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Partitioning of variance in true metabolizable energy determinations: an example using wheat data
Animal Feed Science and Technology 80 (1999) 29±41
Partitioning of variance in true metabolizable energy determinations: an example using wheat data T.H. D'Alfonsoa, H.B. Manbeckb, W.B. Roushc,* a
Rhone Poulenc Animal Nutrition 42, Avenue Aristide Briand, BP 100 92164, Antony Cedex, France b Department of Agricultural and Biological Engineering, Penn State University, University Park PA 16802, USA c Department of Poultry Science, Penn State University, University Park PA 16802, USA Received 16 June 1998; received in revised form 20 October 1998; accepted 25 February 1999
Abstract A variance statistic was used to partition the total variance into that attributable to each step of a TMEN assay procedure. Estimation of the TMEN of wheat was used as an example. The variance statistic can also be used to optimize the design of a TMEN experiment with respect to cost of the experiment and desired accuracy of the result. Experimental design optimization is accomplished by providing a functional relationship between the accuracy of the estimate and the number of replicates of feed, the number of birds used in the experiment, and the cost of each step. The variance statistic is also a useful tool for identifying and removing outliers and highly variable measurements. This feature was demonstrated with the chosen example data. Gross energy of the feed will explain approximately 50% of the variance of the TMEN estimate depending on how many replicates are evaluated. Nitrogen content of the feed sample will explain approximately 40% of the total variance. It is recommended to replicate this measurement as many times as possible. Ten replicates were recommended for the example data. The energy content of excreta from fed birds represented the next largest source of variance, at approximately 4% of the total variance, respectively. If within-bird variance is large, better homogenization of the sample and more replicates are recommended. If among-bird variance is significantly different, more birds should be used. Nitrogen content of excreta from fed birds represented less than 2.5% of the total variance. Energy and nitrogen content of excreta from unfed birds combined represented less than 2% of the total variance, suggesting that the number of unfed birds and the amount of excreta sub-samples may be reduced without adversely affecting the accuracy of the TMEN estimate. Variance due to the amount of excreta collected from the fed birds, and variance due to the amount of feed consumed by the birds, are expected to be small. This result suggested
* Corresponding author. Tel.: +1-814-863-0655; fax:+1-814-865-5691; e-mail: [email protected] 0377-8401/99/$ ± see front matter # 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 8 4 0 1 ( 9 9 ) 0 0 0 3 5 - 8
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that force-feeding may not be necessary for accurate TMEN estimates. # 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Metabolizable energy; Variance; Error estimation; Quality control
1. Introduction The accuracy of TMEN estimates has been investigated in a number of papers (for example, Wolynetz and Sibbald, 1984; Pesti et al., 1988; Dale and Fuller, 1986), particularly as it relates to the improvement in the assay by including the nitrogen correction. A variance statistic may be used to partition the total variance into that attributable to each step of a TMEN assay procedure. The appropriate variance statistic can also be used to optimize the design of a TMEN experiment with respect to cost of the experiment and desired accuracy of the result. Experimental design optimization is accomplished by providing a functional relationship between the accuracy of the estimate and the number of replicates of feed, the number of birds used in the experiment, and the cost of each step. A variance statistic can also be a useful tool for identifying and removing outliers and highly variable measurements. 2. Mathematical background Although the true metabolizable energy (TMEN) Eq. (1) appears deterministic, every term in the equation must be estimated and is subject to variance. Feed and excreta are sampled for bomb calorimetry and multiple birds are used in the assay. By understanding the sources of variance in estimating TMEN, decisions related to experimental procedures and the number of samples of each random variable to be taken can be enhanced. True metabolizable energy (TME) is derived from the principle of conservation of energy. Energy stored in the system equals energy entering the system minus energy leaving the system. TME of poultry feed equals the energy of the feed provided (GEF) minus the energy of the feed excreted (EO). Because not all of the excreta energy is attributed to the feed provided, EO cannot be directly measured. A common method of estimating TME is to deprive feed from a sample of roosters long enough to empty the gastrointestinal tract, provide a known amount of feed to be tested to one group of roosters, continue to deprive food from another group, and collect excreta from both groups for an energy assay (Sibbald, 1986). The energy excreted from the feed-deprived birds (EU) is not attributable to the feed sample. An estimate of EO is the energy excreted from the birds fed a sample of the feed (EF) minus EU. Another problem in estimating EO is that, when the birds are deprived feed, body protein tends to be broken down creating energy. Because this source of excreta energy is not attributed to the feed, TME has been estimated by factoring nitrogen out of each energy term: TMEN
GEF ÿ NGEF ÿ f EF ÿ NEF ÿ EU ÿ NEU g X
(1)
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whereTMEN is the nitrogen corrected true metabolizable energy (J/g, commonly expressed as MJ/kg), GEF the gross energy of the feed provided (J), NGEF the energy attributed to nitrogen in the feed sample (J), EF the energy in the excreta of the fed bird (J), NEF the energy attributed to nitrogen in the excreta of the fed bird (J), EU the energy in the excreta of the feed-deprived bird (J), NEU the energy attributed to nitrogen in the excreta of the feed-deprived bird (J) and X the amount of feed sample provided (g). The variance of a function of multiple random variables, f(X1, X2,. . .,Xn), can be estimated as follows (Beers, 1990). jÿ1 n n X X X @f 2 @f @f VAR Xj 2 COV Xj ; Xk VARf X1 ; X2 ; . . . ; Xn @Xj @Xj @Xk j1 j2 k1 (2) The total variation of a function of random variables is due to variability of each random variable, the covariances among them, and the sensitivity of the function to each random variable (the partial derivative). The proportion of the total variance attributed to variance of each random variable and to the covariances can be expressed by dividing the appropriate term in Eq. (2) by the total variance estimate obtained. This statistic will be used to determine sources of variance in estimating TMEN of common poultry feed ingredients. To accomplish this, terms in Eq. (1) must be made mathematically explicit. Assume there are p feed samples assayed by bomb calorimetry, n fed birds and m feeddeprived birds. Feed samples are evaluated in a bomb calorimeter and an estimate of the gross energy per unit mass of dry matter, EMF (J/g), is calculated. The variable Xj is defined as the amount of feed provided to the bird and GEFj, the amount of energy provided in the feed to bird j, is then estimated by GEFj EMFj Xj
(3)
The total nitrogen energy consumed by fed bird j is estimated by measuring the amount of nitrogen per unit mass of feed sample j, NXFj (g N/g feed), and multiplying by the amount of feed consumed by bird j, Xj, and multiplying by 36 530 J/g N (Titus et al., 1959). That is, NGEF j 36530 NXFj Xj
