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Variability in in situ ruminal degradation parameters causes imprecision in estimated ruminal digestibility

      Abstract

      The ruminal digestibility of dietary ingredients is frequently estimated with degradation parameters (e.g., rate and extent of degradation). Mean degradation parameters (e.g., those in a feed library) are often used, but limited data suggest considerable variation around these means, potentially leading to imprecise digestibility estimates. This experiment quantified degradation parameter variation for forage hays and determined the impact of this variation on the precision of ruminal digestibility estimates. Degradation data were those previously published by our laboratory and included degradation rate, λd (h−1), fraction instantly degraded, a (g·g−1), potential extent of degradation, (a + b) (g·g−1), and fraction not instantly degraded that is potentially degradable, b (g·g−1) of alfalfa, grass, and grass-legume hays. Ruminal digestibilities of chemical fractions (dry matter, neutral detergent fiber, acid detergent fiber, hemicellulose, crude protein) were estimated using these data. Ninety-five percent confidence limits of digestibility were determined using propagation of uncertainty with measured standard deviations of degradation parameters. Values for coefficients of variation of degradation parameters were large; averaged across chemical fractions, they were 24.8, 28.6, 20.7, and 12.6% for λd, a, b, and (a + b). Ninety-five percent confidence limits of digestibility were large (80.5% of digestibility means) and often overlapped each other, even when digestibility means differed greatly numerically. Consequently, digestibility values computed with mean degradation parameters may have little biological and practical significance. When uncertainty in all parameters but λd was set to zero (λd alone had uncertainty), 95% confidence limits still encompassed 54.5% of digestibility means. Thus, uncertainty in λd alone caused considerable imprecision in estimated digestibility. These results caution against using mean degradation parameters to estimate digestibility.

      Key words

      Introduction

      Grass and legume forages are key components of ruminant diets. The quality of these forages and other components of ruminant diets is largely influenced by their digestibility (
      • Minson D.J
      Forage in Ruminant Nutrition.
      ), making a measure of digestibility useful in diet formulation. Because measurement of digestibility in vivo is laborious and requires large amounts of forage, digestibility is frequently estimated. Many current feed evaluation systems [e.g., Cornell Net Carbohydrate and Protein System (
      • Sniffen C.J.
      • O’Connor J.D.
      • Van Soest P.J.
      • Fox D.G.
      • Russell J.B.
      A net carbohydrate and protein system for evaluating cattle diets: II. Carbohydrate and protein availability.
      ); Molly (
      • Baldwin R.L
      Modeling Ruminant Digestion and Metabolism.
      ); Beef NRC (
      NRC
      Nutrient Requirements of Beef Cattle.
      ); Dairy NRC (
      NRC
      Nutrient Requirements of Dairy Cattle.
      )] use ruminal degradation parameters (e.g., rate and extent of digestion) to estimate ruminal digestibility, which in turn is used with estimated postruminal digestion to calculate total tract digestibility. These feeding systems use simple averages of degradation parameters, such as those compiled in a feed library, to estimate ruminal digestibility. However, a limited number of sources suggest that considerable variability exists in these mean values for forages and other feeds (
      • von Keyserlingk M.A.G.
      • Swift M.L.
      • Puchala R.
      • Shelford J.A.
      Degradability characteristics of dry matter and crude protein of forages in ruminants.
      ;
      • Hvelplund T.
      • Weisbjerg M.R.
      In situ techniques for the estimation of protein degradability and postrumen availability.
      ;
      NRC
      Nutrient Requirements of Dairy Cattle.
      ). This variability raises questions about the precision of using simple averages to calculate digestibility. More work is needed to determine the actual variability in degradation parameters of forages and to ascertain whether using average values is valid. The objectives of this study were to 1) determine the variability in ruminal degradation parameters within and across hay forages, and 2) assess the impact of this variability on the precision of digestibility calculations.

      Materials and Methods

      Hay Types and Sampling Procedures

      Hay samples were obtained from entries submitted to the Missouri State Fair Hay Contest in 2002 and 2003. The entries came from across the state of Missouri and included early-cut alfalfa (ECA; n = 20), late-cut alfalfa (LCA; n = 26), cool-season grass (CSG; n = 11), warm-season grass (WSG; n = 4), and grass-legume (GL; n = 20) samples. A detailed description of the species composition of the forages was reported in
      • Hackmann T.J.
      • Sampson J.D.
      • Spain J.N.
      Comparing relative feed value with degradation parameters of grass and legume forages.
      . Samples were collected from hay bales by using a hay probe (Penn State Forage Sampler, Nasco, Ft. Atkinson, CO), ground through a 2-mm screen with a Wiley mill (Arthur H. Thomas Company, Philadelphia, PA), and stored in sealed plastic bags at room temperature for further analysis.

      In Situ and Chemical Analysis

      In situ degradation characteristics were determined for all samples. Dacron bags (10 × 20 cm; 50 ± 15 μm pore size; Ankom Technology, Macedon, NY) were filled with 5 ± 0.1 g (2003 samples) or 4 ± 0.1 g (2002 samples) of sample. Duplicate bags were prepared for insertion into 2 cows, giving a total of 4 bags per sample at each incubation time.
      Two ruminally cannulated, multiparous Holstein cows housed in free-stall facilities at the University of Missouri-Columbia Foremost Dairy Center were selected for in situ procedures. All procedures involving the animals were approved by the Animal Care and Use Committee, University of Missouri-Columbia. Each animal was provided ad libitum access to a lactation diet (240 g of corn silage, 123 g of alfalfa hay, 150 g of alfalfa haylage, 467 g of concentrate; 190 g of CP, 240 g of ADF, 410 g of NDF/kg of DM) formulated to meet its nutrient requirements (
      NRC
      Nutrient Requirements of Dairy Cattle.
      ). Bags containing 2003 hay samples were inserted into 2 cows for 0, 6, 12, 24, and 48 h during one 2-d period. During another 2-d period (approximately 6 mo prior), bags containing 2002 hay samples were inserted into the same 2 cows for 0, 8, 12, 24, and 48 h. However, one of the cows stopped ruminating during the original sampling period for the 2002 samples, and the samples from this cow for that period were discarded. New subsamples of the 2002 hay samples were incubated in this cow for an additional 2-d period (approximately 1 yr after the original sampling period), with incubation times of 0, 6, 12, 24, and 48 h. Zero-hour bags were exposed to rumen fluid briefly (approximately 5 min) to allow hydration. All bags were removed simultaneously, as suggested by
      • Nocek J.E.
      In situ and other methods to estimate ruminal protein and energy digestibility: A review.
      .
      After removal from the rumen, bags were doused with cold water (approximately 15°C) to halt fermentation and were rinsed until wash water ran relatively clear. Bags were stored at <0°C to await further processing. Once removed from storage, bags were thawed to room temperature and washed in a domestic washing machine until the wash water ran completely clear, as suggested by
      • Cherney D.J.R.
      • Patterson J.A.
      • Lemenager R.P.
      Influence of in situ bag rinsing technique on determination of dry matter disappearance.
      . Samples were air-dried in a 55°C convection oven to a constant mass and were then air-equilibrated and weighed to determine residue mass. Residues were then removed and composited by time within cow. Bag residue and original forage samples were ground to pass through a 1-mm screen and subsequently analyzed for DM, NDF, ADF, hemicellulose (HEM), and CP, as described in
      • Hackmann T.J.
      • Sampson J.D.
      • Spain J.N.
      Comparing relative feed value with degradation parameters of grass and legume forages.
      .

