Tài liệu Quantitative aspects of ruminant digestion and metabolism - Phần 16 pdf

Total absorbedamino acids TAbsAa Absorbed propionic acid AbsPr Absorbed glucose AbsGl Absorbed acetate AbsAc Absorbed fatty acids AbsFa Butyrate AbsBu Absorbed nutrients Oxidation BuCd A

21 Lactation: Statistical and Genetic

Aspects of Simulating Lactation Data from Individual Cows using a Dynamic, Mechanistic Model of Dairy Cow Metabolism

H.A Johnson, T.R Famula and R.L Baldwin

Department of Animal Science, University of California, Davis,

CA 95616-8521, USA

Introduction

Empirical models are fitted to experimental data to describe relationshipsbetween dependent and independent variables By definition, they are bestrepresentations of the input:output data from which they were created Also bydefinition, testing predictions of empirical models against data not used informulating the models often leads to failures Thus, it is generally recognizedthat empirical models are only valid for specific situations and do not generalizebecause they do not capture underlying reasons for relationships betweendependent and independent variables In contrast with empirical models, mech-anistic models are derived from theories about the nature of the system mod-elled and, as a result, are based upon our understanding of underlyingmechanisms, which drive the system (France and Thornley, 1984; Baldwin,1995) Also, parameter values in mechanistic equations are derived fromexperimental data on each mechanism and, thus, are not derived from statis-tical analyses of input:output observations on the total system For example, amechanistic model of dairy cow functions would incorporate data on nutrientuptake, nutrient utilization by tissues, metabolic pathways, enzyme activities,nutrient concentrations, regulatory systems, etc while an empirical modelwould use data on intake of nutrients and amount of milk or milk componentsoutput and body weight changes Failure of a mechanistic model to simulatenew, long-term data shows where understanding incorporated into the model

is lacking and what old or new knowledge and experimental data must beincorporated to further refine and develop the model

The testing of mechanistic models in biology must consider two tives when evaluating the suitability of a computer model to serve as a proxy for

perspec-ß CAB International 2005 Quantitative Aspects of Ruminant Digestion

and Metabolism, 2nd edition (eds J Dijkstra, J.M Forbes and J France) 551

physiological processes The first is a statistical perspective, an evaluation ofthe fit of the predictive results of the model to the observed physiologicalphenomena Techniques for such statistical evaluation abound, relying uponloss functions, likelihood surfaces and measures of ‘goodness-of-fit’ (Diggle

et al., 1994) In addition, investigators are asked to evaluate the statisticalmeans by which they will draw conclusions, such as the use of computingalgorithms, the parametric form of distributions to consider and the distinctionbetween classical and Bayesian procedures (Robert and Casella, 1999) Thesecond form, a biological perspective, concerns the assessment of the behav-iour of the predictions, whether the estimates of parameters and the ensuingpredictions from such models, make biological ‘sense’ In other words, doanimals, or populations of animals, display the same properties in nature(in vivo) that the model would have them display ‘in silico’? Our model testingprocess must, of needs, consider both In this chapter, the underlying relation-ships between diet, intake, milk production and genetic potential to produce

are examined First, the main equations representing the metabolic dairy cowmodel are described, previous evaluations of the model are presented andtechniques to evaluate models are explained Second, the sensitivity of themodel to certain parameters used in model evaluation is considered Finally alarge data set of production data is used to evaluate model predictions

Overview ofMOLLYEquations

lactating dairy cow described in detail by Baldwin (1995) and earlier tions The digestion element of the model (Fig 21.1) is comprised of 15differential equations descriptive of transactions associated with the state vari-ables: starch (St), hemicellulose (Hc), cellulose (Ce), soluble carbohydrate (Cs)equivalents arising from the diet and hydrolysis of insoluble carbohydrates,microbes (Mi), acetate (Ac), propionate (Pr), butyrate (Bu), insoluble protein(Pi), amino acids and peptides (Aa), ammonia (Am), ash (soluble as As, insoluble

publica-as Ai), lignin (Lg) and large (Lp) and small feed particles (Sp) Chemical position of the diet is represented by St, Hc, Ce, Lg, Cs (also as Sc), Ac, Pr, Bu,

com-Pi, Ps (soluble protein), As, Ai, Li (lipid), Oa (organic acids), La (lactate), Pe(pectin), Nn (non-protein nitrogen), Ur (urea) and fat Lp and Sp representphysical attributes of the diet that influence the digestion process In general,feed particles pass from the large particle pool to the small particle pool asdigestion proceeds Passage rates of nutrients associated with feed particles areinfluenced by water flow through the digestion process After hydrolysis andmicrobial attachment, the rumen model uses fermentation coefficients to con-vert starch, soluble carbohydrates and amino acids into volatile fatty acids.Microbial growth is dependent on pH, ATP, dietary fat, rumen amino acids,ammonia and particle size

