The specific growth rate,

La tasa específica de crecimiento,

The specific growth rate,

Whatever the topic under investigation may be, it is necessary to calculate

We would like to address the present comment to researchers mainly concerned with experimental designs on growth of aquatic organisms. No new growth index is proposed in this note. Instead, it is intended to briefly show our point of view about two types of biases potentially affecting the calculation of the well-known specific growth rate (biases that cannot be corrected during the statistical analysis of data) and to suggest a few recommendations in the calculation of this growth index, particularly in experimentation on juvenile animals kept in small groups. Let us now consider two situations of increasing complexity with untagged animals reared in the same population.

Suppose a group of

$$G\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{n}{\displaystyle \sum _{\text{i}}\text{\hspace{0.17em}}{G}_{\text{i}}}=\text{\hspace{0.17em}}\frac{1}{n}{\displaystyle \sum _{\text{i}}\text{\hspace{0.17em}}\frac{1}{t}\left(\mathrm{ln}{W}_{\text{i},t}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\mathrm{ln}{W}_{\text{i},0}\right)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{t}\text{\hspace{0.17em}}\left(\frac{1}{n}{\displaystyle \sum _{\text{i}}\mathrm{ln}{W}_{\text{i},t}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{1}{n}{\displaystyle \sum _{\text{i}}\mathrm{ln}{W}_{\text{i},0}}\right)$$

Since the summation of the logarithms of

$$G\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{t}\text{\hspace{0.17em}}\left[\frac{1}{n}\text{\hspace{0.17em}}\mathrm{ln}\left({\displaystyle \prod _{\text{i}}{W}_{\text{i},t}}\right)\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{1}{n}\mathrm{ln}\left({\displaystyle \prod _{\text{i}}{W}_{\text{i},0}}\right)\right]\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{t}\text{\hspace{0.17em}}\left[\mathrm{ln}\left(\sqrt[\text{n}]{{\displaystyle \prod _{\text{i}}{W}_{\text{i},t}}}\right)\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\mathrm{ln}\left(\sqrt[\text{n}]{{\displaystyle \prod _{\text{i}}{W}_{\text{i},0}}}\right)\right]$$

By definition, the n-root of the product of i numbers is the geometric mean of those _{t} and _{0} denote the geometric means of the final and initial individual weights, the final expression of G becomes,

$$G\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{\mathrm{ln}{\mu}_{\text{t}}-\mathrm{ln}{\mu}_{0}}{t}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{\mathrm{ln}\left({\mu}_{\text{t}}/{\mu}_{0}\right)}{t}$$

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, _{i}_{i}

$$\mathrm{ln}\mu \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{ln}M\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\displaystyle \sum _{\text{i}=\text{1}}^{\infty}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\left(-1\right)}^{\text{i}+\text{1}}}{\text{i}\text{\hspace{0.17em}}{M}^{\text{i}}}E\left[{\left({W}_{i}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}M\right)}^{\text{i}}\right]}$$

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 (

$$\mathrm{ln}\mu \text{\hspace{0.17em}}\approx \text{\hspace{0.17em}}\mathrm{ln}M\text{\hspace{0.17em}}-\text{\hspace{0.17em}}0.5\text{\hspace{0.17em}}\frac{{V}_{\text{W}}}{{M}^{2}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{ln}M\text{\hspace{0.17em}}-\text{\hspace{0.17em}}0.5\text{\hspace{0.17em}}{C}^{2}$$

In the above approximation, _{w} denotes the variance in fish weights and

$$G\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{\mathrm{ln}{\mu}_{\text{t}}-\mathrm{ln}{\mu}_{0}}{t}\text{\hspace{0.17em}}\approx \text{\hspace{0.17em}}\frac{\mathrm{ln}{M}_{\text{t}}-\mathrm{ln}{M}_{0}-0.5\text{\hspace{0.17em}}\left({C}_{t}^{2}-{C}_{0}^{2}\right)}{t}$$

$${G}_{M}\text{\hspace{0.17em}}\approx \text{\hspace{0.17em}}G\text{\hspace{0.17em}}+\text{\hspace{0.17em}}0.5\text{\hspace{0.17em}}\frac{\Delta {C}^{2}}{t}$$

where _{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:

∆_{M} =

∆_{M} >

∆_{M} <

The relative bias, 100×(_{M} –

$$\text{\hspace{0.17em}}100\text{\hspace{0.17em}}\frac{{G}_{M}-G}{G}\text{\hspace{0.17em}}\approx \text{\hspace{0.17em}}100\text{\hspace{0.17em}}\frac{\Delta {C}^{2}}{2\mathrm{ln}\left({M}_{\text{t}}/{M}_{0}\right)-\Delta {C}^{2}}$$

For example, _{t}/_{0} 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

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 _{0} (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

$$\mathrm{ln}{\mu}_{0}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{ln}\sqrt[n]{{\displaystyle \prod _{\text{i}}{W}_{\text{i},0}}}=\text{\hspace{0.17em}}\frac{1}{n}\text{\hspace{0.17em}}\mathrm{ln}\left({\displaystyle \prod _{j\text{\hspace{0.17em}}\in \text{\hspace{0.17em}}surviving}{W}_{\text{j},0}}\right)+\text{\hspace{0.17em}}\frac{1}{n}\text{\hspace{0.17em}}\mathrm{ln}\left({\displaystyle \prod _{k\text{\hspace{0.17em}}\in \text{\hspace{0.17em}}dead}{W}_{\text{k},0}}\right)$$

Let _{s} and _{d} the geometric mean weights at time 0 of the surviving and dead individuals in the interval [0,

$$\begin{array}{c}\mathrm{ln}{\mu}_{0}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{s}{s}\text{\hspace{0.17em}}\frac{1}{n}\mathrm{ln}\left({\displaystyle \prod _{\text{j}\text{\hspace{0.17em}}\in \text{\hspace{0.17em}}\text{surviving}}{W}_{\text{j},0}}\right)\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{d}{d}\text{\hspace{0.17em}}\frac{1}{n}\mathrm{ln}\left({\displaystyle \prod _{\text{k}\text{\hspace{0.17em}}\in \text{\hspace{0.17em}}\text{dead}}{W}_{\text{k},0}}\right)\\ \\ \mathrm{ln}{\mu}_{0}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}(1-m)\mathrm{ln}{\mu}_{\text{s}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}m\text{\hspace{0.17em}}\mathrm{ln}{\mu}_{\text{d}}\end{array}$$

Therefore, it is possible to obtain an analytical expression for the bias, _{m}–_{m} denotes the mortality-biased growth rate) as a function _{s} and _{d}. Firstly, consider the expression for

$${G}_{m}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\mathrm{ln}{\mu}_{\text{t}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ln}{\mu}_{\text{0}}}{t}$$

Secondly, substitute the expression of ln _{0} in _{m},

$${G}_{m}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\mathrm{ln}{\mu}_{\text{t}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ln}{\mu}_{\text{s}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}m\text{\hspace{0.17em}}\left(\mathrm{ln}{\mu}_{\text{s}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ln}{\mu}_{\text{d}}\right)}{t}$$

At last, after some rearrangements, the relationship between _{m} and

$${G}_{m}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}G\text{\hspace{0.17em}}+\text{\hspace{0.17em}}m\text{\hspace{0.17em}}\frac{\mathrm{ln}{\mu}_{\text{s}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ln}{\mu}_{\text{d}}}{t}$$

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} ⇒ _{m} =

μ_{s} > μ_{d} ⇒ _{m} >

μ_{s} < μ_{d} ⇒ _{m} <

In practice, because the values of _{s} and _{d} remained unknown in experiments with untagged fish, the size of the selective mortality bias, _{m}–

$$G=\frac{{G}_{\mathrm{max}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}{G}_{\mathrm{min}}}{2}$$

Except for the introduction of geometric means, the above expression is an extension of the correction of

$$Error=\pm \text{\hspace{0.17em}}\frac{{G}_{\mathrm{max}}-\text{\hspace{0.17em}}\text{\hspace{0.17em}}{G}_{\mathrm{min}}}{2}$$

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 _{max} and _{min} will in addition require knowing the true distribution of the population. However, these complications are beyond the scope of this comment.