      Calculations and Statistical Analysis of In Situ Data

      All degradation data were expressed as fractional disappearance. Using an equation from

      Weisbjerg, M. R., P. K. Bhargava, T. Hvelplund, and J. Madsen. 1990. Anvendelse af nedbrydningsprofiler i fodermiddelvurderingen. Beret. Statens Husdyrbrugsfors. Report no. 679. National Institute of Animal Science, Foulum, Denmark.

      , we corrected NDF, HEM, and ADF degradation data at each incubation time for insoluble material washed out of the bag. Because NDF, ADF, and HEM are insoluble entities, it was assumed that the truly soluble fraction was zero when the equation was applied.
      Aberrant bag observations were identified and eliminated using the method reported in
      • Hackmann T.J.
      • Sampson J.D.
      • Spain J.N.
      Comparing relative feed value with degradation parameters of grass and legume forages.
      . When the criteria for eliminating aberrant bag observations were used, nearly all NDF, ADF, and HEM bag observations had to be eliminated for one 2003 ECA sample. So few bag observations were left after this elimination that NDF, ADF, and HEM degradation data could not be fitted to a model for this forage sample.
      In situ data were described using a single, gamma 2-distributed pool model without a lag phase (G2), from
      • Pond K.R.
      • Ellis W.C.
      • Matis J.H.
      • Ferreiro H.M.
      • Sutton J.D.
      Compartment models for estimating attributes of digesta flow in cattle.
      and
      • Ellis W.C.
      • Matis J.H.
      • Hill T.M.
      • Murphy M.R.
      Methodology for estimating digestion and passage kinetics of forages.
      , as follows:
      Y(t)=a+b[1exp(λdt)(1+λdt)],


      where Y(t) is fractional disappearance (g·g−1), λd is degradation rate (h−1), a is the fraction escaping the bag and assumed degraded at t = 0 (g·g−1), (a + b) is the potential extent of degradation (g·g−1), b is the fraction not degraded at t = 0 that is potentially degradable (g·g−1), and t is time (h). The age-dependent G2 model was chosen over the often-used age-independent models (
      • Ørskov E.R.
      • McDonald I.
      The estimation of protein degradability in the rumen from incubation measurements weighted according to rates of passage.
      ) because the G2 model yielded lower values for Akaike's information criterion relative to the age-independent models, indicating the G2 model provided a fit superior to that of first-order kinetic models for this data set (see
      • Hackmann T.J.
      • Sampson J.D.
      • Spain J.N.
      Comparing relative feed value with degradation parameters of grass and legume forages.
      ).
      The NLIN procedure of SAS (SAS Institute Inc., Cary, NC) was used to estimate parameters in the models for DM, NDF, ADF, HEM, and CP degradation data. Because NDF, ADF, and HEM degradation data had been corrected to make disappearance zero at t = 0, the value of a was constrained to zero for fitting the procedures for these fractions.
      Most degradation rates reported in the literature are age independent, whereas λd is an age-dependent rate (
      • Pond K.R.
      • Ellis W.C.
      • Matis J.H.
      • Ferreiro H.M.
      • Sutton J.D.
      Compartment models for estimating attributes of digesta flow in cattle.
      ;
      • Ellis W.C.
      • Matis J.H.
      • Hill T.M.
      • Murphy M.R.
      Methodology for estimating digestion and passage kinetics of forages.
      ). Comparison between an age-independent rate, kd (h−1), and an age-dependent rate, λd, is not commensurate; an age-independent rate is less than an age-dependent one when describing the same degradation data. To allow for a more commensurate comparison, the mean degradation rate of λd, k, was calculated as 0.5964·λd (
      • Pond K.R.
      • Ellis W.C.
      • Matis J.H.
      • Ferreiro H.M.
      • Sutton J.D.
      Compartment models for estimating attributes of digesta flow in cattle.
      ). This mean rate is effectively an age-independent equivalent of an age-dependent rate and can be compared with age-independent rates commonly reported in the literature.
      At first, the degradation data for each cow were kept separate for the fitting procedure. However, after a preliminary ANOVA indicated that degradation parameter values did not differ consistently across cow × period, the cow × period data were pooled by year of forage (2002 or 2003), necessitating that the data for 2 cows (n = 2 for 2002; n = 2 for 2003) be used to construct each degradation curve.
      Values for standard deviations and coefficients of variation of degradation parameter estimates were calculated for each chemical fraction within a forage class. Because values for standard deviations and coefficients of variation were similar across forage classes, the values were averaged across forage classes before they were tabulated. These tabulated values should not be mistaken as standard deviations and coefficients of variation of data pooled by forage class; data were separated by forage class when values for standard deviations and coefficients of variation were calculated, and only thereafter were these values averaged.

      Statistical Analysis with Other Previously Published Degradation Data

      The data of

      Mertens, D. R. 1973. Application of theoretical mathematical models to cell wall digestion and forage intake in ruminants. PhD Diss. Cornell Univ., Ithaca, NY.

      ,
      • von Keyserlingk M.A.G.
      • Swift M.L.
      • Puchala R.
      • Shelford J.A.
      Degradability characteristics of dry matter and crude protein of forages in ruminants.
      ,
      NRC
      Nutrient Requirements of Dairy Cattle.
      , and
      • Hvelplund T.
      • Weisbjerg M.R.
      In situ techniques for the estimation of protein degradability and postrumen availability.
      were used to calculate coefficients of variation of degradation parameter estimates for comparison with our own results. Values for coefficients of variation for (a + b) and kd in

      Mertens, D. R. 1973. Application of theoretical mathematical models to cell wall digestion and forage intake in ruminants. PhD Diss. Cornell Univ., Ithaca, NY.

      were calculated by compiling data in the appendix of that study, which listed estimates of (a + b) and kd individually for each experimental forage. When not reported in the original publication, values for coefficients of variation for a, b, and kd in
      • von Keyserlingk M.A.G.
      • Swift M.L.
      • Puchala R.
      • Shelford J.A.
      Degradability characteristics of dry matter and crude protein of forages in ruminants.
      were calculated directly by using reported means and standard deviations pooled by forage class.
      Because
      • Hvelplund T.
      • Weisbjerg M.R.
      In situ techniques for the estimation of protein degradability and postrumen availability.
      reported standard deviations by forage species and not by forage class, standard deviations for forage class were calculated by pooling variances of degradation parameter estimates across forage species within a class. Similarly, because the
      NRC
      Nutrient Requirements of Dairy Cattle.
      reported standard deviations by forage maturity and not by forage class, standard deviations for forage class were calculated by pooling variances of degradation parameter estimates across forage maturities within a class. Variances were pooled using the equation
      s2=k=1n(nj1)sj2k=1n(nj1),


      where s2 is the pooled variance, and sj2 and nj refer to the variance and sample size of the jth forage class or maturity, respectively. Once values of standard deviations were calculated from these pooled variances, values for the coefficients of variation were then computed as with the data from
      • von Keyserlingk M.A.G.
      • Swift M.L.
      • Puchala R.
      • Shelford J.A.
      Degradability characteristics of dry matter and crude protein of forages in ruminants.
      .

