## Abstract

*λ*

_{d}(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

*λ*

_{d},

*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

*λ*

_{d}was set to zero (

*λ*

_{d}alone had uncertainty), 95% confidence limits still encompassed 54.5% of digestibility means. Thus, uncertainty in

*λ*

_{d}alone caused considerable imprecision in estimated digestibility. These results caution against using mean degradation parameters to estimate digestibility.

## Key words

## Introduction

## Materials and Methods

### Hay Types and Sampling Procedures

**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

### In Situ and Chemical Analysis

**HEM**), and CP, as described in

### Calculations and Statistical Analysis of In Situ Data

**G2**), from

where

*Y*(

*t*) is fractional disappearance (g·g

^{−1}), λ

_{d}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 (

*t*= 0, the value of

*a*was constrained to zero for fitting the procedures for these fractions.

_{d}is an age-dependent rate (

*k*

_{d}(h

^{−1}), and an age-dependent rate, λ

_{d}, 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

*λ*

_{d},

*k*, was calculated as 0.5964·

*λ*

_{d}(

### Statistical Analysis with Other Previously Published Degradation Data

*a*+

*b*) and

*k*

_{d}in

*a*+

*b*) and

*k*

_{d}individually for each experimental forage. When not reported in the original publication, values for coefficients of variation for

*a*,

*b*, and

*k*

_{d}in

where

*s*

^{2}is the pooled variance, and ${s}_{j}^{2}$ and

*n*

_{j}refer to the variance and sample size of the

*j*th 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

### Calculation of Ruminal Digestibility and 95% Confidence Limits

where

*digestibility*

_{i,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);

*a*

_{i,j}is

*a*for chemical fraction

*i*and forage class

*j*;

*b*

_{i,j}is

*b*for chemical fraction

*i*and forage class

*j*;

*λ*

_{d|i,j}is

*λ*

_{d}for chemical fraction

*i*and forage class

*j*; and

*k*

_{p}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

developed by

*λ*

_{d}in equation [1] in place of the independent rate

*k*

_{d}in equation [2] (see the Appendix for the derivation of equation [1]).

*a*,

*b*,

*λ*

_{d}) 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

*x*

_{1},

*x*

_{2},…,

*x*

_{n}is the following:

where ∂

*y*/∂

*x*

_{k}and ∂

*y*/∂

*x*

_{l}are the partial derivatives of

*y*with respect to

*x*

_{k}and

*x*

_{l}, respectively; Δ

*x*

_{k}and Δ

*x*

_{l}are the uncertainties of

*x*

_{k}and

*x*

_{l}, respectively; and ${\rho}_{{x}_{k}{x}_{l}}$ is the correlation coefficient between variables

*x*

_{k}and

*x*

_{l}. When

*k*=

*l*, the term $\Delta {x}_{k}\Delta {x}_{l}{\rho}_{{x}_{k},{x}_{l}}=\Delta {x}_{k}^{2}$ and when

*k*≠

*l*, $\Delta {x}_{k}\Delta {x}_{l}{\rho}_{{x}_{k},{x}_{l}}={\sigma}_{{x}_{k},{x}_{l}}\text{,}$ the covariance between

*x*

_{k}and

*x*

_{l}; from these relations and from equations [1] and [3], the corresponding uncertainty in digestibility is expressed as