The animal element of the model (Fig 21.2) begins with absorbed nutrients(from Fig 21.1) and defines transactions associated with ten state variables:

Fat Ur

Water (Passage)

Cs, RAa,Am, Fl, As, Mi, RLa, RPr, RBu, RAc

Insoluble protein (Pi)

Rumen amino acids (RAa) from Ps, Pi, Mi

Saliva Ammonia (Am)

Microbes (Mi)

Fermentation producing volatile fatty acids (RAc,RPr, RBu, RLa)

(Ha)

Soluble carbohydrates (Cs) from

St, Sc, Ha, Pe, Oa, La, Hc, Ce

Long chain fatty acids (Fl)

Ce Hc

Faeces Ot, As, Pi, Hc, Ce, Fl, Mi

Absorbed nutrients

As, Am, RAa (TAbsAa), RAc (AbsAc), RBu (AbsBu),RPr (AbsPr), RLa (AbsRLa), Glucose (AbsGlfrom Cs, La and Mi), Fl (AbsFaand GyGlV)

Large feed particles (Lp) with

Hc, Ce, Ha, Pi, Lg, Ai

Fig 21.1 Basic flux relationships in the digestive element of theMOLLYmodel Solid lines indicate digestion processes associated with chemicalcharacteristics of the diet Dashed lines represent physical processes associated with digestion

Total absorbed

amino acids (TAbsAa)

Absorbed propionic acid (AbsPr)

Absorbed glucose (AbsGl)

Absorbed acetate (AbsAc)

Absorbed fatty acids (AbsFa)

Butyrate (AbsBu)

Absorbed nutrients

Oxidation (BuCd)

Ammonia (Am)

Urea

Milk lactose (Lm)

Saliva

Pregnancy (PRG)

Plasma glucose (Gl)

Oxidation (PrCd)

O2

CO2

Urea TAaDEG TAaGlV

PrGlV

Absorbed lactate (AbsRLa)

Lactate in body (LaGlB) Lactate in adipose

(LaGlF) LaGlV

GlLmV

Storage triglyceride (Ts) triose phosphate (TpF), NADPH2 (HyF), fatty acids (TsF)

Viscera triose phosphate (TpV) NADPH2 (HyV)

Oxidation (GlCd)

O 2

CO2

TAaPm, TAaLAV

Plasma acetate (Ac) Oxidation (AcCd)

O 2

CO2

Glycerol (Gy) GlHyV GlTpV GyGlV

AcTsF GlTpF GlHyF

GlLaB

Plasma fatty acids (Fa) TsFaF

AcTmV FaTmV

Oxidation (FaCd)

O2

CO2Urea

Fig 21.2 Basic flux relationships in the animal element ofMOLLY State variables are outlined by heavy black lines

total amino acids (TAa), glucose (Gl), acetate (Ac) and lipids (Fa), body protein(Pb), visceral protein (Pv), storage triacylglycerol (Ts), milk protein (Pm), milklactose (Lm) and milk fat (Tm) Concentration of nutrients in blood is denoted

by a lower case c (i.e cGl is concentration of glucose in blood, cAc is tration of acetate in plasma, cTAa is concentration of total amino acids) and iscalculated by dividing the state variable by the distribution volume of glucose.Inputs into nutrient pools are influenced by absorption of the nutrient fromdiet and/or digestion, and conversion from other nutrients or metabolic inter-mediates by deamination, fermentation or synthetic processes Outputs fromnutrient pools are oxidation, synthesis of metabolic intermediates, synthesis ofbody tissues or secreted products (milk, milk fat, etc.) Algebraic equations areused in the model to calculate body weights, weight of viscera, weight of bodyfat, milk production, excretions, respiratory exchange, energy costs of individ-ual nutrient transactions, ration metabolizable energy values, total heat produc-tion, income over feed costs and other outputs Therefore the model predictsmilk lactose (total volume milk produced), protein and milk fat based on themetabolic state of the cow, nutrients available to the udder and potential of thecow to produce milk through the parameters number of udder cells (UCELLS)and maximal velocities for milk fat and milk protein synthesis There are alsoequations for the demands of pregnancy To simulate a lactation, diet compos-ition, daily dry matter intake, initial body weight, body fat per cent (or bodycondition score), length of the simulation (days) and UCELLS must be input tothe model

