Productive Life! It is what every dairyman wants out of his cows! It is the reason why we are in the business we are in! But what is it exactly? I can not find any one spot where they show the calculation in this “number”….so I am going to keep an ongoing blog on this subject. My goal is to better understand what Productive Life is in a bulls proof….Here you go!

## Simple definition of Productive Life (PL)

Productive life measures how long dairy cows survive in a herd after they calve for the first time. It is based on calving dates, culling or death dates, and days in milk (based on dry dates) in each lactation for cows on DHI test. Cows receive credit for each month in milk, including time beyond 305 days of lactation, starting with their first calf and continuing until they die or are culled from the herd, regardless of age. This approach differs from genetic evaluations for milk production, which include only the first five records, even if cows continue to make additional records. Each month in production receives a slightly different weight based on a standard lactation curve, so that months around peak yield receive more weight than months in late lactation. The heritability of PL is low at 0.085, and cows express this trait only once in their lifetime.

PL is a difficult trait to improve through selection because of low heritability and expression of the trait late in life. Genetic evaluations for PL in AI bulls rely on genetically correlated traits when progeny are too young for complete lifetimes. Traits used for predicting PL on younger cows include yield traits, fertility, somatic cell score, the calving difficulty traits, and the three type composites shown in Table 1. Proofs are expressed in months of PL.

If you seem to be scratching your head…I will simplify it for you with the calculations!

**Now for the calculations that I found:**

**Development of Diminishing Credits for PL**

The diminishing credits approach to redefine PL imposed no restriction on age or lactation length of the cow. The credits were based on population lactation curves across 999 d of lactation. Test-day data, available at AIPL-USDA for Holstein cows that had calved from 1997 to 2003, were used to obtain the suitable prediction formulas for lactation curves for the population. After editing, there were 903,579 lactation records of 305,202 cows with lactation lengths varying from 5 to 999 d. Initially, descriptive statistics such as means and frequency distributions of DIM were investigated Journal of Dairy Science Vol. 89 No. 8, 2006 for each parity to determine the variability in length of extended lactations of Holsteins. Three parity groups were defined as first, second, and third or greater (up to 9 lactations) based on the differences in shape of the lactation curves. Curves may differ for very old cows, but few records were available to estimate these shapes.

Lactation curves for each parity group were fitted to the average daily yield of cows that remained in milk for each given day of lactation. Lactation curves described in the literature (Wood, 1967; Rook et al., 1993; Dijkstra et al., 1997; Pollott, 2000) and the multiphasic curves of Grossman and Koops (1988) were tested using the PROC NLIN procedure in SAS (SAS Institute, 2000). However, the fit was poor for production beyond 305 d, as seen earlier by Grossman and Koops (2003). During the later stages of extended lactations, the observed yields exceeded the predictions. This is in part due to the prediction curves in this study being based on only the cows still in milk at each day of lactation. Therefore, the following empirical model (a modification of the model of Dijkstra et al., 1997) was used to estimate the curves for each of the 3 parity groups:

Y*i,t*= *β*0,*i *+ *β*1,*i **e*[*β*2,*i*(1 − *e*−*β*3,*i **t*)/*β*3,*i *−*β*4,*i**t*]

where *Y**i,t *is the average test-day yield on the *t*th day in milk (*t *= 1, 2, . . ., 999) for the *i*th parity group (*i *=1, 2, 3), and *β *are curve parameters.

Cows must produce at a certain level to cover costs, and only when they exceed these costs can they produce a profit. Analogous to this concept, a baseline (*β*5) was imposed for all parities, and only the yield above the baseline (*Y**i,t *− *β*5) was credited in PL. Introduction of a baseline altered the credits assigned to each day of lactation and consequently improved the statistical properties of the PL derived. The baseline (*β*5) of 13.62 kg (= 30 lb) was a compromise that slightly improved heritability of PL and was used for all parity groups. The USDA recently changed from a mature equivalent basis to a 36-mo equivalent in adjusting for age. Thus, we chose the average daily yield during the first 305 d of the second parity (*Y*2,305= 35.09 kg) as the base milk yield for deriving credits. The credit for the *t*th day in milk of the *i*th parity group (*ω**i,t*) was equal to the yield deviation from the baseline on the *t*th day in milk (*Y**i,t *− *β*5), proportional to *Y*2,305 − *β*:

ωi,t = Yi,t − β5/Y2,305 − β5 =

β0,i + β1,ie[β2,i(1−e−β3,it)/β3,i − β4,it] − β5 / ∫ 305 1 β0,2 + β1,2 e[β2,2(1−e−β3,2t)/β3,2 − β4,2 t] dt /305 − β5

Note that

ω*i,t*> 1.0 for *Y**i,t *> *Y*2,305

ω*i,t*= 1.0 for *Y**i,t *= *Y *2,305

and

ω*i,t *<1.0 for *Y**i,t *< *Y*2,305.