      Calculation of Ruminal Digestibility and 95% Confidence Limits

      Predicted ruminal digestibilities of DM, ADF, HEM, and CP by forage class were calculated using the equation
      digestibilityi,j=ai,j+bi,jλd|i,j2(λd|i,j+kp)2,
      [1]


      where digestibilityi,j (g·g−1) is the ruminal digestibility of chemical fraction i (DM, ADF, HEM, CP) and forage class j (ECA, LCA, CSG, WSG, GL); ai,j is a for chemical fraction i and forage class j; bi,j is b for chemical fraction i and forage class j; λd|i,j is λd for chemical fraction i and forage class j; and kp is the fractional rate of passage from the rumen (h−1), set to a constant value of 0.06 h−1. This equation is conceptually analogous to the often-used equation
      digestibility=a+bkdkd+kp,
      [2]


      developed by
      • Ørskov E.R.
      • McDonald I.
      The estimation of protein degradability in the rumen from incubation measurements weighted according to rates of passage.
      . Equations [1] and [2] are derived using the same biological principles and compute the same entity, although their precise mathematical forms differ because of the use of the age-dependent rate λd in equation [1] in place of the independent rate kd in equation [2] (see the Appendix for the derivation of equation [1]).
      Upper and lower 95% confidence limits of calculated digestibility were determined using the law of propagation of uncertainty (e.g.,
      • Taylor J.R
      An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements.
      ;
      • Heuvelink G.B.M
      Error Propagation in Environmental Modelling with GIS.
      ). Frequently used in the physical and quantitative sciences, the law of propagation of uncertainty is used to compute the amount of uncertainty (analogous to standard deviation) expected in a calculated quantity (e.g., digestibility) originating from uncertainty in one or more measured quantities (e.g., a, b, λd) used in the calculations. In general, when a first-order Taylor approximation is used, uncertainty in a dependent variable y that is a function of independent variables x1, x2,…, xn is the following:
      Δy=k=1nl=1nyxkyxlΔxkΔxlρxk,xl2,
      [3]


      where ∂y/∂xk and ∂y/∂xl are the partial derivatives of y with respect to xk and xl, respectively; Δxk and Δxl are the uncertainties of xk and xl, respectively; and ρxkxl is the correlation coefficient between variables xk and xl. When k = l, the term ΔxkΔxlρxk,xl=Δxk2 and when kl, ΔxkΔxlρxk,xl=σxk,xl, the covariance between xk and xl; from these relations and from equations [1] and [3], the corresponding uncertainty in digestibility is expressed as
      Δdigestibilityi,j=digestibilityi,jai,jΔai,j2+digestibilityi,jbi,jΔbi,j2+digestibilityi,jai,jΔλd|i,j2+2digestibilityi,jai,jdigestibilityi,jbi,jσai,j,bi,j+2digestibilityi,jai,jdigestibilityi,jλd|i,jσai,j,λd|i,j+2digestibilityi,jbi,jdigestibilityi,jλd|i,jσbi,k,λd|i,j2
      [4]


      where Δdigestibilityi,j is uncertainty in digestibilityi,j; Δai,j is uncertainty in ai,j; Δbi,j is uncertainty in bi,j; Δλd|i,j is uncertainty in λd|i,j; σai,j,bi,j is covariance between ai,j and bi,j; σai,j,λd|i,j is covariance between ai,j and λd|i,j; σbi,j,λd|i,j is covariance between bi,k and λd|i,j; ∂digestibilityi,j/∂ai,j is the partial derivative of digestibilityi,jwith respect to ai,j, equal to 1; ∂digestibilityi,j/∂bi,j is the partial derivative of digestibilityi,j with respect to bi,j, equal to λd|i,j2/(λd|i,j+kp)2 and ∂digestibilityi,j/∂λd|i,j is the partial derivative of digestibilityi,jwith respect to λd|i,j, equal to
      2bλd|i,j(λd|i,j+kp)2λd|i,j2(λd|i,j+kp)3.


      As mentioned below, the value of Δkp was set to 0; for simplicity, equation [4] has already been rendered with Δkp = 0.
      Examination of equations [3] and [4] reveals the key result that increasing uncertainty in a measured quantity (e.g., degradation parameters) increases uncertainty in a calculated quantity (e.g., digestibility). This point is illustrated by Figure 1, which shows how uncertainty in b translates into uncertainty in digestibility. The solid black line represents values of digestibility (computed with equation [1]) when b varies from 0 to 1 (and with a and λd set to 0 and 0.075, values similar to those for NDF of WSG). The value of Δb is 0.08 and is centered around b = 0.60 (values similar to those for NDF of WSG); covariances and all other uncertainties are set to 0. As shown in Figure 1, a vertical line extending from the lower bound of Δb on the x-axis intersects with a horizontal line extending from the lower bound of Δdigestibility on the y-axis, where the point of intersection is a point on the graph of digestibility. A similar relationship is observed between the upper bounds of Δb and Δdigestibility (Figure 1). As such, when Δb is increased, Δdigestibility increases in kind. Note that the exact relationship between b and digestibility shown in Figure 1 exists only when covariances and all other uncertainties are equal to 0, but the general principle illustrated (Δdigestibility increases with Δb) holds true under all conditions.
      Figure thumbnail gr1
      Figure 1Graphical illustration of how increasing uncertainty in a measured quantity (e.g., degradation parameters) increases uncertainty in a calculated quantity (e.g., digestibility) according to the law of propagation of uncertainty. Digestibility (solid black line) is computed using equation in the text, with b [fraction not degraded at t = 0 that is potentially degradable (g·g−1)] varying from 0 to 1 and a [fraction degraded at t = 0 (g·g−1)] and λd [degradation rate (h−1)] held constant at 0 and 0.075. The value of Δb (uncertainty in b) is 0.08 and is centered around b = 0.60, with covariances and all other uncertainties set to zero. Dashed lines mark the upper and lower bounds of Δb and Δdigestibility (uncertainty in digestibility). The point of intersection between vertical and horizontal lines extending from the lower bounds of Δb and Δdigestibility is a point on the graph of digestibility; a similar relationship exists between the upper bounds of the 2 variables. Increasing Δb hence increases Δdigestibility.
      Further examination of equations [3] and [4] also reveals that increasing the partial derivative of the measured quantity relative to the calculated quantity (i.e., ∂y/∂xk) increases uncertainty, as does increasing the covariances between the measured quantities (i.e.,σxk,xk).
      At first, Δai,j and Δbi,j were set to the standard deviations of ai,j and bi,j measured in the study; so too were covariances set to their measured values, to determine the composite effect of Δai,j, Δbi,j, and Δλd|i,j on Δdigestibilityi,j. Next, Δai,j and Δbi,j were set to zero to study the effect of Δλd|i,j on Δdigestibilityi,j in isolation. In all analyses, the values of Δλd|i,j and Δkp were set to the standard deviations of λd|i,j and 0, respectively.
      To calculate the composite effect of Δai,j, Δbi,j, and Δλd|i,j on Δdigestibilityi,j, methods other than the law of propagation of uncertainty may be used; one alternative is to determine the digestibility of each sample by using equation [1] and then to compute the standard deviations of these digestibility values to yield Δdigestibilityi,j when Δai,j, Δbi,j, and Δλd|i,j are equal to their standard deviations. However, only the law of propagation of uncertainty can be used for the more complex analysis in which the effect of Δλd|i,j on Δdigestibilityi,j is studied in isolation. For this reason, the law of propagation of uncertainty was used for all analyses.
      Upper and lower 95% confidence limits were computed as the quantity digestibilityi,j± Δdigestibilityi,j·t0.025,n−1, where t0.025,n−1 is the critical value of the right tail of the t distribution with α/2 = 0.025 and with n − 1 degrees of freedom (n = number of digestibilityi,j observations).

