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Pooling Excreta Prior to Calorimetry in the Bioassay for True Metabolizable Energy: The Effect on Estimates of Variance1
METABOLISM AND NUTRITION Pooling Excreta Prior to Calorimetry in the Bioassay for True Metabolizable Energy: The Effect on Estimates of Variance1 I. R. SIBBALD2 and P. M. MORSE3 Animal Research Centre and Engineering and Statistical Research Institute, Agriculture Canada, Ottawa, Ontario, Canada K1A 0C6 (Received for publication October 12, 1981)
1982 Poultry Science 61:1853-1858
INTRODUCTION M e a s u r e m e n t of t h e gross energy p e r u n i t weight of air-dry e x c r e t a (G) is an integral part of t h e bioassay for true metabolizable energy (TME) (Sibbald, 1976). O x y g e n b o m b calorimetry, die usual p r o c e d u r e for measuring G, is tedious, t i m e consuming, and, therefore, expensive. C o n s e q u e n t l y , there is interest in t h e analysis of pooled rather than individual excreta samples. E d m u n d s o n ( 1 9 8 0 ) p o i n t s o u t t h a t pooling excreta before calorimetry has t h e effect of ignoring t h e variation in G a m o n g individual birds and t h a t this can result in overestimation or u n d e r e s t i m a t i o n of t h e standard deviation (SD) of t h e TME. H e argues t h a t because " s o m e 2 5 0 replicates spread over a variety of feedingstuffs" gave an average reduction of only 3% in t h e a p p a r e n t SD, it is an acceptable risk t o treat G as a c o n s t a n t .
'Contribution Numbers 1028 Animal Research Centre and 1-336 Engineering and Statistical Research Institute. 2 Animal Research Centre. 3 Statistical Research Institute.
In the assay of a single feedingstuff, h o w ever, or in a single comparative e x p e r i m e n t , t h e bias in t h e SD may be considerable if t h e variation in G is ignored. Moreover, this bias m a y n o t b e i n d e p e n d e n t of t h e diet n o r of t h e c o n d i t i o n s of t h e assay (e.g. age and t y p e of bird), because it d e p e n d s on t h e correlation b e t w e e n t h e weight of air-dry e x c r e t a (D) and G and also on t h e relative size of their coefficients of variation. To investigate t h e p o t e n tial bias, results of T M E bioassays of 2 0 feedingstuffs were used t o simulate t h e effect of pooling excreta. An additional factor is- t h e effect of t h e sampling error when calorimetry is p e r f o r m e d o n samples from an actual p o o l e d aggregate of excreta. Advocates of pooling (Dale and Fuller, 1981) cite advantages in reducing time and labor and imply t h a t only one sample need be t a k e n from t h e p o o l e d excreta for calorimetric m e a s u r e m e n t . Even with careful mixing, t h e potential difference between pooled and u n p o o l e d excreta as regards sampling error c a n n o t b e ignored, a n d t h e question arises as t o h o w m a n y samples t o t a k e . This aspect was also investigated.
1853
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ABSTRACT Pooling excreta, within treatments, prior to calorimetry reduces the time and cost of the bioassay for true metabolizable energy (TME) and does not alter mean TME values. However, by ignoring the among-bird variance of the gross energy per unit weight of excreta (G) and its covariance with the weight of excreta (D), the standard errors of mean TME values are subject to bias. An additional factor for consideration is the effect of the sampling error from pooled aggregates. These factors were examined using data from TME bioassays of 20 feedingstuffs and by measuring the standard deviations of gross energy values for subsamples from pooled excreta. The standard deviation of G for each unit subsample from pooled excreta samples was .0381 kj/g with 20 degrees of freedom. The resulting contribution to the standard errors of mean TME values was so small that replicate subsampling from the bulked excreta is of minor importance, although a useful precaution. However, ignoring the among-bird variation in G as a consequence of pooling excreta caused the standard errors of mean TME values to be incorrectly estimated by amounts ranging from —62 to + 64%. Although for these particular feedingstuffs the average discrepancy was small, the potential magnitude of the bias for an individual feedingstuff is too large to be discounted in a research laboratory. (Key words: true metabolizable energy, bioassay, pooling samples, calorimetry, sampling)
SIBBALD AND MORSE
1854 MATERIALS AND METHODS
RESULTS AND DISCUSSION
The TME value of a feedingstuff is estimated as:
the same considerations as for excreta from fed birds, but the influence on the SD of the TME may perhaps be less. Here attention will be confined to the variance arising from the remaining term Eef, i.e., the estimate of energy voided by fed birds. Let the excreta voided by the i t n bird on a particular diet have weight d; and energy concentration gj as measured by calorimetry on a single unit-weight sample. Let there be nt, birds on the diet. Energy Excreted Estimated from Individual Excreta. When gj is measured for each bird without pooling excreta, the mean energy per bird is estimated as Ej, where: nb Ei = 2 (gi di), the summation being over all n D birds. The standard error (SE) of Ei is estimated as: SE(E I ) = S D ( g d ) / V n b where SD (gd) may be calculated in the usual way from squared deviations of the individual products g:d- about their mean. The corresponding standard errors of the mean TME values of 20 feedingstuffs are given in column 4 of Table 1 ("Standard"). In calculating these, E e f was the only contributing factor, because the energy per unit of feed (Ef) and the metabolic plus endogenous energy out (E e u ) were assumed to be constant for all birds within a diet. Simulation of Estimation from Pooled Excreta. Assuming that a weighted mean of the gi values simulates the energy concentration of the pooled excreta, the mean energy excretion per bird is calculated as E ^ , where: nbEW = g w £ d i
TME = Ef - (E e f - E e u )/I where Ef is the gross energy per unit weight of the feedingstuff, I is the weight of feedingstuff placed in the bird, and E e f and E e u are the amounts of excreta energy voided by fed and unfed birds, respectively. Of these variables I can be assumed to have negligible error under precision feeding and Ef is usually sufficiently well-estimated that its error variance may be assumed to be relatively unimportant. The value of E e u may be estimated from the same birds as those fed or from different birds. Pooling excreta from fasted birds is subject to
and:
g\V = 2 fei d i ) / S d i so that Eyj is exactly the same as E,. Hence, the true standard errors of the two estimates should be exactly the same. However, in actual pooling only g w and individual dj values are known; the individual products g-d- are unknown. Let Var (d) and Var (g) denote the estimates of among-bird variance of d and g, respectively, and let Cov (d,g)
Downloaded from http://ps.oxfordjournals.org/ at University of North Dakota on May 23, 2015
Data from TME bioassays of 20 feedingstuffs form the basis of the report. There were from 5 to 10 Single Comb White Leghorn cockerels in each assay; all calculations were based on air-dry excreta. Standard TME variances were calculated using the G and D values appropriate to individual birds. "Pooled" TME variances were calculated for each feedingstuff using individual D values and a single weighted mean value for G (the sum of the gross energy voided by birds receiving a, particular feedingstuff divided by the sum of their excreta weights). Excreta remaining from the TME bioassays of five feedingstuffs (5, 7, 10, 12, and 14 of Table 1) were used to estimate the variability associated with actual subsampling from a pooled aggregate of excreta. For each of these feedingstuffs, excreta from the 10 birds were combined in the proportions voided during the assay; the absolute amounts were slightly less than those voided because of prior removal of samples for calorimetry. Excreta were mixed in a Waring blendor and then transferred to a plastic tub from which five 1-g subsamples were drawn for calorimetry; excreta were stirred between removal of each subsample. When pooling excreta, no correction was made for any changes in moisture content since the original TME analyses; consequently, it was anticipated that the mean G based on the analyses of the subsamples from the pooled excreta would be slightly different from the calculated weighted mean G.
POOLING EXCRETA FOR CALORIMETRY
O O O ^ t « M ' COOCO'+ON'OO'-OI^QOmm OmNOO NMJOI CO S N ^ W * W >
c o r ^ O ,< ^ o o \ < N, C "v] < N O o o r * ' - t t > o o u - > r ^ * H © ( S r-*i-< ooocoo\ + oooinvot-*\Oo\mt"*-0\0'
tnr-O'n0\rJC0'« rHT-cm^nwr-)O^|-0'-<'-iOi-ti-<^H(S'-irf>»n'n
coOMOo\r-iNOco»n-+'*Omo\":t'*000,r> G\O\NH0sOC\C00\miH00M^N00'+'l-inC0 i n N f i m m m N N " ! ! » rn ^ •* N m rg O +
C N H ^ - W O N N i - ^ i A -
rs,oo\^'-;'AO\»nr^ "nr^mMi-finiviOTH H H N fn >)• m >0 I I I I I ! I I I
tONOiWvOHON'J'OinhCO^'-nninOO + O m N H t N M X ^ C A M i S O H O O C M n H H M
m'nmt^.oON'ncot^'^i'n^-OovOr^W'Or-lON in>not^.o\oiX'*co'^r^i-"«%mtN'^-r^o\foO
t ^ m f t N i
i>Omfn^-t^f««nc«l»ni
\oocoooNo\r*<
H ( s m * » n o ^ C O O \ O H N i
i t
m « N
MONO
Downloaded from http://ps.oxfordjournals.org/ at University of North Dakota on May 23, 2015
OO'-<'OOt^0\,0000\Nm>Oi-i'-tOO-^--*O m Q\ h « w M h w * ff; H w co in ^ in co O CO m O CO H N ^ N in N N CN N O 6 ^ N ON H 6 tf> H
SIBBALD AND MORSE
1856
denote the corresponding covariance. Calculating the SE only from the deviations of the g«f d- about their mean is equivalent to treating giy as constant, so that this naive estimate is g w V K V a r (d) } / % ] • The effect of ignoring bird-to-bird variation in g may be assessed as follows. Since gd is a product of two random variates, a first-order approximation to its standard error is (Kendall and Stuart, 1958):
PREDICTED
• 60
• 40
•
-60
-40
-20
V [ { g Var (d) + 2 gd Cov (g,d) + d2 V a r ( g ) } / n b ]
.