(4)
Excreta samples of the fed birds are evaluated in the calorimeter and an estimate of the excreta energy per unit mass, EMC (J/g) is used. If Yj is the dry matter mass (g) of excreta produced by bird j and EMCj is the energy per unit mass of the excreta from bird j, and NYFj is the amount of nitrogen per unit mass in excreta sample j (g N/g excreta), then EFj EMCj Yj
(5)
NEF j 36530 NYFj Yj
(6)
and Excreta samples of each of the feed-deprived birds are evaluated in a bomb calorimeter and an estimate of the excreta energy per unit mass, EMU (J/g), is derived. If Gi is the
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dry matter mass (g) of excreta from bird i, EMUi is the energy per unit mass of the excreta from bird i, and NYUi is the amount of nitrogen in excreta sample i (g N/g excreta), then m 1X EMUi Gi m i1
(7)
m 36530 X NYUi Gi m i1
(8)
EU and NEU
The number of replicates for each TMEN estimate is equal to the number of fed birds; therefore, TMEN
n 1X TMEN j n j1
(9)
Eqs. (1),(3),(4)±(9) were algebraically manipulated to yield TMEN EMF ÿ 36530NXF ÿ
n n 1X EMCj Yj 36530 X NYFj Yj Xj n j1 n j1 Xj
n n EU X 1 NEU X 1 ÿ n j1 Xj n j1 Xj
(10)
In order to employ Eq. (2), the variance of every variable in Eq. (10) was estimated and the partial derivatives of Eq. (10) with respect to every variable were calculated. Every variable in Eq. (10) was estimated by using the average value measured in the assay. The variance of the mean is obtained by dividing the sample variance by the number of replicates. An important result of the procedures described in measuring TMEN is that there are no covariances among variables because each measurement is independent of the others. VAR(EMF) can be estimated from replicates of the feed sample calorimetry. VAR(NXF) can be estimated by repeated nitrogen assays of the feed sample. Again, because EMF and NXF are averages, it is important to remember that the variance of the mean is used. Therefore, as the number of replicates increases, variance of the estimate of the average decreases. The variance of excreta energy content for each of the fed birds, VAR(EMCj) is due to the variability in the bomb calorimetry process and heterogeneity among subsamples, the latter of which can be minimized by grinding and mixing each excreta sample. VAR(Yj) represents the error associated with collecting, drying, and weighing the excreta produced by each bird. This variance is expected to be very small, provided care is taken in collecting the sample. Error in estimating how much each bird consumed, VAR(Xj), is related to the accuracy of the measuring technique and uncertainty of the amount of feed the bird was actually provided. Birds may occasionally regurgitate some of the feed. Variance of the amount of nitrogen in each excreta sample, VAR(NYFj), was estimated by repeated assays of each excreta sample. VAR(EU) was estimated from replicates of unfed birds. This term was further partitioned by applying the statistic in
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Eqs. (2)±(7). That is, VAR EU
m 1 X G2 VAR EMUi EMU2i VAR Gi m2 i1 i
(11)
Again, replicates of each EMUi are conducted to produce an average, so the variance of each EMUi equals the sample variance divided by the number of replicates. Similarly, the variance of nitrogen content in the excreta of unfed birds, VAR(NEU), can be estimated by applying the statistic in Eqs. (2)±(8). That is, VAR NEU
m 1334 106 X G2i VAR NYUi NYU2i VAR Gi 2 m i1
(12)
The application of the statistic in Eq. (2) to the estimate of TMEN (Eq. (10)) yields 2 X n 2 1 Yj VAR EMCj VAR TMEN VAR EMF 1334 106 VAR NXF n j1 Xj 2 X n n 1 EMCj 2 36530 2 X NYFj 2 VAR Yj VAR Yj Xj Xj n j1 n j1 n 2 2 X n 2 36530 2 X Yj 1 1 VAR NYFj VAR EU Xj n n j1 Xj j1 !2 2 X 2 X n 2 n 1 1 1 EMCj Yj VAR NEU VAR Xj n j1 Xj n j1 Xj2 !2 !2 n 2 X n 36530 2 X NYFj Yj EU 1 VAR Xj VAR Xj n n Xj2 Xj2 j1 j1 !2 n NEU 2 X 1 VAR Xj (13) n Xj2 j1 Variables EMF and NXF are mean values; therefore, their variances are estimated by the experimental variance divided by the number of observations (p feed samples). Variables EU and NEU have variance estimates obtained from Eqs. (11) and (12), respectively. All other variables are single observations whose variances are described by the uncertainty of each measurement, as will be described in Section 4. 3. Methodology As an example, five replicates of the feed ingredient sample, wheat, were analyzed for gross energy content by employment of a bomb calorimeter. Five replicates of the feed sample were analyzed for nitrogen content by the combustion method (A.O.A.C., 1980). Six adult white leghorn roosters were fasted for 24 h after which four were fed 30 g of feed sample. Excreta were collected from all six birds for the next 48 h, during which
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time no feed was provided. Excreta were dried and weighed, four replicates of each excreta sample were analyzed for energy content and four replicates of each excreta sample were analyzed for nitrogen content. Weights of feed and excreta were determined using a scale accurate to one tenth of a gram. The presence of an outlier can seriously change the outcome of the analysis. The cause of an outlier can be a mistake in laboratory procedures or calculations. A common method for identifying outliers is to calculate the standardized residuals for each observation (dij) and to remove the observation associated with the largest absolute value of dij (Montgomery, 1991). eij (14) dij p MSE where eij is the difference between each observation and the mean value, and MSE is the mean square error of the set of observations. An alternative to this criterion for removing outliers is to evaluate the change in variance of the estimate. 4. Results Table 1 contains the energy and nitrogen contents for replicates of the feed sample. Sample standard deviation and variance is computed. Table 2 contains data associated with the fed birds including amount of excreta produced, and energy and nitrogen replicates. Table 3 contains the same data for the fasted birds. TMEN is computed from Eq. (1) with the following parameters: X 30 g, GEF 17 350 (J/g) 30 g 520 496 J, NGEF 36 530 J/g N 2.954% N 30 g 32 377 J, EF (14 392 11.39) (14 357 11.54) (13 343 14.86) (14 573 12.03) 175 803 J, N E F (36 530 J/g N) {(8.989% 11.39) (7.383% 11.54) (12.102% 14.86) (8.312% 12.03)} 42 686 J, EU (10 906 11.1) (10 815 10.57) 117 683 J, N EU (36 530 J/g N) {(24.98% 11.1) (25.22% 10.57)} 99 325 J, and, therefore, TME N {(520 496 ÿ 32 337) ÿ (175 803 ÿ 42 686) (117 683 ÿ 99 325)}/30 12 445 J/g 12.45 MJ/kg. The range of accuracy for the scale was one tenth of a gram. It is common to assume that three standard deviations approximate the range (0.1 g range in this case); therefore, Table 1 Feed sample energy and nitrogen content measurements and variance Energy (J/g)
Nitrogen (%)
Observations
17 588 17 161 17 538 17 430 17 032
2.880 2.462 3.283 3.851 2.296
Average Standard deviation Variance (J2/g2)
17 350 243 58 954
2.954 0.631 0.004
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Table 2 Excreta data for fed birds used in the TMEN assay Fed Bird 1 Amount of excreta (g) Replicates of energy content (J/g)
Average Standard deviation Variance (J2/g2)
Fed Bird 2
Fed Bird 3
Fed Bird 4
11.5 14263 14391 14585 14188
14.9 13077 13228 13498 13571
12.0 14744 14299 14858 14392
14393
14357
13343
14573
1754 3075841
173 30090
231 53379
270 72710
11.4 14746 14535 12029a 16262
Replicates of nitrogen content(%)
8.522 9.523 9.640 8.270
7.921 6.770 7.514 7.328
12.395 12.434 11.742 11.835
8.288 8.280 8.286 8.394
Average
8.989
7.383
12.102
8.312
Standard deviation Variance (%)
0.694 0.005
0.478 0.002
0.364 0.001
0.055 0.003
a
Outlier, as determined by having the maximum standard residual of all of the observations.