model has been rewritten to accommodate four amino acid pools: sulphuramino acids (SAa), lysine (Lys), histidine (His) and remaining amino acids (Aa).Equations for the uptake of individual amino acids by mammary tissue (Hanigan

et al., 1992) were incorporated This revision allows either SAa, Lys, His or

Aa to limit the synthesis of milk (Pm), body (Pb) and visceral (Pv) proteins anda-lactalbumin and, as a result, lactose synthesis The stoichiometric parameters,which define amino acid degradation in the model, have become dynamicvariables dependent on the amount of individual amino acids entering andleaving the several pools The sources of entering amino acids are the digestion

of microbial protein, rumen bypass and abomasally infused proteins, aminoacids and degradation of body and visceral proteins Individual amino acidsleave the pools for the synthesis of milk, body, visceral, salivary, fetal andplacental proteins, and via amino acid degradation Stoichiometries are calcu-lated based upon the metabolic pathways for degradation of individual aminoacids Figure 21.3 shows in detail the equations presented in Fig 21.2, whichare of primary importance to the discussion presented in this chapter

where k is a rate constant in units of per minute, A is amount or concentration

of substrate, B is amount of product, e.g moles) or Michaelis–Menten form({vA,B¼ VA,B=(1 þ kA,B=A)}; where vA,Bis velocity of reaction A to B, VA,B is

which half maximal velocity is reached) For example, a mass action equation in

AbsGl, going directly to plasma glucose, UpGl is 10% An example of

dAc/dt (mol/day) ¼ absAc þ TAaAc
AcCd
AcTsF
AcTmV

Ac ¼ Total acetate in plasma (mol)

absAc ¼ Acetate absorption (mol/day)

TAaAc ¼ Portion of total amino acids degraded (TAaDEG) that result in the

formation of acetate (mol/day)

AcCd ¼ Acetate oxidation (mol/day)

AcTsF ¼ Acetate to triglyceride synthesis in adipose (mol/day)

AcTmV ¼ Acetate to milk fat synthesis in viscera – mammary (mol/day)

dFa/dt (mol/day) ¼ absFa þ TsFaF
FaCd
FaTsF
FaTmV

Fa ¼ Total fatty acids in plasma (mol)

AbsFa ¼ Fatty acid absorption (mol/day)

TsFaF ¼ Triglyceride breakdown to fatty acids in adipose (mol/day)

FaCd ¼ Fatty acid oxidation (mol/day)

FaTsF ¼ Fatty acids to triglyceride synthesis in adipose (mol/day)

FaTmV ¼ Fatty acids to milk fat synthesis in viscera – mammary (mol/day)

dGl/dt (mol/day) ¼ PrGlV þ UpGl þ TAaGlV þ LaGlV þ GyGlV

GlLmV
GlHyF
GlTpF
GlLaB
GlHyV
GlTpV
GlCd

Gl ¼ Total glucose in plasma (mol)

PrGlV ¼ Portion of absorbed propionate that results in glucose formation (mol/day) UpGl ¼ Portion of absorbed glucose that contributes to plasma glucose (mol/day)

(Note: PrGlV þ UpGl ¼ absorbed glucose (absGl)

TAaGlV ¼ Total amino acids going to glucose in viscera – liver (mol/day)

LaGlV ¼ Lactate to glucose in viscera – liver (mol/day)

GyGlV ¼ Glycerol to glucose in viscera – liver (mol/day)

GlLmV ¼ Glucose to milk lactose in viscera – mammary (mol/day)

GlHyF ¼ Glucose oxidized via pentose phosphate path for

NADPH production in adipose (mol/day) GlTpF ¼ Glucose to triose phosphate (glycerol) in adipose (mol/day)

GlLaB ¼ Glucose to lactate in the body – muscle, etc (mol/day)

GlHyV ¼ Glucose oxidized via pentose phosphate path for

NADPH production in viscera – mammary (mol/day) GlTpV ¼ Glucose to triose phosphate in viscera – mammary (mol/day)