where Δ

*digestibility*

_{i,j}is uncertainty in

*digestibility*

_{i,j}; Δ

*a*

_{i,j}is uncertainty in

*a*

_{i,j}; Δ

*b*

_{i,j}is uncertainty in

*b*

_{i,j}; Δ

*λ*

_{d|i,j}is uncertainty in

*λ*

_{d|i,j}; ${\sigma}_{{a}_{i,j},{b}_{i,j}}$ is covariance between

*a*

_{i,j}and

*b*

_{i,j}; ${\sigma}_{{a}_{i,j},{\lambda}_{d|i,j}}$ is covariance between

*a*

_{i,j}and

*λ*

_{d|i,j}; ${\sigma}_{{b}_{i,j},{\lambda}_{d|i,j}}$ is covariance between

*b*

_{i,k}and

*λ*

_{d|i,j}; ∂

*digestibility*

_{i,j}/∂

*a*

_{i,j}is the partial derivative of

*digestibility*

_{i,j}with respect to

*a*

_{i,j}, equal to 1; ∂

*digestibility*

_{i,j}/∂

*b*

_{i,j}is the partial derivative of

*digestibility*

_{i,j}with respect to

*b*

_{i,j}, equal to ${\lambda}_{d|i,j}^{2}/{({\lambda}_{d|i,j}+{k}_{p})}^{2}$ and ∂

*digestibility*

_{i,j}/∂

*λ*

_{d|i,j}is the partial derivative of

*digestibility*

_{i,j}with respect to

*λ*

_{d|i,j}, equal to

As mentioned below, the value of Δ

*k*

_{p}was set to 0; for simplicity, equation [4] has already been rendered with Δ

*k*

_{p}= 0.

*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

*λ*

_{d}set to 0 and 0.075, values similar to those for NDF of WSG). The value of Δ

*b*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 Δ

*b*on the x-axis intersects with a horizontal line extending from the lower bound of Δ

*digestibility*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 Δ

*b*and Δ

*digestibility*(Figure 1). As such, when Δ

*b*is increased, Δ

*digestibility*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 (Δ

*digestibility*increases with Δ

*b*) holds true under all conditions.

*y*/∂

*x*

_{k}) increases uncertainty, as does increasing the covariances between the measured quantities $(\text{i}.\text{e}.,{\sigma}_{{x}_{k},{x}_{k}})$.

*a*

_{i,j}and Δ

*b*

_{i,j}were set to the standard deviations of

*a*

_{i,j}and

*b*

_{i,j}measured in the study; so too were covariances set to their measured values, to determine the composite effect of Δ

*a*

_{i,j}, Δ

*b*

_{i,j}, and Δ

*λ*

_{d|i,j}on Δ

*digestibility*

_{i,j}. Next, Δ

*a*

_{i,j}and Δ

*b*

_{i,j}were set to zero to study the effect of Δ

*λ*

_{d|i,j}on Δ

*digestibility*

_{i,j}in isolation. In all analyses, the values of Δ

*λ*

_{d|i,j}and Δ

*k*

_{p}were set to the standard deviations of

*λ*

_{d|i,j}and 0, respectively.

*a*

_{i,j}, Δ

*b*

_{i,j}, and Δ

*λ*

_{d|i,j}on Δ

*digestibility*

_{i,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 Δ

*digestibility*

_{i,j}when Δ

*a*

_{i,j}, Δ

*b*

_{i,j}, and Δ

*λ*

_{d|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 Δ

*λ*

_{d|i,j}on Δ

*digestibility*

_{i,j}is studied in isolation. For this reason, the law of propagation of uncertainty was used for all analyses.

*digestibility*

_{i,j}± Δ

*digestibility*

_{i,j}·

*t*

_{0.025,n−1,}where

*t*

_{0.025,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

*digestibility*

_{i,j}observations).

## Results

Item, g·kg DM^{−1} | n | Mean | Minimum | Maximum | SD |
---|---|---|---|---|---|

