Conventional and alternative dose–response models to estimate nutrient requirements of aquaculture species

Aquaculture 292 (2009) 207–213

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Aquaculture j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / a q u a - o n l i n e

Conventional and alternative dose–response models to estimate nutrient requirements of aquaculture species Alfredo Hernandez-Llamas ⁎ Programa de Acuicultura, Centro de Investigaciones Biológicas del Noroeste (CIBNOR), Mar Bermejo 195, Colonia Playa Palo de Santa Rita, La Paz, B.C.S. 23090, Mexico

a r t i c l e

i n f o

Article history: Received 5 November 2008 Received in revised form 14 April 2009 Accepted 14 April 2009 Keywords: Nutrient requirements Dose–response models Equivalence tests

a b s t r a c t Despite criticism in the literature, the conventional broken-line model (BLM) and the four-parameter saturation kinetics model (SKM) are the dose–response models most frequently used to estimate nutrient requirements of aquaculture species. This study combines the advantages of both conventional models to produce models that more accurately estimate requirements, the broken saturation kinetics model (BSKM) and the broken-convex curve model (BCCM). Additionally, the feasibility of using other alternative dose–response models is addressed and an equivalence test is introduced as a method to deal with statistical results that are not significant at the plateau of the response curve. The models were evaluated using 24 published cases, considering weight gain and feed efficiency as response parameters. There were significant differences (P b 0.05) in model-fitting performance and parsimony. The BSKM and BCCM fitted better, while some evidence supported conventional models as being more parsimonious. There was significant evidence that fitting performance and parsimony of the models tended to coincide among the cases. There was also significant evidence that estimates of nutrient requirements differed, depending on the model used, and that the models tended to consistently yield low, intermediate, or high requirements among the cases. The BLM produced the lowest estimates and the SKM produced an erratic performance. When the BLM was fitted to a published case, the slope at the plateau was not significant; yet an equivalence test, using the two-one sided procedure, with a similarity level of 5%, indicated that it was not adequate to assume that fish were non-responsive to nutrient inputs. The results indicate that combining the advantages of the conventional models resulted in models that were easy to implement and that more accurately estimated nutrients requirements. Additionally, equivalence tests were shown to be useful for analyzing results that were not significant at the plateau. © 2009 Elsevier B.V. All rights reserved.

1. Introduction The choice of statistical methods is of major importance for estimating nutrient requirement (Baker, 1986; Shearer, 2000). According to Shearer (2000), ANOVA and the broken-line model are the methods most frequently used to estimate dietary nutrient requirements of aquaculture species. There are concerns, however, that ANOVA is inadequate for dose–response analysis (Morris, 1999; Shearer, 2000). The main criticism of ANOVA is that the best estimate of the response is by a fitted curve, not the differences among the means of treatments. On the other hand, Curnow (1973), Fisher et al. (1973), Gold (1977), Robbins et al. (1979, 2006), Morris (1989, 1999), and Shearer (2000) have discredited the broken line method. The main criticism of the broken-line is that, while it adequately describes the response of an individual to nutrient inputs, it is inadequate to describe the response of a population because it assumes an unrealistic, abrupt shift at the plateau, when a smooth transition actually occurs. Consequently, nutrient requirements are usually underestimated. ⁎ Tel.: +52 612 123 8416; fax: +52 612 125 3625. E-mail address: [email protected]. 0044-8486/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.aquaculture.2009.04.014

In preparing for this study, a review of the literature covering 80 articles published during 1993–2009 was undertaken. The review confirmed the conclusions of Shearer (2000) that ANOVA and the broken-line were the most common methods to estimate nutrient requirements of aquaculture species (Table 1). Regardless of the methods used, most (68%) of the reviewed cases analyzed the left side of the response curve and the plateau section. The declining right side of the curve was not considered. The broken-line and the four-parameter saturation kinetics model were the dose–response models most frequently used to study the left side and the plateau section for a wide variety of species, response parameters, and nutrients. However, Robbins et al. (1979), Baker (1986), and Morris (1989, 1999) explained that two major problems arise when using curvilinear models with upper asymptotes, including the saturation kinetics model. These models attempt to predict continuing response at high inputs, when the real response has ceased and concomitantly, a requirement is arbitrarily established, usually at 95% of the maximum asymptotic response. There are no biological or statistical arguments for such a definition of a requirement. The broken-line has been frequently used in conjunction with ANOVA. For the left side of the response curve, ANOVA normally yields

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Table 1 Results of survey of methods that studied nutrient requirements of aquaculture species. Method

Referencesa (%)