« •
2
•
[1]
•
•
20
• • • >
20
40
60 OBSERVED
-20
_ 100R (2r + R) 1 + R (2r + R)
•
•
-40
-60
FIG. 1. Percentage bias in standard error of mean TME estimated from pooled excreta and as predicted from formula [2].
[2]
where R = Cg/Cd- Since the denominator of [2] cannot be negative, the bias in the variance, and therefore in the SE, is positive or negative according to whether r is less than or greater than - R / 2 . Column 5 in Table 1 ("Pooled") shows the standard error of the mean TME calculated by ignoring the among-bird variation associated with g,y. Column 6 shows the discrepancy from the "Standard" SEM expressed as a percentage to eliminate the influence of design parameters (number of birds, weight of feed input, etc.). The good agreement between the observed bias in SEM of TME and the bias predicted from expression [2] is shown in Figure 1. The bias extended beyond 60% in both directions; for the most part, the size and sign of r were dominant factors. For this particular set of feedingstuffs the average bias was only —1.8%, but it need not follow that the population value is close to zero nor that the deviation from this value is independent of diet and assay conditions. In any case, discrepancies potentially as large as these are too great to be discounted in individual assays. Estimates from Actual Pooled Excreta: Variation Among Birds. The question of practical importance is, what happens when the excreta are actually pooled before calorimetry? Assume that np unbiased subsamples of unit weight are taken from die bulked excreta,
giving a mean estimate gp of energy concentration from which the mean energy excretion is estimated as: E P = g p 2 dj/nb Because there is now no information on the among-bird variance of gp and its covariance with Ed-, the estimate of the SE of Ep is bound to be subject to bias of approximately the same order as for Ey, given in [ 2 ] . The estimate of mean energy excretion, however, is theoretically unbiased. The bias in the among-bird component of variance for the pooled excreta may be overcome by incorporating the among-bird variation in g. In the absence of prior estimates of variancejmd covariance, this could be done by dividing the birds on a diet into subgroups, and pooling the excreta for each subgroup before calorimetry. Some of the advantage of pooling is thereby lost, because the amount of calorimetry is increased. In an experiment that is designed in replicated blocks, with several birds to a block, pooling excreta within blocks would serve the purpose. In quality control work, a solution would be to set up the control limits using preliminary data from unpooled excreta. Sampling Error. Differing sampling procedures may also cause different estimates of die variance. To investigate this, let the variance of an estimate of mean energy voided be
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• Hence, the squared ratio of the two estimates of standard error is approximately Cgl(c\ + 2rC(jCg + C|), where C
POOLING EXCRETA FOR CALORIMETRY expressed as V^j (E) + Vg (E), where Vw is the variance associated with the model (i.e., incorporating inter-bird variation and any design restrictions) and Vg is the variance due to sampling excreta for calorimetry. Then Var (Ep)/Var (E,) may be written as: VM(Ep) V
M(EI>
+
Vg(Ep)
+ V
S< E I>
n b 2 V S (E P ) = V S (gp 2 di) ( 2 di) 2 V S (g p ) and: n b 2 V S (Ei) = V S { S(d; g i ) } = (Sdi2)VS(gi) Since ( £ d i ) 2 > 2 d i 2 , then V S (E p ) will be greater than VS(ET) when Vs(gp)>Vs(g;), which should be true in the special case where np is 1 and there is no differential effect of sampling from different quantities. While a full mathematical comparison of Vs(Ep) and V§(Ej) is beyond the scope of this communication, a rough evaluation is obtained by assuming that the dj all have the same value, D. Allowing for sampling from a finite population: V S ( E P ) = D2 a
SP (1
Dn b
)V
and: VS(El) = D 2 a
1 ^~D~""b
2 ( 1
SI
where ffgp and CTg,are the standard deviations for subsamples from the pooled and unpooled excreta, respectively. Because it seems reasonable to assume that: aSp>ffSI,letaSp2=(l+X)aSI2 where X>0. Then the percentage by which the expected variance of Ep exceeds that of Ej is Py, where:
n =
nt nb np
b ~ nP n
bnP
+
X ^ -
Dn b [3]
np