GlCd ¼ Glucose oxidation (mol/day)

dTAa/dt (mol/day) ¼ TabsAa þ TPbAaB þ TPvAaV

TAaPbB
TAaPvV
TAaPmV
TAaSAL
TAaDEG
TAaPRG TAa ¼ Total amino acids in plasma (mol)

TabsAa ¼ Total amino acid absorption (mol/day)

TPbAaB ¼ Protein degradation to total amino acids in the body – muscle (mol/day) TPvAaV ¼ Protein degradation to total amino acids in viscera (mol/day)

TAaPbB ¼ Total amino acids to protein synthesis – muscle (mol/day)

TAaPvV ¼ Total amino acids to protein synthesis – viscera (mol/day)

TAaPmV ¼ Total amino acids to milk protein synthesis – mammary (mol/day)

TAaSAL ¼ Total amino acids to salivary protein synthesis (mol/day)

TAaDEG ¼ Total amino acids degraded i.e to glucose and acetate in viscera (mol/day) TAaPRG ¼ Total amino acids to support fetal growth/pregnancy (mol/day)

and cGl is the concentration of circulating glucose (A) The factor (EBW0:75) hasbeen added as a scalar to modify the equation for empty body weight (EBW)differences between cows and is not included in the classical Michaelis–Mentenequation form.

Previous Evaluations of MOLLY

evaluations were qualitative or, at best, semi-quantitative in nature Theseevaluations were directed to the question, are specific equations or systems ofequations adequate in direction and magnitude of responses to perturbations toallow simulations of reality (Baldwin, 1995) In these tests, the answers wereoften no and indicated that our understanding of specific functions was inad-equate to the simulation of reality For example, model failures led to experi-mental studies of factors, which cause variations in rumen microbial growthrates and yields These studies led to the identification of amino acids (and laterpeptides), microbial maintenance requirements and ammonia availability asimportant determinants of growth yields and led to parameterization of equa-tions to represent these effects (Maeng et al., 1976; Argyle and Baldwin,1989) Current representations of digestion products and amino acid absorp-tion from the rumen produced the results depicted in Table 21.1 Cottrill et al.(1982) fed maize silage-based diets to calves weighing approximately 100 kg.The simulated data presented in Table 21.1 were produced by resetting the

intended to be used to simulate calf data, the magnitude and direction of changebetween observed and predicted values in Table 21.1 are similar

Results of model simulations presented in Table 21.2 agree very well tatively with those reported by Clark (1975), Polan et al (1991), Rulquin et al.(1993) and Whitelaw et al (1986) In Table 21.2, responses to supplementation

quali-Table 21.1 Simulated responses to urea and fishmeal supplementation of a maize

AaSI

(mol/day)

AaSIpredicted(mol/day)

MiAa

(mol/day)

MiAapredicted(mol/day)

with SAa alone or with Lys alone were relatively minor because both were veryclose to limiting (reference diet) As a result, when the concentration of one ofthese amino acids in blood was increased by supplementation, the other aminoacid became limiting and effects upon milk (DMILK) and protein (PPM) wererelatively minor When the availabilities of both SAa and Lys were increased,milk production increased (5.8% at 84 days) and daily milk protein increased10% When a maize-based diet with maize gluten meal as the protein supple-ment was input into the model, Lys was limiting and supplementation with Lysresulted in a 7.9% increase in predicted daily milk and an 8.5% increase in dailymilk protein yield at 84 days in milk.

Supplementation of the maize-based diet with Lys and SAa resulted in a30% increase in milk and protein yields at 84 days in milk These responses toSAa and Lys supplementation are higher than those reported by Clark (1975)and Polan et al (1991), however the rates of SAa and Lys supplementationwere higher than those used in the cited experiments Polan et al (1991)reported no significant increases in milk and milk protein with rumen protectedmethionine supplementation alone (0.11 mol/day) and 7.4% increase in milk(kg/day) and 2.4% increase in milk protein with rumen protected methionine(0.11 mol/day) and lysine (0.16 mol/day) supplementation over 22–112 days

in milk Clark (1975) showed data from two studies with an increase of 3.1% kg

Table 21.2 Effects of base diets and supplements on model outputs

Treatment

DMILK(kg/day)

PPM(%)

cTAa

PmLim Aa

TVMLK(kg)

TDMIN(kg)