ECA | |||||

DM | 20 | 870 | 842 | 896 | 14 |

NDF | 20 | 431 | 329 | 502 | 45 |

ADF | 20 | 304 | 218 | 369 | 45 |

HEM | 20 | 127 | 95 | 184 | 28 |

CP | 20 | 208 | 150 | 293 | 38 |

LCA | |||||

DM | 26 | 859 | 825 | 880 | 15 |

NDF | 26 | 384 | 268 | 463 | 51 |

ADF | 26 | 265 | 195 | 354 | 43 |

HEM | 26 | 119 | 66 | 185 | 37 |

CP | 26 | 222 | 194 | 260 | 20 |

CSG | |||||

DM | 11 | 876 | 867 | 892 | 7 |

NDF | 11 | 658 | 452 | 772 | 81 |

ADF | 11 | 338 | 299 | 380 | 28 |

HEM | 11 | 320 | 123 | 392 | 73 |

CP | 11 | 123 | 60 | 174 | 33 |

WSG | |||||

DM | 4 | 867 | 845 | 886 | 17 |

NDF | 4 | 623 | 395 | 732 | 155 |

ADF | 4 | 266 | 233 | 343 | 52 |

HEM | 4 | 357 | 155 | 484 | 154 |

CP | 4 | 180 | 104 | 233 | 54 |

GL | |||||

DM | 20 | 868 | 819 | 890 | 18 |

NDF | 20 | 453 | 355 | 613 | 52 |

ADF | 20 | 304 | 241 | 384 | 36 |

HEM | 20 | 149 | 101 | 272 | 41 |

CP | 20 | 204 | 124 | 308 | 40 |

*λ*

_{d},

*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

*λ*

_{d}and

*k*were numerically higher for alfalfa (ECA, LCA) than for grasses (CSG, WSG). Within alfalfa,

*λ*

_{d}and

*k*were higher for LCA than for ECA, and within grasses, these parameters were higher for CSG than for WSG. Differences in

*λ*

_{d}and

*k*were consistently preserved across chemical fractions (DM, NDF, ADF, HEM, CP).

Item^{,}^{,} | Chemical fraction | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

DM | NDF | ADF | HEM | CP | ||||||

Mean | SEM | Mean | SEM | Mean | SEM | Mean | SEM | Mean | SEM | |

ECA (n = 20) | ||||||||||

λ_{d}, h^{−1} | 0.198 | 0.008 | 0.155 | 0.012 | 0.147 | 0.014 | 0.208 | 0.019 | 0.231 | 0.010 |

k, h^{−1} | 0.118 | 0.005 | 0.093 | 0.007 | 0.088 | 0.008 | 0.124 | 0.011 | 0.138 | 0.006 |

a, g·g^{−1} | 0.341 | 0.023 | — | — | — | — | — | — | 0.412 | 0.030 |

b, g·g^{−1} | 0.419 | 0.021 | 0.554 | 0.019 | 0.535 | 0.018 | 0.626 | 0.026 | 0.483 | 0.027 |

(a + b), g·g^{−1} | 0.760 | 0.011 | 0.554 | 0.019 | 0.535 | 0.018 | 0.626 | 0.026 | 0.894 | 0.009 |

LCA (n = 26) | ||||||||||

λ_{d}, h^{−1} | 0.229 | 0.008 | 0.173 | 0.009 | 0.154 | 0.010 | 0.234 | 0.017 | 0.263 | 0.008 |

k, h^{−1} | 0.136 | 0.005 | 0.103 | 0.005 | 0.092 | 0.006 | 0.140 | 0.010 | 0.157 | 0.005 |

a, g·g^{−1} | 0.387 | 0.022 | — | — | — | — | — | — | 0.442 | 0.025 |

b, g·g^{−1} | 0.407 | 0.018 | 0.537 | 0.016 | 0.514 | 0.015 | 0.584 | 0.025 | 0.481 | 0.022 |

(a + b), g·g^{−1} | 0.794 | 0.008 | 0.537 | 0.016 | 0.514 | 0.015 | 0.584 | 0.025 | 0.924 | 0.005 |

CSG (n = 11) | ||||||||||

λ_{d}, h^{−1} | 0.109 | 0.007 | 0.098 | 0.005 | 0.098 | 0.006 | 0.104 | 0.011 | 0.130 | 0.010 |

k, h^{−1} | 0.065 | 0.004 | 0.058 | 0.003 | 0.058 | 0.004 | 0.062 | 0.007 | 0.078 | 0.006 |

a, g·g^{−1} | 0.246 | 0.018 | — | — | — | — | — | — | 0.341 | 0.037 |

b, g·g^{−1} | 0.451 | 0.020 | 0.583 | 0.026 | 0.589 | 0.032 | 0.576 | 0.035 | 0.507 | 0.032 |

(a + b), g·g^{−1} | 0.697 | 0.018 | 0.583 | 0.026 | 0.589 | 0.032 | 0.576 | 0.035 | 0.848 | 0.035 |