      Results

      The chemical composition of the forages is presented in Table 1. Numerically, both alfalfa classes (ECA, LCA) had higher mean concentrations of CP and lower concentrations of fiber (NDF, ADF, HEM) than did grass classes (CSG, WSG). Within alfalfa, LCA had a numerically higher mean CP concentration and had lower fiber concentrations than did ECA. Within grasses, CSG had a numerically higher mean CP concentration and had lower fiber concentrations than did WSG.
      Table 1Chemical composition of forages
      ECA=early-cutting alfalfa; LCA=late-cutting alfalfa; CSG=cool-season grass; WSG=warm-season grass; GL=grass-legume mix; HEM=hemicellulose.
      Item,
      ECA=early-cutting alfalfa; LCA=late-cutting alfalfa; CSG=cool-season grass; WSG=warm-season grass; GL=grass-legume mix; HEM=hemicellulose.
      g·kg DM−1
      nMeanMinimumMaximumSD
      ECA
       DM2087084289614
       NDF2043132950245
       ADF2030421836945
       HEM201279518428
       CP2020815029338
      LCA
       DM2685982588015
       NDF2638426846351
       ADF2626519535443
       HEM261196618537
       CP2622219426020
      CSG
       DM118768678927
       NDF1165845277281
       ADF1133829938028
       HEM1132012339273
       CP111236017433
      WSG
       DM486784588617
       NDF4623395732155
       ADF426623334352
       HEM4357155484154
       CP418010423354
      GL
       DM2086881989018
       NDF2045335561352
       ADF2030424138436
       HEM2014910127241
       CP2020412430840
      1 ECA = early-cutting alfalfa; LCA = late-cutting alfalfa; CSG = cool-season grass; WSG = warm-season grass; GL = grass-legume mix; HEM = hemicellulose.
      Table 2 reports the means of degradation parameter estimates λd, k, a, b, and (a + b). Mean values of (a + b) of DM and CP were numerically higher for alfalfa (ECA, LCA) than for grass samples (CSG, WSG). Values of (a + b) of fiber (NDF, ADF, HEM) were similar across classes. Mean values of λd and k were numerically higher for alfalfa (ECA, LCA) than for grasses (CSG, WSG). Within alfalfa, λd and k were higher for LCA than for ECA, and within grasses, these parameters were higher for CSG than for WSG. Differences in λd and k were consistently preserved across chemical fractions (DM, NDF, ADF, HEM, CP).
      Table 2Degradation parameter estimates of forages by forage class and chemical fraction
      Item
      ECA=early-cutting alfalfa; LCA=late-cutting alfalfa; CSG=cool-season grass; WSG=warm-season grass; GL=grass-legume mix.
      ,
      k, λd=degradation rate; a=fraction degraded at t=0; (a + b)=potential extent of degradation; b=fraction not degraded at t=0 that is potentially degradable.
      ,
      k=0.59635·λd.
      Chemical fraction
      DMNDFADFHEM
      HEM=hemicellulose.
      CP
      MeanSEMMeanSEMMeanSEMMeanSEMMeanSEM
      ECA (n = 20)
      Value of n for DM and CP. Value of n for NDF, ADF, and HEM is 19 because NDF, ADF, and HEM degradation data of 1 sample could not be fitted to a degradation model (see Materials and Methods section).
      λd, h−10.1980.0080.1550.0120.1470.0140.2080.0190.2310.010
      k, h−10.1180.0050.0930.0070.0880.0080.1240.0110.1380.006
      a, g·g−10.3410.0230.4120.030
      b, g·g−10.4190.0210.5540.0190.5350.0180.6260.0260.4830.027
       (a + b), g·g−10.7600.0110.5540.0190.5350.0180.6260.0260.8940.009
      LCA (n = 26)
      λd, h−10.2290.0080.1730.0090.1540.0100.2340.0170.2630.008
      k, h−10.1360.0050.1030.0050.0920.0060.1400.0100.1570.005
      a, g·g−10.3870.0220.4420.025
      b, g·g−10.4070.0180.5370.0160.5140.0150.5840.0250.4810.022
       (a + b), g·g−10.7940.0080.5370.0160.5140.0150.5840.0250.9240.005
      CSG (n = 11)
      λd, h−10.1090.0070.0980.0050.0980.0060.1040.0110.1300.010
      k, h−10.0650.0040.0580.0030.0580.0040.0620.0070.0780.006
      a, g·g−10.2460.0180.3410.037
      b, g·g−10.4510.0200.5830.0260.5890.0320.5760.0350.5070.032
       (a + b), g·g−10.6970.0180.5830.0260.5890.0320.5760.0350.8480.035
      WSG (n = 4)
      λd, h−10.0930.0220.0710.0030.0580.0060.0710.0190.0910.009
      k, h−10.0560.0130.0430.0020.0350.0040.0420.0110.0540.005
      a, g·g−10.2500.0160.3590.068
      b, g·g−10.4210.0180.5670.0170.5350.0290.5150.0390.4240.055
       (a + b), g·g−10.6700.0080.5670.0170.5350.0290.5150.0390.7830.016
      GL (n = 20)
      λd, h−10.1810.0120.1020.0080.1310.0100.0810.0270.2260.012
      k, h−10.1080.0070.0610.0050.0780.0060.0480.0160.1350.007
      a, g·g−10.3110.0200.3940.025
      b, g·g−10.4520.0210.4200.0170.5640.0210.4350.0190.4930.025
       (a + b), g·g−10.7630.0080.4200.0170.5640.0210.4350.0190.8870.008
      1 ECA = early-cutting alfalfa; LCA = late-cutting alfalfa; CSG = cool-season grass; WSG = warm-season grass; GL = grass-legume mix.
      2 k, λd = degradation rate; a = fraction degraded at t = 0; (a + b) = potential extent of degradation; b = fraction not degraded at t = 0 that is potentially degradable.
      3 k = 0.59635·λd.
      4 HEM = hemicellulose.
      5 Value of n for DM and CP. Value of n for NDF, ADF, and HEM is 19 because NDF, ADF, and HEM degradation data of 1 sample could not be fitted to a degradation model (see Materials and Methods section).
      Table 3 reports the standard deviations and coefficients of variation of degradation parameter estimates [λd, k, a, b, and (a + b)]. Values in Table 3 were averaged across forage classes for brevity because standard deviations and coefficients of variation did not vary appreciably across classes. Values of standard deviations were large relative to the mean, as illustrated by large values for coefficients of variation. Values for coefficients of variation ranged from 17.3 to 36.2% for λd and k, 24.8 to 32.4% for a, 18.0 to 23.4% for b, and 5.5 to 16.4% for (a + b). Values for coefficients of variation for (a + b) were systematically smaller than for any other parameter, whereas values for coefficients of variation for other parameters (λd, k, a, and b) were similar in magnitude. Similarly, values for coefficients of variation for HEM were consistently larger than for any other chemical fraction, whereas coefficients of variation for CP were systematically smaller.
      Table 3Standard deviations and CV of degradation parameter estimates, averaged by chemical fraction
      k, λd,=degradation rate (h−1); a=fraction degraded at t=0 (g·g−1); (a + b)=potential extent of degradation (g·g−1); b=fraction not degraded at t=0 that is potentially degradable (g·g−1); HEM=hemicellulose.
      ,
      Values of SD and CV are averaged across forage class. Units for SD are per hour for λd and k and are gram per gram for a, b, and (a + b).
      ItemDegradation parameter
      λdkab(a + b)
      DM
       SD0.0310.0190.0790.0770.041
       CV, %17.317.324.818.05.5
      NDF
       SD0.0320.0190.0730.073
       CV, %21.721.712.912.9
      ADF
       SD0.0380.0230.0880.088
       CV, %29.129.115.715.7
      HEM
       SD0.0690.0410.0970.097
       CV, %36.236.216.416.4
      CP
       SD0.0370.0220.1270.1110.065
       CV, %19.719.732.423.410.1
      1 k, λd, = degradation rate (h−1); a = fraction degraded at t = 0 (g·g−1); (a + b) = potential extent of degradation (g·g−1); b = fraction not degraded at t = 0 that is potentially degradable (g·g−1); HEM = hemicellulose.
      2 Values of SD and CV are averaged across forage class. Units for SD are per hour for λd and k and are gram per gram for a, b, and (a + b).
      Table 4 shows calculated digestibility means and their associated upper and lower 95% confidence limits when Δλd, Δa, and Δb were set to the standard deviation values of λd, a, and b measured in the study. The mean digestibilities of CP ranged from 0.513 to 0.761 g·g−1 for CP. Numerically, these were generally higher than those for DM digestibility, with values of 0.406 to 0.642 g·g−1. These values were, in turn, numerically higher than those for NDF, ADF, and HEM, which were themselves similar, with digestibilities ranging from 0.193 to 0.288 g·g−1 for NDF, 0.192 to 0.270 for ADF, and 0.217 to 0.396 g·g−1 for HEM.
      Table 4Calculated digestibility of chemical fractions by forage class and upper and lower 95% confidence limits (CL), using uncertainty values of λd, a, and b equal to their SD
      λd=degradation rate (h−1); a=fraction degraded at t=0 (g·g−1); b=fraction not degraded at t=0 that is potentially degradable (g·g−1); (a + b)=potential extent of degradation; ECA=early-cut alfalfa; LCA=late-cut alfalfa; CSG=cool-season grass; WSG=warm-season grass; GL=grass-legume mixture; HEM=hemicellulose.
      ,
      Values calculated using equations [1] and [4] in text and a passage rate of 0.06h−1.
      Item, g·g−1Forage class
      ECALCACSGWSGGL
      DM
       Digestibility
      All means within rows have overlapping 95% CL; corresponding superscript symbols have been omitted for brevity.
      0.588
      Means with different superscripts within column have nonoverlapping 95% CL.
      0.642
      Means with different superscripts within column have nonoverlapping 95% CL.
      0.436
      Means with different superscripts within column have nonoverlapping 95% CL.
      0.406
      Means with different superscripts within column have nonoverlapping 95% CL.
      0.570
      Means with different superscripts within column have nonoverlapping 95% CL.
       Lower 95% CL0.4560.5080.3010.3240.434
       Upper 95% CL0.7200.7770.5710.4870.707
      NDF
       Digestibility
      All means within rows have overlapping 95% CL; corresponding superscript symbols have been omitted for brevity.
      0.288
      Means with different superscripts within column have nonoverlapping 95% CL.
      0.296
      Means with different superscripts within column have nonoverlapping 95% CL.
      0.225
      Means with different superscripts within column have nonoverlapping 95% CL.
      0.193
      Means with different superscripts within column have nonoverlapping 95% CL.
      0.296
      Means with different superscripts within column have nonoverlapping 95% CL.
       Lower 95% CL0.1570.1770.1130.1230.192
       Upper 95% CL0.4200.4150.3370.2640.401
      ADF
       Digestibility
      All means within rows have overlapping 95% CL; corresponding superscript symbols have been omitted for brevity.