EBW(kg)

or casein per abomasum The NRC (1989) equation was used to calculate feed intakes for these tions Column codes are daily milk yield (DMILK), percentage of protein in milk (PPM), total dry matter intake (TDMIN), empty body weight (EBW), total concentrations of amino acids (cTAa), the amino acid pool most limiting to milk protein synthesis (Pm Lim Aa) and total milk yield (TVMLK) It should be noted that simulated day 84 values were different when the supplement treatments were simulated beginning on day

simula-70 of lactation rather than beginning the simulation at initiation of lactation due to carryover effects like those illustrated in Fig 21.5.

milk per day, 6% milk protein and a decrease of 8.1% kg milk per day and

an increase of 1% milk protein in response to methionine supplementation.With lysine supplementation, milk yield increased 5.9% and 3.3% kg/day andprotein increased 8% and 5%, respectively Supplementation was over 7 daysand levels of methionine and lysine supplementation were not listed Rulquin

et al (1993) developed dose–response curves for lysine and methionine effects

on milk yield They found very little response to supplementation with ing amounts of lysine or methionine (less than 1 kg milk per day) and changes

0.15 to þ0.15 for methionine supplementation

Responses to abomasal infusion of casein were also greater than thosereported by Whitelaw et al (1986) With casein infusions of 1.9, 3.7 and5.6 mol/day for 14 days, milk yield increased 17%, 27% and 32%, respect-ively Milk protein increased 3.0%, 4.3% and 5.4%, respectively Clark (1975)also summarized data from casein infusion studies in which casein infusions of2.8, 4.05 and 8.04 mol/day resulted in increases in milk yield of 6.6%, 8.3%and 12.5% kg/day, respectively With 2.8 mol casein infused per day, milkprotein also increased 9%

Additional comparisons of model outputs with detailed data for a 40%barley diet and a 40% maize diet showed that a large number of model outputswere within the standard errors (10%) of observed values (Baldwin and Bau-man, 1984) Experimental studies of adipose tissue metabolism to define andparameterize improved equations to represent metabolite interactions, theregulation of lipogenesis, energy storage and lipolysis (Yang and Baldwin,1973a,b), experimental studies of cow liver metabolism (Knapp et al., 1992),mammary gland metabolism and nutrient uptake (Miller et al., 1991; Hanigan

et al., 1992; Hanigan and Baldwin, 1994) were undertaken to better defineand parameterize equations for the metabolism of these tissues Detailedmodels of metabolism in adipose tissues (Baldwin, 1995), liver (Freetly et al.,1993) and mammary glands (Hanigan and Baldwin, 1994) were constructed tosupport the formulation and parameterization of aggregated equations incorp-orated into the cow model These are also used in formulating changes inexisting cow model equations

Two quantitative evaluations are presented in Figs 21.4 and 21.5 Themetabolizable energy values of feeds are dependent upon many digestiveand animal functions Values predicted by the model agree with observedvalues, within experimental errors, for a wide range of feeds (ME values of7.5–13 MJ/kg; Baldwin et al., 1994) with no systematic errors (Fig 21.4).Rumen and total tract digestion coefficients for starch, hemicellulose, celluloseand protein agree closely with observed values (Baldwin, 1995) Several ex-ceptions to close agreements with data in simulations of digestion have beenreported (Baldwin et al., 1994) The most notable is that rumen starch diges-tion is significantly overestimated for cracked maize diets (20%) and at highfeed intakes (20–30% at 25 kg feed per day)

Broster and Broster (1984) summarized the results of a comprehensiveseries of full lactation studies with cows fed a variety of diets These studiesdefined very significant ‘carryover’ effects after feeding high-energy and

high-protein diets during early lactation These observations prompted thesimulation analyses presented in Fig 21.5 The model responses to low andhigh intakes of energy and protein during early lactation were simulated verywell More importantly, the carryover effects noted by Broster and Broster(1984) were simulated very well in terms of magnitude and duration.

