A COMPARISON OF ACCURACY AND EXTRAPOLATION RANGE OF THE VON BERTALANFFY EQUATION AND THE MODIFIED POWER FUNCTION

The paper presents results of a comparative analysis of two mathematical models of fish length grmvth: the von Berta­ lanffy equation and the modified power :function. The accuracy of mathematical modelling of empirical data and a potential for extrapolation of theoretical growth range beyond the empirical data are compared. The data on accuracy and extrapolation potential, obtained with the two models tested and averaged for a species, were subject to the Student's t test. Additionally, an attempt was made to detennine a relationship between accu­ racy and exirapolation potential of each model; linear regres­ sion was used to describe the relationship. The generalised (ignoring between-specific differences) description of the rela­ tionship involved exponential, logarithmic, and power regres­ sions in addition to linear regression.


INTRODUCTION
The accuracy of mathematical models offish length growth and their potential for ex trapolation of the results beyond the range of empirical data are problems which have not attracted much attention.At the same time, these problems, once solved, may significantly contribute to the improvement of methodology of research on fish growth.The results of such studies may also have a practical application; for example, knowledge on the potential for extrapolation of results obtained with a model can verify back-read data for the oldest age groups (which ate not fully representative due to a low, as a rule, sample size) and/or those for the first year of life offish (poor legibility of the first annual ring).
To secure the maximum possible comparability of the data when calculating parame ters of the growth models used, lengths attained by the fish species in their first eight years of life were included.The data set on cisco, due to the short life cycle of the species which made it impossible to collect length data on 8 years, was limited to the first 6 years.
The von Bertalanffy equation parameters were detennined from empirical data using the Ford-Walford method.Parameters A, B, and C of the modified power fonction (L 1 =Al+ C) were calculated as proposed by S zy pula (1991).Should the increment in the first year oflife turn out to be substantially different from increments in subsequent years, it was disregarded when calculating Loo in the von Bertalanffy equation and exponent B in the modified power function, a procedure resulting in a higher accuracy of a model calculated.
Accuracy of the models tested was compared by calculating a mean absolute differ ence (LI) between the empirical and theoretical ( calculated from the model) results.As the species studied showed very diverse growth rates (from less than 20 cm in the sixth year of life, as in roach and rudd, to 60-70 cm in that year, as in cod, pike, and pikeperch), the mean absolute difference was also converted to percentage (Llw) of the mean length calcu lated across 8 ( 6 in cisco) years of life.Values of LI and Llw were calculated separately for eacli case; the Tables 9ontain averaged data calculated from all the differences obtained for a species.Szypula (1987b} us�d' an identical methodology to determine the accuracy of the model he was studying. A' pot6ntial fo:r extiapol�ti8n of growth rate beyond the empirical data range was de teffilin.edby d.lcclafing pai��ete�s' ;fa �odel from a half of the real data; as a rule, lengths attainecl'd1iring the't1 rn {, r ( J in cis; c6) years o flife were used.Subsequently, empirical data for years 5�8 (4..'..6 in cisco ); �ere cohlpJed: with those extrapolated with a model, assuming / .' J. '; .
.,. •. :::• .. _ .:,-.. .: • .. :-: .• i ' .: the extrapolation to be permissible within the range in which a difference between the empirical and extrapolated lengths did not exceed 5% of the first.The extrapolation range limit was determined in the following way: lengths, attained by a species in two consecutive years and selected so that the difference in the first year would b� less than 5% and the difference in the second year would exceed 5% were compared and the extrapolation (with the differ ence of exactly 5%) range limit was determined by interpolation.The absolute value of ex trapolation range (Zo.05) was expressed in years and supplemented with a relative value (Zw) of the range; the relative value, expressed as a percentage, was a ratio between Za.o5 and the range of empirical data used for the growth model (Zv = ZE/2, where ZE is the total range of empirical data).The method used was described in detail by Szypula (1987a).
Relationships between the model's accuracy and its extrapolation range (both for the absolute and the relative values) were studied with .theaid of the linear regression (y = a + bx), calculated with the least squares method.The relationships were described separately for each species from data on accuracy and extrapolation range of the cases ana lysed for the species.Due to a substantial scatter in the data (cf Tabs.1-4), the data were initially transformed by grouping the argument of the function (A or Aw) into classes at 0.1 cm intervals and calculating mean class values and mean values of the function (Za.05 or Zw).The transformed data were used to calculate values oflinear regression terms.
The final stage of the calculations was an attempt to generalise the nature of the rela tionships studied, regardless of the species-specific characteristics.At that stage, the rela tionships were described from mean values of and Zo.os (for absolute values) contained in Tables 1 and 3, and from Aw and Zw (for percentages), given in Tables 2 and 4. In addition to the linear regression, the exponential (y = a•l x ), logarithmic (y =a+ b•ln x) and power (y = a, x b ) regression analyses were used.
Both the accuracy of each model and its extrapolation potential and regression pa rameters were determined separately for each growth model compared.