since 1/Dn b is relatively small and can be neglected. Obviously [3] is reduced by increasing n p above the customary value of 1, although this is counter to the object of pooling, which is to have n p as small as possible. To assess the practical impact of Py, an estimate of (Jgp was obtained by subsampling from the pooled excreta of feedstuffs 5, 7, 10, 12, and 14, with the results shown in Table 2. As expected, the mean TME values differ slightly from those presented in Table 1, because no correction was made for changes in moisture since the original analyses. The variances within feedingstuffs were not heterogeneous (P>.05) and the combined SD estimate was .0381 with 20 degrees of freedom. Assuming X = 0 (i.e., CTgp =CTg,,and taking (Jgj to be .0381, application of [3] to the feedingstuffs listed in Table 1 showed that in terms of TME units the greatest expected increase in the SEM was for feedingstuff 1. For this material the SEM using pooled excreta would be expected to exceed the "standard" value calculated from individual excreta samples by about 2.5%, i.e. by about .004 kj/g of dry matter. Such a small difference is trivial. Now consider the case of X>0 (i.e., CTgpXTgr) and note from [3] that P y increases with X, for fixed n^ and np. Because the increment arises solely from the term X/n p , there is more incentive to consider increasing n p . Taking a rather extreme case, suppose CTgp = 10 CTgj, so that X = 99 instead of zero. Equation [3] may be rewritten as: PyVar(Ej)
^
nb - np
100 D 2 CTgp2 ( 1 + X ) n b n p
X [4] (1+X)np
Taking CTgp = .0381 and considering again the example of feedingstuff 1 (for which n b = 6), when np = 1 the right-hand side of [4] instead of being .8333 is .0083 + .99, i.e., .9983. This represents an overall increase in the SEM of the TME amounting to about 2.7%, i.e., still only just over .004 kj/g of dry matter. If np were increased to 2, the corresponding values would be 1.5% and .002 kj/g.
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If the gp are accorded their components of among-bird error, the Vjvi terms are the same, so that the difference between the numerator and denominator lies in the V§ terms. Now:
P y Var (ET) 100 D1 a,SI
1857
SIBBALD AND MORSE
1858
TABLE 2. Gross energy of subsamples from pooled excreta (kj/g) Feedingstuff 5 7 10 12 14
Mean
SD (4 df)
14.09 12.93 13.68 13.79 14.55 Combined
.048 .032 .028 .046 .032 .038 (20 df)
ACKNOWLEDGMENTS
The authors thank I. C. Edmundson for stimulating correspondence and discussion about analyses of pooled samples. Thanks are extended to M. R. Binns for useful discussion on sampling theory and comments on an earlier draft and to S. Tobin for her able technical assistance. REFERENCES Dale, N. M., and H. L. Fuller, 1981. The use of true metabolizable energy (TME) in formulating poultry rations. Pages 50—57 in Proc. Georgia Nutr. Conf. Edmundson, I. C , 1980. The true metabolizable energy of meat and bone meal determined at different dose levels. Pages 20—25 in Proc. South Pacific Poultry Sci. Conv., Auckland, New Zealand. Kendall, M. G., and A. Stuart, 1958. Page 232 in The Advanced Theory of Statistics. Vol. 1. Griffin, London. Sibbald, I. R., 1976. A bioassay for true metabolizable energy in feedingstuffs. Poultry Sci. 55: SOSSOS.
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Under the conditions of these assays, there appears to be little need to take n„ greater than 1 to improve precision, although some replication is always useful as a precaution and check of technique. To summarize, pooling excreta prior to calorimetry is attractive because it reduces the time and cost of the TME bioassay. However, unless the among-bird variation in gross energy estimates is incorporated, pooling can give erroneous estimates of the SE's of mean TME values. The magnitudes of these discrepancies depend upon the variances and covariances of the individual components of the assay, and may vary among feedingstuffs and laboratories. In situations where speed and cost are im-
portant, and where statistical comparisons or interval estimates are of little concern, pooling has distinct advantages. However, because scientific advances often depend on comparisons involving variances, the technique of pooling may have disadvantages in a research laboratory.