      0.270
      Means with different superscripts within column have nonoverlapping 95% CL.
      0.266
      Means with different superscripts within column have nonoverlapping 95% CL.
      0.226
      Means with different superscripts within column have nonoverlapping 95% CL.
      0.192
      Means with different superscripts within column have nonoverlapping 95% CL.
      0.266
      Means with different superscripts within column have nonoverlapping 95% CL.
       Lower 95% CL0.1230.1670.1140.1200.158
       Upper 95% CL0.4170.3660.3380.2630.374
      HEM
       Digestibility
      All means within rows have overlapping 95% CL; corresponding superscript symbols have been omitted for brevity.
      0.377
      Means with different superscripts within column have nonoverlapping 95% CL.
      0.370
      Means with different superscripts within column have nonoverlapping 95% CL.
      0.231
      Means with different superscripts within column have nonoverlapping 95% CL.
      0.217
      Means with different superscripts within column have nonoverlapping 95% CL.
      0.396
      Means with different superscripts within column have nonoverlapping 95% CL.
       Lower 95% CL0.1740.1340.1070.0450.170
       Upper 95% CL0.5810.6060.3550.3890.623
      CP
       Digestibility
      All means within rows have overlapping 95% CL; corresponding superscript symbols have been omitted for brevity.
      0.716
      Means with different superscripts within column have nonoverlapping 95% CL.
      0.761
      Means with different superscripts within column have nonoverlapping 95% CL.
      0.597
      Means with different superscripts within column have nonoverlapping 95% CL.
      0.513
      Means with different superscripts within column have nonoverlapping 95% CL.
      0.702
      Means with different superscripts within column have nonoverlapping 95% CL.
       Lower 95% CL0.5440.6400.3480.2400.573
       Upper 95% CL0.8770.8820.8470.7870.831
      a–c Means with different superscripts within column have nonoverlapping 95% CL.
      1 λd = degradation rate (h−1); a = fraction degraded at t = 0 (g·g−1); b = fraction not degraded at t = 0 that is potentially degradable (g·g−1); (a + b) = potential extent of degradation; ECA = early-cut alfalfa; LCA = late-cut alfalfa; CSG = cool-season grass; WSG = warm-season grass; GL = grass-legume mixture; HEM = hemicellulose.
      2 Values calculated using equations 1 and 4 in text and a passage rate of 0.06 h−1.
      3 All means within rows have overlapping 95% CL; corresponding superscript symbols have been omitted for brevity.
      Within ECA, the 95% confidence limits of DM and CP were distinct from those of NDF and ADF, and all other 95% confidence limits overlapped each other. Within LCA, the 95% confidence limits of CP digestibility were distinct from those of HEM, ADF, and NDF digestibility; the 95% confidence limits of DM digestibility were distinct from those of NDF and ADF digestibility; and all other 95% confidence limits overlapped each other. Within CSG, the 95% confidence limits of CP were distinct from those of all other chemical fractions (DM, NDF, ADF, HEM), and all other 95% confidence limits overlapped each other. Within WSG, the 95% confidence limits of DM were distinct from all other chemical fractions (NDF, ADF, HEM, CP), and all other 95% confidence limits overlapped each other. Within GL, the 95% confidence limits of DM and CP digestibility were distinct from those of NDF and ADF, and all other 95% confidence limits overlapped each other.
      When compared within the same chemical fraction, numerically, mean digestibilities for WSG and CSG were systematically lower than those for other forage classes, which were themselves similar to each other (Table 4). As shown in Table 4, the 95% confidence limits for DM, NDF, ADF, HEM, and CP digestibilities overlapped each other across forage classes, with only one exception (the 95% confidence limits for DM digestibility of WSG were distinct from those of LCA).
      Table 5 shows digestibility means and their associated upper and lower 95% confidence limits when Δλd was set to the standard deviations of λd measured in this study (and Δa and Δb were set to zero). Mean digestibilities were the same as those reported in Table 4. Within ECA, LCA, and CSG, the 95% confidence limits of CP and DM digestibilities were distinct from each other and from all other chemical fractions, and the 95% confidence limits of NDF, ADF, and HEM digestibilities overlapped each other. A similar pattern was found for WSG, except that the 95% confidence limits of HEM were not distinct from those of DM. Within GL, the 95% confidence limits of CP digestibility were distinct from those of NDF, ADF, and HEM digestibilities; the 95% confidence limits of DM digestibility were distinct from those of NDF and ADF digestibilities; and all other 95% confidence limits overlapped each other.
      Table 5Calculated digestibility of chemical fractions by forage class and upper and lower 95% confidence limits (CL), using uncertainty values of λd equal to its SD
      λd=degradation rate (h−1); ECA=early-cut alfalfa; LCA=late-cut alfalfa; CSG=cool-season grass; WSG=warm-season grass; GL=grass-legume mixture; HEM=hemicellulose.
      ,
      Values calculated using equations [1] and [4] in text and a passage rate of 0.06h−1.
      Item, g·g−1Forage class
      ECALCACSGWSGGL
      DM
       Digestibility0.588
      Means with different superscripts within a column have nonoverlapping 95% CL.
      ,
      Means with different superscripts within a row have nonoverlapping 95% CL.
      0.642
      Means with different superscripts within a column have nonoverlapping 95% CL.
      ,
      Means with different superscripts within a row have nonoverlapping 95% CL.
      0.436
      Means with different superscripts within a column have nonoverlapping 95% CL.
      ,
      Means with different superscripts within a row have nonoverlapping 95% CL.
      0.406
      Means with different superscripts within a column have nonoverlapping 95% CL.
      ,
      Means with different superscripts within a row have nonoverlapping 95% CL.
      0.570
      Means with different superscripts within a column have nonoverlapping 95% CL.
      Means with different superscripts within a column have nonoverlapping 95% CL.
      ,
      Means with different superscripts within a row have nonoverlapping 95% CL.
       Lower 95% CL0.5390.6020.3910.3870.497
       Upper 95% CL0.6360.6830.4810.4240.644
      NDF
       Digestibility0.288
      Means with different superscripts within a column have nonoverlapping 95% CL.
      0.296
      Means with different superscripts within a column have nonoverlapping 95% CL.
      0.225
      Means with different superscripts within a column have nonoverlapping 95% CL.
      0.193
      Means with different superscripts within a column have nonoverlapping 95% CL.
      0.296
      Means with different superscripts within a column have nonoverlapping 95% CL.
       Lower 95% CL0.1730.2120.1750.1460.209
       Upper 95% CL0.4040.3810.2750.2410.383
      ADF
       Digestibility0.270
      Means with different superscripts within a column have nonoverlapping 95% CL.
      0.266
      Means with different superscripts within a column have nonoverlapping 95% CL.
      0.226
      Means with different superscripts within a column have nonoverlapping 95% CL.
      0.192
      Means with different superscripts within a column have nonoverlapping 95% CL.
      0.266
      Means with different superscripts within a column have nonoverlapping 95% CL.
       Lower 95% CL0.1360.1690.1430.1040.145
       Upper 95% CL0.4040.3640.3090.2800.387
      HEM
       Digestibility0.377
      Means with different superscripts within a column have nonoverlapping 95% CL.
      0.370
      Means with different superscripts within a column have nonoverlapping 95% CL.
      0.231
      Means with different superscripts within a column have nonoverlapping 95% CL.
      0.217
      Means with different superscripts within a column have nonoverlapping 95% CL.
      0.396
      Means with different superscripts within a column have nonoverlapping 95% CL.
       Lower 95% CL0.2390.2580.1720.0110.227
       Upper 95% CL0.5160.4830.2900.4230.566
      CP
       Digestibility0.716
      Means with different superscripts within a column have nonoverlapping 95% CL.
      ,
      Means with different superscripts within a row have nonoverlapping 95% CL.
      0.761
      Means with different superscripts within a column have nonoverlapping 95% CL.
      ,
      Means with different superscripts within a row have nonoverlapping 95% CL.
      0.597
      Means with different superscripts within a column have nonoverlapping 95% CL.
      ,
      Means with different superscripts within a row have nonoverlapping 95% CL.
      0.513
      Means with different superscripts within a column have nonoverlapping 95% CL.
      ,
      Means with different superscripts within a row have nonoverlapping 95% CL.
      0.702
      Means with different superscripts within a column have nonoverlapping 95% CL.
      ,
      Means with different superscripts within a row have nonoverlapping 95% CL.
       Lower 95% CL0.6640.7220.5300.4360.640
       Upper 95% CL0.7670.8010.6650.5900.763
      a–c Means with different superscripts within a column have nonoverlapping 95% CL.
      x–z Means with different superscripts within a row have nonoverlapping 95% CL.
      1 λd = degradation rate (h−1); ECA = early-cut alfalfa; LCA = late-cut alfalfa; CSG = cool-season grass; WSG = warm-season grass; GL = grass-legume mixture; HEM = hemicellulose.
      2 Values calculated using equations 1 and 4 in text and a passage rate of 0.06 h−1.
      Compared across forages, the 95% confidence limits of DM digestibility for CSG and WSG were distinct from those for ECA, LCA, and GL. Additionally, the 95% confidence limits for CP digestibility of WSG were distinct from those of ECA, LCA, and GL, and the 95% confidence limits for CP digestibility of CSG were distinct from those of LCA. All other 95% confidence limits of forage classes overlapped each other.

