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Estimation of Milk Production in LactatingSows by Determination of Deuterated WaterTurnover in Three Piglets per LitterPeter Kappel Theil , Thomas Thymann Nielsen , Niels Bastian Kristensen ,Rodrigo Labouriau , Viggo Danielsen , Charlotte Lauridsen & Kirsten JakobsenPublished online: 05 Nov 2010.

To cite this article: Peter Kappel Theil , Thomas Thymann Nielsen , Niels Bastian Kristensen , RodrigoLabouriau , Viggo Danielsen , Charlotte Lauridsen & Kirsten Jakobsen (2002) Estimation of Milk Production inLactating Sows by Determination of Deuterated Water Turnover in Three Piglets per Litter, Acta AgriculturaeScandinavica, Section A — Animal Science, 52:4, 221-232, DOI: 10.1080/090647002762381104

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Estimation of Milk Production inLactating Sows by Determination ofDeuterated Water Turnover in ThreePiglets per Litter

Theil, P. K., Nielsen, T. T., Kristensen, N. B., Labouriau, R.,Danielsen, V., Lauridsen, C. and Jakobsen, K. (Departments of 1AnimalNutrition and Physiology, and 2Agricultural Systems, Biometry Re-search Unit Danish Institute of Agricultural Sciences, P.O. Box 50,DK-8830 Tjele, Denmark). Estimation of milk production in lactatingsows by determination of deuterated water turnover in three piglets perlitter. Accepted August 1, 2002. Acta Agric. Scand., Sect. A, AnimalSci. 52: 221–232, 2002. © 2002 Taylor & Francis.

The milk intake in piglets from four lactating sows was measured inthree 48 h periods, starting on days 3, 10 and 17 of lactation, usingthe deuterium oxide (D2O) dilution technique. The milk intake wasestimated from the water turnover in the piglets corrected for theproduction of metabolic water. The water turnover was estimatedfrom the water dilution space and the fractional turnover of bodywater, determined by the rate of D2O dilution. To determine themagnitude of isotopic recycling, a randomly selected piglet in eachlitter was not enriched with D2O. Milk production of the four lactat-ing sows was also measured by the weigh–suckle–weigh (WSW)method on days 4, 11 and 18 of lactation. The average daily milkproduction determined by the D2O dilution technique (9.9190.69 kg)was higher (PB0.05) than that determined by the uncorrected WSWmethod (8.6590.81 kg). Different procedures for selecting pigletswere evaluated to obtain the best possible estimation of total milkproduction from three piglets in each litter. The total milk productionof a sow based on three piglets in the litter was obtained from aquadratic model of the relationship between milk intake and dailygain. When the daily gain of all piglets were known the Pearsoncorrelation between milk production determined for the whole litter(n¼1) and only three median piglets was 0.98.

Peter Kappel Theil1,*, ThomasThymann Nielsen1, NielsBastian Kristensen1,Rodrigo Labouriau2,Viggo Danielsen1,Charlotte Lauridsen1 andKirsten Jakobsen1

Departments of 1Animal Nutritionand Physiology, and 2AgriculturalSystems, Biometry Research UnitDanish Institute of AgriculturalSciences, P.O. Box 50, DK-8830 Tjele,Denmark

Key words: data imputation,deuterium oxide, isotope recycling,milk composition, total body water,water turnover, weigh–suckle–weigh.

Introduction

Traditionally, weighing the litter before and aftereach of a number of consecutive suckling bouts hasbeen applied to measure the milk production of sows.

However, different protocols for the weigh–suckle–weigh (WSW) method have been used (Den Hartog etal., 1983). The method has the advantages of easeand low costs, but there may be drawbacks in termsof lack of accuracy and precision (Den Hartog et al.,1983; Pettigrew et al., 1985). This may partly be dueto evaporation as well as losses of urine, faeces and* Corresponding author. E-mail: Peter.Theil@agrsci.dk

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saliva during the suckling period (Klaver et al., 1981).In addition, because the piglets is isolated from thesow between suckling bouts, an arti� cial sucklingfrequency is imposed, which may impose a stresssituation and consequently affect the milk productiondetermined by the WSW method. As an alternative tothe WSW method, several experiments have con-cerned the measurement of milk production withisotope dilution techniques, using either tritiated wa-ter or deuterium oxide (D2O) (e.g. Dove, 1988; Glen-cross et al., 1997; Pluske et al., 1997). Higher milkproduction (in the range of 12–18%) was found forthe D2O dilution technique than for the WSWmethod (Pettigrew et al., 1985; Prawirodigdo et al.,1990a). Using D2O dilution in all littermates to deter-mine milk production is costly owing to labour andcosts associated with isotopes, analyses and process-ing. The aims of the present study were: (i) to com-pare milk production measured by the WSW methodand the D2O dilution technique, and (ii) to evaluatethe accuracy and precision of milk production esti-mated by D2O dilution in only a few piglets per litter.

Materials and methods

Animals and feeding

Four multiparous sows (Danish Landrace½York-shire) were randomly selected from the herd at theDanish Institute of Agricultural Sciences. Sows withpiglets were kept individually in farrowing pens. Aheating lamp was temporarily installed in the pen tokeep the piglets warm. Sows were fed twice daily witha standard diet based on barley, wheat and soyabeanmeal, containing 9.34 MJ net energy (NE) kg¼ 1 drymatter (DM) and 197, 9.9 and 2.8 g kg¼ 1 DM ofcrude protein, lysine and methionine, respectively.The feeding level for the lactating sows was adjustedrelative to the litter size according to Danish recom-mendations (Danielsen, 1988), and was graduallyraised until day 10 of lactation, reaching the level of15.44 MJ NE »additionally 3.86 MJ NE for eachsuckling piglet per litter. The sows had access towater ad libitum. The sows were housed in pens witha partly slatted � oor, and sows and piglets wereprovided with moderate quantities of straw bedding.In the pens there was a special set-up for obtaining ahigh accuracy and a high precision of the WSWmethod. The set-up included a grate that could belifted and lowered from the outside of the pen. Thegrate allowed separation and reunion of sow andpiglets when appropriate. The set-up allowed thepiglets to be gently pushed onto a scale with aminimum of stress. Milk production was determinedin weeks 1, 2 and 3 of lactation by means of D2Odilution and WSW. Milk samples were collected after

intravenous injection (10 IU) of oxytocin (Leo Phar-maceutical Products, Ballerup, Denmark) and equalamounts of milk were obtained from each teat andpooled by sow and week. To assess the chemicalcomposition of the sow’s milk during the lactationperiod, milk samples were taken after the sequencesof WSW measurements had been completed. Milkchemical composition is necessary for converting wa-ter intake to milk intake, when using the D2O dilu-tion technique. The experiments were approved bythe Danish Experimental Inspectorate, Copenhagen,Denmark.