1982 Poultry Science 61:1853-1858
INTRODUCTION M e a s u r e m e n t of t h e gross energy p e r u n i t weight of air-dry e x c r e t a (G) is an integral part of t h e bioassay for true metabolizable energy (TME) (Sibbald, 1976). O x y g e n b o m b calorimetry, die usual p r o c e d u r e for measuring G, is tedious, t i m e consuming, and, therefore, expensive. C o n s e q u e n t l y , there is interest in t h e analysis of pooled rather than individual excreta samples. E d m u n d s o n ( 1 9 8 0 ) p o i n t s o u t t h a t pooling excreta before calorimetry has t h e effect of ignoring t h e variation in G a m o n g individual birds and t h a t this can result in overestimation or u n d e r e s t i m a t i o n of t h e standard deviation (SD) of t h e TME. H e argues t h a t because " s o m e 2 5 0 replicates spread over a variety of feedingstuffs" gave an average reduction of only 3% in t h e a p p a r e n t SD, it is an acceptable risk t o treat G as a c o n s t a n t .
'Contribution Numbers 1028 Animal Research Centre and 1-336 Engineering and Statistical Research Institute. 2 Animal Research Centre. 3 Statistical Research Institute.
In the assay of a single feedingstuff, h o w ever, or in a single comparative e x p e r i m e n t , t h e bias in t h e SD may be considerable if t h e variation in G is ignored. Moreover, this bias m a y n o t b e i n d e p e n d e n t of t h e diet n o r of t h e c o n d i t i o n s of t h e assay (e.g. age and t y p e of bird), because it d e p e n d s on t h e correlation b e t w e e n t h e weight of air-dry e x c r e t a (D) and G and also on t h e relative size of their coefficients of variation. To investigate t h e p o t e n tial bias, results of T M E bioassays of 2 0 feedingstuffs were used t o simulate t h e effect of pooling excreta. An additional factor is- t h e effect of t h e sampling error when calorimetry is p e r f o r m e d o n samples from an actual p o o l e d aggregate of excreta. Advocates of pooling (Dale and Fuller, 1981) cite advantages in reducing time and labor and imply t h a t only one sample need be t a k e n from t h e p o o l e d excreta for calorimetric m e a s u r e m e n t . Even with careful mixing, t h e potential difference between pooled and u n p o o l e d excreta as regards sampling error c a n n o t b e ignored, a n d t h e question arises as t o h o w m a n y samples t o t a k e . This aspect was also investigated.
1853
Downloaded from http://ps.oxfordjournals.org/ at University of North Dakota on May 23, 2015
ABSTRACT Pooling excreta, within treatments, prior to calorimetry reduces the time and cost of the bioassay for true metabolizable energy (TME) and does not alter mean TME values. However, by ignoring the among-bird variance of the gross energy per unit weight of excreta (G) and its covariance with the weight of excreta (D), the standard errors of mean TME values are subject to bias. An additional factor for consideration is the effect of the sampling error from pooled aggregates. These factors were examined using data from TME bioassays of 20 feedingstuffs and by measuring the standard deviations of gross energy values for subsamples from pooled excreta. The standard deviation of G for each unit subsample from pooled excreta samples was .0381 kj/g with 20 degrees of freedom. The resulting contribution to the standard errors of mean TME values was so small that replicate subsampling from the bulked excreta is of minor importance, although a useful precaution. However, ignoring the among-bird variation in G as a consequence of pooling excreta caused the standard errors of mean TME values to be incorrectly estimated by amounts ranging from —62 to + 64%. Although for these particular feedingstuffs the average discrepancy was small, the potential magnitude of the bias for an individual feedingstuff is too large to be discounted in a research laboratory. (Key words: true metabolizable energy, bioassay, pooling samples, calorimetry, sampling)
SIBBALD AND MORSE
1854 MATERIALS AND METHODS
RESULTS AND DISCUSSION
The TME value of a feedingstuff is estimated as:
the same considerations as for excreta from fed birds, but the influence on the SD of the TME may perhaps be less. Here attention will be confined to the variance arising from the remaining term Eef, i.e., the estimate of energy voided by fed birds. Let the excreta voided by the i t n bird on a particular diet have weight d; and energy concentration gj as measured by calorimetry on a single unit-weight sample. Let there be nt, birds on the diet. Energy Excreted Estimated from Individual Excreta. When gj is measured for each bird without pooling excreta, the mean energy per bird is estimated as Ej, where: nb Ei = 2 (gi di), the summation being over all n D birds. The standard error (SE) of Ei is estimated as: SE(E I ) = S D ( g d ) / V n b where SD (gd) may be calculated in the usual way from squared deviations of the individual products g:d- about their mean. The corresponding standard errors of the mean TME values of 20 feedingstuffs are given in column 4 of Table 1 ("Standard"). In calculating these, E e f was the only contributing factor, because the energy per unit of feed (Ef) and the metabolic plus endogenous energy out (E e u ) were assumed to be constant for all birds within a diet. Simulation of Estimation from Pooled Excreta. Assuming that a weighted mean of the gi values simulates the energy concentration of the pooled excreta, the mean energy excretion per bird is calculated as E ^ , where: nbEW = g w £ d i
TME = Ef - (E e f - E e u )/I where Ef is the gross energy per unit weight of the feedingstuff, I is the weight of feedingstuff placed in the bird, and E e f and E e u are the amounts of excreta energy voided by fed and unfed birds, respectively. Of these variables I can be assumed to have negligible error under precision feeding and Ef is usually sufficiently well-estimated that its error variance may be assumed to be relatively unimportant. The value of E e u may be estimated from the same birds as those fed or from different birds. Pooling excreta from fasted birds is subject to
and:
g\V = 2 fei d i ) / S d i so that Eyj is exactly the same as E,. Hence, the true standard errors of the two estimates should be exactly the same. However, in actual pooling only g w and individual dj values are known; the individual products g-d- are unknown. Let Var (d) and Var (g) denote the estimates of among-bird variance of d and g, respectively, and let Cov (d,g)
Downloaded from http://ps.oxfordjournals.org/ at University of North Dakota on May 23, 2015
Data from TME bioassays of 20 feedingstuffs form the basis of the report. There were from 5 to 10 Single Comb White Leghorn cockerels in each assay; all calculations were based on air-dry excreta. Standard TME variances were calculated using the G and D values appropriate to individual birds. "Pooled" TME variances were calculated for each feedingstuff using individual D values and a single weighted mean value for G (the sum of the gross energy voided by birds receiving a, particular feedingstuff divided by the sum of their excreta weights). Excreta remaining from the TME bioassays of five feedingstuffs (5, 7, 10, 12, and 14 of Table 1) were used to estimate the variability associated with actual subsampling from a pooled aggregate of excreta. For each of these feedingstuffs, excreta from the 10 birds were combined in the proportions voided during the assay; the absolute amounts were slightly less than those voided because of prior removal of samples for calorimetry. Excreta were mixed in a Waring blendor and then transferred to a plastic tub from which five 1-g subsamples were drawn for calorimetry; excreta were stirred between removal of each subsample. When pooling excreta, no correction was made for any changes in moisture content since the original TME analyses; consequently, it was anticipated that the mean G based on the analyses of the subsamples from the pooled excreta would be slightly different from the calculated weighted mean G.
POOLING EXCRETA FOR CALORIMETRY
O O O ^ t « M ' COOCO'+ON'OO'-OI^QOmm OmNOO NMJOI CO S N ^ W * W >
c o r ^ O ,< ^ o o \ < N, C "v] < N O o o r * ' - t t > o o u - > r ^ * H © ( S r-*i-< ooocoo\ + oooinvot-*\Oo\mt"*-0\0'
coOMOo\r-iNOco»n-+'*Omo\":t'*000,r> G\O\NH0sOC\C00\miH00M^N00'+'l-inC0 i n N f i m m m N N " ! ! » rn ^ •* N m rg O +
C N H ^ - W O N N i - ^ i A -
rs,oo\^'-;'AO\»nr^ "nr^mMi-finiviOTH H H N fn >)• m >0 I I I I I ! I I I
tONOiWvOHON'J'OinhCO^'-nninOO + O m N H t N M X ^ C A M i S O H O O C M n H H M
m'nmt^.oON'ncot^'^i'n^-OovOr^W'Or-lON in>not^.o\oiX'*co'^r^i-"«%mtN'^-r^o\foO
t ^ m f t N i
i>Omfn^-t^f««nc«l»ni
\ooco
H ( s m * » n o ^ C O O \ O H N i
i t
m « N
MONO
Downloaded from http://ps.oxfordjournals.org/ at University of North Dakota on May 23, 2015
OO'-<'OOt^0\,0000\Nm>Oi-i'-tOO-^--*O m Q\ h « w M h w * ff; H w co in ^ in co O CO m O CO H N ^ N in N N CN N O 6 ^ N ON H 6 tf> H
SIBBALD AND MORSE
1856
denote the corresponding covariance. Calculating the SE only from the deviations of the g«f d- about their mean is equivalent to treating giy as constant, so that this naive estimate is g w V K V a r (d) } / % ] • The effect of ignoring bird-to-bird variation in g may be assessed as follows. Since gd is a product of two random variates, a first-order approximation to its standard error is (Kendall and Stuart, 1958):
PREDICTED
• 60
• 40
•
-60
-40
-20
V [ { g Var (d) + 2 gd Cov (g,d) + d2 V a r ( g ) } / n b ]
.