Table 3 Excreta data for fasted birds used in the TMEN assay Fasted Bird 1
Fasted Bird 2
Amount of excreta (g) Replicates of energy content (J/g)
11.1 10709 10711 11177 11026
10.6 10655 10757 10836 11012
Average
10906
10815
Standard deviation Variance (J2/g2)
234 54904
151 22801
Replicates of nitrogen content (%)
25.41 25.11 24.75 24.63
24.88 25.57 25.49 24.94
Average
24.98
25.22
Standard deviation (%) Variance (%)
0.35 0.001
0.36 0.001
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variance of the scale was (0.1/6)2 (0.0167)2 0.0003 g2. Tables 1±3 contain the other needed variance estimates. Variance of the TMEN estimate was determined by Eq. (13) to be 30 296 J2/g2. The coefficient of variation for this estimate is the standard deviation of the estimate divided by the mean of the estimate, or (30 296)0.5/12 445 1.4%. The greatest source of variance was energy content of the feed, representing 39% of the variance, followed by the energy due to nitrogen composition of the feed, representing 35% of the total variance. Energy content of the excreta produced by Fed Bird 1 represented 24% of the total variance (see Table 2). Every other parameter represented less than 2% of the total variance. One reason for the large variance in the amount of energy in the feed sample may be the presence of an outlier measurement (12 029 J/g, see Table 2). By removing the outlier value (12 029 J/g), the TMEN estimate changed to a mean value of 12.37 MJ/kg with a variance of 23 036 J2/g2, for a reduction in variance of 14%. By removing this outlier, the number of replicates of excreta energy content from Fed Bird 1 was reduced to three, and over 10% of the variance of the TMEN estimate still came from the variance of excreta energy from Fed Bird 1 (2668 J2/g2). By completely eliminating energy content from excreta produced by Fed Bird 1, the TMEN estimate changed to a mean value of 12.31 MJ/kg with a variance of 23 368 J2/g2. This represents a reduction in variance from the original estimate of 23%. As a result of removing the outlier measurements, variance of the TMEN estimate attributable to feed energy content (11 791 J2/g2) was approximately 51% of the total variance. After considering the contribution of gross energy of the feed to the total variance of TMEN and noticing that the CV of this measurement was over 1% (Table 1:243/17 350 1.4%), additional replicates of feed samples were generated to double the number of feed samples assayed for energy and nitrogen content from five to ten. The amount of variance contributed to the TMEN estimate by variance in feed energy content became 5895 J2/g2, and the amount of variance contributed to the TMEN estimate by variance in feed nitrogen content became 5316 J2/g2, reducing the total variance to 12 157 J2/g2 and changing the average only slightly to 12.32 MJ/kg. By only increasing the number of replicates of the feed sample to ten, the variance of the TMEN estimate was reduced by nearly 60%. Table 4 summarizes the sources of variance for estimating TMEN, with and without outliers, and with five or ten feed energy replicates. Fig. 1 is a graph illustrating the contribution of each random variable to the total variance of the TMEN estimate, with outliers removed and the recommended 10 feed sample replicates used. Energy content of the feed (EMF) still represented over 48% of the total variance, followed by the nitrogen content of the feed samples (NXF) at 44%. Energy content of excreta from the fed birds (EMCj) represented 3.8% of the total variance, nitrogen content of the excreta from fed birds (NYFj) represented 2.3%, and the energy content of excreta from the unfed birds (EU) 1.4%. The amount of excreta collected from the fed birds (Yj), the amount of feed consumed by the birds (Xj), and nitrogen content of excreta from unfed birds each made negligible contributions to the total variance. 5. Discussion The intention of this paper was to demonstrate with an example how the statistic contained in Eq. (1) could be used to partition variance of TMEN estimation for a given
Table 4 Sources of variance in estimating TMEN with and without outliers and with 5 or 10 replicates of feed gross energy content 5 feed energy reps with outlier observation
5 feed energy reps without outlier observation
5 feed energy reps without outlier measurement
11791 10631 2668 70 205 183 5 4 3 4 145 70 68 1 120 46 9 9 5 26036 161 12371 12.37
11791 10631
10 feed energy reps without outlier measurement
Value (J2/g2) % of total Feed sample energy (EMF) Feed sample N (NXF) Fed birds' excreta energy (EMC)
11791 10631 6928 70 205 183 Amt. of excreta collected from fed birds (Y) 4 4 3 4 Fed birds' excreta N (NYF) 145 70 68 1 Unfed birds' excreta energy (EU) 120 46 Unfed birds' excreta N (NEU) 9 9 Amt. feed consumed (X) 5 Total variance (J2/g2) 30296 s.d. (J/g) 174 TMEN (J/kg) 12445 TMEN (MJ/kg) 12.45
38.0 35.1 24.0
0.1
0.9
0.5 0.1 0.02
45.3 40.8 12.0
0.1
1.1
0.6 0.1 0.02
70 205 183 5 4 3 4 145 70 68 1 120 46 9 9 4 23368 153 12314 12.31
50.5 45.5 2.0
0.1
1.2
0.7 0.1 0.02
5895 5316 70 205 183 5 4 3 4 145 70 68 1 120 46 9 9 4 12157 110 12323 12.32
48.5 43.7 4.0
0.1
2.3
1.4 0.2 0.04
T.H. D'Alfonso et al. / Animal Feed Science and Technology 80 (1999) 29±41
Source of variance
37
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Fig. 1. Sources of variance in estimating TMEN.