Weigh–suckle–weigh technique

The procedure for WSW involved isolation of thepiglets 70 min before the � rst suckling. Before re-union, piglets were weighed to the nearest 0.01 kg ona digital scale, on a whole-litter basis, after whichthey were allowed to nurse. Having completed nurs-ing the piglets were again isolated from the sow andthe litter was weighed again. The sequence of isola-tion followed by weight measurements before andafter nursing periods was repeated in six consecutivesuckling bouts, each cycle lasting for 70 min. If anattempt to suckle was unsuccessful for a maximum of15 min, the piglets were reisolated and a new attemptwas made 70 min later. Measurement of milk produc-tion using the WSW method took place on days 4, 11and 18 of lactation. The WSW method used in thepresent experiment was the standard procedure ap-plied at the Danish Institute of Agricultural Sciences.

Deuterium oxide dilution technique

Blood samples were obtained from the jugular veinusing hypodermic needles (1.2½40 mm) and 5 mlvacutainers for serum collection (Hemograd Z; Bec-ton Dickinson, Meyland, France). Piglets were fastedfor 2 h before the � rst blood sample was drawn inorder to determine the background level of D2O.Time and piglet live weight (LW) were registeredwhen blood samples were collected. D2O (99.9%) wasdiluted in an isotonic sodium chloride solution (9 gNaCl l¼ 1) to form either a 10% or a 20% solution. Inthe � rst week of lactation, the 10% solution was usedand injected at 1 g kg¼ 1 LW. To decrease theamount of D2O solution injected in the second andthird weeks, the 20% solution was used and injectedat 0.5 g kg¼ 1 LW. The piglets were enriched on days3, 10 and 17 of lactation by an intramuscular D2Oinjection in the neck. After D2O injection the pigletswere kept isolated from the sow for another hour, toallow D2O to equilibrate (Rudolph et al., 1988). Asecond blood sample was taken after equilibration,allowing determination of the total body water on theday of enrichment. After a 48 h experimental period,

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piglets were again isolated from the sow for a 2 hperiod and a third blood sample was collected. Theprocedure allowed determination of total waterturnover in three 48 h periods, i.e. on days 3–5,10–12 and 17–19. All piglets except one in each litterwere injected with D2O. The non-enriched animal ineach litter was chosen randomly on day 3 and used todetermine isotopic recycling throughout the experi-ment. In terms of blood sampling, non-enrichedpiglets followed the same schedule as enriched litter-mates. Vacutainers with blood were kept overnight at5°C, after which they were centrifuged (1558½g, 25min, 5°C). When estimating the sow’s milk produc-tion the control piglet in each litter was regarded as amissing observation and this was corrected for byusing a statistical imputation technique (Dempster etal., 1977) described in the Statistics section.

Chemical analyses

Milk samples were analysed for DM, ash, fat, lactoseand total nitrogen (N). DM and ash were determinedaccording to AOAC (1990). Protein (N½6.38) wasmeasured by the Kjeldahl method using a Kjell-foss16200 autoanalyser (Foss Electric, Hillerød, Den-mark). Fat was determined after hydrolysis with hy-drochloric acid and extraction with diethyl ether(Stoldt, 1952). Lactose was determined using a spec-trophotometric method, in which lactose was hy-drolysed with b-galactosidase, yielding glucose andgalactose (Beutler, 1984). Galactose was determinedafter enzymic reaction with b-galactose dehydroge-nase by determination of NADH. The atomic frac-tion (AF) of deuterium in water hydrogen wasmeasured in serum ultra� ltrate obtained using Cen-trisart-C4 microcentrifuge � lters (5 kDa; SartoriusAG, Gottingen, Germany), centrifuged at 6000½g at20°C for 2 h. One microlitre of ultra� ltrate wasreduced to free hydrogen in a chromium reactoroperated at 900°C (H:Device; Finnigan MAT, Bre-men, Germany). The AF of deuterium in the liber-ated hydrogen was measured by isotope ratio massspectrometry (Delta S; Finnigan MAT, Bremen, Ger-many). A pure hydrogen gas (99.9999%) was used asreference gas (Iso-Top 0‰; Messer Griesheim,Krefeld, Germany). For further details see Brand etal. (1996).

Calculations

When using the WSW method the daily milk produc-tion was calculated as average milk intake by thelitter in all suckling bouts, multiplied by 24 h day¼ 1

and by 60 min h¼ 1 and divided by 70 min persuckling. The average milk intake by the litter wasbased only on successful suckling bouts. Milk pro-duction determined by the WSW was corrected for

metabolic, salivary and evaporative losses accordingto Klaver et al. (1981) and Den Hartog et al. (1983).Klaver et al. (1981) suggested that weight loss duringsuckling due to activity (grams per animal perminute) could be estimated by the equation 0.072½age (days) »2.2, whereas Den Hartog et al. (1983)suggested the equation 0.019½age (days) »0.443.Correction of milk production according to Klaver etal. (1981) and to Den Hartog et al. (1983) wasincluded in the present experiment for evaluating themilk production assessed by different methods. How-ever, correction of milk production for additionallosses of weight due to defecation and urination werenot performed.

When using the D2O dilution technique to measurethe sow’s milk production, the milk intake of individ-ual littermates was summed. The milk intake of apiglet was calculated as the sum of the water turnoverand the potential metabolic water stored (PMWS),divided by the potential water fraction of the milk(Equation (1)):

Milk intake in piglet (g day¼ 1)¾

Water turnover»PMWSPotential water fraction of milk

(1)

The piglets were enriched with isotopes every week.Re-enrichment of piglets in weeks 2 and 3 was donebefore the natural background level of D2O wasreached. A detailed description of the modi� ed calcu-lations leading to Eqn. (1) is shown in Eqns (2)–(9).

The dilution space of individual piglets (expressedas g of body water) was determined at the beginningof each 48 h experimental period. Dilution space (Q1)was calculated according to Eqn. (2). Dividing theinitial water dilution space by the LW (in g) yieldedthe body water fraction.