RESULTS
According to the theoretical assumptions of the modified power function, the expo nent B describes the nature of growth of a species studied (asymptotic growth at B<O and unlimited growth at B>O).Analysis of the detailed data on individual species, used to con struct the modified power function in order to assess the accuracy of the model showed B>O in all cases analysed for a species (blue bream, bream, roach, rudd, perch, and cod), which theoretically evidences the unlimited growth of those species.Out of 21 cases analysed for pike, B<O was obtained in one case only (4.8%); out of 21 cases analysed for pikeperch, B<O occurred in 5 cases (23.8%).Thus both species can also be regarded as having in most cases the unlimited growth.Cisco, on the other hand, can be regarded as a species whose grnwth is poorly detennined (B<O occurred in 10 out of25 cases, or in 40.0%), while most cases of herring data (70.4%)pointed to the asymptotic growth.
Accuracy of the mathematical description of , growth, as measured by a mean differ ence between empirical data and those calculated from the two models compared is pre sented in Table 1.The data show that the modified power function is a more accurate model for half of the species studied (roach, pike, cisco, herring, and cod); the two models had identical accuracy when applied to bream data, while a higher accuracy (i.e., lower) of de scription of growth of the remaining 4 species (blue bream, rudd, perch, and pikeperch) was obtained with the von Bertalanffy equation.Moreover, the weighted mean of values, calcu lated for all the species studies, turned out to be lower with the modified power function.Noteworthy is the extensive scatter of data obtained for different cases within a spe cies.The coefficient of variation (v) was, as a rule, higher than 50%; in the extreme case (accuracy of the von Bertalanffy equation applied to perch) v was even 100%.
The data on accuracy of the models analysed are also given in Table 2; they are, how ever, expressed as percentages (Aw).The data were obtained by converting the values re ported in Table 1 (d) to the mean length calculated from 8 (6 in cisco) empirical data used for construction of the models compared.A pattern reverse in relation to that in Table 1 is observed: the von Bertalanffy equation turned out to be more accurate for 5 species, the accuracy of the two models was identical in one species, and in the remaining 4 • species lower values of Aw were obtained with the modified power function.However, the weighted mean Aw was lower in the modified power function as well.The scatter of the data obtained in different cases characterising growth of a species was very extensive, too, al though not so as in the case of A.