ANOVA 59 Broken-line 40 4 and 5-parameter saturation kinetics models 8 Quadratic equation 18 Third-order polynomial 4 Other (logistic and exponential equations; diphasic and nitrogen 7 deposition models) Total 136

43.4 29.4 5.9 13.2 2.9 5.2

2.1.1. The broken-line Following Zeitoun et al. (1976), the broken-line analysis “assumes a positive linear relation between weight gain (y) and the dietary level of the nutrient (x), which breaks instantly to horizontal at the minimum requirement estimated by the abscissa of the breakpoint” (xbp). Accordingly, the following parameterization of the broken-line model was used: if x b xbp ;

ð1Þ

y = ymax + b2 x if x z xbp ;

ð2Þ

y = i + b1 x

100

a Most articles (91%) used more than one method. Surveyed journals were Aquaculture, Aquaculture Nutrition, Aquaculture Research, British Journal of Nutrition, Comparative Biochemistry and Physiology, Journal of Nutrition, and Journal of the World Aquaculture Society.

significant results for the response parameter. However, on the plateau section, significant results are normally not observed. A steady state or homeostatic response of organisms at the plateau is the foundation for using the broken-line where a flat, horizontal line is used to indicate that the organisms do not respond to nutrient inputs. In contrast, curvilinear properties in the saturation kinetics model are used to describe smooth responses, but do not consider the possibility of homeostasis at the plateau. Instead, it assumes that an infinitesimal and significant response is ultimately produced. The “curve-and-plateau” is a modeling approach combining the advantages of curvilinear models and the broken-line (Morris, 1999). Curvilinear models describe a smooth transition to the plateau section, while a breakpoint in the broken-line is useful to estimate a nutrient requirement. Robbins et al. (2006) and Heger et al. (2008) used a curve-and-plateau approach to estimate the isoleucine requirement and the growth response to sulphur amino acid intake of growing pigs, employing a quadratic equation in combination with a flat straight line. Fitting the model, however, requires using the NLIN procedure that is only available in SAS statistical software. There is no antecedent in the reviewed literature for the use of curve-and-plateau models to estimate nutrient requirements of aquaculture species. This study introduces the use of broken-curves based on such an approach. Similar to the conventional models, the broken-curves are intended for general use in estimating requirements following the recommendations by Mercer (1980), who emphasizes the model's ability to describe nutritional response over a wide range of nutrient input for many different types of nutrients and species. The broken-curves can be fitted using standard statistical packages generally used to fit the conventional models. Of the studies listed in Table 1, none considered a statistical test to assume reliably that the response of the organism at the plateau is homeostatic. Instead, the broken-line was fitted directly, assuming the slope at the plateau is equal to zero. In other cases, the zero assumption is based on results that were not significant and obtained from regression analysis or ANOVA at the plateau. However, it is erroneous to use such results as statistical evidence that the organisms are not responding to inputs because only a Type-I statistical error is considered (protection against falsely rejecting the null hypothesis). Protection against falsely accepting the corresponding null hypothesis (Type-II statistical error; Zar, 1999) is also necessary. In this study, an example is presented of how an equivalence test can be used for that purpose.

where i and b1 are parameters describing the positive linear relation and ymax is the maximum response. To assume a constant response, the slope at the plateau (b2) was set at zero. 2.1.2. The saturation kinetics model The saturation kinetics model proposed by Mercer et al. (1989) was used. Accordingly: n

y = bK0:5 + Rmax x

n


n n
= K0:5 + x ;

ð3Þ

where b is the intercept on the y-axis, Rmax is maximum theoretical response, n is the apparent kinetic order, and K0.5 is the nutrient level for (Rmax + b ) /2. Nutrient requirement was estimated as conventionally (95% of Rmax). 2.1.3. The broken saturation kinetics model The saturation kinetics model (Eq. (3)) was used in combination with the straight-line equation at the plateau (Eq. (2)):

n n y = bK0:5 + Rmax xn = K0:5 + xn if x b xbp ; if x z xbp ;

y = ymax + b2 x

ð4Þ

where parameters are as defined previously. 2.1.4. The broken-convex curve model A convex curve (Eq. (5)), the modified Freundlich model (Ratkowsky, 1990), was used in combination with the straight-line equation at the plateau (Eq. (2)): c

y = a + mx y = ymax + b2 x

if x b xbp ; if x z xbp ;

ð5Þ

where a, m, and c are parameters of the convex curve, and the parameters of the straight-line are as defined previously. The nutrient requirement was estimated from Eq. (5) using: xbp = [(ymax −a) /m]1 / c. Note that this model is the same as the broken-line model, except that the c exponent is incorporated into the independent variable, hence, the broken-line is a particular case of the broken-convex curve when c = 1. Values of c within the range N0–b1 produce convex curves; values of c N 1 produce concave curves and indicate that the broken-convex curve should not be used because it describes an abrupt, rather than a smooth, shift to the plateau. In the following, the models will be denominated BLM (brokenline model), SKM (saturation kinetics model), BSKM (broken saturation kinetics model), and broken-convex curve model (BCCM).