WSG (n = 4) | ||||||||||

λ_{d}, h^{−1} | 0.093 | 0.022 | 0.071 | 0.003 | 0.058 | 0.006 | 0.071 | 0.019 | 0.091 | 0.009 |

k, h^{−1} | 0.056 | 0.013 | 0.043 | 0.002 | 0.035 | 0.004 | 0.042 | 0.011 | 0.054 | 0.005 |

a, g·g^{−1} | 0.250 | 0.016 | — | — | — | — | — | — | 0.359 | 0.068 |

b, g·g^{−1} | 0.421 | 0.018 | 0.567 | 0.017 | 0.535 | 0.029 | 0.515 | 0.039 | 0.424 | 0.055 |

(a + b), g·g^{−1} | 0.670 | 0.008 | 0.567 | 0.017 | 0.535 | 0.029 | 0.515 | 0.039 | 0.783 | 0.016 |

GL (n = 20) | ||||||||||

λ_{d}, h^{−1} | 0.181 | 0.012 | 0.102 | 0.008 | 0.131 | 0.010 | 0.081 | 0.027 | 0.226 | 0.012 |

k, h^{−1} | 0.108 | 0.007 | 0.061 | 0.005 | 0.078 | 0.006 | 0.048 | 0.016 | 0.135 | 0.007 |

a, g·g^{−1} | 0.311 | 0.020 | — | — | — | — | — | — | 0.394 | 0.025 |

b, g·g^{−1} | 0.452 | 0.021 | 0.420 | 0.017 | 0.564 | 0.021 | 0.435 | 0.019 | 0.493 | 0.025 |

(a + b), g·g^{−1} | 0.763 | 0.008 | 0.420 | 0.017 | 0.564 | 0.021 | 0.435 | 0.019 | 0.887 | 0.008 |

*k*,

*λ*

_{d}= 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·

*λ*

_{d}.

*λ*

_{d},

*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

*λ*

_{d}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 (

*λ*

_{d},

*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.

^{,}

Item | Degradation parameter | ||||
---|---|---|---|---|---|

λ_{d} | k | a | b | (a + b) | |

DM | |||||

SD | 0.031 | 0.019 | 0.079 | 0.077 | 0.041 |

CV, % | 17.3 | 17.3 | 24.8 | 18.0 | 5.5 |

NDF | |||||

SD | 0.032 | 0.019 | — | 0.073 | 0.073 |

CV, % | 21.7 | 21.7 | — | 12.9 | 12.9 |

ADF | |||||

SD | 0.038 | 0.023 | — | 0.088 | 0.088 |

CV, % | 29.1 | 29.1 | — | 15.7 | 15.7 |

HEM | |||||

SD | 0.069 | 0.041 | — | 0.097 | 0.097 |

CV, % | 36.2 | 36.2 | — | 16.4 | 16.4 |

CP | |||||

SD | 0.037 | 0.022 | 0.127 | 0.111 | 0.065 |

CV, % | 19.7 | 19.7 | 32.4 | 23.4 | 10.1 |

*k*, λ

_{d}, = 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.

*λ*

_{d}and

*k*and are gram per gram for

*a*,

*b*, and (

*a*+

*b*).

*λ*

_{d}, Δ

*a*, and Δ

*b*were set to the standard deviation values of

*λ*

_{d},

*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.