--
A-mean relative difference (in %) between empirical data and results calculated with growth model analysed.
Although mean values of A and Aw calculated for the species studied differed between the two models compared, sometimes distinctly so, the differences within a species were, as a rule; non-significant (Student's t test, 0.99 confidence level).It was only in cod that the differences between values of LI. and.Llw obtained with the von Bertalanffy equation and the modifieq .power function differ�d significantly.
The extrapolation ra�g� of length growth rate in different fish species was determined (with 5% tolerance), as described in the previous chapter, by calculating the analysed growth models from lengths attained in the first 4 years of life and comparing, for the sub sequent 4 years, results calculated from a model with the empirical data (the models for ciscowere calculated for the first 3 years and compared for the subsequent 3 years).The number of cases analysed for roach, rudd, and.pike was lower (by.n than that used for model accuracy determination (Tabs, 1 and 2).The difference resulted from the fact that in some cases it was impossib\e to construct the von Bertalanffy mo4�I from data .on the first 4 years (no decreasing trend ip , l e¥ gth increments, such trend being necessary to apply the model).
Table 3 summarises d<J,ta o n tl}e absolute values of the extrapolation range (Z0.05), bro ken down by fish species, model used, and basic statistical parameters ( arithmetic mean, standard deviation, coefficient of variation).
The analysis of different cases µsed to calculate arithmetic means reported in Table 3 showed that the difference between the empirical and extrapolated lengths in the final, eighth ( sixth in cisco) year was frequently (in 63 and 72 cases analysed • with the von Berta lanffy equation and the modified power function, respectively, i.e., in 25.5 and 29.1% of all the 247 cases, respectively) lower (sometimes much lower) than the assumed 5% tolerance.The lowest number of such low differences (2 out of28 cases, i.e., 7.1%) was observed in cod data, while the highest (15 out of26 cases, i.e., 57.7%) was recorded in perch; these results concern both models tested.The results discussed are well correlated with species specific mean values of Zo.05: the highest values (3.0 and 3.26 for the von Bertalanffy equa tion and the modified power function, respectively) were recorded in the perch data as well.The lowest value obtained with the von Bertalanffy equation (1.40) was recorded in cod.The modified power function applied to the cod data resulted in a relatively low value (1.92), too; however, still lower values (1.90 and 1.69) were observed in pikeperch and cisco.Similarly to L1 and L1w, an extensive scatter of data was recorded as well (the coeffi cient of variation ranged from more than 30 to more than 70%).The weighted mean Z0.05 values calculated for all the species studied demonstrated that the modified power function provides a slightly wider e:x"trapolation range.In separate species-specific analyses, a higher range of Zo. 05 with the modified power function was found in 8 cases, while only two cases of the von Bertalanffy equation yielded higher range of Z 0 _05.4, arranged similarly to the previous one, shows values of the relative extrapo lation range Zw,obtained by converting Z0. 05 to the range of empirical data used to calculate the growth models compared (ZM).The range of ZM was 4 years (3 in cisco ).Similarly to the Z0.05 analysis, application of the von Bertalanffy equation yielded the highest values of Zwfor perch (77.59 and 81.42%), the lowest value (35.09%) being typical of cod.It was only in pikeperch that the modified power function resulted in a Zw value slightly lower than that found for cod.On the other hand, Zw of cisco, the species having the lowest Zo.05 in the modified power function analysis, was.clearly higher than that of cod and pikeperch.This happened because Zo.os was referred to 3 years rather than to 4, as was the case in the re mait1ing species ... Coefficients of variation varied within a range similar to those of .Zo.os.Weighted meat1 Zw values were also slightly higher when the modified power function was used, Similarly to Z0.05, the modified power function yielded better results in 8 species, while the von Bertalanffy equation supplied better results in 2 species only.

(percentages}
All the relation��ipsthe terms of which are given in Tables 5 and 6, showed negative values of both the regn::ssioll @yfn,cient (b) and the correlation coefficient (r), which points to a reverse relationship between the v;ariables compared.In other words, the less accurate a model (a larger difference between A 9r_Aw), usually the narrower the extrapolation range with the 5% tolerance assumed (Zo.05 or Zw).The relationships are illustrated in Figs. 1 and 2.
Fig. 1, in which the real and extrapolated length growth rates of the Szczecin Lagoon perch are presented (based on data reported by Szypula 1994b ), shows that the extrapolated growth curve almost ideally fits the empirical data; the empirical and calculated lengths for the final year of life are identical, which suggests that the true extrapolation range is likely to be by a few years wider than that calculated with the model used.At the same time, the av erage difference between the empirical and model data was in this case minimal for the first 4 years of life (A= 0.02 cm; Llw = 0.14%).Fig. 2 shows a reverse pattern, found for the Szczecin Lagoon pikeperch (data reported by Nagi�c 1961).The average differences in this case were large (Li = 2.49 cm; Llw = 4. 71 % ), the extrapolation range being relatively narrow (Z0_ 05 = 0.97 yr).As of the fifth year oflife, the extrapolated growth curve runs much higher than that based on the empirical data (particularly in the seventh and eighth years oflife ).