2. Methods

2.2. Selection of cases for model comparison

2.1. Models evaluated

From the reviewed articles (Table 1), 24 cases were selected according to the following criteria. (1) There was no evidence of a declining right side in the response curve; (2) Authors report using the BLM and/or the SKM, and/or ANOVA; (3) At least six nutrient levels were reported, allowing sufficient degrees of freedom for fitting the models used in this study.

Based on Shearer (2000) and the survey summarized in Table 1, the broken-line and the saturation kinetics model were selected as the most commonly used conventional models; two broken-curve models are introduced here.

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Table 2 Studies used to obtain cases for model comparison. Study description Case

Nutrient

Species

1 2

Threonine (% dry weight) Threonine (g/kg diet)

3

Threonine (g/kg diet)

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Protein (% crude protein) Vitamin C (mg/kg diet) Choline (g/kg diet) Sulfur amino acid (g/100 g diet) Phosphorus (% diet) Folic acid (mg/kg diet) Lysine (% protein) Thiamine (mg/kg diet) Vitamin B6 (mg/kg diet) Lysine (%), 16 MJ/kg digestible energy Lysine (%), 20 MJ/kg digestible energy Protein (g/kg crude protein) Inositol (mg/kg diet) Vitamin E (mg/kg diet) (4% lipid) Vitamin E (mg/kg diet) (9% lipid) L-methionine (g/100 g dry weight) Lysine (g/kg crude protein) Lysine (g/100 g dry diet) Lysine (% protein) Threonine (g/kg dry matter) Threonine (g/kg dry matter)

juvenile Sciaenops ocellatus juvenile Morone chrysops× M. saxatilis (9–10 g initial weight). juvenile Morone chrysops× M. saxatilis (2–3 g initial weight). Oreochromis niloticus × O. aureus Sciaenops ocellatus Perca falvescens Perca flavescens Chanos chanos Penaeus monodon Mystus nemurus Haliotis discus Penaues monodon Oncorhynchus mykiss Oncorhynchus mykiss Epinephelus coioides Penaeus monodon Epinephelus malabaricus Epinephelus malabaricus Epinephelus coioides juvenile Sparus aureata Lateolabrax japonicus Rhamdia quelen Oncorhynchus mykiss Salmo salar

a b c d e f g

na

tb

Parameters

Models

Source

6 6

56 56

WGc, FEd WG, FE

ANOVA, BLMe ANOVA, BLM

Boren and Gatlin (1995) Keembiyehetty and Gatlin (1997)

6

56

WG, FE

ANOVA, BLM

Keembiyehetty and Gatlin (1997)

6 8 6 8 7 8 7 7 8 6 6 6 8 7 7 6 6 6 7 11 11

70 70 77 70 112 56 90 112 70 84 84 56 42 56 56 56 42 70 119 24 36

WG, FE WG, FE WG WG, FE WG, FE WG, FE WG WG WG WG, FE WG, FE WG WG WG, FE WG, FE WG WG, FE WG, FE WG FWg, FE FWg, FE

ANOVA, BLM ANOVA, BLM ANOVA, BLM BLM ANOVA, BLM ANOVA, BLM ANOVA ANOVA, BLM ANOVA, BLM BLM BLM ANOVA ANOVA, BLM ANOVA, BLM ANOVA, BLM ANOVA, BLM ANOVA, BLM, SKMf ANOVA, BLM ANOVA BLM BLM

Twibell and Brown (1998) Aguirre and Gatlin (1999) Twibell and Brown (2000) Twibell et al. (2000) Borlongan and Satoh (2001) Shiau and Huang (2001) Tantikitti and Chimsung (2001) Zhu et al. (2002) Shiau and Wu (2003) Encarnação et al. (2004) Encarnação et al. (2004) Luo et al. (2004) Shiau and Su (2004) Lin and Shiau (2005) Lin and Shiau (2005) Luo et al. (2005) Marcouli et al. (2006) Mai et al. (2006) Montes-Girao and Fracalossi (2006) Bodin et al. (2008) Bodin et al. (2008)

Number of nutrient levels. Experimental trial duration. Weight gain. Feed efficiency. Broken-line model. Four-parameter saturation kinetics model. For this study, weight gain was calculated from reported initial and final weight.

