Nonlinear Mixed Effects Model for Swine Growth

Transcription

1 Nonliner Mixed Effects Model for Swine Growth A. P. Schinckel nd B. A. Crig Deprtment of Animl Sciences nd Deprtment of Sttistics, Purdue University Introduction Severl nonliner growth functions model weight s function of dys of ge. Typiclly these functions re solved by itertive procedures tht minimize the residul vrince. The residul vlues (the observed minus predicted live weights) re ssumed to be independent with constnt vrince. Growth curves re often fit to seril live weight dt typiclly every dys from 50 lbs. to 250 lbs. live weight. This seril dt usully hs underlying reltionships or correltions mongst the seril live weight observtions. Hevier pigs t birth nd wening usully hve competitive dvntge nd remin hevier thn the remining pigs of the group. Also, the vrition mongst the pigs for live weight increses s ge increses. This typicl result contrdicts the ssumption tht the residul vlues re independent nd re constnt vrince t ech ge. Also, mrketing pigs in specific live weight rnge or ending reserch trils t specific live weight my bis the growth curves. To void discounts for pigs outside the specific pork processor s optiml crcss weight rnges, the fstest growing pigs re mrketed erly nd the slowest pigs re mrketed lter. By including live weight dt fter the fstest gining pigs hve been mrketed, under predicts the true growth rtes especilly fter 200 lbs. live weight (Schinckel nd DeLnge, 1996; Smith et l., 1999). The objective of this study ws to evlute the use of mixed effects version of commonly used live weight growth curve. Mterils nd Methods Swine growth dt ws fit to the Bridge s function, t = C ( 1 exp (- mt ) + eit WTGAIN where WTGAIN t is the expected live weight gin from birth to ge t, C is mture weight, m is the exponentil growth decy constnt, is the kinetic order constnt, nd e it is the residul weight gin for the i th pig t time t. The e it re ssumed to be normlly distributed with men zero nd vrince Vr(e). The mixed effects model is WTGAIN it = ( C + ci )( exp (- mt ) + eit 1 where c i is the rndom effect for the i th pig. The c i s re ssumed to be normlly distributed with men of zero nd vrince Vr(c). This model simply llows the mture weight C to vry mongst the pigs. 2 The vrince in live weight t specific ge is Vr( )( 1- exp( - mt ) Vr( e) c +. The totl vrince t specific ge cn be prtitioned into the mount cused by the vrition between pigs ( c) ( 1 exp( mt ) 2 Vr or (Vr c/c 2 )(WTGAIN t ) 2 nd the residul vrince (Vr(e), or vrince of ech pig s live weights from their predicted vlues from the mixed model) is ssumed to be constnt. This model ssumes constnt between pig coefficient of vrition nd predicted s [(Vr (c))/c 2 ] 1/2. The covrince of seril live weights t two ges (t = 1 nd t = 2) cn be expressed s = (Vr (c)/c 2 ) WTGAIN 1 * WTGAIN 2. These expressions ccount for the 20 Purdue University

2 incresing vrince in live weight s ge increses nd greter seril correltions t the lter ges (greter live weights) thn the erly ges. Dt Anlysis To evlute the improved fit of the mixed effects model, seril live weight dt from 93 pigs tken t 54, 68, 82, 96, 110, 124, nd 138 dys of ge were used. To demonstrte the mixed model s decresed sensitivity to selective smpling, both the trditionl fixed nd mixed effects model were fit to dt set in which the heviest 15% were removed fter 110 dys of ge nd the next heviest pigs removed t 124 dys of ge. The dt ws nlyzed using NLINMIX procedures of SAS. Since the exponentil growth decy prmeter ws close to zero, the model ws re-prmeterized C( 1 exp( exp( ) t )) + e where (m' = log(m)) to help with the convergence of the solution of the nonliner regression prmeters nd pproximte stndrd errors. Results Fixed Effects Model The prmeters estimtes re summrized in Tble 1. To demonstrte the violtion of ssumptions, Tble 2 summrizes the covrinces nd correltions (observed minus predicted vlues) mong the residuls t different ges. The residul vrince increses over time (min digonl) nd the covrinces mong the sequentil residuls increse s the ge of the pig increses. The correltions hve distinct pttern. The correltions between residuls decrese s the time spn between the observtions increse. The correltions between subsequent observtions (14 d prt or 28 d prt) re much lrger t the lter ges thn erly ges. This is due to the fct tht the prt-whole reltionships between biweekly weights increse s live weight increses. Mixed Effects Model The between pig vribility is modeled by vrying the mture weight. This builds the observed incresing vrince nd covrince reltionships with ge directly into the model. The mixed model prmeter estimtes re lso presented in Tble 1. The estimtes do not chnge due to the fct tht the men of the mixed effects model is the sme s the fixed effects nd ll pigs were observed t ech ge. The overll vrition hs prtitioned into the within pig vrition (i.e., VAR e) nd the between pig vrition (i.e., VAR c). The pproximte stndrd error for ech prmeter hs been significntly reduced, lmost hlved. This improvement in precision is lso pprent when estimting the growth rte t ech ge. These estimtes nd stndrd errors re presented in Tble 3. The difference in stndrd errors is most noticeble t the older ges s the between pig vrition increses. Selective Smpling Another importnt feture of the mixed effects model is its bility to hndle the potentil bis tht comes with the erly removl of hevier pigs. To demonstrte this, the heviest 15% (14 pigs) is removed fter d 110 nd gin fter d 124 (12 pigs). Both the fixed nd mixed model prmeter estimtes re summrized in Tble 4. The mture weight shifted substntilly downwrd using the fixed model but only slightly in the mixed effects model. In the mixed model, becuse the weights of ech pig vry bout pig-specific growth curve, the selective smpling does not hve lrge effect on the verge popultion curve. Purdue University 21