Q1 ¾minj(AFinj ¼AF0)Mdist H2O

Minj(AF1 ¼AF0)1.031(2)

where Q1 is the initial dilution space (g), minj is themass of injectate (inj) in g, AFinj is the atomic frac-tion of D2O in the injectate, AF0 is the atomicfraction of D2O in serum before the actual injectionof D2O, AF1 is the atomic fraction of D2O in serumafter equilibration (1 h after injection), Mdist H2O isthe molecular weight of distilled water (18.01499 gmol¼ 1), Minj is the molecular weight of the injectate,and 1.031 is the correction factor used to correct forthe non-water dilution space of deuterium (Haggartyet al., 1994).

The enrichment of the injectate was too high to bemeasured with the same instrument settings as usedwhen measuring the samples. Measuring the atomicfraction in diluted injectate (AFdil inj) and solving

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Equation (3) for AFinj and ninj iteratively, were per-formed to determine the atomic fraction and theapparent molecular weight of the injectate. Prepara-tion and measurements of the atomic fraction of thediluted injectate (AFdil inj) and the atomic fractionof the distilled water used to dilute the injectate(AFdist H2O) were performed as described for serumsamples.

AFdil inj ½ndil inj ¾AFinj ½ninj »AFdist H2O ½ndist H2O

(3)

where n is the amount of water (in moles) and AF isthe atomic fraction in the respective solutions.

The calculations of ndist H2O, ninj and ndil inj, areshown in Equations (4)–(6), respectively:

ndist H2O ¾ (mdil inj ¼minj):Mdist H2O (4)

where ndist H2O is the amount of distilled water in thediluted injectate (moles), mdil inj is the total mass ofinjectate and distilled water used for dilution (g), andminj is the mass of the diluted injectate (g);

ninj ¾0.991minj

AFinj ½MD2O »(1¼AFinj)½MD2O free H2O

(5)

where ninj is the number of moles of D2O and distilledH2O in the injectate, 0.991 is the amount of water inthe isotonic saline used, MD2O is the molecular weightof D2O (20.02748 g mol¼ 1), and MD2O free H2O is themolecular weight of distilled H2O corrected for natu-ral background level of D2O (18.01499 g mol¼ 1).

For analysing the isotopic content of the injectatethe following overall mass balance was set up (Eqn.(6)):

ndil inj ¾ninj »ndist H2O (6)

The actual body water fraction was calculated bydividing the water dilution space (in g) by the pigletlive weight (LW1). Assuming a constant body waterfraction over the 48 h experimental period, the � nalquantity of body water (Q48) was calculated by multi-plying � nal live weight (LW48) by the body waterfraction. The turnover of water was calculated usingEqn. (7), which was a modi� cation of the equationreported by Coward et al. (1982). AF0 was substi-tuted with AFb to account for re-enrichment of theanimals. Fifty per cent of the total water loss wasassumed to be fractionated. The correction factor forwater fractionation (0.9235) was obtained from Hag-garty et al. (1988):

Water turnover¾

Q1 ¼Q48

logQ1

Q48

½ logAF1 ¼AFb

AF48 ¼AFb

2½Dt

»

Q1 ¼Q48

logQ1

Q48

½ logAF1 ¼AFb

AF48 ¼AFb

2½Dt½0.9235(7)

where water turnover is in g day¼ 1, AFb is theatomic fraction in serum before the � rst injection,AF1 and AF48 are the atomic fraction in serum 1 hand 48 h postinjection, respectively, Q1 and Q48 arethe total body water 1 h and 48 h postinjection (g),and Dt is the time difference (in days) between thetwo blood samples. The number 2 in the denominatorshows that 50% of the water loss was assumed to befractionated. The factor 0.9235 was used to correctfor fractionation (Haggarty et al., 1988).

The digested organic compounds were either de-posited or oxidized. Oxidized DM yielded metabolicwater, whereas retained DM also stored potentialmetabolic water. The DM gain of each piglet wascalculated as the LW gain minus the increase in bodywater dilution space. The DM gain was assumed tocontain 48% fat, 48% protein and 4% ash (De Passile& Hartsock 1979). The assumption of equal retentionof fat and protein was in accordance with Noblet &Etienne (1987). Complete oxidation of 1 g of fat,protein and lactose yielded 1.07, 0.42 and 0.58 g ofwater, respectively (De Passile & Hartsock, 1979).Adding the fractions of retained fat and proteinyielded an overall factor of 0.71 g water g¼ 1 DM.The factor re� ected the amount of water that wouldbe liberated if 1 g of piglet DM gain was oxidized.Multiplying the factor by DM gain yielded theamount of potential metabolic water (PMWS), whichwas stored as fat or protein in the experimentalperiod (Eqn. (8)):

PMWS¾ ((LW48 ¼LW1)¼ (Q48 ¼Q1))½0.71c:Dt(8)

where PMWS is the potential metabolic water storedas fat or protein (g day¼ 1), and c is the overall factorfor converting DM gain to potential metabolic waterstored (0.71¾0.42½0.48 »1.07½0.48) (De Passile& Hartsock, 1979).

Equivalent to the calculation of PMWS, potentialmetabolic water content of milk was calculated usingthe same coef� cients for oxidation of fat, protein andlactose, corrected for the respective apparent digesti-bilities of protein, fat and lactose:

Potential water fraction of milk

¾Water fraction»((Crude protein½0.42½ADp)

»(Fat½1.07½ADf)»(Lactose½0.58½ADl)

*DM fraction:100) (9)

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where crude protein, fat and lactose are in percentageof DM, and 0.42, 1.07 and 0.58 (g) are the amountsof water liberated from oxidation of 1 g of protein,fat and lactose, respectively. The apparent digestibil-ity of crude protein (ADp) of 88% was obtained fromMavromichalis et al. (2001), and the apparent digesti-bility of fat (ADf) of 98% was obtained from Braude& Newport (1973). The apparent digestibility of lac-tose (ADl) was assumed to be 100%.