laboratory and its inherent practices. The statistic is a good tool to determine which procedures may need to be improved or require additional replicates. It is common in any laboratory practice to remove outliers or suspect data, but criteria for making these decisions are often rules of thumb rather than rigorous statistical techniques. The results presented here demonstrated a means of accomplishing this by observing the effect on the estimate and its variance. Criteria for removing outliers are often ambiguous; however, the procedure outlined herein, in which the reduction in variance was 14% by removing one outlier observation, is straightforward. Another procedure that was demonstrated was the ability to estimate reduction in variance by increasing the number of replicates. A cost can be associated with the number of birds, use of the bomb calorimeter, and the nitrogen assay to optimize experimental design. This is accomplished by either minimizing variance while keeping cost below a fixed amount, or by minimizing experiment cost while keeping variance below a fixed amount. It is very important to take many representative samples of the feed to determine gross energy content. It is possible to reduce this source of variance by producing a more homogeneous and representative sample of feed. Even if ten replicates were conducted, over 48% of the variance on the TMEN estimate was attributable to uncertainty of feed energy content. This is due to the mathematical impact of this variable on the estimate of TMEN and to the variability (albeit small) of the sample itself. If this variance value were higher, one might suspect malfunctions in the bomb assay (e.g. a sample being blown out of the retainer cup). No other variable in the TMEN estimate will reduce the variance as dramatically as increasing the number of feed sample replicates. The second largest source of variance in the TMEN estimate is the nitrogen content of the feed. Nitrogen correction is useful and helps to reduce the total variance of the TMEN
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estimate. To reduce experiment costs, reducing replicates for determining nitrogen content might be considered; however, the total variance of the TMEN estimate will increase. A scientist should consider the cost of the assay vis-aÁ-vis the desired accuracy of the estimate. The third greatest contributor to variance was the energy content of excreta from the fed birds, representing approximately 3.8% of the variance. There are two methods to reduce this variance, first by increasing the number of replicates and second by increasing the number of birds. The latter option is more expensive and should be considered only if the among-bird variability of excreta energy content is large. This may result if birds of different genetic strain, age, weights, health, etc. are used. If the within-bird excreta energy content variance is large, methods to produce more homogenous samples should be practiced and more replicates should be conducted. The fourth greatest source of variance was the nitrogen composition of the excreta from fed birds, representing approximately 2.3% of the total variance. The energy content of excreta from the unfed birds, which comprises of both the amount collected and the results of bomb calorimetry represented 1.4% of the total variance. The remaining sources of variance each represent less than 1% of the total variance. This includes the amount of feed consumed by the birds, the nitrogen content of the excreta from unfed birds, and the amount of excreta collected from the fed birds. It is possible for these factors to represent higher proportions of the total variance during other assays. The amount of excreta collected is a difficult source of variance to improve. Excreta samples, which must be dried and homogenized thoroughly, are often contaminated with dander and feathers. One technique for obtaining even greater accuracy in the collection of excreta is to clip feathers from, and harness a bag over the bird's posterior (Sibbald, 1986). Variance of the amount of feed consumed by the birds contributes very little to the total variance of the estimate. This result suggests that the necessity of force feeding versus voluntary feeding should be further investigated. Any differences in the TMEN estimate between force feeding and voluntary feed intake must be large enough to be detected through the total variance. It is important that the amount of feed consumed be accurately measured and that the feed consumed be representative. Parsons et al. (1984) introduced a voluntary feed intake bioassay for determining dietary ME content of poultry feeds and D'Alfonso (1994), as a precursor to the experiment described in the current paper, demonstrated that variability in the amount of feed consumed by the birds contributed less than 3% of the total variation of a TME estimate for five different feed samples assayed when the birds were allowed voluntary intake. There are methods to improve homogeneity of feed samples, including pelleting and fine grinding. Birds deprived feed will eat finely-ground feed. If this procedure is not applicable to a particular ingredient, TMEN can be estimated by changing the ingredient's proportion in a mixture with a homogenous feed that is suitable for grinding or pelleting. In any case, if birds consume different amounts of feed (the Xj variables), the variance of TMEN is impacted only by the variance of each measurement, which in this case is simply the accuracy of weighing the amount of feed consumed by each bird. The statistic described in this paper has been applied to determine TME variance of multiple feed samples conducted with various differences to the experimental procedures
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in a work by D'Alfonso (1994). These differences included voluntary feed intake, a 30 h excreta collection period, and no nitrogen correction. A longer collection period and a nitrogen correction are typically recommended; however, the main conclusions regarding variance of the energy content of the feed and of the amount of feed consumed by the birds remain the same. This earlier work is useful for describing more of the mathematics of the variance statistic and comparing the statistical results among multiple samples. A variety of procedures and laboratory practices will produce different rankings for sources of variance from excreta energy content and the amount of excreta collected. Nevertheless, it appears that feed sample gross energy content was the largest source of variance and nitrogen content of feed was the next largest source of variance.The energy and nitrogen of the excreta and the amount of feed consumed by the birds will contribute negligible amounts of variance as long as sound laboratory practices are followed and no mistakes are made in the experiment. The statistic described herein is also a useful tool for detecting errors. 6. Conclusions The conclusions are based on the data set present, as the study was intended to show an example of how scientists can be guided in improving an existing assay method from the status quo. The following conclusions can be drawn from this experiment: 1. The variance statistic described in this paper can be used to optimize the design of a TMEN experiment with respect to cost of the experiment and desired accuracy of the result. 2. The variance statistic described in this paper is a useful tool for identifying and removing outliers and highly variable measurements. 3. Gross energy of the feed will explain between 40% and 50% of the variance of the TMEN estimate depending on how many replicates are conducted. It is recommended to replicate the gross energy measurement as many times as possible. 4. The nitrogen composition of the feed sample represented the second largest source of variance for these experimental methods. At least ten sub-samples of feed nitrogen composition are recommended. 5. The energy content of excreta from fed birds was reduced from 24% to approximately 4% of the total variance by identifying and removing an outlier. The fed bird's excreta nitrogen composition tended to contribute less than 2.3% of the total variance of the estimate. Additional replicates are recommended to reduce these sources of variance. If among-bird variance is significantly different, more birds should be used. 6. The energy and nitrogen content of excreta from unfed birds combined represented less than 2% of the total variance. It may be possible to reduce the number of unfed birds and the number of energy and nitrogen assays on the excreta from unfed birds, without significantly increasing the variance of the TMEN estimate. 7. The amount of excreta collected from the fed birds represented only approximately 0.1% of the total variance. This source of variance could contribute more to the total
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variance than energy content of the excreta, depending on the given laboratory and its practices. It is recommended to use an excreta collection bag in this case. 8. Variance due to the amount of feed consumed by the birds is expected to be small. Force feeding may not be necessary for accurate TMEN estimates. Acknowledgements This research was supported by the USDA National Needs Fellowship. We wish to thank Drs. Theron Rumsey and Barbara Glenn of the USDA Ruminant Nutrition Laboratories in Beltsville, MD, and Dr. Jeff Firman of the University of Missouri Department of Animal Science, for permitting use of their laboratories and lending expertise during various stages of this project. References Association of Official Analytical Chemists, 1980. Official Methods of Analysis, 13th ed. Assoc. Offic. Anal. Chem., Washington, DC. Beers, Y., 1990. Introduction to the Theory of Error. Addison Wesely Publishers, New York, NY. Dale, N.M., Fuller, H.L., 1986. Repeatability of true metabolizable energy versus nitrogen corrected true metabolizable energy values. Poultry Sci. 63, 352±354. D'Alfonso, T.H., 1994. Quality Control of Nutritional and Environmental Factors in Laying Hen Production. Ph.D. Dissertation: Agricultural and Biological Engineering and Operations Research. The Pennsylvania State University, University Park, PA. Montgomery, D.C., 1991. Design and Analysis of Experiments, 3rd ed. Wiley, New York, NY. Parsons, C.M., Potter, L.M., Bliss, B.A., 1984. A modified voluntary feed intake bioassay for determination of metabolizable energy with leghorn roosters. Poultry Sci. 63, 1610±1616. Pesti, G.M., Dale, N.M., Ware, G.O., 1988. Critique of methods of estimating the variability of metabolizable energy from assays with fasted roosters. Poultry Sci. 67, 1188±1191. Sibbald, I.R., 1986. The T.M.E. System of Feed Evaluation: Methodology, Feed Composition Data and Bibliography. Agriculture Canada Research Branch Technical Bulletin 1986-4E. Ottawa, Canada. Titus, H.W., Mehring Jr., A.L., Johnson Jr., D., Nesbitt, L.L., Thomas, T., 1959. An evaluation of MCF (microcel-fat), a new type of fat product. Poultry Sci. 38, 1114±1119. Wolynetz, M.S., Sibbald, I.R., 1984. Relationship between apparent and true metabolizable energy and the effects of nitrogen correction. Poultry Sci. 63, 1386±1399.