6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5

6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5

Observed ME (MJ/kg)

Fig 21.4 Comparisons of predicted vs observed estimates of metabolizable energy (ME)

literature for diets including high- and low-quality legumes, maize silage, maize meal, soybeanmeal and high- and low-quality grass hays (Baldwin et al., 1994)

Fig 21.5 Effects of different feeding strategies upon lactation performance Diets were 50%forage, 50% concentrate with fishmeal added to 15% or 18% crude protein HHHH was fed 18%crude protein diet at a feeding rate of 10 kg/day plus 1 kg feed per 3 kg milk for 180 days HHHMwas fed 18% crude protein diet for 84 days and then fed 15% crude protein diet for the last 96days Feeding rate of both diets was 10 kg/day plus 1 kg feed per 3 kg milk averaged over theprevious 3 weeks HMHM was fed 15% crude protein diet at a feeding rate of 10 kg/day plus 1 kgfeed per 3 kg milk averaged over the previous 3 weeks for 180 days HMLM was fed 15% crudeprotein diet for 180 days For the first 84 days, feeding rate was 10 kg/day plus 1 kg feed per 3 kgmilk averaged over the previous 3 weeks For the last 96 days, feeding rate was 13 kg/day plus 1 kgfeed per 3 kg milk averaged over the previous 3 weeks From Baldwin (1995)

The relationship illustrated in Fig 21.6 indicates that theMOLLYmodel cansimulate the classical effect of changes in the availability of a limiting amino acidupon rates of degradation of other amino acids In these simulations, the modelpredicted that processes such as milk protein production and body proteinsynthesis would be limited by the amount of sulphur amino acids available tothe cow According to the limiting amino acid theory, as more sulphur (i.e.limiting) amino acids become available to the cow, use of other amino acids forprotein processes should increase resulting in decreases in degradation of otheramino acids Figure 21.6 shows that as the absorption of the limiting aminoacids (sulphur amino acids) is increased in sequential simulations, lysine degrad-ation is decreased due to an increase in lysine use for protein synthesis.

individual animal feed intakes were specified as input, an important cumulativeerror became evident (Johnson et al., 1999; McNamara and Baldwin, 2000).This error was a net accumulation of body fat during lactation on several but notall diets The errors are relatively small on a daily basis (1–4% of MEI) but thecumulative error is large; up to 100 kg EBW When the diets were fed according

to the NRC equations, this accumulation of fat did not occur This may indicateerrors attributable to overestimates of feed intakes in the several experiments orthe equation Errors in model predictions of ME for the feeds (0.1–0.2 Mcal/kg)

NRC are in close agreement for the experiments simulated McNamara andBaldwin (2000) examined the observed and simulated dynamics of lipid metab-olism and identified some systematic errors in the regulation of adipose tissuelipogenesis and lipolysis Modifications have been made in the model and will betested However, these will not correct the problem since if too much energy isinput or not enough output, fat gain is the essential result Milk energy outputs inthe simulations were essentially as observed Thus, the remaining source of error

to consider is heat production Possible errors in estimates of heat production

Absorption of sulphur amino acids (mol/day)

Fig 21.6 Simulated effect of sulphur amino acids (SAa) on whole-body degradation of Lys Thereference diet, containing 50% lucerne hay and 50% concentrate with 15% crude protein, wassupplemented with Lys at 0.3 mol/day

attributable to physiological work, protein turnover and ion (nutrient) transportwere mentioned above Whether or not changes in these are adequately simu-lated, particularly at high feed intakes, is an issue Another possibility is thatchanges in the relative weights of high vs low energy requiring tissues duringlactation are not adequately simulated.

Statistical Methods for Model Evaluation

Traditional problems in statistics begin with the design of an experiment, thecollection of relevant data and proceed to methods of parameter estimation andhypothesis testing Investigators in these settings build simple-to-understand linearstatistical models, under assumptions of normally distributed random variables,using analytical techniques like regression and the analysis of variance (McCullochand Searle, 2001) These methods permit estimation of unknown parameters,the variances of these estimates, and accordingly, investigators typically contrastthese parameter estimates in an effort to declare ‘significance’ of effects.The mechanistic model considered in this chapter is of such complexity that weturn to more computer-intensive strategies to evaluate their effectiveness (Hjorth,1994) Such methods eventually appeal to bootstrap techniques (Efron, 1979;Efron and Tibshirani, 1993), a strategy made possible by advances in computerhardware and software This technique should become a critical component ofstrategies for model validation, selection and the evaluation of parameter estimates

as a complement to the more traditional methods of statistical evaluation.Error and loss functions