Statistics

The experiment was carried out to represent sow’smilk production on days 4, 11 and 18 of lactation,the same days as the milk production was measuredusing WSW. The milk intake of the non-enrichedcontrol piglet was estimated by an adapted version ofthe imputation technique proposed in a more generalcontext by Dempster et al. (1977), as follows. Aquadratic normal mixed covariance analysis model,in which the piglet weight gain and the effect of theperiod of observation were the explanatory variables,was used to produce a prediction of the milk intake.The mixed covariance analysis model also containedtwo random components: one related to the litter andone representing the piglet. In this way the naturalcorrelation structure of the data was taken into ac-count. The missing values of the milk intake (i.e. milkintake in the non-enriched piglets) were then replacedby the predicted values based on the covarianceanalysis model and the respective weight gains. Anew correlation analysis model was � tted using thedata with the estimated missing observations and theobservations for which the milk intake was available.The observations that were originally missing werethen replaced by the predicted values obtained fromthe new covariance analysis model. The estimationprocedure continued in the same way: alternatingadjustments of the covariance model and substitu-tions of the originally missing observations by therespective predicted values based on their weightgains. This process iterates until convergence, in thesense that no more signi� cant changes in the parame-ters and the maximums of the likelihood functionwere observed. This algorithm was implemented inSAS (Version 8e) as a macro that evoked the proce-dure MIXED (as described in the Appendix). Sum-ming the observed and the estimated milk intake ofthe piglets estimated the milk production for a sow ina given period.

The normal mixed covariance analysis model ap-plied for estimating milk intake in non-enrichedpiglets was as follows. Denote the milk intake and theweight gain of the k th piglet in the jth litter in the ithperiod by Yijk and Gijk, respectively. According to themodel, for all k, j and i,

Yijk ¾m»ai »bGijk »g(Gijk )2 »yj »tjk »oijk (10)

where m is the overall mean, ai is the � xed effect ofthe period (week), b and g are the regression parame-ters, Gijk is the individual gain (in kg day¼ 1), yj andtjk are normal-distributed random components andoijk are normal-distributed residuals. The two randomcomponents and the residuals are independent andtheir expectations are zero. Standard techniques formodel control were applied to this mixed covarianceanalysis model. The presence of interactions betweenthe gain effect and the period was investigated butfound to be not statistically signi� cant.

The conversion ratio, de� ned as the ratio of themilk intake and the mean weight gain, was estimatedthrough the basic mixed model used to predict themilk intake from the weight gain (Eqn. (10)). Moreprecisely, the milk intake at the mean weight gain inthe ith period (i¾1, 2 and 3), Mi, was then estimatedby

M*i ¾m »ai »bGi »g(Gi )2 (11)

where Gi is the mean weight gain in the ith periodand m, ai, b and g are estimates of the parameters inEqn. (10). The conversion ratio (E*i ) was estimatedby

E*i ¾M*i :Gi (12)

The standard deviation of the estimates of theconversion ratio were calculated using the asymptoticvariances and the covariances for the estimates of theparameters m, ai, b and g.

For evaluating the precision of the D2O dilutionmethod when only three piglets were enriched, thesame data were employed. In this case piglets withineach litter were ranked according to LW. Two ap-proaches for systematic selection of piglets and arandomized selection procedure (the Monte Carlostudy) were evaluated. The � rst approach includedthree median piglets in the ranking order in eachlitter, and the second approach included the smallest,the median and the largest piglet. In case the controlpiglet (which was not enriched with D2O) was in-cluded in the selected piglets, the next piglet in theranking order was selected. The estimation procedurebased on the imputation technique described abovewas applied to estimate the milk intake when the milkintake of non-selected piglets was regarded as a miss-ing observation for which only the gain in weight wasknown.

A Monte Carlo (Davison & Hinkley, 1997) simula-tion study was performed by sampling 2000 combina-tions of milk intake and piglet gain based on threedifferent piglets per litter, among the possible combi-nations. The samples were taken randomly with repo-sition by attributing the same probability of selection

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Table 1. Data on sucking piglets in weeks 1, 2 and 3 of lactation

Day in lactation

3 10 17

Number of piglets 51 43 4312.890.8 10.890.8Litter size 10.890.81713 992c 31139101bLive weight (g) 46209101a

Water dilution space (g H2O) 1293 968c 2185974b 3210974a

0.768 90.003a 0.70690.003b 0.69290.003cBody water fraction166 910b 211 911aLive weight gain (g piglet¼1 day¼1)1 204911a

Milk intake (g piglet¼1 day¼1)1,2 644 933b 994 937a 1054937a

3.7890.11c 4.5890.09bConversion ratio (g milk g gain¼1)1,2,3 4.8990.10a

Data are LSmeans9SEM.a,b,c Superscript letters a row indicate a signi� cant difference between days in lactation. All parameters exceptfor number of piglets and litter size were based on enriched (n¼1) piglets.1 Live weight gain, milk intake and conversion ratio were determined for 48 h periods, i.e. at days 3–5, 10–12 and17–19 for the three columns, respectively.2 Milk intake measured by the D2O dilution technique.3 Conversion ratio was de� ned as grams of milk required for each gram of piglet gain. Estimate and asymptoticSEM of conversion ratio were calculated using the mean rate of gain in each period and the covariances of theparameters m, ai, b and g (from Eqn. (12)).

for each piglet in the litter. The Monte Carlo simula-tion allowed evaluation of the variance of the wholeprocedure for estimating the milk production.

Unless otherwise stated, data are reported asmeans 9SEM. Differences in animal performanceand chemical composition between weeks were testedusing the MEANS procedure in SAS. A signi� cantdifference was declared at PB0.05. A paired two-tailed t-test was conducted to compare milk produc-tion determined by the D2O dilution technique andall other measurements of milk production. The Pear-son correlations were investigated between litter gainand milk production determined by the WSWmethod and the D2O dilution technique or derived bydifferent approaches.

Results

On days 3 and 17 after farrowing, litter sizes of thefour sows amounted 12, 14, 14 and 11, and 10, 13, 11and 9, respectively, with mean9SEM as shown inTable 1. Water dilution space (PB0.001) and pigletweight (PB0.001) increased from week to week,whereas the body water fraction decreased from 0.768in week 1 to 0.692 in week 3 (PB0.001). The bodywater fraction dropped curve-linearly with increasingLW (Fig. 1). There was an individual effect (ai ) ofpiglets (PB0.01).

The gain in body weight was lower in week 1(PB0.001) than in weeks 2 and 3 (Table 1). A similarpattern was observed for the milk intake, whichincreased from 644 g piglet¼ 1 day¼ 1 in week 1 to 994

and 1054 g piglet¼ 1 day¼ 1 in weeks 2 and 3, respec-tively (PB0.001). The performance of sow no. 608was poor in the third week of lactation. During the48 h period seven littermates had daily LW gains ofapproximately 0 g, and the rest (six piglets) had adaily gain of approximately 200 g. The mean conver-sion ratio (g of milk intake g¼ 1 of gain), based on theD2O dilution technique, increased as lactation pro-gressed. The conversion ratio was estimated to 3.78,4.58 and 4.89 g milk g¼ 1 piglet gain (PB0.001) atdays 3–5, 10–12 and 17–19, respectively (Table 1).