3 Discussion Accurtely chrcterizing the growth potentil for popultion of pigs is necessry to identify lterntive strtegies to improve the efficiency nd dily nutrients required for swine production. It ws shown for these pig, dt tht simply llowing the mture weight prmeter to vry, provided n dequte explntion for the incresing vrince nd seril correltion, nd thus fit the dt much better thn the fixed effects model. Recently, pork producers hve become concerned with quntifying nd reducing vrition in the growth of pigs (Schinckel et l., 1998; Smith et l., 1999). This version of mixed model produces n incresing vrince, decresing coefficient of vrition, nd incresing seril correltions mongst sequentil live weights s ge increses. These chnges in prmeters re commonly found in pig growth dt (King, 1999; Smith et l., 1999). Stochstic modeling is lso strightforwrd within the mixed effects version since only one dditionl vrible (c i ) needs to be simulted to model the growth of ech pig. With fixed effects models, ny removl of the fst-gining pigs t mrketing or live weight in reserch trils bises the growth curves. In the pst, one option hs been to only include dt prior to the mrketing of ny nimls (Smith et l., 1999). Producers could purposely not remove pigs, however tht could crete greter effective stocking density thn their norml sequentil mrketing procedures. The use of the live weight dt only up to men weight of 240 lbs. resulted in lrge bootstrp stndrd errors for the predicted growth rtes from lbs. live weight (Schinckel et l., 1998). The use of mixed effects model llows the mrketing of nimls sequentilly t their idel mrket weight rnge, s normlly would be done, nd include ll the live weight dt without severely bising the predicted growth curve prmeters. It my be importnt to note tht the mixed model uses mximum likelihood solution versus some nonliner solutions tht utilized minimum lest-squres. The prmeters C, Vr(c), nd VAR e re the set of vlues which best describe the reltive mgnitude of the between pig nd within pig vrition. If the correltions between the seril live weights re high, indicting tht the heviest pigs sty hevier nd light pigs lighter, then the rtio of [Vr(c i )/(C 2 ] to Vr e will be high. This lso implies tht the use of the individul c i produces growth curve with only smll devitions from the pig s individul growth curve. The lrger the bsolute nd reltive mgnitude of the Vr e, the lower the correltions mongst the seril live weights nd greter the devitions of ech pig s live weights from it s individul curve using the c i for tht specific pig. These sttistics my be used to prmeterize both the mount nd type of between pig nd within pig vrition tht produces the totl mount of vrition in live weight t ny specific ge. Implictions Animl growth models hve been developed with gol of optimizing production systems. These models require prmeteriztion of niml growth. Nonliner mixed effects models llow more precise evlution of niml growth functions thn the trditionl fixed effects models. Mixed effects models cn lso reduce the impct of potentil bises of selective smpling nd provide n dditionl prmeter tht describes niml to niml vrition. 22 Purdue University

4 References King, R. H A Review - Nutritionl constrints to pig performnce nd pig vribility. Mnipulting Pig Production VII. Agriculture Victori, Victorin Institute of Animl Science, Privte Bg 7, Werribee, Vic pp Schinckel, A. P. nd C. F. M. delnge Chrcteriztion of growth prmeters needed s inputs for pig growth models. J. Anim. Sci. 74: Schinckel, A. P., J. W. Smith, M. D. Tokch, M. E. Einstein, J. L. Nielsen, nd R. D. Goodbnd Smpling errors ssocited with protein ccretion nd mino cids requirements predicted from seril live weight nd rel-time ultrsound mesurements Purdue Swine Dy. West Lfyette, IN pp Smith, J. W., M. D. Tokch, A. P. Schinckel, S. S. Dritz, M. Einstein, J. L. Nelssen, R. D. Goodbnd Developing frm-specific lysine requirements using ccretion curves: Dt collection procedures nd techniques. Swine Helth Prod. 7(6): Tble 1. Prmeter Estimtes nd Approximte Stndrd Errors for Bridges Model bsed on seril growth dt (54, 68, 82, 96, 110, 124, nd 136 d) from 93 pigs Prmeter Estimte Approximte std. error Approximte 95% confidence limits The Fixed Effects Model C σ The Mixed Effects Model C Vr e Vr c The fixed effects version is live weight gin (kg) from birth equls C( 1 exp( exp( ) t )) + e, where e is Norml with vrince σ 2. The mixed effects model is ( C + di )( 1 exp( exp( ) t )) + e, where e is Norml with vrince Vr (e) nd c i ; the i th pig s rndom effect, is Norml with vrince (Vr c) Purdue University 23

5 Tble 2. Vrinces, covrinces nd correltions of the residuls (lb 2 fixed model, N=93) Age Cov r Cov r Cov r Cov r Cov r Cov r Cov r Vrinces on the digonl, covrinces (COV) nd correltions (r) on the off digonl Tble 3. Rte of growth ADG (lb/d), estimtes nd stndrd errors Age Estimte Fixed Std Err Mixed Std Err Estimte bsed on fixed effects model 24 Purdue University

6 Tble 4. Prmeter estimtes under seril mrketing Prmeter Estimte Fixed effects Mixed effects C Vr (e) Vr (c) The fixed effects version is live weight gin (lb) from birth equls C( 1 exp( exp( ) t )) + e, where e is Norml with vrince Vr (e). The mixed effects model is ( C + ci )(1 exp( exp( ) t )) + e, where e is Norml with vrince Vr (e) nd c i the i th pig s rndom effect, is Norml with vrince Vr (c) Purdue University 25