from

KELLY R. ROBBINS, HORACE W. NORTON ANDDAVID H. BAKER Department of Animal Science, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 ABSTRACT Two least squares methods of estimating nutrient require ments from growth data were compared. One method involved fitting a broken line by the method of least squares. The requirement was taken as the abscissa of the breakpoint in the curve. The other method involved fitting an appropriate exponential function to the growth data and estimat ing the requirement as the abscissa of the point on the fitted curve whose ordinate was 95% of the upper asymptote. For the nine sets of data studied, the broken line provided adequate fits for only six. The nonlinear models provided adequate fits for all the data studied. When both the broken line and the chosen nonlinear model provided adequate fits, the estimated re quirements were nearly the same. However, the consistently good fits ob tained with the nonlinear models suggest that this approach may generally be more useful. J. Nutr. 109: 1710-1714, 1979. dietary requirement â€¢amino acids INDEXING KEY WORDS vitamins â€¢growth The broken-line model with a horizontal branch, fitted by the method of least squares, was adopted ( 1956 ) because it yielded an objective estimate of a nutrient "requirement," that dosage which is fully adequate ( 1 ). Some responses seem to conform nearly to a broken line, and any response which approaches an asymptote can be fitted by a broken line for a suit able narrow range of doses and a large enough error variance. However, many re sponses are not fitted satisfactorily. Fur thermore, the implicit notion that the response to a complex physiological pro cess, exhibited (as it must be) by data involving many experimental animals of differing experience and genetic constitu tion, should have a discontinuity in its first derivative cannot be realistic. A more sophisticated approach is needed. Some responses are evidently sigmoidal (2-5). Of the innumerable functions of this form, we have used (i)

y = u + v(l â€”ewi)

(u)

may suffice. It has the desired properties and is simpler by having only three param eters. Further, it is a limiting form of (i), ensuring that (i) will also fit any set of data fitted by (ii). Received for publication

1710

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where y is the response to dose x. This model has a lower asymptote, p, an upper asymptote, p + q, and continuous deriva tives. As it has rotational symmetry about its inflection point, it cannot represent re sponses which do not (nearly) have such symmetry. If need be, the exponent can be generalized by adding terms involving powers of the dose. Other responses, not evidently sigmoidal, rise most steeply from the lowest dose tried. Perhaps they are high-dose portions of sigmoidal responses for which lower doses were not tested (or are unattainable). For such responses

March 1, 1979.

NUTRIENT

REQUIREMENT

If a curve which approaches an asymp tote is chosen, we confront the task of choosing a definition for the estimated "requirement." This choice can be objec tive. For example, it might be the dose at which the line tangent to (i) at its inflec tion point intersects the asymptote. How ever, there can be an important response to still higher doses, making this estimate un satisfactory for practical use by ignoring economic considerations. For scientific use, such as testing for the effect of age or of sex or of another component of diet, it may prove to be adequate to adopt a dose cor responding to a relatively near approach to the asymptote, or to a relatively large fraction of the total response (from lower to upper asymptote). We have arbitrarily chosen the dose at which the response reaches 95% of the total response, for model (i), or 95% of the response from lowest dose to asymptote, for model (ii). Given this arbitrary choice, the procedure is entirely objective. Our objective was to compare the two suitable nonlinear models to the brokenline model as a means of estimating nu trient requirements from growth data. PROCEDURE Broken-line model. The general equation of the broken line ' is Y = L + U(R - XLR), where L is the ordinate and R is the ab scissa of the breakpoint (6). R is taken as the estimated requirement. XLR means X less than R, and U is the slope of the line for XLR.By definition R â€”XLRis zero when X> R. This model was fitted by obtaining the least squares estimates of L and U for several values of R, using ordinary least squares techniques. The maximum likeli hood estimate of R was taken as that value of R which maximized the model sum of squares. Approximate sampling errors for L, U and R were calculated using the matrix of sums of squares and products of first deriv atives and Se2,given the least squares esti mates of each parameter. The matrix is: nLRU XLR) S(R-XLR)2 nLRU

US (R- XLR)

U2(R-XLR) nLRlP

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1711

ESTIMATION

200-

O1 OZ L-Histidine

O3 O4 ( of diet)

OS

0.6

Fig. 1 Fitted sigmoid curve describing the gain response (n = 21) of young chicks to in creasing levels of dietary L-histidine. The re quirement was estimated to be 0.316 Â±0.009% L-histidine. Parameter estimates of the fitted sig moid were p + q â€”184, r = 4.74 and s = â€”24.34. The deviation mean square was 48 with 6 de grees of freedom, and the pooled SEM for gain was 4.0 with 20 degrees of freedom.