The chemical composition of the milk (Table 2) didnot change with progress of lactation (PB0.05). The

Fig. 1. Body water fraction of piglets plotted against live weight(LW). The body water fraction of each piglet was determined inweeks 1, 2 and 3 of lactation. There was an individual effect (ai ) ofpiglets (PB0.01).

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Table 2. Chemical composition of milk from experi-mental lactating sows

18.690.4DM (g 100 g¼1 milk)29.990.7Protein (6.38½N), (g 100 g¼1 DM)38.691.3Fat (g 100 g¼1 DM)

Lactose (g 100 g¼1 DM) 27.690.9Ash (g 100 g¼1 DM) 4.090.1

0.941 90.001Potential water fraction1

Data are means9SEM.1 Potential water fraction was the water content of themilk corrected for the metabolic water that was pro-duced if all of the digested dry matter (DM) wasoxidized. The digestibility of protein, fat and lactosewas assumed to be 0.88, 0.98 and 1.00, respectively.

performed based on milk intake of three medianpiglets and of three extreme piglets (Table 3) usingthe described statistical imputation technique. Themilk production estimated by three median pigletswas in accordance with the milk production mea-sured by the D2O technique (Table 3, Fig. 3a),whereas milk production estimated by three extremepiglets overestimated the milk production by 2.4%(Table 3, Fig. 3b).

The Pearson correlations were high (0.98–0.99)between all estimates of milk yield found by the D2Odilution technique (Table 4). When all observationswere included, the Pearson correlation between theD2O method and litter gain was 0.82–0.85, but only0.60–0.65 between the WSW method and litter gain.When one observation was omitted (a sow with poorperformance in week 3), the correlations betweenmilk production and litter gain were elevated to 0.90for both methods. The Pearson correlation betweenmilk yield found by D2O dilution in all littermatesand found by the WSW method was 0.86, but it waselevated to 0.91 when one observation was omitted.

Discussion

D2O dilution technique

Theoretically, recycling of D2O between enrichedpiglets and the sow would cause an underestimationof the milk intake (Baverstock & Green, 1975), be-cause the background level of D2O in milk water wasassumed to be constant throughout the measuringperiod. In principle, recycling can occur by two dif-ferent routes, either from piglets to sow by consump-tion of faeces and urine and back to the piglets viathe milk, or from piglet to piglet, for example, viasaliva. For piglets, the increased level of D2O in thecontrol piglets probably originated from saliva on theteats when littermates sucked other teats immediatelyafter cessation of milk ejection. In a similar study, nochange was found in the level of D2O in sow’s milk aslactation progressed (present group, unpubl.). Conse-quently, the increment of isotopic level in the controlpiglets was probably due to exchange of D2O be-tween littermates, and did not involve recyclingthrough the sow’s milk. Therefore, recycling of iso-topes was not taken into consideration in the calcula-tion. When calculating the milk intake in weeks 2and 3 it was necessary to modify the equation pro-posed by Coward et al. (1982) owing to an increasingbackground level of D2O (Eqn. (7)) as a conse-quence of re-enrichment. When calculating the D2Odilution space, the level of D2O immediately before(re-)enrichment was used and when calculating thewater turnover the background level of D2O beforethe � rst enrichment was used. An alternative way

mean potential water fraction of sow’s milk was0.941 90.001.

The milk production determined by the WSWmethod was on average 8.65 kg sow¼ 1 day¼ 1, whichwas lower (PB0.05) than when measured by theD2O dilution technique (9.91 kg sow¼ 1 day¼ 1). Thecalculated deviation of mean WSW (raw) estimatesfrom corresponding mean D2O dilution estimatesamounted to –12.7% on day 4 (P\0.05), ¼21.3%on day 11 (PB0.05) and ¼4.7% on day 18 (P\0.05) (Table 3). Overall, WSW underestimated themilk production found by the D2O dilution techniqueby ¼12.7% (PB0.05). Using correction factors ob-tained from the literature, the evaporative andmetabolic losses can be accounted for using the WSWmethod. Table 3 shows the effect of corrections basedon factors obtained from Den Hartog et al. (1983)and Klaver et al. (1981). The WSW estimates cor-rected for metabolic and evaporative losses werelower than D2O estimates, although not signi� cantlylower (Table 3).

Using the D2O dilution technique, the daily milkintake (in g day¼ 1) in individual piglets (Yijk ) wasrelated to rate of gain (Gijk ) in g day¼ 1 by thefollowing relationship (see Fig. 2):

Yijk ¾m »ai »1.168*Gijk »0.00425* (Gijk )2 »yj »tjk

»oijk (13)

where (m»ai )¾317939 g day¼ 1, 531942 g day¼ 1

and 582939 g day¼ 1 for days 3–5, 10–12 and17–19, respectively, the variance within the litter was1964, the variance between litters was 681 and theresidual variance was 4778.

The D2O level of the control piglet increased frombackground to 2% of the deuterium level of theenriched littermates 48 h postinjection. If correctingfor this, the estimated sow’s milk production was2.890.7% higher in the � rst week of lactation. Esti-mation of daily milk production of the sows was

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228

Tab

le3.

Milk

pro

duct

ion

inso

ws

mea

sure

db

yth

ed

eute

riu

mox

ide

(D2O

)d

iluti

on

tech

niq

ue

(all)

and

wei

gh

–su

ckle

–wei

gh(W

SW

)m

eth

od

(raw

),an

dm

ilkp

rod

uct

ion

(kg

day

¼1)

esti

mat

edby

diff

eren

tap

pro

ach

es

D2O

D2O

-MC

1W

SW

Ran

ge

Raw

DH

3K

L4S

ow

IDD

ayof

lact

atio

nA

ll3

med

ian

23

extr

eme2

3ra

ndo

m2

Std

3.73

8.95

9.64

9.70

0.52

10.4

610

.68

10.6

110

.69

447

147

10.

584.

3710

.93

11.6

112

.03

1112

.56

13.1

913

.09

12.6

814

.10

0.62

5.05

14.8

115

.63

16.4

318

14.3

714

.08

471

14.3

52.

636.

787.

377.

420.

367.

337.