      Discussion

      Chemical Composition and Degradation Parameter Means

      The means and standard deviations of the chemical composition data in Table 1 were generally similar to those summarized by the Dairy NRC (
      NRC
      Nutrient Requirements of Dairy Cattle.
      ), although our samples of WSG had lower NDF, ADF, and HEM and higher CP than reported by the
      NRC
      Nutrient Requirements of Dairy Cattle.
      . These results suggest that a representative range of forages was included in this study. With the exception of λd (see below), degradation parameter means in Table 2 were similar to those presented in other reports (
      • Smith L.W.
      • Goering H.K.
      • Gordon C.H.
      Relationships of forage compositions with rates of cell wall digestion and indigestibility of cell walls.
      ;

      Mertens, D. R. 1973. Application of theoretical mathematical models to cell wall digestion and forage intake in ruminants. PhD Diss. Cornell Univ., Ithaca, NY.

      ;
      • von Keyserlingk M.A.G.
      • Swift M.L.
      • Puchala R.
      • Shelford J.A.
      Degradability characteristics of dry matter and crude protein of forages in ruminants.
      ), indicating they were suitable for the digestibility analyses below. The values of λd generated in this study were numerically greater than the degradation rate values reported by other investigators because λd is an age-dependent degradation rate, whereas most degradation rates reported in the literature are age-independent rates. For commensurate comparisons between degradation rates in this and prior studies, the mean degradation rate k should be used for reference (see the Materials and Methods section).