Denoting the inverse of this matrix as C, the approximate sampling errors of L, U and R were calculated as CnSe2, c22Se2and CasSe2,respectively. Nonlinear models. Both models were fit ted iteratively using the matrix of sums of squares and products of first derivatives and a four- or three-element vector of con stant terms (7). The approximate sampling error of the requirement was estimated as (Zx' CZx)Se2, where Zx is the column vector of the par tial derivatives of X with respect to the model parameters (i.e., with respect to p, 1 The broken line consists of two parts : a straight line with an Increasing or decreasing slope and a horizontal line. Their point of Intersection Is the breakpoint. The broken-line model Is often adequate for flttlng growth data, especially If the levels tested cover a rather narrow range. For other types of biologlcnl data, a broken-line equation describing two Intersecting straight lines, both with non-zero slopes, may be needed. The general equation for this type of broken line Is Y = L + U(R â€”XLR) + V(Xon â€”R). For this case, XOR means X greater than R, and V Is the slope of the line for XOB. Also, XORâ€”R is zero when X < R.

1712

ROBBINS, NORTON

AND BAKER

TABLE 1 Effect of dietary copper on the sulfur amino acid requirement of the chick (experiment /)' (mg/kg)32502104160789296980.57Â±0.0113097-3120.60 copper DietarySAA diet)20.250.300.400.450.500.550.600.70Method (% of

usedBroken lineRequirement Â±SKDMS df)4L>uÂ»SigmoidRequirement (5

1274118-3960.53Â±0.019211207.76-20.03Added Â±0.0

Â±SEDMS df)4p (4

Â±0.02991007.00-16.68500-2112034445660670.63Â±0.0293567-1800.69Â±

q6rÂ§8*0152371951061191161200.52 + 1 Data are mean weight gains (g) of triplicate groups of six chicks fed the experimental diets from day 8 to day 16 posthatching. Average initial weight was 90 g. Pooled SEM for gain was 4.2 with 48 degrees of freedom. 2Consisted of a 50:50 (w : w) mixture of Dir-methionine and L-cystine. 3Provided as CuSOÂ«-5HjO. * Deviation mean square and degrees of freedom. 5 Estimates of the model parameters.

q, r and s for the sigmoid curve, and u, v and w for the asymptotic curve), and C is the appropriate variance-covariance matrix

(7). RESULTS

The growth response of young chicks (fig. 1) to graded additions of L-histidine (6) was fitted with a broken line yielding a requirement estimate of 0.338$: L-histi dine, but deviations from the fitted model are significant (P 0.10), yield ing a requirement of 0.316 Â±0.009% L-his tidine when estimated at 95% of the upper asymptotic value. Data in tables 1 and 2 are the results of two experiments designed to assess the effect of supplemental copper on the sulfur

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amino acid requirement of young chicks. The basal diet and experimental procedure have been described (6). In the first ex periment (table 1) eight levels of sulfur amino acids were fed with supplemental copper at 0, 250 or 500 mg/kg diet. In each case, both the broken line and the sigmoid fit the data adequately 2 (P > 0.10). How ever, for the data for supplemental copper at zero and at 250 mg/kg diet, the sigmoid accounted for more of the variation due to amino acid dose, indicating that this model may provide a better description of the responses. The two methods yield similar requirement estimates consistent with a linear increase in the sulfur amino acid requirement with the addition of copper. In the second experiment (table 2) only five levels of sulfur amino acids were fed at each copper level. The broken line did not fit the copper-unsupplemented data2 (P < 0.05), precluding estimation of a re quirement. The sigmoid provided a good â€¢ Probability that the ratio oÃ- the deviation mean square to the error mean square is greater than tabu lar F.

NUTRIENT

REQUIREMENT

1713

ESTIMATION

TABLE 2 Effect of dietary copper on the sulfur amino acid requirement of the chick (experiment 2)' (mg/kg)s03464109113120â€”i110â€”â€”0.52Â±0.017 copper DietarySAA diet)20.300.400.500.600.700.80Method (% of

usedBroken lineRequirement SEDMS Â± df)'LÂ«UÂ«Sigmoid (2

Requirement Â±SE df)Â«P+q6rÂ«8Â«Added DMS (1

Â±0.01550105-3120.70 Â±0.0241476-1830.72Â±0.048

Â±0.056 6311711.28-26.91250185681103106â€”0.57 601133.07-9.18500_30496678780.65 0.9805.40-11.36

1Data are mean weight gains (g) of triplicate groups of six chicks fed the experimental diets from day 8 to day 16 posthatching. Average initial weight was 75 g. Pooled SE for gain was 3.1 with 30 degrees of freedom. 2 Consisted of a 50:50 (w:w) mixture of DL-methionine and i/-cystjne. 3Provided as CuCOi. * Deviation mean square and degrees of freedom. 6 For this model the deviation mean square was signifi cant (P < 0.05), so that a requirement estimate would be invalid. â€¢ Estimates of the model parameters. TABLE 3 The choline requirement of chicks during the seventh week of life1 cholinemg/kg058116232464696Method Dietary

fit- (P>0.10) to each set of data and indicated that supplemental copper in creased the sulfur amino requirement. Figure 2 depicts the growth response of chicks to graded additions of biotin.3 The data are not visibly sigmoidal, therefore the simpler curve having only an upper asymp tote was fitted, and was found to be ade quate 2 (P > 0.10). The fitted curve yielded a requirement estimate of 63 Â±9.7 /*g/kg biotin. The broken line did not adequately fit these data 2 (P < 0.10). The chick's response to dietary choline