Partitioning of variance in true metabolizable energy determinations: an example using wheat data T.H. D'Alfonsoa, H.B. Manbeckb, W.B. Roushc,* a
Rhone Poulenc Animal Nutrition 42, Avenue Aristide Briand, BP 100 92164, Antony Cedex, France b Department of Agricultural and Biological Engineering, Penn State University, University Park PA 16802, USA c Department of Poultry Science, Penn State University, University Park PA 16802, USA Received 16 June 1998; received in revised form 20 October 1998; accepted 25 February 1999
Abstract A variance statistic was used to partition the total variance into that attributable to each step of a TMEN assay procedure. Estimation of the TMEN of wheat was used as an example. The variance statistic can also be used to optimize the design of a TMEN experiment with respect to cost of the experiment and desired accuracy of the result. Experimental design optimization is accomplished by providing a functional relationship between the accuracy of the estimate and the number of replicates of feed, the number of birds used in the experiment, and the cost of each step. The variance statistic is also a useful tool for identifying and removing outliers and highly variable measurements. This feature was demonstrated with the chosen example data. Gross energy of the feed will explain approximately 50% of the variance of the TMEN estimate depending on how many replicates are evaluated. Nitrogen content of the feed sample will explain approximately 40% of the total variance. It is recommended to replicate this measurement as many times as possible. Ten replicates were recommended for the example data. The energy content of excreta from fed birds represented the next largest source of variance, at approximately 4% of the total variance, respectively. If within-bird variance is large, better homogenization of the sample and more replicates are recommended. If among-bird variance is significantly different, more birds should be used. Nitrogen content of excreta from fed birds represented less than 2.5% of the total variance. Energy and nitrogen content of excreta from unfed birds combined represented less than 2% of the total variance, suggesting that the number of unfed birds and the amount of excreta sub-samples may be reduced without adversely affecting the accuracy of the TMEN estimate. Variance due to the amount of excreta collected from the fed birds, and variance due to the amount of feed consumed by the birds, are expected to be small. This result suggested
* Corresponding author. Tel.: +1-814-863-0655; fax:+1-814-865-5691; e-mail: [email protected] 0377-8401/99/$ ± see front matter # 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 8 4 0 1 ( 9 9 ) 0 0 0 3 5 - 8
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that force-feeding may not be necessary for accurate TMEN estimates. # 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Metabolizable energy; Variance; Error estimation; Quality control
1. Introduction The accuracy of TMEN estimates has been investigated in a number of papers (for example, Wolynetz and Sibbald, 1984; Pesti et al., 1988; Dale and Fuller, 1986), particularly as it relates to the improvement in the assay by including the nitrogen correction. A variance statistic may be used to partition the total variance into that attributable to each step of a TMEN assay procedure. The appropriate variance statistic can also be used to optimize the design of a TMEN experiment with respect to cost of the experiment and desired accuracy of the result. Experimental design optimization is accomplished by providing a functional relationship between the accuracy of the estimate and the number of replicates of feed, the number of birds used in the experiment, and the cost of each step. A variance statistic can also be a useful tool for identifying and removing outliers and highly variable measurements. 2. Mathematical background Although the true metabolizable energy (TMEN) Eq. (1) appears deterministic, every term in the equation must be estimated and is subject to variance. Feed and excreta are sampled for bomb calorimetry and multiple birds are used in the assay. By understanding the sources of variance in estimating TMEN, decisions related to experimental procedures and the number of samples of each random variable to be taken can be enhanced. True metabolizable energy (TME) is derived from the principle of conservation of energy. Energy stored in the system equals energy entering the system minus energy leaving the system. TME of poultry feed equals the energy of the feed provided (GEF) minus the energy of the feed excreted (EO). Because not all of the excreta energy is attributed to the feed provided, EO cannot be directly measured. A common method of estimating TME is to deprive feed from a sample of roosters long enough to empty the gastrointestinal tract, provide a known amount of feed to be tested to one group of roosters, continue to deprive food from another group, and collect excreta from both groups for an energy assay (Sibbald, 1986). The energy excreted from the feed-deprived birds (EU) is not attributable to the feed sample. An estimate of EO is the energy excreted from the birds fed a sample of the feed (EF) minus EU. Another problem in estimating EO is that, when the birds are deprived feed, body protein tends to be broken down creating energy. Because this source of excreta energy is not attributed to the feed, TME has been estimated by factoring nitrogen out of each energy term: TMEN
GEF ÿ NGEF ÿ f EF ÿ NEF ÿ EU ÿ NEU g X
(1)
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31
whereTMEN is the nitrogen corrected true metabolizable energy (J/g, commonly expressed as MJ/kg), GEF the gross energy of the feed provided (J), NGEF the energy attributed to nitrogen in the feed sample (J), EF the energy in the excreta of the fed bird (J), NEF the energy attributed to nitrogen in the excreta of the fed bird (J), EU the energy in the excreta of the feed-deprived bird (J), NEU the energy attributed to nitrogen in the excreta of the feed-deprived bird (J) and X the amount of feed sample provided (g). The variance of a function of multiple random variables, f(X1, X2,. . .,Xn), can be estimated as follows (Beers, 1990). jÿ1 n n X X X @f 2 @f @f VAR Xj 2 COV Xj ; Xk VARf X1 ; X2 ; . . . ; Xn @Xj @Xj @Xk j1 j2 k1 (2) The total variation of a function of random variables is due to variability of each random variable, the covariances among them, and the sensitivity of the function to each random variable (the partial derivative). The proportion of the total variance attributed to variance of each random variable and to the covariances can be expressed by dividing the appropriate term in Eq. (2) by the total variance estimate obtained. This statistic will be used to determine sources of variance in estimating TMEN of common poultry feed ingredients. To accomplish this, terms in Eq. (1) must be made mathematically explicit. Assume there are p feed samples assayed by bomb calorimetry, n fed birds and m feeddeprived birds. Feed samples are evaluated in a bomb calorimeter and an estimate of the gross energy per unit mass of dry matter, EMF (J/g), is calculated. The variable Xj is defined as the amount of feed provided to the bird and GEFj, the amount of energy provided in the feed to bird j, is then estimated by GEFj EMFj Xj
(3)
The total nitrogen energy consumed by fed bird j is estimated by measuring the amount of nitrogen per unit mass of feed sample j, NXFj (g N/g feed), and multiplying by the amount of feed consumed by bird j, Xj, and multiplying by 36 530 J/g N (Titus et al., 1959). That is, NGEF j 36530 NXFj Xj
(4)