Moreover, the credits given for each day a cow is in milk were always positive, because the baseline used was lower than the asymptotes of the 3 lactation curves. For the second lactation, a cow will earn a total of 305 d of PL credits if she has exactly 305 DIM. Finally, the PL with diminishing credits for a cow was defined as the summation of the credits earned for the DIM in all her lactations.

Note: You can find more details HERE on this ONE calculation of PL.

**Full Model for Indirect Prediction**

An indirect estimate of PTA for PL (uˆind) was obtained from PTA for milk, fat, and 14 linear type traits using the following equation (3).

uˆind = Cov[uPL,**u**]¢[Var(**u**)] –1**u**

where uPL = transmitting ability for PL, **u **= vector of true transmitting abilities for production and type traits, and **uˆ**= vector of multiple-trait BLUP predictions of **u**. The method is approximate in the current situation because PTA for linear type traits are calculated using a multiple-trait BLUP, but PTA for milk and fat are currently calculated using a single-trait BLUP. The REL of uˆ ind was calculated using the following expression:

RELind = Cov[uPL,**u**]¢[Var(**u**)]–1[Var(**uˆ**)][Var(**u**)]–1Cov[uPL,**u**]/Var(uPL)

and

max(RELind) = Cov[uPL,**u**]¢[Var(**u**)]–1Cov[uPL,**u**]/Var(u PL) ,

which occurs when Var(**uˆ **) = Var(**u**) (i.e., with an infinite amount of data on traits in **u)** .

Direct and indirect PL predictions were combined in a weighted mean as follows;

uˆcomb = wdiruˆ dir + winduˆind

where

wdir = (1 – RELind ´c)/(1 – RELindRELdir ´ c2) ,

wind = (1 – RELdir ´c)/(1 – RELindRELdir ´ c2) ,

c = 1 + [DEboth/DEdirDEind]´Ö[(4 – hdir) (4 – ) / ( ) ]

and DE = daughter equivalent (14). Note that c, which is a measure of the lack of independence between direct and indirect evaluations, is a function of direct and indirect trait heritabilities and of the proportion of progeny evaluated for type and production traits that also have direct culling data available. The quantity c can be derived as 1 + [Co(edir,eind) /Cov(udir,uind)](DEboth /DEdirDEind) . The covariance of direct and indirect daughter means equals the genetic covariance multiplied by (DEboth/DEdirDEind) , and we assumed that direct and indirect daughter means are regressed toward the parent average by REL dir and RELind/max (RELind) , respectively. Only daughters with both direct and indirect observations contribute to the error covariance. The c term represents the covariance of the daughter means divided by Co(udir,uind) . If genetic and phenotypic correlations are assumed to be equal, then [Cov(edir,eind)/Cov(udir,uind) ] equals the square root of [(4 hdir) ( 4 – ) / ( )], which is equal 2hind2hdir2hind2 to 26.3 for direct and indirect trait heritabilities of 0.085 and 0.25, respectively. The weights, wdir and wind, are then determined as follows. We know that the covariance of a genetic effect with its BLUP predictor equals REL times genetic variance, which is also the variance of the predictor. Therefore, Cov(uˆdir,uPL) = Var(uˆ dir) = RELdirVar(uPL) , and Cov(uˆind,uPL) = Var(uˆ ind) = RELindVar(uPL) . Furthermore, Cov(uˆdir,uˆ ind) RELdirRELindcVar(uPL) . Then, w = Cov[(uˆdir,uˆ ind),uPL]¢[Var(uˆ dir,uˆ ind)]–1. Simple rules for inverting 2´2 matrices allow the weights to be reexpressed as wdir=(1 – RELind ´ c)/(1 – RELindRELdir ´ c2) and wind = (1 – RELdir ´ c)/(1 – RELindRELdir ´ c2) . Finally, approximate REL of the weighted average was calculated as RELcomb = (RELdir + RELind–2RELindRELdir ´ c)/(1 – RELindRELdir ´ c2) .

Note: You can find more details HERE on this ONE calculation of PL.

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