usedBroken lineRequirement Â±SEDMS df)2LÂ»UÂ»AsymptoticRequirement (3

(8) resembles the response to biotin, i.e., the growth response from zero to excess choline is asymptotic (table 3). Again, the asymptotic curve fits the data well2 ( P > 0.10), providing a requirement estimate of 209 Â±73 mg/kg. However, the broken line (P > 0.10) Â±SEDMS Â±7337436-4.34 also fits the data adequately2 df)2u (3 yielding a requirement estimate of 355 Â± v3W3Gain9381396404413432432355Â±12764432-0.13209 + 127 mg/kg.

1Data from Molitpris and Baker (9), and repre sent mean weight gains (g) of quadruplicate groups of six chicks. Pooled SEM for gain was 10 with 18 degrees of freedom. 2 Deviation mean square and degrees of freedom. 3 Estimates of the model parameters.

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DISCUSSION

The popularity of the broken-line method of estimating nutrient requirements lies in â€¢ Scott, H. M. (1969)

UnpublisheÃ¤data.

1714

BOBBINS, NOBTON

285

255

225

e

5 195

50

75

100

125

150

AND BAKER

simpler curve, which exponentially ap proaches an upper asymptote may be ade quate. With minerals, growth is generally not the criterion of choice, although exami nation of the potassium requirement studies of Bieri ( 11 ) indicated that growth rate as a function of dietary potassium con centration also fits a sigmoid curve. Our results indicate that when both the chosen nonlinear model and the brokenline model fit the data adequately, the esti mated requirements are nearly the same. Even in these cases, the broken line, with its discontinuous first derivative, cannot be more than a rough approximation. The consistently good fits obtained with the non linear models presented here indicate that they are generally preferable, despite the subjectivity that seems to enter the choice of the definition for "requirement."

Blotln (ug/kg died

Fig. 2 Fitted asymptotic curve describing the gain response ( n = 20 ) of young chicks to in creasing levels of dietary biotin. The requirement was estimated as 63 Â±9.7 yag/kg. Parameter esti mates of the fitted asymptotic curve were u + v â€”268 and w = â€”32.17. The deviation mean square was 302 with 4 degrees of freedom, and the pooled SEM for gain was 8.5 with 7 degrees of freedom.

two assumptions: (a) a growing animal will respond linearly to additions of a limit ing indispensable nutrient until the exact requirement is met, after which (b) no further growth response will be observed. Of course, there is indeed a portion of the growth response which is nearly linear and also a portion where growth is near a maxi mum. Depending on the design of the experiment, a broken line may describe the response adequately, as illustrated by the example data presented herein. However, the failure of the broken line to fit some data adequately, and a priori evidence that the response curve is not a broken line (2-5) indicate that a curvilinear model is needed. The selected sigmoid model is an obvious choice because of its mathematical sim plicity. The growth response to graded levels of a vitamin (8-10) usually do not exhibit the full sigmoidal shape, so the

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LITERATURE

CITED

1. Mameesh, M. S., Schendel, H. E., Norton, H. W. & Johnson, B. C. (1956) The effect of penicillin on the thiamine requirement of the rat. Br. J. Nutr. 10, 23-27. 2. Parks, J. R. (1970) Growth curves and the physiology of growth. II. Effects of dietary energy. Am. J. Physiol. 219, 837-839. 3. Parks, J. R. (1970) Growth curves and the physiology of growth. III. Effects of dietary protein. Am. J. Physiol. 219, 840-843. 4. Parks, J. R. (1971) Growth curves and the physiology of growth. IV. Effects of dietary methionine. Am. J. Physiol. 221, 1845-1848. 5. Mercer, L. P., Flodin, N. W. & Morgan, P. H. ( 1978 ) New methods for comparing the bio logical efficiency of alternate nutrient sources. J. Nutr. 108, 1244-1249. 6. Bobbins, K. R., Baker, D. H. & Norton, H. W. ( 1977 ) Histidine status of the chick as mea sured by growth rate, plasma free histidine and breast muscle carnosine. J. Nutr. 107, 2055-2061. 7. Draper, N. R. & Smith, H. (1966) Applied Regression Analysis. John Wiley & Sons, Inc., New York, pp. 263-304. 8. Molitoris, B. A. & Baker, D. H. (1976) The choline requirement of broiler chicks during the seventh week of life. Poultry Sci. 55, 220-224. 9. Andrews, J. W. & Murai, T. (1978) Dietary niacin requirements for channel catfish. J. Nutr. Ã•08, 1508-1511. 10. Murai, T. & Andrews, J. W. (1978) Riboflavin requirement of channel catfish fingerlings. J. Nutr. 108, 1512-1517. 11. Bieri, J. G. (1977) Potassium requirement of the growing rat. J. Nutr. 107, 1394-1395.