Excreta samples of the fed birds are evaluated in the calorimeter and an estimate of the excreta energy per unit mass, EMC (J/g) is used. If Yj is the dry matter mass (g) of excreta produced by bird j and EMCj is the energy per unit mass of the excreta from bird j, and NYFj is the amount of nitrogen per unit mass in excreta sample j (g N/g excreta), then EFj EMCj Yj
(5)
NEF j 36530 NYFj Yj
(6)
and Excreta samples of each of the feed-deprived birds are evaluated in a bomb calorimeter and an estimate of the excreta energy per unit mass, EMU (J/g), is derived. If Gi is the
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dry matter mass (g) of excreta from bird i, EMUi is the energy per unit mass of the excreta from bird i, and NYUi is the amount of nitrogen in excreta sample i (g N/g excreta), then m 1X EMUi Gi m i1
(7)
m 36530 X NYUi Gi m i1
(8)
EU and NEU
The number of replicates for each TMEN estimate is equal to the number of fed birds; therefore, TMEN
n 1X TMEN j n j1
(9)
Eqs. (1),(3),(4)±(9) were algebraically manipulated to yield TMEN EMF ÿ 36530NXF ÿ
n n 1X EMCj Yj 36530 X NYFj Yj Xj n j1 n j1 Xj
n n EU X 1 NEU X 1 ÿ n j1 Xj n j1 Xj
(10)
In order to employ Eq. (2), the variance of every variable in Eq. (10) was estimated and the partial derivatives of Eq. (10) with respect to every variable were calculated. Every variable in Eq. (10) was estimated by using the average value measured in the assay. The variance of the mean is obtained by dividing the sample variance by the number of replicates. An important result of the procedures described in measuring TMEN is that there are no covariances among variables because each measurement is independent of the others. VAR(EMF) can be estimated from replicates of the feed sample calorimetry. VAR(NXF) can be estimated by repeated nitrogen assays of the feed sample. Again, because EMF and NXF are averages, it is important to remember that the variance of the mean is used. Therefore, as the number of replicates increases, variance of the estimate of the average decreases. The variance of excreta energy content for each of the fed birds, VAR(EMCj) is due to the variability in the bomb calorimetry process and heterogeneity among subsamples, the latter of which can be minimized by grinding and mixing each excreta sample. VAR(Yj) represents the error associated with collecting, drying, and weighing the excreta produced by each bird. This variance is expected to be very small, provided care is taken in collecting the sample. Error in estimating how much each bird consumed, VAR(Xj), is related to the accuracy of the measuring technique and uncertainty of the amount of feed the bird was actually provided. Birds may occasionally regurgitate some of the feed. Variance of the amount of nitrogen in each excreta sample, VAR(NYFj), was estimated by repeated assays of each excreta sample. VAR(EU) was estimated from replicates of unfed birds. This term was further partitioned by applying the statistic in
T.H. D'Alfonso et al. / Animal Feed Science and Technology 80 (1999) 29±41
33
Eqs. (2)±(7). That is, VAR EU
m 1 X G2 VAR EMUi EMU2i VAR Gi m2 i1 i
(11)
Again, replicates of each EMUi are conducted to produce an average, so the variance of each EMUi equals the sample variance divided by the number of replicates. Similarly, the variance of nitrogen content in the excreta of unfed birds, VAR(NEU), can be estimated by applying the statistic in Eqs. (2)±(8). That is, VAR NEU
m 1334 106 X G2i VAR NYUi NYU2i VAR Gi 2 m i1
(12)
The application of the statistic in Eq. (2) to the estimate of TMEN (Eq. (10)) yields 2 X n 2 1 Yj VAR EMCj VAR TMEN VAR EMF 1334 106 VAR NXF n j1 Xj 2 X n n 1 EMCj 2 36530 2 X NYFj 2 VAR Yj VAR Yj Xj Xj n j1 n j1 n 2 2 X n 2 36530 2 X Yj 1 1 VAR NYFj VAR EU Xj n n j1 Xj j1 !2 2 X 2 X n 2 n 1 1 1 EMCj Yj VAR NEU VAR Xj n j1 Xj n j1 Xj2 !2 !2 n 2 X n 36530 2 X NYFj Yj EU 1 VAR Xj VAR Xj n n Xj2 Xj2 j1 j1 !2 n NEU 2 X 1 VAR Xj (13) n Xj2 j1 Variables EMF and NXF are mean values; therefore, their variances are estimated by the experimental variance divided by the number of observations (p feed samples). Variables EU and NEU have variance estimates obtained from Eqs. (11) and (12), respectively. All other variables are single observations whose variances are described by the uncertainty of each measurement, as will be described in Section 4. 3. Methodology As an example, five replicates of the feed ingredient sample, wheat, were analyzed for gross energy content by employment of a bomb calorimeter. Five replicates of the feed sample were analyzed for nitrogen content by the combustion method (A.O.A.C., 1980). Six adult white leghorn roosters were fasted for 24 h after which four were fed 30 g of feed sample. Excreta were collected from all six birds for the next 48 h, during which
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time no feed was provided. Excreta were dried and weighed, four replicates of each excreta sample were analyzed for energy content and four replicates of each excreta sample were analyzed for nitrogen content. Weights of feed and excreta were determined using a scale accurate to one tenth of a gram. The presence of an outlier can seriously change the outcome of the analysis. The cause of an outlier can be a mistake in laboratory procedures or calculations. A common method for identifying outliers is to calculate the standardized residuals for each observation (dij) and to remove the observation associated with the largest absolute value of dij (Montgomery, 1991). eij (14) dij p MSE where eij is the difference between each observation and the mean value, and MSE is the mean square error of the set of observations. An alternative to this criterion for removing outliers is to evaluate the change in variance of the estimate. 4. Results Table 1 contains the energy and nitrogen contents for replicates of the feed sample. Sample standard deviation and variance is computed. Table 2 contains data associated with the fed birds including amount of excreta produced, and energy and nitrogen replicates. Table 3 contains the same data for the fasted birds. TMEN is computed from Eq. (1) with the following parameters: X 30 g, GEF 17 350 (J/g) 30 g 520 496 J, NGEF 36 530 J/g N 2.954% N 30 g 32 377 J, EF (14 392 11.39) (14 357 11.54) (13 343 14.86) (14 573 12.03) 175 803 J, N E F (36 530 J/g N) {(8.989% 11.39) (7.383% 11.54) (12.102% 14.86) (8.312% 12.03)} 42 686 J, EU (10 906 11.1) (10 815 10.57) 117 683 J, N EU (36 530 J/g N) {(24.98% 11.1) (25.22% 10.57)} 99 325 J, and, therefore, TME N {(520 496 ÿ 32 337) ÿ (175 803 ÿ 42 686) (117 683 ÿ 99 325)}/30 12 445 J/g 12.45 MJ/kg. The range of accuracy for the scale was one tenth of a gram. It is common to assume that three standard deviations approximate the range (0.1 g range in this case); therefore, Table 1 Feed sample energy and nitrogen content measurements and variance Energy (J/g)
Nitrogen (%)
Observations
17 588 17 161 17 538 17 430 17 032
2.880 2.462 3.283 3.851 2.296
Average Standard deviation Variance (J2/g2)
17 350 243 58 954
2.954 0.631 0.004
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Table 2 Excreta data for fed birds used in the TMEN assay Fed Bird 1 Amount of excreta (g) Replicates of energy content (J/g)
Average Standard deviation Variance (J2/g2)
Fed Bird 2
Fed Bird 3
Fed Bird 4
11.5 14263 14391 14585 14188
14.9 13077 13228 13498 13571
12.0 14744 14299 14858 14392
14393
14357
13343
14573
1754 3075841
173 30090
231 53379
270 72710
11.4 14746 14535 12029a 16262
Replicates of nitrogen content(%)
8.522 9.523 9.640 8.270
7.921 6.770 7.514 7.328
12.395 12.434 11.742 11.835
8.288 8.280 8.286 8.394
Average
8.989
7.383
12.102
8.312
Standard deviation Variance (%)
0.694 0.005
0.478 0.002
0.364 0.001
0.055 0.003
a
Outlier, as determined by having the maximum standard residual of all of the observations.