The statistical evaluation of models typically focuses on concern for uncertaintyand error Generically error can be thought of originating from any of severalcategories: uncertainty rooted in limitations of sample size, systematic error asthe result of inadequacies of a statistical model to accommodate effects present

in the data and prediction error, as occurs when predicting observations intothe future when residuals cannot be evaluated Another form of uncertainty,often neglected, is model selection uncertainty, that form of error which canoccur when one set of data is evaluated under a variety of models, with onemodel selected, under some criterion, as ‘best’ For example, in a regressionmodel, which predicts milk production from dry matter intake, sample sizelimits would be the error associated with number of cows used to represent thetrue population mean for milk production Systematic error is consistent under

or over prediction of milk production by the regression equation and might

be represented by not including diet composition or initial body weight in theregression equation of milk production on dry matter intake Prediction error isthe error associated with using dry matter intake to predict the cow’s future milkproduction Model uncertainty is error associated with the prediction ofmilk production using several models (milk production regressed on dry matterintake, milk production regressed on dry matter intake and diet composition, or

As referred to in the prior passage, a criterion for model selection andevaluation must be agreed upon as a measure of error (described as a lossfunction) Though not an exhaustive list, there are several ways to measurestatistical error (Mood et al., 1974) Perhaps simplest is absolute error, taken asthe absolute value of the difference between predicted and observed, thoughthe statistical properties of such a measure are algebraically difficult to evaluate.

easy to interpret and is nearly universally accepted, along with variations (e.g.Hanigan et al., 1998) Other strategies have been suggested (e.g Huber,1964), usually as a variation on squared error, in which some weightingprocedure is used, such that residuals are weighted differentially for differentobserved values For example, a prediction error of 1 kg causes greater con-cern in a cow producing 10 kg of milk vs a cow producing 50 kg and one maydesire that model evaluation reflects that concern

Validation of models

Methods of model validation find their widest application in complex models of

a multivariate character Several strategies are available, many of which arecomputationally intensive Typically they involve dismantling the entire data set(defined as DS, with notation borrowed from Hjorth, 1994) into two subsets;the estimation set (ES) from which model parameters can be estimated, and thetesting set (TS), where the model parameters estimated with ES can be used topredict the observations in TS Often, though not always, DS is the union of ES

The simplest strategy divides DS randomly into two halves for ES and TS,though there is little to recommend this strategy beyond simplicity More com-monly used is the ‘take one out strategy’, used iteratively where one data point is

process is repeated for all N observations in DS in a strategy called PRESS, forprediction sum of squares (Allen, 1971) Moreover, there are extensions of thistechnique that permit parameter estimation as a component of the modelvalidation procedure (Stone, 1974) Parenthetically, there is an extension ofthe PRESS technique that deals with the model selection process as well, thosecircumstances where the ultimate size of the model is yet to be determined.For data on lactation, which involves time series models, we can takeadvantage of methods of forward validation as a means of fitting an entire

points (i.e y1, y2, , yt
1) with the testing set containing the data point from

observation at time t, in a strategy otherwise the same as cross model ation This is an ideal method for fitting a model to time course data as inlactation curve fitting (Diggle et al., 1994)

valid-Implementation of these model selection/validation processes for complexmodels is also an issue Though several strategies are available (e.g maximumlikelihood), a growing number of investigators are turning to the bootstrap

methods first developed by Efron (1979) This technique has grown in larity because of its simplicity in application as well as its theoretically appealingproperties In essence, the technique permits evaluation of uncertainty and bias

popu-in parameter estimation, free of any distribution assumptions For clarity popu-in thispresentation, model validation was discussed as separate and distinct fromestimation of parameter uncertainty via the bootstrap However, these pro-cesses can all be combined to allow simultaneous evaluation of models, estima-tion of unknown parameters, the validation of models and the estimation ofuncertainty and bias in parameter estimation, all through intensive computertechniques (Hjorth, 1994)

Example of the bootstrap technique

To demonstrate, on a simpler scale, the techniques that comprise the strap, Table 21.3 contains a sample data set of eight observations For thissimple example, the goal is to use linear regression to predict total milkproduction from initial body weight and udder cells Obviously the analysis ofeight observations with linear regression does not require a bootstrap, but theprocess can be easily illustrated

boot-Using an eight-sided die, bootstrap samples 1 to 4 (also in Table 21.3) werecreated by 32 successive rolls of the die The first eight results decided whichobservations of the original data set would appear in bootstrap sample 1 Thesecond eight rolls of the die selected the observations for bootstrap sample 2,and so on Thus, for example, bootstrap sample number 1 is based on threesamples of observation 5 (of the original sample) and none of observation 1.Table 21.4 presents the results of fitting the data of Table 21.3 in amultiple regression model, presenting first the results provided by least squaresfrom the complete set of eight observations In addition, the estimated interceptand slopes for the four bootstrap samples are also presented in Table 21.4