Table5
Relat ionship between e>..irapolation range (Z0_   Correlation coefficients concerning the von Bertalanffy equation, repmted in Table 5, ranged from -0.0963 to -0.8989.In 5 species, the absolute values of r exceeded 0.6, evi dencing a rather strong correlation of the variables analysed.In the remaining 5 species, the absolute values of r were lower than 0.5 (weaker correlation between model accuracy and its extrapolation range), virtually no correlation existing in cisco (r =; -:c-0.0963).As far as the modified power function is concerned, the range of variation in r was-narrower (from -0.2272 to -0.8441); however, in as many as 6 species the absolute value ofr exceeded 0.6.Usually (in 7 out of the 10 species studied), the absolute value of r was higher when the modified power function was used, compared to the correlation coefficient obtained when using the von Bertalanffy equation.The stronger correlations between the model accuracy and its extrapolation range, obtained for the modified power function, are evidenced also by the values of r, averaged across species separately for the "VOn Bertalanffy equation (-0.5483) and the modified power function (-0.6151).
Very similar were the relationships determined for the percentages (Tab.6): negative regression and correlation coefficients were obtained both for all the species and in both models compared.Similarly to the absolute values, in 7 out of 10 cases higher absolute val ues of correlation coefficients resulted from using the modifi ed power function.Moreover, absolute values of correlation coefficients were, both in the von Bertalan:ffy equation and in the modifi ed power function, slightly higher than those obtained when analysing the abso lute values (-0.6353 and -0.7317 were the respective means for all the fish species used in the analysis).
Finally, an attempt was made to determine the general nature of relationships between the extrapolation range and the accuracy.of the models compared, regardless of species.Data (arithmetic means) in Tables 1 and 3 (absolute data) and in Tables 2 and 4 (percent , ages) served as the material for the analyses.To determine the nature of the relationships studied as accurately as possible, the exponential, logarithmic, and power regressions were used in addition to the linear regression.Parameters of t he relationships studied and the .,relevant correlation coefficients are summarised in Table 7.
The correlation coefficients were relatively high; as•. a rule, they exceeded 0. 6 • (except • for the exponential regression calculated on absolute values obtained by using the modified .power function).As a rule, curvilinear regressions (the logarithmic regression in particular), allowed to more accurately correlate the values analysed than did the linear regression: In contrast to linear regressions calculated for individual species (Tabs.5 and 6), the data in Table 7 demonstrate that the von Bertalanffy equation provided a closer correlation be tween the extrapolation range and the model accuracy.The difference in favour of the von Bertalanffy model (higher absolute values of correlation coefficients) is particularly evident when comparing the relationships calculated on absolute values (Zo.os and "1).General relationship between extrapolation range (Z o .os,Zw) and accuracy (4 Llw) of model tested, described by linear regression (j1 = a +b,x), expoptential regression (y = a•i "' ), logarithmic regres sion (y = a + !J. ln x) and power regression (y = a The differences in accuracy between different methods used for a general description of the extrapolation range vs. model accuracy relationship ( differences between the means in Tables 3 and 4 and values calculated with regression equations given in Table 7) were slight only and, as a rule, ranged from 0.1 to 0.6 yr and within 2-8% for absolute values and percentages, respectively.