7358

44

7.51

7.38

0.40

584

2.42

6.01

6.46

7.02

118.

469.

028.

778.

250.

4858

43.

068.

589.

3310

.05

1810

.39

11.0

411

.37

10.7

52.

868.

899.

599.

640.

458.

958.

678.

418.

254

608

12.7

60.

503.

649.

3910

.20

10.6

911

12.7

160

812

.78

12.7

97.

0811

.00

11.9

712

.91

0.83

608

9.57

8.95

9.07

9.55

180.

4461

13.

844.

054.

604.

644

6.38

5.44

6.55

6.33

0.40

611

3.83

6.53

7.15

7.53

118.

038.

028.

668.

273.

217.

858.

599.

3261

10.

4110

.04

10.4

89.

4110

.02

18

–7.

179

1.31

7.80

91.

357.

859

1.36

Mea

n4

8.21

91.

117.

969

1.18

8.41

91.

098.

279

1.09

––

8.22

91.

31b

8.86

91.

35b

9.32

91.

36a

–10

.49

91.

09a

10.8

39

1.09

aM

ean

1110

.44

91.

11a

10.7

59

1.18

a

–M

ean

–10

.56

91.

3111

.38

91.

3512

.18

91.

3618

11.0

89

1.11

10.9

09

1.18

11.2

99

1.09

11.1

29

1.09

9.96

90.

68b

––

8.65

90.

81c

9.35

90.

84b

9.78

90.

89b

Ove

rall

9.91

90.

69b

9.87

90.

74b

10.1

79

0.68

aM

ean

Dat

aar

eLS

mea

ns 9

SE

M.

a,b

,cD

iffer

ent

sup

ersc

rip

tle

tter

sin

aro

win

dic

ate

sign

i�ca

nt

dif

fere

nce

s(t

wo-

taile

dp

aire

dt-

test

;P

¾0.

05)

bet

wee

nm

ilkpr

od

ucti

on

mea

sure

db

yth

eD

2O

dilu

tion

tech

niq

ue

and

milk

pro

du

ctio

nfo

und

by

the

WS

Wm

eth

od

or

estim

ated

by

dif

fere

nt

app

roac

hes

.1

Mo

nte

Car

lo:

std

¾st

and

ard

devi

atio

no

fth

e20

00M

on

teC

arlo

sim

ula

tion

s;R

ang

e¾

max

–min

of

the

2000

Mo

nte

Car

losi

mu

lati

on

s;3

ran

dom

¾es

tim

ated

mea

nm

ilkp

rod

uct

ion

of

the

2000

Mo

nte

Car

losi

mu

lati

on

s.2

3m

edia

n,

3ex

trem

ean

d3

rand

om

:se

lect

ion

pro

ced

ure

of

thre

epi

gle

ts,

afte

rp

igle

tsw

ere

ran

ked

acco

rdin

gto

live

wei

ght

.3

Cor

rect

edfo

rev

apo

rati

vean

dm

etab

olic

wei

gh

tlo

ssac

cord

ing

toD

enH

arto

get

al.

(198

3).

4C

orre

cted

for

evap

ora

tive

and

met

abo

licw

eig

ht

loss

acco

rdin

gto

Kla

ver

etal

.(1

981)

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Milk intake in piglets

Fig. 2. Milk intake in piglets estimated by means of the D2Odilution technique plotted against piglet gain for days 3–5 ( ),10–12 ( ) and 17–19 (*) in lactation. The solid lines representsthe � tted covariance model for the respective days in lactation.

water in week 1) indicated that piglets were bornpractically without fat, as also found by Nielsen(1973). The water content decreased to approximately70% in week 3, indicating that part of the retained DMwas fat. The � ndings are in close agreement with thewater content in newborn piglets found by otherresearchers. Rudolph et al. (1988) reported 80.5%water (LW¾1.2 kg) by the D2O dilution technique,and Nielsen (1973) found 79.9% water (LW¾1.3 kg)by desiccation.

Sow’s milk production increased from 8.2 kg day¼ 1

on days 3–5 to 10.4 kg day¼ 1 on days 10–12 and 11.1kg day¼ 1 on days 17–19, which was consistent withother studies performed with modern high-yieldingsows during the 1990s (Mackenzie & Revell, 1998).The increasing milk production as lactation pro-gressed was consistent for all sows except for sow 608,where milk production decreased from 12.7 kg day¼ 1

in week 2 to 9.6 kg day¼ 1 in week 3 (Table 3). Thelitter of sow 608 was inhomogeneous in week 3, witha mean daily gain of 95 g day¼ 1. Overall, the dailygain for the piglets was 166 g day¼ 1 in week 1 (Table1), increasing to 211 g day¼ 1 and 204 g day¼ 1 inweeks 2 and 3, respectively. The average amount ofmilk required for 1 g of LW gain (conversion ratio)increased as lactation progressed. The chemical com-position of milk did not differ signi� cantly amongweeks of lactation and was similar to the compositionreported by Darragh & Moughan (1998). A plausibleexplanation for the increasing conversion ratio wouldbe that the energy requirement for maintenance in-creased with increasing LW of piglets. In week 1, 317g day¼ 1 of milk was required to maintain the LWconstant (Eqn. (13)), whereas 531 g day¼ 1 and 582 gday¼ 1 of milk were required in weeks 2 and 3,respectively. Consequently, as the conversion ratioincreased with the progress of lactation, the ef� ciencyof utilizing the milk for retention dropped.

From the D2O dilution data, milk production wasestimated by means of three piglets in each litter.Selection of the three piglets can be done in severalways, either randomly or systematically. The correla-tion between LW gain and milk intake allows selectionof representative piglets based on LW. Random selec-tion of three piglets (the Monte Carlo simulation) andtwo different approaches for systematic selection ofthree piglets were tested: median or extreme piglets.This study showed that milk intake in three piglets canpredict the sow’s milk production estimated by enrich-ing n¼1 littermates (r¾0.98–0.99). Estimation ofmilk production based on three extreme piglets overes-timated the total milk production (Table 3, Fig. 3b),whereas estimation of milk production based on thethree median piglets was in accordance with themeasured milk production (Table 3, Fig. 3a). A likelyexplanation could be that the described imputation

would have been to analyse the D2O level in the milkevery week.

As the results in Fig. 1 and Table 1 indicate, thebody water fraction decreased markedly during thelactation period as the LW of piglets increased. Thehigh content of water in newborn piglets (77% of body

Fig. 3. Milk production estimated by means of the D2O dilutiontechnique based on the milk intake of (a) three median piglets and(b) three extreme piglets plotted against milk production deter-mined using n¼1 littermates. Each data point represents a sow ina given week.