« •
2
•
[1]
•
•
20
• • • >
20
40
60 OBSERVED
-20
_ 100R (2r + R) 1 + R (2r + R)
•
•
-40
-60
FIG. 1. Percentage bias in standard error of mean TME estimated from pooled excreta and as predicted from formula [2].
[2]
where R = Cg/Cd- Since the denominator of [2] cannot be negative, the bias in the variance, and therefore in the SE, is positive or negative according to whether r is less than or greater than - R / 2 . Column 5 in Table 1 ("Pooled") shows the standard error of the mean TME calculated by ignoring the among-bird variation associated with g,y. Column 6 shows the discrepancy from the "Standard" SEM expressed as a percentage to eliminate the influence of design parameters (number of birds, weight of feed input, etc.). The good agreement between the observed bias in SEM of TME and the bias predicted from expression [2] is shown in Figure 1. The bias extended beyond 60% in both directions; for the most part, the size and sign of r were dominant factors. For this particular set of feedingstuffs the average bias was only —1.8%, but it need not follow that the population value is close to zero nor that the deviation from this value is independent of diet and assay conditions. In any case, discrepancies potentially as large as these are too great to be discounted in individual assays. Estimates from Actual Pooled Excreta: Variation Among Birds. The question of practical importance is, what happens when the excreta are actually pooled before calorimetry? Assume that np unbiased subsamples of unit weight are taken from die bulked excreta,
giving a mean estimate gp of energy concentration from which the mean energy excretion is estimated as: E P = g p 2 dj/nb Because there is now no information on the among-bird variance of gp and its covariance with Ed-, the estimate of the SE of Ep is bound to be subject to bias of approximately the same order as for Ey, given in [ 2 ] . The estimate of mean energy excretion, however, is theoretically unbiased. The bias in the among-bird component of variance for the pooled excreta may be overcome by incorporating the among-bird variation in g. In the absence of prior estimates of variancejmd covariance, this could be done by dividing the birds on a diet into subgroups, and pooling the excreta for each subgroup before calorimetry. Some of the advantage of pooling is thereby lost, because the amount of calorimetry is increased. In an experiment that is designed in replicated blocks, with several birds to a block, pooling excreta within blocks would serve the purpose. In quality control work, a solution would be to set up the control limits using preliminary data from unpooled excreta. Sampling Error. Differing sampling procedures may also cause different estimates of die variance. To investigate this, let the variance of an estimate of mean energy voided be
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• Hence, the squared ratio of the two estimates of standard error is approximately Cgl(c\ + 2rC(jCg + C|), where C
POOLING EXCRETA FOR CALORIMETRY expressed as V^j (E) + Vg (E), where Vw is the variance associated with the model (i.e., incorporating inter-bird variation and any design restrictions) and Vg is the variance due to sampling excreta for calorimetry. Then Var (Ep)/Var (E,) may be written as: VM(Ep) V
M(EI>
+
Vg(Ep)
+ V
S< E I>
n b 2 V S (E P ) = V S (gp 2 di) ( 2 di) 2 V S (g p ) and: n b 2 V S (Ei) = V S { S(d; g i ) } = (Sdi2)VS(gi) Since ( £ d i ) 2 > 2 d i 2 , then V S (E p ) will be greater than VS(ET) when Vs(gp)>Vs(g;), which should be true in the special case where np is 1 and there is no differential effect of sampling from different quantities. While a full mathematical comparison of Vs(Ep) and V§(Ej) is beyond the scope of this communication, a rough evaluation is obtained by assuming that the dj all have the same value, D. Allowing for sampling from a finite population: V S ( E P ) = D2 a
SP (1
Dn b
)V
and: VS(El) = D 2 a
1 ^~D~""b
2 ( 1
SI
where ffgp and CTg,are the standard deviations for subsamples from the pooled and unpooled excreta, respectively. Because it seems reasonable to assume that: aSp>ffSI,letaSp2=(l+X)aSI2 where X>0. Then the percentage by which the expected variance of Ep exceeds that of Ej is Py, where:
n =
nt nb np
b ~ nP n
bnP
+
X ^ -
Dn b [3]
np