Selected cases considered the requirements of protein, amino acids, vitamins, and phosphorus for several fish, crustacean, and mollusk species. The cases used 6–11 nutrient levels in trials lasting 24–119 days, and weight gain (92% of the cases) and feed efficiency (71% of the cases) as response parameters (Table 2). A few reports used other response parameters. Results in the form of weight gain (or percentage weight gain) and feed efficiency (as g gain g− 1 dry feed or g gain g− 1 dry feed ×100) provided larger sample sizes and, together with the nutrient levels reported by the authors, were used to compare the models and estimate nutrient requirements for each case.

Values of RSS, RV, and estimates of nutrient requirements were transformed to arcsine values (Zar, 1999), yet the Kolgomorov– Smirnov test showed that transformed values were not normally distributed. Friedman's nonparametric repeated-measures ANOVA by ranks, post-hoc Tukey-type multiple comparison, and the Kendall's coefficient of concordance were used to analyze differences in RSS, RV, and estimates of nutrient requirements among the models (Zar, 1999). Procedures for nonlinear regression and Friedman's ANOVA from STATISTICA 6.0 (StatSoft, Tulsa, OK) were used. Post-hoc tests and Kendall's coefficient of concordance were calculated according to Zar (1999) with spreadsheets prepared in Excel (Microsoft, Bellevue, WA). Significance was set at P b 0.05.

2.3. Model comparison

2.4. Equivalence test

Models were compared in terms of fitting and model parsimony. Since three of the models are nonlinear in their parameters, residual sum of squares (RSS) and residual variance (RV) were used for fitting and model parsimony, respectively (Ratkowsky, 1990; Quinn and Keough, 2002; Hernandez-Llamas and Ratkowsky, 2004). For weight gain, all the cases were used to compare the BLM, SKM, and BCCM. Only nine studies contained sufficient degrees of freedom to apply the BSKM. For feed efficiency, 17 cases were suitable for the first three models and six cases were suitable for testing the four models. To combine data from the different cases and compare the results obtained with the models, the tabled values of response and nutrient input levels (dose–response pairs) presented in the cases were standardized to percentages. For this, the minimum and maximum values of the tabled values of the variables were set at 0 and 100%, respectively, and the rest of the values were standardized to the corresponding percentages.

This test was developed to interpret and report studies that do not find statistically significant differences. A possible outcome of an equivalence test is, for example, the conclusion that, at the 5% level, two means do not differ by more than some specified amount (Hauck and Anderson, 1986). A case was prepared containing estimates of replicate values of the specific growth rate that is graphically presented in Mai et al. (2006). These estimates do not necessarily coincide with the actual data; it is used to show how an equivalence test can be employed when studying the plateau response section. As the first step, the BLM was fitted using Eqs. (1) and (2), letting the slope corresponding to the plateau (b2) being estimated, rather than set at zero and observing that the slope was not significant. As the second step, the equivalence test was performed to determine whether it would be reasonable to assume that the slope did not differ from zero by more than a specified amount. In this case, 5% of the estimated slope was used as the similarity level (Garret, 1997).

For assessment of equivalence, Chow and Liu (2004) recommend using the two one-sided procedure to test the following interval hypotheses: (1) H0: b2 − k ≤ −d or b2 − k ≥ d versus Ha: −d b b2 − k b d, where b2 represents the slope value at the plateau obtained in the first step, k represents the value of the slope to be compared with zero (in this case), and d is the specified amount of difference. The hypotheses in (1) were deconstructed into two sets of onesided hypotheses:

198.3 ± 249.6 (2.2)a 0.89 (b0.001)

All models 3 models

153.8 ± 184.8 (2.0)a 116.6 ± 120.3 (1.7)a

RV

61.4 ± 46.0 (2.5)a 73.2 ± 63.7 (2.5)a 102.7 ± 125.0 (3.1)a 48.5 ± 31.6 (1.8)a 0.2 (N 0.5–0.75 b)

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(2) H02: b2 − k ≤ −d versus Ha2: b2 − k N −d

154.9 ± 218.3 (2.4)ab 0.72 (b 0.001) 383.0 ± 521.6 (1.5)b 0.79 (b0.001)

130.0 ± 155.7 (1.8)a 137.4 ± 206.8 (1.7)a

408.5 ± 364.3 (3.0)a 479.8 ± 417.7 (3.5)a 276.8 ± 315.0 (1.4)b 283.6 ± 318.8 (2.0)bc 0.34 (N 0.1–0.25 b) 493.7 ± 598.7(2.3)a 464.7 ± 695.8 (2.1)ab

BLM SKM BSKM BCCM KC

(3) H03: b2 − k ≥ d versus Ha3: b2 − k b d.