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Table 4. Pearson correlation coef� cients between litter gain and all combinations of determined and estimatedmilk yield

D2O LitterD2O WSWD2O WSWD2OCorrelation WSWAll 3 median 3 extreme KLcoef� cient (r) DH3 random gainRaw

D2O, all – 0.98 0.99 0.99 0.91 0.91 0.92 0.900.98 – 0.99 0.98 0.89D2O, 3 median 0.89 0.90 0.890.99 0.99 – 0.99D2O, 3 extreme 0.90 0.90 0.92 0.88

D2O, 3 random 0.99 0.98 0.99 – 0.91 0.92 0.93 0.92WSW, raw 0.86 0.83 0.81 0.87 – 1.00 1.00 0.91

0.86 0.82 0.80 0.86WSW, DH 1.00 – 1.00 0.910.86 0.82 0.81 0.86 0.99WSW, KL 1.00 – 0.890.82 0.83 0.85 0.84 0.65 0.64Litter gain 0.60 –

n¾12 in the lower left corner, whereas one observation was omitted in the upper right corner (n¾11).Abbreviations as in Table 3.

technique was sensitive when the runt was included toestimate the milk intake of the whole litter. The studyshowed that the approach with selection of threelittermates did not seem to be sensitive to inhomoge-neous litters. Evaluating the variance components ofthe mixed model showed that variance within the litterwas three times higher than the variance among litters.This indicated that it would be advantageous to enrichmany piglets per litter. Nevertheless, enrichment ofonly three piglets per litter was suf� cient to estimatethe sow’s milk production when the milk intake wasmeasured repeatedly in three consecutive weeks, andthe LW gain was recorded for all piglets. This drasticreduction of the sample was possible because theweight gain carried much information about the milkintake. The covariance analysis model successfullycaptured this information. Note, however, that sincethe reduced data kept observations for the differentlitters and the different observational periods, therewas still enough information to allow a reasonableadjustment of the covariance analysis model.

The Monte Carlo study showed that a randomizedselection procedure was also an applicable approach.The Monte Carlo variance of milk production esti-mate based on the 2000 simulations showed thatthe litter of sow 608 (week 3) was inhomogeneous(Table 3).

Comparison between D2O dilution techniqueand weigh–suckle–weigh method

In this study the milk production found by the WSWmethod was 12.7% lower than that found by the D2Odilution technique (Table 3). This was in accordancewith the data of Pettigrew et al. (1985) andPrawirodigdo et al. (1990a), who found 11.9% and18.2% lower milk production, respectively, when usingthe WSW compared with the D2O dilution technique.Pettigrew et al. (1987) and Prawirodigdo et al. (1990b)validated the D2O dilution technique for measuring

milk intake in piglets, by measuring milk disappear-ance from arti� cial drinkers. Both research groupsreported signi� cant correlations (r¾0.99), with devia-tions between measured and estimated milk intake ofless than 93%. Den Hartog et al. (1983) and Klaveret al. (1981) described the inaccuracy of the WSWmethod owing to loss of weight during suckling, andprovided correction factors that account formetabolic, salivary and evaporative losses. Correctingthe WSW data according to Den Hartog et al. (1983)and Klaver et al. (1981) was a step in the rightdirection. Additional losses of weight due to defeca-tion and urination could be accounted for by adding36.9 g per urination and 22.51 g per defecation (DenHartog et al., 1983). However, as the piglet LW mayaffect the urinary and faecal losses, this was notaccounted for. The frequency with which urinationand defecation occurred was in the range of 40% ofsucking piglets for urination, and 20% of suckingpiglets for defecation (Den Hartog et al., 1983). In thepresent experiment, the weighing procedures includeda special set-up and were carried out by trainedpersonnel, which minimized defecation and urination,although the extent of defecation and urination wasnot registered. The non-signi� cant difference foundhere between the corrected WSW method and the D2Otechnique may be due to the lack of correction fordefecation and urination.

Using all observations, the Pearson correlation co-ef� cients indicated that the D2O dilution techniquewas superior to the WSW method when evaluatingcorrelations with the litter gain. However, omitting theinhomogeneous litter raised the correlation coef� -cients from approximately 0.84 to 0.90 (D2O dilution)and from approximately 0.63 to 0.90 (WSW method).The data suggest that the WSW method might be moreimprecise when the performance of the sow was poor.Correcting the WSW data according to Den Hartog etal. (1983) or Klaver et al. (1981) improved the accu-racy, but did not improve the precision.

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Milk intake in piglets

Conclusions

This study has described a modi� cation of the isotopedilution technique for estimating milk intake inpiglets that negates the need for enrichment of allpiglets within a litter. This will reduce the labour andcosts associated with isotopes, analyses and process-ing. The procedure for selecting piglets for enrich-ment seemed not to be very important, although itmay be advantageous to avoid the runt. An impor-tant issue was that gain for all littermates should berecorded and that the milk intake should be mea-sured repeatedly.

References

AOAC. 1990. Of� cial Methods of Analysis. 15th edn. AOAC,Arlington, VA, 1298 pp.

Baverstock, P. & Green, B. 1975. Water recycling in lactation.Science (Wash.) 187, 657–658.

Beutler, H. O. 1984. Lactase and d-galactase. In: Bergmeyer, H.U., Bergmeyer, J. & Grassl, C. (eds.) Methods of EnzymaticAnalysis, Vol. VI, 3rd edn. Verlag Chemie, Weinheim, pp.104–112.

Brand, W. A., Avak, H., Seedorf, R., Hofmann, D. & Conradi, T.1996. New methods for fully automated isotope ratio determina-tion from hydrogen at natural abundance level. Isotopes Envi-ron. Health Stud. 32, 263–373.

Braude, R. & Newport, M. J. 1973. Arti� cial rearing of pigs. 4.The replacement of butter-milk in a whole-milk diet by eitherbeef tallow, coconut oil or soyabean oil. Br. J. Nutr. 29, 447–455.

Coward, W. A., Cole, T. J., Gerber, H., Roberts, S. B. & Fleet, I.1982. Water turnover and the measurement of milk intake.P� ugers Arch. 393, 344–347.

Danielsen, V. 1988. Nye energinormer til søer. HyologiskTidsskrift Svinet 8, 27–28.