since 1/Dn b is relatively small and can be neglected. Obviously [3] is reduced by increasing n p above the customary value of 1, although this is counter to the object of pooling, which is to have n p as small as possible. To assess the practical impact of Py, an estimate of (Jgp was obtained by subsampling from the pooled excreta of feedstuffs 5, 7, 10, 12, and 14, with the results shown in Table 2. As expected, the mean TME values differ slightly from those presented in Table 1, because no correction was made for changes in moisture since the original analyses. The variances within feedingstuffs were not heterogeneous (P>.05) and the combined SD estimate was .0381 with 20 degrees of freedom. Assuming X = 0 (i.e., CTgp =CTg,,and taking (Jgj to be .0381, application of [3] to the feedingstuffs listed in Table 1 showed that in terms of TME units the greatest expected increase in the SEM was for feedingstuff 1. For this material the SEM using pooled excreta would be expected to exceed the "standard" value calculated from individual excreta samples by about 2.5%, i.e. by about .004 kj/g of dry matter. Such a small difference is trivial. Now consider the case of X>0 (i.e., CTgpXTgr) and note from [3] that P y increases with X, for fixed n^ and np. Because the increment arises solely from the term X/n p , there is more incentive to consider increasing n p . Taking a rather extreme case, suppose CTgp = 10 CTgj, so that X = 99 instead of zero. Equation [3] may be rewritten as: PyVar(Ej)
^
nb - np
100 D 2 CTgp2 ( 1 + X ) n b n p
X [4] (1+X)np
Taking CTgp = .0381 and considering again the example of feedingstuff 1 (for which n b = 6), when np = 1 the right-hand side of [4] instead of being .8333 is .0083 + .99, i.e., .9983. This represents an overall increase in the SEM of the TME amounting to about 2.7%, i.e., still only just over .004 kj/g of dry matter. If np were increased to 2, the corresponding values would be 1.5% and .002 kj/g.
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If the gp are accorded their components of among-bird error, the Vjvi terms are the same, so that the difference between the numerator and denominator lies in the V§ terms. Now:
P y Var (ET) 100 D1 a,SI
1857
SIBBALD AND MORSE
1858
TABLE 2. Gross energy of subsamples from pooled excreta (kj/g) Feedingstuff 5 7 10 12 14
Mean
SD (4 df)
14.09 12.93 13.68 13.79 14.55 Combined
.048 .032 .028 .046 .032 .038 (20 df)
ACKNOWLEDGMENTS
The authors thank I. C. Edmundson for stimulating correspondence and discussion about analyses of pooled samples. Thanks are extended to M. R. Binns for useful discussion on sampling theory and comments on an earlier draft and to S. Tobin for her able technical assistance. REFERENCES Dale, N. M., and H. L. Fuller, 1981. The use of true metabolizable energy (TME) in formulating poultry rations. Pages 50—57 in Proc. Georgia Nutr. Conf. Edmundson, I. C , 1980. The true metabolizable energy of meat and bone meal determined at different dose levels. Pages 20—25 in Proc. South Pacific Poultry Sci. Conv., Auckland, New Zealand. Kendall, M. G., and A. Stuart, 1958. Page 232 in The Advanced Theory of Statistics. Vol. 1. Griffin, London. Sibbald, I. R., 1976. A bioassay for true metabolizable energy in feedingstuffs. Poultry Sci. 55: SOSSOS.
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Under the conditions of these assays, there appears to be little need to take n„ greater than 1 to improve precision, although some replication is always useful as a precaution and check of technique. To summarize, pooling excreta prior to calorimetry is attractive because it reduces the time and cost of the TME bioassay. However, unless the among-bird variation in gross energy estimates is incorporated, pooling can give erroneous estimates of the SE's of mean TME values. The magnitudes of these discrepancies depend upon the variances and covariances of the individual components of the assay, and may vary among feedingstuffs and laboratories. In situations where speed and cost are im-
portant, and where statistical comparisons or interval estimates are of little concern, pooling has distinct advantages. However, because scientific advances often depend on comparisons involving variances, the technique of pooling may have disadvantages in a research laboratory.