Values within the same column with different superscripts were significantly different. Models include: broken-line model (BLM), four-parameter saturation kinetics model (SKM), broken-SKM (BSKM), and broken-convex curve model (BCCM).

279.2 ± 222.2 (3.3)a 330.8 ± 284.1 (3.1)ab 245.7 ± 274.1 (1.5)ab 181.7 ± 163.2 (2.0)b 0.23 (N 0.25–0.5 b) 370.4 ± 361.0 (2.5)a 307.0 ± 268.4 (1.9)ab

102.2 ± 108.0 (1.8)a 130.0 ± 122.6 (2.6)a 147.1 ± 247.1 (3.0)a 91.8 ± 122.3 (2.4)a 0.32 (N0.1–0.25 b)

255.1 ± 239.1 (1.4)b 0.93 (b 0.001)

All models RSS

3 models 3 models All models 3 models Model

RSS

RV

All models

Feed efficiency

and

Weight gain

Table 3 Mean values (± SD) and average rank (in parentheses) of standardized (as percentages) residual sum of squares (RSS) and residual variance (RV) obtained from regression analyses, with the Kendall's coefficient of concordance (KC) and the corresponding p-level (in parenthesis).

210

If the confidence interval for the difference b2 − k (set at 97.5% for this test) is contained within − d and d, the null hypotheses in (3), (2), and (1) are rejected and equivalence is accepted with the conclusion that fish are not responding to nutrient inputs with a tolerance as negligible as 5% of the estimated slope. The confidence limits for the difference, b2 − k, were estimated with a linear combination of regression coefficients. Procedures from Stata 10 (StataCorp, College Station, TX) were used to fit the broken-line model and perform the equivalence test, setting significance at P b 0.05. 3. Results 3.1. Comparison of models In terms of regression ANOVA, significant results were obtained for all the cases and models tested. Requirements estimated by the conventional models generally coincided, as reported by the authors. Friedman's ANOVA and post-hoc tests indicated that there were significant differences among the models in terms of RRS and RV when weight gain and feed efficiency data were fitted (Table 3). For weight gain, the BCCM and BSKM resulted in a better fit when three or four models were compared. When three models were compared, significantly lower RV values indicated that the conventional models were more parsimonious, but no differences were found when the four models were considered. When only six input levels were employed, the conventional models showed being more parsimonious. In cases with seven or more input-levels, however, the BCCM performed better in 33.3% of the cases for weight gain and feed efficiency. Fewer parameters in the conventional models, together with reduced number of levels of nutrient input, produced some evidence that these models were more parsimonious. When feed efficiency data were analyzed for three or four models, the BCCM and BSKM produced better results in terms of RSS (Table 3). Regarding parsimony, RV did not indicate any model as more parsimonious than the other models. Table 4 Mean values (± SD) and average rank (in parentheses) of standardized (as percentages) nutrient requirement estimates obtained from regression analyses, with the Kendall's coefficient of concordance (KC) and the corresponding p-level (in parenthesis). Weight gain Model 3 models BLM SKM BSKM BCCM KC

Feed efficiency All models

48.1 ± 28.1 (1.7)a 36.5 ± 26.4 (2.0)a 54.2 ± 34.7 (1.8)ab 48.4 ± 42.7 (3.0)a 39.9 ± 28.1 (2.4)a 51.3 ± 28.2 (2.4)b 40.2 ± 25.8 (2.6)a 0.94 (b0.001) 0.34 (N 0.1–0.25b)

3 models

All models

49.8 ± 30.0 (1.8)a 27.4 ± 29.1 (2.0)a 49.0 ± 31.7 (1.7)ab 31.8 ± 35.0 (2.8)a 29.9 ± 31.4 (2.6)a 51.2 ± 31.0 (2.5)b 29.5 ± 29.7 (2.6)a 0.95 (b 0.001) 0.23 (N 0.5–0.75b)

Values within the same column with different superscripts were significantly different. Models include: broken-line model (BLM), four-parameter saturation kinetics model (SKM), broken-SKM (BSKM), and broken-convex curve model (BCCM).