      Variation in Degradation Parameter Estimates

      For comparison with values in Table 3, we summarized coefficient of variation values for degradation parameter estimates from prior studies. In the data of
      • von Keyserlingk M.A.G.
      • Swift M.L.
      • Puchala R.
      • Shelford J.A.
      Degradability characteristics of dry matter and crude protein of forages in ruminants.
      , coefficient of variation values for a, b, and kd for CP were 11.4, 20.1, and 35.1% for alfalfa hay (n = 16) and 26.3, 24.1, and 34.9% for grass hay (n = 14). For DM, coefficient of variation values for a, b, and kd were 7.9, 7.4, and 36.7% for alfalfa hay (n = 16) and 16.3, 25.6, and 36.8% for grass hay (n = 14). In the data of
      • Hvelplund T.
      • Weisbjerg M.R.
      In situ techniques for the estimation of protein degradability and postrumen availability.
      , coefficient of variation values for a, b, and kd were 11.1, 16.0, and 27.2% for alfalfa CP (n = 22). In the same data set, coefficient of variation values for a, b, and kd were 21.8, 17.6, and 24.7% for CSG CP (n = 119). In the
      NRC
      Nutrient Requirements of Dairy Cattle.
      data, coefficient of variation values for a, b, (a + b), and kd were 34.8, 32.2, 11.2, 46.5% for alfalfa CP (n = 83). In the same data set, coefficient of variation values for a, b, (a + b), and kd were 22.7, 20.4, 4.8, and 35.5% for CSG CP (n = 113). In

      Mertens, D. R. 1973. Application of theoretical mathematical models to cell wall digestion and forage intake in ruminants. PhD Diss. Cornell Univ., Ithaca, NY.

      , coefficient of variation values for (a + b) and kd for NDF were 18.1 and 21.5% for alfalfa (n = 40), 14.8 and 19.6% for GL, 10.9 and 29.3 for CSG (n = 171), and 12.8 and 25.5% for WSG (n = 8).
      In sum, the literature values for coefficients of variation for a ranged from 7.9 to 34.8%, those for b ranged from 7.4 to 32.2%, those for (a + b) ranged from 4.8 to 18.1, and those for kd ranged from 19.6 to 36.5%, with no clearly detectable differences across forage classes (alfalfa, GL, CSG, WSG) or chemical fractions (DM, CP, NDF). Coefficient of variation values for DM, CP, and NDF in our own data set (Table 3) generally fell within these ranges. The large amount of variability we found in degradation parameter values is thus reasonable. Note that in prior reports, coefficient of variation values for degradation rate were larger than those for other parameters, unlike in this experiment.
      In general, the variation in degradation parameter estimates was due to a combination of 1) variation truly attributable to the chemical fraction and forage class and 2) procedural variation. For in situ studies (such as the current one), common sources of procedural variation arise from factors such as bag characteristics (material, size, pore size, sample size to surface area), sample preparation (grind size), incubation procedure (presence or absence of preincubation, reticuloruminal region in which bags are incubated, incubation times, order of bag removal and insertion), replications (number of animals, days, and bag replications), rinsing technique (hand vs. mechanical rinsing), and animal-related parameters (species, diet, temporal and individual variation in the reticulorumen environment;
      • Nocek J.E.
      In situ and other methods to estimate ruminal protein and energy digestibility: A review.
      ;
      • Vanzant E.S.
      • Cochran R.C.
      • Titgemeyer E.C.
      Standardization of in situ techniques for ruminant feedstuff evaluation.
      ). Most of these factors (bag characteristics, sample preparation, incubation technique, rinsing technique, animal species, diet) were controlled in this study by adopting a standardized procedure. Other sources of variation, such as variation among animals, days, and replicate bags (
      • Mehrez A.Z.
      • Ørskov E.R.
      A study of the artificial fibre bag technique for determining the digestibility of feeds in the rumen.
      ;
      • Vanzant E.S.
      • Cochran R.C.
      • Titgemeyer E.C.
      Standardization of in situ techniques for ruminant feedstuff evaluation.
      ) were reduced by using multiple animals (2 per forage) and bags (2 per animal).
      Although an extensive effort was made to minimize procedural error relative to variation attributable to chemical fraction and forage class, procedural error may still have been appreciable. In this context, it is difficult to explain systematic differences in coefficients of variation across degradation parameters and chemical fractions because these differences are confounded by the 2 sources of variation defined above. For example, the consistently smaller coefficients of variation for (a + b) compared with other degradation parameters could be caused by lower measurement error because of some feature of the procedure, or by lower true variation in this parameter across chemical fractions and forages, or both. Whatever its source, variation in degradation parameters was substantial and was comparable with studies using a wide range of in situ and in vitro methodologies.

      Calculated Digestibilities and Their 95% Confidence Limits

      Many feed analysis systems [e.g., Cornell Net Carbohydrate and Protein System (
      • Sniffen C.J.
      • O’Connor J.D.
      • Van Soest P.J.
      • Fox D.G.
      • Russell J.B.
      A net carbohydrate and protein system for evaluating cattle diets: II. Carbohydrate and protein availability.
      ); Molly (
      • Baldwin R.L
      Modeling Ruminant Digestion and Metabolism.
      ); Beef NRC (
      NRC
      Nutrient Requirements of Beef Cattle.
      ); Dairy NRC (
      NRC
      Nutrient Requirements of Dairy Cattle.
      )] use degradation parameter estimate means to calculate ruminal digestibility or TDN. Because of the considerable variability in degradation rate and other degradation parameter estimates, as discussed above, the precision in using simple means may be questioned. To determine variability in ruminal digestibility calculated by using mean values, the 95% confidence limits of digestibility were computed by using the law of propagation of uncertainty.
      Values in Table 4 (where λd, a, and b equal their standard deviations) represent the case in which digestibility is calculated from estimated a, b, and λd means, such as those from a feed library. The 95% confidence limits of digestibility means in Table 4 frequently overlapped each other, both within and across forage classes, even when there were large numerical differences between the means. As may be anticipated by their frequent overlapping, the 95% confidence limits spanned a wide range of values, equal to 80.5% of their associated digestibility means on average. These findings, together with those discussed above, suggest that digestibility values calculated by using mean values of a, b, and λd may have limited practical and biological meaning.
      In Tables 4 and 5, we indicate which 95% confidence limits overlapped, mainly to illustrate the width of the 95% confidence limits, rather than to report the results of a formal pair-wise test of digestibility means. We did not use a suitable methodology for pair-wise comparison of digestibility means (notably, a Tukey or Bonferroni correction for multiple comparisons), and it is inappropriate to interpret the results above as such comparisons.
      Note that kp and chemical composition of the forages were assumed to have no uncertainty—that is, there was assumed to be no error in their estimation. Because there was considerable variability in kp and because published equations predict kp with low precision [the best equation for predicting kp of forages in
      • Seo S.
      • Tedeschi L.O.
      • Schwab C.G.
      • Lanzas C.
      • Fox D.G.
      Development and evaluation of empirical equations to predict feed passage rate in cattle.
      had a root mean square prediction error of 0.011 h−1 with R2 = 0.39], 95% confidence limits would be much greater in more practical cases (e.g., in which kp was not known and thus had to be estimated) than in Table 4, where it is assumed to be known with certainty. In all likelihood, the 95% confidence limits reported in Table 4 are greatly underestimated. This result further cautions against using the mean values of a, b, and λd to calculate digestibility.

