proportional mortality. Then,

$$\begin{array}{c}\ln {\mu }_0=\frac{s}{s}\frac{1}{n}\ln \left({\displaystyle \prod _{\text{j}\in \text{surviving}}{W}_{\text{j},0}}\right)+\frac{d}{d}\frac{1}{n}\ln \left({\displaystyle \prod _{\text{k}\in \text{dead}}{W}_{\text{k},0}}\right)\\ \\ \ln {\mu }_0=(1-m)\ln {\mu }_{\text{s}}+m\ln {\mu }_{\text{d}}\end{array}$$

Therefore, it is possible to obtain an analytical expression for the bias, G_m–G (where G_m denotes the mortality-biased growth rate) as a function m, µ_s and µ_d. Firstly, consider the expression for G calculated regardless of the mortality,

$$G_m=\frac{\ln {\mu }_{\text{t}}-\ln {\mu }_{\text{0}}}{t}$$

Secondly, substitute the expression of ln µ₀ in G_m,

$$G_m=\frac{\ln {\mu }_{\text{t}}-\ln {\mu }_{\text{s}}+m\left(\ln {\mu }_{\text{s}}-\ln {\mu }_{\text{d}}\right)}{t}$$

At last, after some rearrangements, the relationship between G_m and G becomes,

$$G_m=G+m\frac{\ln {\mu }_{\text{s}}-\ln {\mu }_{\text{d}}}{t}$$

It is now apparent that the potential bias is dependent on the quotient of geometric mean weights of surviving and dying subpopulations. To our knowledge, this is a new result. Under positive mortality, three statements can be derived from the above expression,

μ_s = μ_d ⇒ G_m = G

μ_s > μ_d ⇒ G_m > G

μ_s < μ_d ⇒ G_m < G

In practice, because the values of µ_s and µ_d remained unknown in experiments with untagged fish, the size of the selective mortality bias, G_m–G, cannot be precisely estimated. Nevertheless, in the case of a researcher interested in getting an approximation to the actual value of G and acquainted with the exact mortality in the experiment, it can be suggested to calculate a maximum G value assuming that only the largest animals at the initial time died during the interval [0, t], and also a minimum G value under the assumption that only the smallest animals at the initial time died during [0, t]. Thus the following estimation for the arithmetic mean of G can be useful for such an experimenter,

$$G=\frac{G_{\max }+G_{\min }}{2}$$

Except for the introduction of geometric means, the above expression is an extension of the correction of Folkvord and Otterå (1993), but it would be more appropriate when the validity of the “mortality of the smaller” principle is not clear. If the researcher is also interested in evaluating the goodness of the experimental measurement, the length of the semi-interval can be proposed as the maximal value of the error:

$$Error=\pm \frac{G_{\max }-G_{\min }}{2}$$

It should be noted that in experimental scenarios where it is not possible to sample the whole population, for example when working with larval stages, the uncertainty of the calculated specific growth rate is also affected by the sampling error. Therefore, the uncertainty in G will also depend on the statistical distribution of the selected metric variable—more specifically, on the shape of the distribution of lnW or lnL. On the other hand, when the mortality rate is above zero, the calculation of G_max and G_min will in addition require knowing the true distribution of the population. However, these complications are beyond the scope of this comment.