Table 21.3 Bootstrap example data set, including the original sample data set

of eight data points, followed by four randomly selected sample data sets

Of course the purpose behind this example is actually found in the finalcolumns of Table 21.4, which are the mean and standard deviation of thefour bootstrap samples Clearly, there is a wide discrepancy between the meanbootstrap estimate of the unknown parameters and the estimates provided byordinary least squares However, in a limited fashion, Tables 21.3 and 21.4illustrate the simplicity of the bootstrap process, a simplicity that will becomemore apparent given the complexity of non-linear models considered in thischapter Finally, Table 21.5 provides estimates of these simple regressionparameters from a sample of 10,000 bootstrap data sets As expected, theresults more closely approximate those of the full data set, though certainly arenot identical.

Biological Validation: What is Genetic Potential?

In one sense, the concept of genetic potential is contrary to the foundations ofthe mechanistic models considered in this chapter ‘Genetic potential’, even asused by quantitative geneticists, is treated as an amalgam of effects, represent-ing theorized but not identified genetic mechanisms That is, we know thatgenes are active in the expression of nearly all physiological functions andproduction traits, but the exact genes that turn ‘on’ and ‘off’, and the quantity

of their products, are only just being identified through the advancing gies of molecular biology Accordingly, the genetic merit or potential of ananimal is often treated as a residual, the value an animal brings to its phenotype

technolo-Table 21.4 Parameter estimates from the full data set, along with the parameter

estimates from each bootstrap sample, along with the mean and standard deviation of thefour bootstrap estimates

Combined bootstrapsamples

BW, body weight; UCELLS, arbitrary number representing secretory cell number

Table 21.5 Parameter estimates based on the meanand standard deviation of 10,000 bootstrap estimates

after other terms (e.g diet, gender and age) are accounted for, with no care orconcern for the individual genetic factors that contribute to the final value Inother words, genetic potential, as treated by animal breeders, is a ‘black box’.Nevertheless, the theory of quantitative genetics does stipulate howgenetic potential must behave, at least statistically In fact the foundations ofquantitative genetics are built upon these well-known, quantifiable results(Lynch and Walsh, 1998) One such measure of genetic potential is heritability,the fraction of phenotypic variance that can be traced to variation in additivegenetic contributions The milk production traits evaluated in this model havewell-established estimates of heritability (Ensminger, 2002) These estimatesserve as a guide for how variable we can expect genetic potential to behave in apopulation of cows if it is to accommodate our thoughts of ‘genetic potential’.Similarly, the very nature of quantitative genetics demands that the phenotypes

of relatives share a similarity not found when contrasting two independentlysampled unrelated individuals Thus, we would expect our estimates of geneticpotential to be correlated among relatives, and equally as important, that themagnitude of this correlation has boundaries that can be identified by ourknowledge of the quantitative genetics of production traits

To differentiate simulations of data from different cows, genetic differencesmust be expressed in the model Genetic potential to produce milk is the variableUCELLS, which is an arbitrary number representing secretory cell number(or DNA/udder) that differentiates udder enzyme synthetic capacity or uddermetabolic capacity between cows (UENZ) Udder enzyme synthetic capacitydetermines daily milk yield depending on diet and intake Milk synthesis equa-

UCELLS and Lhor (lactation hormone) have the greatest effect on uddersynthetic capacity (Usyn) UCELLS acts as a constant multiplier to increaseUsyn for the whole udder It is an arbitrary value that remains constantthroughout a simulation and can be used to distinguish difference in the geneticpotential of a cow to produce milk Lhor causes a sequential decrease Usyn andresults in UENZ following the classic shape of the lactation curve Udegrepresents the effect of retained milk on udder enzymes based on a 21 daysaverage of retained milk in the udder and UENZ Therefore an increase inUCELLS increases total potential udder enzyme synthetic capacity betweensimulations whereas UENZ changes within a simulation and represents dailychanges in udder enzyme synthetic capacity Figure 21.8 shows how Usyn andUdeg change at two different levels of UCELLS over a lactation

Milk component production such as milk fat, milk protein and milk lactose,which affects total volume of milk, is affected directly through UENZ (Fig 21.9).Since UENZ represents the activity of all enzymes to produce milk fat, proteinand lactose, other parameters are used to account for genetic differencesbetween cows’ ability to produce milk components For instance, other milkproduction parameters within the model that could be altered to represent