Darragh, A. J. & Moughan, P. J. 1998. The composition ofcolostrum and milk. In: Verstegen, M. W. A., Moughan, P. J. &Schrama, J. W. (eds.) The Lactating Sow. Wageningen Pers,Wageningen, pp. 3–21.

Davison, A. C. & Hinkley, D. V. 1997. Bootstrap Methods andTheir Application. Cambridge University Press, Cambridge, 582p.

Dempster, A. P., Laird, N. M. & Rubin, D. B. 1977. Maximumlikelihood from incomplete data via the EM-algorithm. J. R.Statist. Soc. B 39, 1–38.

Den Hartog, L. A., Verstegen, M. W. A., Hermans, H. A. T.,Noordewier, G. J. & Van Kempen, G. J. M. 1983. Some factorsassociated with determination of milk production in sows byweighing of piglets. Z. Tierphysiol. Tierernahr. Futtermittelkde.51, 148–157.

De Passile, A. M. & Hartsock, T. G. 1979. Within- and between-litter variation of proximate composition in newborn and 10-dayold Landrace swine. J. Anim. Sci. 49, 1449–1457.

Dove, H. 1988. Estimate of the intake of milk by lambs, from theturnover of deuterium- or tritium labelled water. Br. J. Nutr. 60,375–387.

Glencross, B. D., Mullan, B. P., Tuckey, R. C. & Hartmann, P. E.1997. A simpli� cation of the deuterium oxide dilution techniqueusing FT-IR analysis of plasma, for estimating piglet milkintake. Aust. J. Agric. Res. 48, 1099–1104.

Haggarty, P., McGaw, B. A. & Franklin, M. F. 1988. Measure-ment of fractionated water loss and carbon dioxide productionusing triply labelled water. J. Theor. Biol. 134, 291–308.

Haggarty, P., Franklin, M. F. & Fuller, M. F. 1994. Validation ofthe doubly labelled water method in growing pigs. Am. J.Physiol. 36, R1574–R1588.

Klaver, J., Van Kempen, G. J. M., De Lange, P. G. B., Verstegen,M. W. A. & Boer, H. 1981. Milk composition and daily yield ofdifferent milk components as affected by sow condition andlactation:feeding regimen. J. Anim. Sci. 52, 1091–1097.

Mackenzie, D. D. S. & Revell, D. K. 1998. Genetic in� uences onmilk quantity. In: Verstegen, M. W. A., Moughan, P. J. &Schrama, J. W. (eds.) The Lactating Sow. Wageningen Pers,Wageningen, pp. 97–112.

Mavromichalis, I., Parr, T. M., Gabert, V. M. & Baker, D. H.2001. True digestibility of amino acids in sow’s milk for 17-day-old pigs. J. Anim. Sci. 79, 707–713.

Nielsen, H. 1973. Growth and development in pigs in the pre- andpostnatal period with special reference to later growth andcarcass composition. 405. Beretning fra forsøgslaboratoriet,København, 152 pp. (In Danish with English summary.)

Noblet, J. & Etienne, M. 1987. Body composition, metabolic rateand utilization of milk nutrients in suckling piglets. Reprod.Nutr. Dev. 27, 829–839.

Pettigrew, J. E., Sower, A. F., Cornelius, S. G. & Moser, R. L.1985. A comparison of isotope dilution and weigh–suckle–weigh methods for estimating milk intake by pigs. Can. J. Anim.Sci. 65, 989–992.

Pettigrew, J. E., Cornelius, S. G., Moser, R. L. & Sower, A. F.1987. A re� nement and evaluation of the isotope dilution tech-nique for estimating milk intake by piglets. Livestock Prod. Sci.16, 163–174.

Pluske, J. R., Fenton, T. W., Lorschy, M. L., Pettigrew, J. E.,Sower, A. F. & Aherne, F. X. 1997. A modi� cation to theisotope-dilution technique for estimating milk intake of pigsusing pig serum. J. Anim. Sci. 75, 1279–1283.

Prawirodigdo, S., King, R. H., Dunkin, A. C. & Dove, H. 1990a.Evaluation of techniques for estimating milk production bysows. 2. Estimating the milk consumption of piglets by thedeuterium dilution and weigh–suckle–weigh methods. Asian–Aust. J. Anim. Sci. 3, 143–148.

Prawirodigdo, S., King, R. H., Dunkin, A. C. & Dove, H. 1990b.Evaluation of techniques for estimating milk production bysows. 1. Deuterium oxide dilution method for estimating milkintake by piglets. Asian Aust. J. Anim. Sci. 3, 135–141.

Rudolph, B. C., Stahly, T. S. & Cromwell, G. L. 1988. Estimationof body composition of neonatal pigs via deuterium oxidedilution: validation of technique. J. Anim. Sci. 66, 53–61.

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Appendix: Algorithm for the imputation-basedestimation

A description of the imputation algorithm used isincluded, to enable the proposed procedure to beimplemented in other similar studies. In the descrip-tion below it was presupposed that a reasonableparametric regression-like model for predicting themilk intake from the weight gain was available. Thismodel was referred to as the predictive model. In thepresent study the predictive model was the normalmixed quadratic covariance analysis model given byEquation (10). Since there was no special reason tobelieve that this speci� c model was appropriate to

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other cases, the particular form of the predictivemodel is not speci� ed in the description of the proce-dure.

The observations for which both the weight gainand the milk intake were known were termed thecomplete observations. The observations for whichonly weight gain (but not milk intake) was knownwere termed the missing observations. The steps of thealgorithm proposed were as follows.

1. Estimate the parameters of the predictive modelbased on the complete observations. Call theseestimates the current values of the parameters.

2. Evaluate the current predicted milk intake of themissing observations by the predicted values ob-tained from the predictive model, using the cur-

rent values of the parameters and the knownweight gains of the missing observations.

3. Construct the current predicted full data by merg-ing the data containing the complete observationswith the data formed by the missing observations,with their unknown milk intake replaced by thecurrent predicted milk intake.

4. If the changes in the parameter estimates were lessthan a prespeci� ed tolerance limit, then it wasdeclared that the algorithm converged and theprocess stopped; otherwise, it returned to step 2.

This algorithm is equivalent to the general imputa-tion algorithm proposed in Dempster et al. (1977)and widely used in many statistical applications. Thealgorithm converges under mild regularity conditions.

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Yor

k U

nive

rsity

Lib

rari

es]

at 1

9:28

17

Nov

embe

r 20

14