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of concordance indicated that the models tended to consistently yield low, intermediate, or high requirements among the cases. No significant differences were detected when all the models were compared. When both response parameters were analyzed for three models, the lowest and highest requirements were produced by the BLM and BCCM, respectively. No significant differences were observed among estimates obtained from the SKM and the other models. The SKM was the only model yielding, in some cases, lower estimates than the BLM. In two cases, it produced extremely high estimates, beyond the experimental ranges reported in the studies. The BSKM did not produce a solution for estimates of requirements in 38% of the cases for weight gain and 24% for the cases for feed efficiency because the curvilinear model did not intersect the flat straight line. The BCCM showed more consistency in estimation and the differences in estimates by the BLM and BCCM were particularly evident. Accordingly and depending on the nutrient under study, an underestimation of 6.2 to 9.2% by the BLM was calculated. These results support the criticism by several authors regarding the tendency of the BLM to underestimate requirements. Examples of large, moderate, and negligible differences in estimates of requirements based on weight gain are shown in Fig. 1. Estimates of nutrient requirements tended to coincide among the models when the section of the response curve approaching the plateau were poorly represented, as in cases where the populations switched to the plateau at low nutrient levels. When the switch occurred at intermediate and higher nutrient levels, the BSKM and BCCM were clearly superior, resulting in a wider variation of response, a reduced plateau region, lower RSS and RV, and higher estimates of nutrient requirements than the BLM (Fig. 1, Tables 3 and 4).

3.2. Equivalence test Fitting the BLM to the case of Mai et al. (2006) produced a significant slope for the first three input levels (b1 = 0.33, P b 0.001), but the slope fitted for three higher levels at the plateau was not significant (b2 = −0.027, P = 0.07) (Fig. 2). The linear combination of coefficients resulted in −0.062 and 0.0072 as confidence limits for the difference b2 − k (Fig. 2). Assuming that b2 does not differ by more than 5% from zero (d = 0.0013), equivalence cannot be concluded because the confidence interval is not contained within the tolerance limits (i.e. −0.0013 and 0.0013). This result indicates that we cannot assume that fish were not responding to nutrient inputs.

Fig. 1. Results of estimates of nutrient requirements using broken-line model (BLM), saturation kinetics model (SKM), broken saturation kinetics model (BSKM), and brokenconvex curve model (BCCM). Large (A), moderate (B), and negligible (C) differences in estimates are indicated. The data correspond to Cases 16, 11, and 9 in Table 2. The estimated requirements from the models are indicated on the nutrient-level axis.

Significant values of Kendall's coefficient of concordance were obtained when three models were tested for weight gain and feed efficiency, although no differences were observed when using the four models (Table 3). High coefficient values indicated that the fitting performance and parsimony of the models among the cases were more consistent than results expected by chance, that is, the results obtained from each case tended to concur regarding performance of the models. Friedman's ANOVA showed that estimates of nutrient requirements differed significantly when three models were tested for weight gain and feed efficiency (Table 4). Significant values for Kendall's coefficient

Fig. 2. Results of the equivalence test at the plateau for a case from Mai et al. (2006). Output from the linear combination of coefficients is included. Confidence intervals for the difference between the slope estimate and zero (b2 − k), and interval for the similarity level (d) are indicated.

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4. Discussion Better-fitting performance to weight gain and feed efficiency data were consistently observed for the BSKM and BCCM. On the other hand, the conventional models showed greater parsimony only when weight gain were analyzed for three models. The results indicate that, when larger number of input levels was used, increased degrees of freedom in the regression tests resulted in improved parsimony of the BCCM. Working with replicate values of the response parameter also increases degrees of freedom and it is reasonable to assume that the BCCM and the BSKM could exhibit greater parsimony whenever replicate values are available. Based on this review, the number of nutrient input levels used by nutritionists appears to be increasing. A more accurate description of the response of aquaculture species, accompanied by better methods for estimating nutrients requirements, will contribute to improved diets for aquaculture species. The BCCM proved to fit better over a wide range of situations and demonstrated its adaptability. The BSKM tended to produce similar results, but using an asymptotic model resulted in several cases where there were no solutions for estimating nutrient requirements. It is reasonable to expect that this could be a frequent problem for brokencurve models using upper asymptotes, as in the case of the BSKM. In contrast, using a convex model for the BCCM consistently produced a solution for estimating requirements. The SKM performed erratically when estimating requirements. This was the consequence of a lack of biological and statistical bases for supporting the arbitrary criterion of using 95% of maximum response to define a requirement. As explained by Morris (1989), when fitting asymptotic models, one temptation that must be resisted at all costs is to choose some arbitrary proportion of maximum output as the “requirement.” The results of this study by fitting the BLM to the case by Mai et al. (2006) and using an equivalence test were inconclusive. However, regression analysis yielded close to significant results at the plateau (P = 0.07), thus suggesting that fish could be responding to nutrient inputs. Investigators can decide to accept the results as significant at that P level and look for a maximum response, but it can also be concluded that it would be convenient to conduct a new trial, looking for increased statistical power to detect a significant response at a lower P value. In general, it is a good practice to fit, as a first step, the slope corresponding to the plateau, rather than assuming it equals zero without previously obtaining evidence to support the assumption. In the example presented here, the specified tolerance for the equivalence test was an arbitrary similarity level (5% of the slope estimated at the plateau). However, it might also be based on knowledge of how much leeway is tolerable in the response being measured (Garret, 1997). The supplementation technique is the method more commonly used to study nutrient requirements. If the nutrient under test is sufficiently deficient in the basal diet, adding graded supplements of the nutrient results in a smooth response curve that is used to estimate the nutrient requirement (Dean and Scott, 1965). According to D'Mello (1982), the technique presents the following limitations (mainly for determining amino acid requirements). (1) Each diet does not have the same amino acid balance. (2) At high levels of supplementation, the test amino acid may no longer be first-limiting. (3) Further response to this amino acid might be prevented by deficiencies of other amino acids, (4) It is difficult to devise a basal diet which is low in the test amino acid, but adequate in all other indispensable amino acid, thus precluding the use of a wide range of input levels of the tests amino acid. The dilution technique was devised by Fisher and Morris (1970) to overcome the limitations of the supplementation technique. In aquaculture nutrition, the diet dilution technique has been used in a form of dose–response studies where an exponential model serves to analyze mathematically amino acid requirements in fish (Liebert, 2005). The method allows concluding requirements for the individual limiting amino acid depending on graded protein deposition (growth