Table 3 Excreta data for fasted birds used in the TMEN assay Fasted Bird 1
Fasted Bird 2
Amount of excreta (g) Replicates of energy content (J/g)
11.1 10709 10711 11177 11026
10.6 10655 10757 10836 11012
Average
10906
10815
Standard deviation Variance (J2/g2)
234 54904
151 22801
Replicates of nitrogen content (%)
25.41 25.11 24.75 24.63
24.88 25.57 25.49 24.94
Average
24.98
25.22
Standard deviation (%) Variance (%)
0.35 0.001
0.36 0.001
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T.H. D'Alfonso et al. / Animal Feed Science and Technology 80 (1999) 29±41
variance of the scale was (0.1/6)2 (0.0167)2 0.0003 g2. Tables 1±3 contain the other needed variance estimates. Variance of the TMEN estimate was determined by Eq. (13) to be 30 296 J2/g2. The coefficient of variation for this estimate is the standard deviation of the estimate divided by the mean of the estimate, or (30 296)0.5/12 445 1.4%. The greatest source of variance was energy content of the feed, representing 39% of the variance, followed by the energy due to nitrogen composition of the feed, representing 35% of the total variance. Energy content of the excreta produced by Fed Bird 1 represented 24% of the total variance (see Table 2). Every other parameter represented less than 2% of the total variance. One reason for the large variance in the amount of energy in the feed sample may be the presence of an outlier measurement (12 029 J/g, see Table 2). By removing the outlier value (12 029 J/g), the TMEN estimate changed to a mean value of 12.37 MJ/kg with a variance of 23 036 J2/g2, for a reduction in variance of 14%. By removing this outlier, the number of replicates of excreta energy content from Fed Bird 1 was reduced to three, and over 10% of the variance of the TMEN estimate still came from the variance of excreta energy from Fed Bird 1 (2668 J2/g2). By completely eliminating energy content from excreta produced by Fed Bird 1, the TMEN estimate changed to a mean value of 12.31 MJ/kg with a variance of 23 368 J2/g2. This represents a reduction in variance from the original estimate of 23%. As a result of removing the outlier measurements, variance of the TMEN estimate attributable to feed energy content (11 791 J2/g2) was approximately 51% of the total variance. After considering the contribution of gross energy of the feed to the total variance of TMEN and noticing that the CV of this measurement was over 1% (Table 1:243/17 350 1.4%), additional replicates of feed samples were generated to double the number of feed samples assayed for energy and nitrogen content from five to ten. The amount of variance contributed to the TMEN estimate by variance in feed energy content became 5895 J2/g2, and the amount of variance contributed to the TMEN estimate by variance in feed nitrogen content became 5316 J2/g2, reducing the total variance to 12 157 J2/g2 and changing the average only slightly to 12.32 MJ/kg. By only increasing the number of replicates of the feed sample to ten, the variance of the TMEN estimate was reduced by nearly 60%. Table 4 summarizes the sources of variance for estimating TMEN, with and without outliers, and with five or ten feed energy replicates. Fig. 1 is a graph illustrating the contribution of each random variable to the total variance of the TMEN estimate, with outliers removed and the recommended 10 feed sample replicates used. Energy content of the feed (EMF) still represented over 48% of the total variance, followed by the nitrogen content of the feed samples (NXF) at 44%. Energy content of excreta from the fed birds (EMCj) represented 3.8% of the total variance, nitrogen content of the excreta from fed birds (NYFj) represented 2.3%, and the energy content of excreta from the unfed birds (EU) 1.4%. The amount of excreta collected from the fed birds (Yj), the amount of feed consumed by the birds (Xj), and nitrogen content of excreta from unfed birds each made negligible contributions to the total variance. 5. Discussion The intention of this paper was to demonstrate with an example how the statistic contained in Eq. (1) could be used to partition variance of TMEN estimation for a given
Table 4 Sources of variance in estimating TMEN with and without outliers and with 5 or 10 replicates of feed gross energy content 5 feed energy reps with outlier observation
5 feed energy reps without outlier observation
5 feed energy reps without outlier measurement
11791 10631 2668 70 205 183 5 4 3 4 145 70 68 1 120 46 9 9 5 26036 161 12371 12.37
11791 10631
10 feed energy reps without outlier measurement
Value (J2/g2) % of total Feed sample energy (EMF) Feed sample N (NXF) Fed birds' excreta energy (EMC)
11791 10631 6928 70 205 183 Amt. of excreta collected from fed birds (Y) 4 4 3 4 Fed birds' excreta N (NYF) 145 70 68 1 Unfed birds' excreta energy (EU) 120 46 Unfed birds' excreta N (NEU) 9 9 Amt. feed consumed (X) 5 Total variance (J2/g2) 30296 s.d. (J/g) 174 TMEN (J/kg) 12445 TMEN (MJ/kg) 12.45
38.0 35.1 24.0
0.1
0.9
0.5 0.1 0.02
45.3 40.8 12.0
0.1
1.1
0.6 0.1 0.02
70 205 183 5 4 3 4 145 70 68 1 120 46 9 9 4 23368 153 12314 12.31
50.5 45.5 2.0
0.1
1.2
0.7 0.1 0.02
5895 5316 70 205 183 5 4 3 4 145 70 68 1 120 46 9 9 4 12157 110 12323 12.32
48.5 43.7 4.0
0.1
2.3
1.4 0.2 0.04
T.H. D'Alfonso et al. / Animal Feed Science and Technology 80 (1999) 29±41
Source of variance
37
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T.H. D'Alfonso et al. / Animal Feed Science and Technology 80 (1999) 29±41
Fig. 1. Sources of variance in estimating TMEN.