DISCUSSION
The results obtained when analysing accuracy of the models compared (Tabs. 1 and 2) point to a rather high accuracy of the mathematical description of length growth.The evi dence is provided by the relatively low average across-species difference between empirical and calculated data (slightly exceeding 0.5 and about 2% for absolute values and percent ages, respectively).The detailed analysis ofresults obtained for individuals species seems to demonstrate some advantage in using the modified power function (in 5 species, the results were lower than those obtained with the von Bertalanffy equation; the reverse was true in 4 species, while the two results were identical in 1 case).The weighted mean calculated for all species was, too, slightly lower when the modified power function was used.However, the difference between the weighted means was small and the differences between the two models in individual species were in 9 cases ( except that of cod) non-significant.If, addi tionally, a wide scatter of individual data is taken into account (as evidenced by both the coefficients of variation and :i;-angei,; of values), the two models can be regarded as having similar accuracies.
Results of earlier studies on the problem, obtained by Szypula (1987b) are difficult to compare due to different assumptions when collecting the data.That study concerned, i.a., effects of the range of empirical data (ZE) on a model's accuracy.For this reason, for each of the 50 species studied, only 1 case of length growth rate was analysed, but growth mod els were calculated from different ZM ranges, starting from age group 3 up to--in some cases-several years of life.Additionally, apart from the von Bertalanffy equation, the Gompertz model, bn:iomial, and the Ford-Walford formula were used.Out of the array of species studied by Szypula (1987b ), 4 only are included in the present study (bream, perch, pikeperch, and herring).A comparison of results concerning those 4 species showed that the earlier study resulted in clearly lower values of Ll and L1w, the difference being most con spicuous in pikeperch and the smallest in perch.Possibly, the discrepan\:ies might have been caused by the fact that, as mentioned above, the von Bertalanffy equation ( obviously this model only could be used for cotnparison) parameters were determined from different ranges of ZE, while the range was � as a rule -8 years in the present work.Somewhat more concordant results were obtained when the all-species average only, calculated with the use of the 8-yr range, was taken into account.In that case, the absolute value.of the difference was 0.5 cm (0.63 cm in the present study), while the difference between percentages was quite substantial (1.09 % in the earlier work vs. 2.12% in this study).
Parenthetically, it is worth mentioning that when calculating the results for different cases, the concordance between the model and the empirical data was usually higher when the data were derived from a large and representative sample.It is suggested that the mathematical description of growth rate be based exclusively on such data and given up when they are not available, since results of theoretical calculations may then be very far from reality.
Results concerning extrapolation ranges (Zo.05 and Zw) obtained in this work are largely close to earlier data of Szypula (1987a).With reservations analogous to those ex pressed when discussing the accuracy of the models compared, an average Zo.o 5 of 2.16 yrs (2.30 in the earlier study) was obtained for the range equal to 8 yrs; Zw was 55.6% in both works: Slightly higher differences occurred in individual species, compared to the data of Suryn (1990) who--using identical methodology-studied a potential for extrapolation of growth of roach, perch and herring.He used, i.a., the von Bertalanffy equation and the modified power function.His results are clearly lower (with respect to perch in particular) than those obtained in this study.Additionally, Suryn contended that clearly better results could have been obtained with the von Bertalanffy equation, while this study demonstrates a slight advantage of the modified power function over the von Bertalanfly model.The negative regression (b) and correlation (r) coefficients obtained for all the species studied when using linear regression to analyse the relationships between and Liw on the one hand and Zo_o5 and Zw on the other fully justify considering the relationships as reverse (although lower's in some case demonstrate poor correlations-or a vi1tual lack of correla tion-between the values analysed).In other words, a more accurate model (lower Li and Liw) allows, as a rule, for a wider extrapolation range (higher Zo.05 and Zw).This conclusion can be of a considerable practical importance when attempting to use the extrapolated lengths instead of tentative (few and non-representative) empirical values, particularly with respect to oldest age groups.
Finally, the various detailed descriptions of general relationships between the ex:trapo lati.on range and the model accuracy (Tab.7) confirm, on the one hand, the nature of the relationships, concerning individual species, as discussed above.On the other hand-as shown by the relatively high correlation coefficients in each case-the relationships can be regarded as rather highly correlated.It should be reminded that, when determining the rela tionships, species-specific means were used as the starting points (given in Tabs.1-"4); thus the relationships have a more general nature than those summarised in Tables 5 and 6.A somewhat closer correlation between the values, produced by the curvilinear regressions, was practically to be expected.It is of interest to note that the best results (highest absolute values of r) were, as a rule, obtained with the logarithmic regression.It has to be remem bered, when assessing and comparing values of r yielded by different types of regression, that correlation coefficients of exponential and power regressions given in Table 7 concern the logarithmic forms of those regressions.

Table4
Relative extrapolation range (Zw) ofvon ]3ertalanffy equation (a) and modified power function (b ranges (with respect to both Zo.os and Zw) deter mined with the two mod'els were non-significant (Student's t test, 0.99 confidence level) for all the species.'.Another pr o, blem tiickled in this work involved relationship between extrapolation range and the accuracy of the growth model used.The problem was studied with respect to both the absoh.i.le (Zo.o; !µ). d A)and percentage (Zw and Aw) values.The relationships were described with 'linear regressions, separately for different fish species and growth models compared.:\:Vithina-species, thi relationships ��e �tudied based unth� analysis of the cases . ... included in ciilcufations of,{ .a;:andfhe corresponding Zoos and Zw.The linear regression terms along with correlation <;:()e{ficients are reported in 05 ) and accuracy (Li) of model tested, described by linear regression (Zo.os =a+ b 0

Table 1
Comparison of accuracy ofvon Bertalanffy equation (a) and modified power function (b) used for mathematical description of length growth in various fi sh species

Table 5 (
absolute valuy�J and Table