performance) and observed dietary amino acid efficiency using physiological based modeling. However, defining model parameters requires reliable experimental data depending on species, genotype and age and measuring the dietary efficiency of individual amino acid in their limiting position (Liebert and Benkendorff, 2007). The criticism of the supplementation technique has been challenged by D'Mello (1982), who compared both techniques and concluded that they are equally acceptable, and the ultimate choice of method depends upon the amino acid under assay. All of the cases selected for this study employed the supplementation technique and no clear tendency was observed for this technique to be replaced by the dilution technique. Other alternative models have been used to estimate nutrient requirements. A modification of the diphasic linear model by Koops and Grossman (1993) was used by Booth et al. (2007) to study nutritional requirements of the Australian snapper. The model simulates a curvilinear transition from an increasing linear phase to the plateau and a breakpoint in the transition curve serves for estimating the requirement. However, the breakpoint is a parameter representing the central point of the transition and it does not correspond to the nutrient level where the response of the population begins to be homeostatic, as in the “curve-and-plateau” approach. The diphasic model represents an improvement over the BLM, yet it also tends to underestimate requirements. Also, the model has difficulties for realistic estimates of the parameters. In most cases, a proper estimation of the parameter for smoothness of the transition is not feasible and the value of this parameter must be arbitrarily fixed. This is a considerable limitation for evaluation and comparison of the fitting performance of the model. Compared to the diphasic model, the BSKM and BCCM assume that a larger fraction of the population is likely to shift to the homeostatic stage. The “Reading model” is a model reaching a genuine plateau when the requirements of the most demanding animal are satisfied Morris (1999). However, it must be noted that this model is designed for terrestrial animals and it determines nutrient requirements based on feed intake, rather than concentration of nutrients. This is a critical distinction because accurate assessment of feed intake by fish is one of the most difficult achievements of nutrition research in aquaculture (Cowey, 1992; Glencrooss et al., 2007). To fit the model, specific software that is not available in standard statistical packages is necessary (R. Gous, pers. comm., University of KwaZulu-Natal). The difficulties of estimating nutrient requirements were recognized and addressed by Rodehutscord and Pack (1999). According to these authors, estimating requirements is an elusive concept, as it involves the definition of what, and at which level, the species is expected to perform. Response parameters, such as protein deposition, have been considered more adequate to estimate amino acid requirements of fish (Bureau and Encarnação, 2006). However, it must be noted that the range of response parameters that can be examined is large and increasing (Glencrooss et al., 2007). Since the definition of animal performance is usually related to criteria of economic significance (Rodehutscord and Pack, 1999), most studies of nutrient utilization use growth as the response parameter (Glencrooss et al., 2007). Weight gain and feed efficiency were used in this study because almost all of the 80 articles surveyed reported using weight gain, with feed efficiency as the second most frequently used parameter. These parameters have direct practical implications and provide an adequate sample size for evaluation of a model. Nutritionists using other types of response parameters could use the same standard statistical packages to fit and compare the conventional models and the broken-curve models presented in this study. 5. Conclusions More consistent and better-fitting performance of the brokencurves allows estimating nutrient requirements more reliably when