laboratory and its inherent practices. The statistic is a good tool to determine which procedures may need to be improved or require additional replicates. It is common in any laboratory practice to remove outliers or suspect data, but criteria for making these decisions are often rules of thumb rather than rigorous statistical techniques. The results presented here demonstrated a means of accomplishing this by observing the effect on the estimate and its variance. Criteria for removing outliers are often ambiguous; however, the procedure outlined herein, in which the reduction in variance was 14% by removing one outlier observation, is straightforward. Another procedure that was demonstrated was the ability to estimate reduction in variance by increasing the number of replicates. A cost can be associated with the number of birds, use of the bomb calorimeter, and the nitrogen assay to optimize experimental design. This is accomplished by either minimizing variance while keeping cost below a fixed amount, or by minimizing experiment cost while keeping variance below a fixed amount. It is very important to take many representative samples of the feed to determine gross energy content. It is possible to reduce this source of variance by producing a more homogeneous and representative sample of feed. Even if ten replicates were conducted, over 48% of the variance on the TMEN estimate was attributable to uncertainty of feed energy content. This is due to the mathematical impact of this variable on the estimate of TMEN and to the variability (albeit small) of the sample itself. If this variance value were higher, one might suspect malfunctions in the bomb assay (e.g. a sample being blown out of the retainer cup). No other variable in the TMEN estimate will reduce the variance as dramatically as increasing the number of feed sample replicates. The second largest source of variance in the TMEN estimate is the nitrogen content of the feed. Nitrogen correction is useful and helps to reduce the total variance of the TMEN
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39
estimate. To reduce experiment costs, reducing replicates for determining nitrogen content might be considered; however, the total variance of the TMEN estimate will increase. A scientist should consider the cost of the assay vis-aÁ-vis the desired accuracy of the estimate. The third greatest contributor to variance was the energy content of excreta from the fed birds, representing approximately 3.8% of the variance. There are two methods to reduce this variance, first by increasing the number of replicates and second by increasing the number of birds. The latter option is more expensive and should be considered only if the among-bird variability of excreta energy content is large. This may result if birds of different genetic strain, age, weights, health, etc. are used. If the within-bird excreta energy content variance is large, methods to produce more homogenous samples should be practiced and more replicates should be conducted. The fourth greatest source of variance was the nitrogen composition of the excreta from fed birds, representing approximately 2.3% of the total variance. The energy content of excreta from the unfed birds, which comprises of both the amount collected and the results of bomb calorimetry represented 1.4% of the total variance. The remaining sources of variance each represent less than 1% of the total variance. This includes the amount of feed consumed by the birds, the nitrogen content of the excreta from unfed birds, and the amount of excreta collected from the fed birds. It is possible for these factors to represent higher proportions of the total variance during other assays. The amount of excreta collected is a difficult source of variance to improve. Excreta samples, which must be dried and homogenized thoroughly, are often contaminated with dander and feathers. One technique for obtaining even greater accuracy in the collection of excreta is to clip feathers from, and harness a bag over the bird's posterior (Sibbald, 1986). Variance of the amount of feed consumed by the birds contributes very little to the total variance of the estimate. This result suggests that the necessity of force feeding versus voluntary feeding should be further investigated. Any differences in the TMEN estimate between force feeding and voluntary feed intake must be large enough to be detected through the total variance. It is important that the amount of feed consumed be accurately measured and that the feed consumed be representative. Parsons et al. (1984) introduced a voluntary feed intake bioassay for determining dietary ME content of poultry feeds and D'Alfonso (1994), as a precursor to the experiment described in the current paper, demonstrated that variability in the amount of feed consumed by the birds contributed less than 3% of the total variation of a TME estimate for five different feed samples assayed when the birds were allowed voluntary intake. There are methods to improve homogeneity of feed samples, including pelleting and fine grinding. Birds deprived feed will eat finely-ground feed. If this procedure is not applicable to a particular ingredient, TMEN can be estimated by changing the ingredient's proportion in a mixture with a homogenous feed that is suitable for grinding or pelleting. In any case, if birds consume different amounts of feed (the Xj variables), the variance of TMEN is impacted only by the variance of each measurement, which in this case is simply the accuracy of weighing the amount of feed consumed by each bird. The statistic described in this paper has been applied to determine TME variance of multiple feed samples conducted with various differences to the experimental procedures
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in a work by D'Alfonso (1994). These differences included voluntary feed intake, a 30 h excreta collection period, and no nitrogen correction. A longer collection period and a nitrogen correction are typically recommended; however, the main conclusions regarding variance of the energy content of the feed and of the amount of feed consumed by the birds remain the same. This earlier work is useful for describing more of the mathematics of the variance statistic and comparing the statistical results among multiple samples. A variety of procedures and laboratory practices will produce different rankings for sources of variance from excreta energy content and the amount of excreta collected. Nevertheless, it appears that feed sample gross energy content was the largest source of variance and nitrogen content of feed was the next largest source of variance.The energy and nitrogen of the excreta and the amount of feed consumed by the birds will contribute negligible amounts of variance as long as sound laboratory practices are followed and no mistakes are made in the experiment. The statistic described herein is also a useful tool for detecting errors. 6. Conclusions The conclusions are based on the data set present, as the study was intended to show an example of how scientists can be guided in improving an existing assay method from the status quo. The following conclusions can be drawn from this experiment: 1. The variance statistic described in this paper can be used to optimize the design of a TMEN experiment with respect to cost of the experiment and desired accuracy of the result. 2. The variance statistic described in this paper is a useful tool for identifying and removing outliers and highly variable measurements. 3. Gross energy of the feed will explain between 40% and 50% of the variance of the TMEN estimate depending on how many replicates are conducted. It is recommended to replicate the gross energy measurement as many times as possible. 4. The nitrogen composition of the feed sample represented the second largest source of variance for these experimental methods. At least ten sub-samples of feed nitrogen composition are recommended. 5. The energy content of excreta from fed birds was reduced from 24% to approximately 4% of the total variance by identifying and removing an outlier. The fed bird's excreta nitrogen composition tended to contribute less than 2.3% of the total variance of the estimate. Additional replicates are recommended to reduce these sources of variance. If among-bird variance is significantly different, more birds should be used. 6. The energy and nitrogen content of excreta from unfed birds combined represented less than 2% of the total variance. It may be possible to reduce the number of unfed birds and the number of energy and nitrogen assays on the excreta from unfed birds, without significantly increasing the variance of the TMEN estimate. 7. The amount of excreta collected from the fed birds represented only approximately 0.1% of the total variance. This source of variance could contribute more to the total
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variance than energy content of the excreta, depending on the given laboratory and its practices. It is recommended to use an excreta collection bag in this case. 8. Variance due to the amount of feed consumed by the birds is expected to be small. Force feeding may not be necessary for accurate TMEN estimates. Acknowledgements This research was supported by the USDA National Needs Fellowship. We wish to thank Drs. Theron Rumsey and Barbara Glenn of the USDA Ruminant Nutrition Laboratories in Beltsville, MD, and Dr. Jeff Firman of the University of Missouri Department of Animal Science, for permitting use of their laboratories and lending expertise during various stages of this project. References Association of Official Analytical Chemists, 1980. Official Methods of Analysis, 13th ed. Assoc. Offic. Anal. Chem., Washington, DC. Beers, Y., 1990. Introduction to the Theory of Error. Addison Wesely Publishers, New York, NY. Dale, N.M., Fuller, H.L., 1986. Repeatability of true metabolizable energy versus nitrogen corrected true metabolizable energy values. Poultry Sci. 63, 352±354. D'Alfonso, T.H., 1994. Quality Control of Nutritional and Environmental Factors in Laying Hen Production. Ph.D. Dissertation: Agricultural and Biological Engineering and Operations Research. The Pennsylvania State University, University Park, PA. Montgomery, D.C., 1991. Design and Analysis of Experiments, 3rd ed. Wiley, New York, NY. Parsons, C.M., Potter, L.M., Bliss, B.A., 1984. A modified voluntary feed intake bioassay for determination of metabolizable energy with leghorn roosters. Poultry Sci. 63, 1610±1616. Pesti, G.M., Dale, N.M., Ware, G.O., 1988. Critique of methods of estimating the variability of metabolizable energy from assays with fasted roosters. Poultry Sci. 67, 1188±1191. Sibbald, I.R., 1986. The T.M.E. System of Feed Evaluation: Methodology, Feed Composition Data and Bibliography. Agriculture Canada Research Branch Technical Bulletin 1986-4E. Ottawa, Canada. Titus, H.W., Mehring Jr., A.L., Johnson Jr., D., Nesbitt, L.L., Thomas, T., 1959. An evaluation of MCF (microcel-fat), a new type of fat product. Poultry Sci. 38, 1114±1119. Wolynetz, M.S., Sibbald, I.R., 1984. Relationship between apparent and true metabolizable energy and the effects of nitrogen correction. Poultry Sci. 63, 1386±1399.