      Analysis Where a and b Are Known with Certainty

      Values of a and b may often be measured or estimated readily by using solubility and chemical assays, such as buffer-soluble N (
      • Krishnamoorthy U.
      • Sniffen C.J.
      • Stern M.D.
      • Van Soest P.J.
      Evaluation of a mathematical model of rumen digestion and an in vitro simulation of rumen proteolysis to estimate the rumen-undegraded nitrogen content of feedstuffs.
      ), which is used to estimate a for CP, and lignin, which is used to predict b for NDF (
      • Smith L.W.
      • Goering H.K.
      • Gordon C.H.
      Relationships of forage compositions with rates of cell wall digestion and indigestibility of cell walls.
      ;

      Mertens, D. R. 1973. Application of theoretical mathematical models to cell wall digestion and forage intake in ruminants. PhD Diss. Cornell Univ., Ithaca, NY.

      ;
      • Traxler M.J.
      • Fox D.G.
      • Van Soest P.J.
      • Pell A.N.
      • Lascano C.E.
      • Lanna D.P.
      • Moore J.E.
      • Lana R.P.
      • Velez M.
      • Flores A.
      Predicting forage indigestible NDF from lignin concentration.
      ). To mimic cases in which a and b are measured, the 95% confidence limits were recalculated, assuming that a and b were known with certainty and that λd alone had uncertainty (Table 5).
      The 95% confidence limits in Table 5 overlapped each other less frequently than those in Table 4, compared both within and across forage classes. However, considerable variation in calculated digestibility was still present when a and b were known with certainty. On average, 95% confidence limits encompassed values 54.4% of their associated digestibility means. This percentage was less than when a and b were uncertain (80.5%), but it was still large enough to limit the practical and biological meaning of the calculated digestibilities.
      As discussed above, kp of the forages were assumed to have no uncertainty; thus, the 95% confidence limits listed in Table 5 are likely to be smaller than realized when kp must be estimated. Note also that it is unrealistic to assume values of a and b with absolute certainty, as was done in calculating the 95% confidence limits reported in Table 5. At the very least, analytical error in measuring a and b contributes uncertainty to the values of these parameters. Further, a and b cannot be measured directly but must be estimated from another chemical or physical measurement; for example, b of NDF is often estimated from lignin content of the NDF (
      • Smith L.W.
      • Goering H.K.
      • Gordon C.H.
      Relationships of forage compositions with rates of cell wall digestion and indigestibility of cell walls.
      ;

      Mertens, D. R. 1973. Application of theoretical mathematical models to cell wall digestion and forage intake in ruminants. PhD Diss. Cornell Univ., Ithaca, NY.

      ;
      • Traxler M.J.
      • Fox D.G.
      • Van Soest P.J.
      • Pell A.N.
      • Lascano C.E.
      • Lanna D.P.
      • Moore J.E.
      • Lana R.P.
      • Velez M.
      • Flores A.
      Predicting forage indigestible NDF from lignin concentration.
      ); errors in this estimation add uncertainty. Overall, because of these sources of uncertainty that were not accounted for in the calculations in this report, the widths of the 95% confidence limits in Table 5 are likely underestimated.
      Here and throughout, one may attribute the large 95% confidence limits we found to some suspected peculiarity in our in situ procedure, kinetic model (equation [1]), uncertainty analysis (equation [4]), or some other aspect of our methodology. One may also suggest that our 95% confidence limits are inflated because sample sizes for some forage classes were limited (n = 4 for WSG). We reemphasize that the amount of variability in degradation parameter values we found was comparable with that of in vitro and other in situ studies using a wide range of methodologies, including those studies using very large sample sizes (n > 100;

      Mertens, D. R. 1973. Application of theoretical mathematical models to cell wall digestion and forage intake in ruminants. PhD Diss. Cornell Univ., Ithaca, NY.

      ;
      • Hvelplund T.
      • Weisbjerg M.R.
      In situ techniques for the estimation of protein degradability and postrumen availability.
      ;
      NRC
      Nutrient Requirements of Dairy Cattle.
      ). Although we carried out the uncertainty analysis fully with our parameter values alone, we would expect similarly large 95% confidence limits when using values from other reports because of this comparable variability (cf., Figure 1 and the Materials and Methods section). Although largely foreign to the animal and nutritional sciences, the uncertainty analysis we used is itself well accepted in the physical sciences. The large 95% confidence limits we found were thus not artifacts of the conditions of our study.
      These results caution against using the mean degradation parameter values to estimate digestibility. As a corollary, caution should be exercised when using digestibility values garnered from feed analysis systems that rely on approaches to calculate digestibility similar to those used in this report. Because uncertainty in λd contributes appreciable uncertainty in calculated digestibility, techniques to measure or effectively estimate λd efficiently should continue to be developed. One promising approach is that proposed by
      • Van Soest P.J.
      • Van Amburgh M.E.
      • Tedeschi L.O.
      Rumen balance and rates of fiber digestion.
      , who developed multiple regression equations that can accurately estimate the degradation rate of forage NDF from chemical composition and in vitro digestibility. The data set from this study could be used to further validate the equations of
      • Van Soest P.J.
      • Van Amburgh M.E.
      • Tedeschi L.O.
      Rumen balance and rates of fiber digestion.
      and to formulate similar equations for non-NDF chemical fractions.

      Appendix

      Digestibility is the sum of ai,j and bi,j that is digested in the rumen:
      digestibilityi,j=ai,j+bi,j


      where ai,j and bi,j are ai,j and bi,j that is digestible (g·g−1). The terms ai,j and bi,j are explicitly defined as
      ai,j=pai,jai,j,


      and
      bi,j=pbi,jbi,j,


      where pai,j and pbi,j are the fractions (g·g−1; ai,j or bi,j) of ai,j and bi,j that are digestible. It is assumed that the ai,j fraction is completely digestible (i.e., pai,j=1 ), following its definition that it is instantly degraded. For the G2 model used in this report, pbi,j is equal to
      pbi,j=λd|i,j2(λd|i,j+kp)2.


      This follows from the definition of fractional digestibility as the amount of material (in this case, bi,j) that disappears by digestion divided by its total disappearance; mathematically, this is equivalent to the integral (over the interval t = 0 to infinity) of ri,j(t), the rate function for digestion of bi,j (h−1), multiplied by Bi,j(t), the amount of bi,j remaining over time (g·g−1; bi,j):
      pbi,j=t=0[ri,j(t)Bi,j(t)]dt=λd|i,j2(λd|i,j+kp)2.


      [For a G2 model, ri,j(t) is
      ri,j(t)=λd|i,j2t1+λd|i,jt


      (
      • Pond K.R.
      • Ellis W.C.
      • Matis J.H.
      • Ferreiro H.M.
      • Sutton J.D.
      Compartment models for estimating attributes of digesta flow in cattle.
      ;
      • Ellis W.C.
      • Matis J.H.
      • Hill T.M.
      • Murphy M.R.
      Methodology for estimating digestion and passage kinetics of forages.
      ) and the term Bi,j(t),
      Bi,j(t)=(1+λd|i,jt)exp[t(λd|i,j+kp)],


      is the solution to the differential equation that describes the change of Bi,j(t) over time by digestion and passage:
      dBi,j(t)dt=[ri,j(t)+kp]Bi,j(t).]


      Making the appropriate substitutions, the final digestibility equation (equation [1] in the text) is then
      digestibilityi,j=ai,j+bi,jλd|i,j2(λd|i,j+kp)2.


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