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compared to conventional models. Broken-curve models can be used on a wide variety of species and nutrients using standard statistical packages with nonlinear regression procedures. The use of higher number of input levels and replicate values is recommended for better representation of the response curve, where the broken-curve models also show more parsimony. Statistical analysis of the response at the plateau is necessary for adequate use of the broken-line model or broken-curve models, and testing for equivalence is an adequate method for that purpose. Acknowledgements This investigation started with discussions with Roberto Civera and Humberto Villarreal at CIBNOR. Additionally, I am particularly grateful to Axel Buchner, Karl Shearer, David Ratkowsky, Rob Gous, Trevor Morris, and Victor Gomez for their helpful comments and observations. I received valuable assistance and suggestions from technical support personnel at StataCorp and Ira Fogel, English editor at CIBNOR. I assume responsibility for what is stated in the study. The comments by two anonymous reviewers were very helpful for the improvement of the manuscript. This investigation was partially supported by a research grant from Consejo Nacional de Ciencia y Tecnología (SAGARPACONACYT 042-C, “Uso del cártamo (Carthamus tinctorius L.) como fuente de proteína en alimentos para organismos acuáticos”). References Aguirre, P., Gatlin III, D.M., 1999. Dietary vitamin C requirement of red drum Sciaenops ocellatus. Aquac. Nutr. 5, 247–249. Baker, D.H., 1986. Problems and pitfalls in animal experiments designed to establish dietary requirements for essential nutrients. J. Nutr. 116, 2339–2349. Bodin, N., Mambrini, N., Wauters, J., Abboudi, T., Ooghe, W., Le Boulenge, E., Larondelle, I., Rollin, X., 2008. Threonine requirements for rainbow trout (Oncorhynchus mykiss) and Atlantic salmon (Salmo salar) at the fry stage are similar. Aquaculture 274, 353–363. Booth, M.A., Allan, G.I., Anderson, A.J., 2007. Investigations of the nutritional requirements of Australian snapper Pagrus auratus (Bloch & Schneider, 1801): effects of digestible energy content on utilization of digestible protein. Aquac. Res. 38, 429–440. Boren, R.S., Gatlin III, D.M., 1995. Dietary threonine requirement of juvenile red drum Sciaenops ocelatus. J. World Aquac. Soc. 26, 279–283. Borlongan, I.G., Satoh, S., 2001. Dietary phosphorus requirement of juvenile milkfish Chanos chanos (Forsskal). Aquac. Res. 32, 26–32. Bureau, D., Encarnação, P., 2006. Adequately defining amino acid requirement of fish: the case example of lysine. In: Cruz-Suárez, E., Ricque-Marie, R., Tapia-Salazar, M., Nieto-López, M.G., Villarreal-Cavazos, D.A., Peullo-Cruz, A.C., García-Ortega, A. (Eds.), Avances en Nutrición Acuícola VIII. VIII Simposium Internacional de Nutrición Acuícola, Nuevo Leon, Mexico. Universidad Autónoma de Nuevo Leon, pp. 29–53. Chow, S.C., Liu, J.P., 2004. Design and Analysis of Clinical Trials. John Wiley & Sons, Hoboken, NJ. 729 pp. Cowey, C.B., 1992. Nutrition: estimating requirements of rainbow trout. Aquaculture 100, 117–189. Curnow, R.N., 1973. A smooth population response curve based on an abrupt threshold and plateau model for individuals. Biometrics 29, 1–10. Dean, W.F., Scott, H.M., 1965. The development of an amino acid reference diet for the early growth of chicks. Poultry Sci. 44, 801. D'Mello, J.P.F., 1982. A comparison of two empirical methods of determining amino acid requirements World Poult. Sci. J. 38, 114–119. Encarnação, P., de Lange, C., Rodehutscord, M., Hoehler, D., Bureau, W., Bureau, D., 2004. Diet digestible energy content affects lysine utilization, but not dietary lysine requirements of rainbow trout (Oncorhynchus mykiss) for maximum growth. Aquaculture 235, 569–586. Fisher, C., Morris, T.R., 1970. The determination of the methionine requirement of laying pullets by a diet dilution technique. Br. Poult. Sci. 11, 67. Fisher, C., Morris, T.R., Jennings, R.C., 1973. A model for the description and prediction of the response of laying hens to amino acid intake. Br. Poult. Sci. 14, 469–484. Garret, K.A., 1997. Use of statistical tests of equivalence (bioequivalence tests) in plant pathology. Pyhtopatholgy 372–374. Glencrooss, B.D., Booth, M., Allan, G.L., 2007. A feed is only as good as its ingredients — a review of ingredient evaluation strategies for aquaculture feeds. Aquac. Nutr. 13, 17–34.

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