*λ*

_{d},

*a*, and

*b*equal to their SD

^{1}

^{,}

Item, g·g^{−1} | Forage class | ||||
---|---|---|---|---|---|

ECA | LCA | CSG | WSG | GL | |

DM | |||||

Digestibility | 0.588 | 0.642 | 0.436 | 0.406 | 0.570 |

Lower 95% CL | 0.456 | 0.508 | 0.301 | 0.324 | 0.434 |

Upper 95% CL | 0.720 | 0.777 | 0.571 | 0.487 | 0.707 |

NDF | |||||

Digestibility | 0.288 | 0.296 | 0.225 | 0.193 | 0.296 |

Lower 95% CL | 0.157 | 0.177 | 0.113 | 0.123 | 0.192 |

Upper 95% CL | 0.420 | 0.415 | 0.337 | 0.264 | 0.401 |

ADF | |||||

Digestibility | 0.270 | 0.266 | 0.226 | 0.192 | 0.266 |

Lower 95% CL | 0.123 | 0.167 | 0.114 | 0.120 | 0.158 |

Upper 95% CL | 0.417 | 0.366 | 0.338 | 0.263 | 0.374 |

HEM | |||||

Digestibility | 0.377 | 0.370 | 0.231 | 0.217 | 0.396 |

Lower 95% CL | 0.174 | 0.134 | 0.107 | 0.045 | 0.170 |

Upper 95% CL | 0.581 | 0.606 | 0.355 | 0.389 | 0.623 |

CP | |||||

Digestibility | 0.716 | 0.761 | 0.597 | 0.513 | 0.702 |

Lower 95% CL | 0.544 | 0.640 | 0.348 | 0.240 | 0.573 |

Upper 95% CL | 0.877 | 0.882 | 0.847 | 0.787 | 0.831 |

_{d}= 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.

*λ*

_{d}was set to the standard deviations of

*λ*

_{d}measured in this study (and Δ

*a*and Δ

*b*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.

*λ*

_{d}equal to its SD

^{,}

Item, g·g^{−1} | Forage class | ||||
---|---|---|---|---|---|

ECA | LCA | CSG | WSG | GL | |

DM | |||||

Digestibility | 0.588^{,} | 0.642^{,} | 0.436^{,} | 0.406^{,} | 0.570^{,} |

Lower 95% CL | 0.539 | 0.602 | 0.391 | 0.387 | 0.497 |

Upper 95% CL | 0.636 | 0.683 | 0.481 | 0.424 | 0.644 |

NDF | |||||

Digestibility | 0.288 | 0.296 | 0.225 | 0.193 | 0.296 |

Lower 95% CL | 0.173 | 0.212 | 0.175 | 0.146 | 0.209 |

Upper 95% CL | 0.404 | 0.381 | 0.275 | 0.241 | 0.383 |

ADF | |||||

Digestibility | 0.270 | 0.266 | 0.226 | 0.192 | 0.266 |

Lower 95% CL | 0.136 | 0.169 | 0.143 | 0.104 | 0.145 |

Upper 95% CL | 0.404 | 0.364 | 0.309 | 0.280 | 0.387 |

HEM | |||||

Digestibility | 0.377 | 0.370 | 0.231 | 0.217 | 0.396 |

Lower 95% CL | 0.239 | 0.258 | 0.172 | 0.011 | 0.227 |

Upper 95% CL | 0.516 | 0.483 | 0.290 | 0.423 | 0.566 |

CP | |||||

Digestibility | 0.716^{,} | 0.761^{,} | 0.597^{,} | 0.513^{,} | 0.702^{,} |

Lower 95% CL | 0.664 | 0.722 | 0.530 | 0.436 | 0.640 |

Upper 95% CL | 0.767 | 0.801 | 0.665 | 0.590 | 0.763 |

_{d}= 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.

## Discussion

### Chemical Composition and Degradation Parameter Means

*λ*

_{d}(see below), degradation parameter means in Table 2 were similar to those presented in other reports (

*λ*

_{d}generated in this study were numerically greater than the degradation rate values reported by other investigators because

*λ*

_{d}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

*a*,

*b*, and

*k*

_{d}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

*k*

_{d}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

*a*,

*b*, and

*k*

_{d}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

*k*

_{d}were 21.8, 17.6, and 24.7% for CSG CP (n = 119). In the

*a*,

*b*, (

*a*+

*b*), and

*k*

_{d}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

*k*

_{d}were 22.7, 20.4, 4.8, and 35.5% for CSG CP (n = 113). In

*a*+

*b*) and

*k*

_{d}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).

*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

*k*

_{d}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.

*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

*λ*

_{d},

*a*, and

*b*equal their standard deviations) represent the case in which digestibility is calculated from estimated

*a*,

*b*, and

*λ*

_{d}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

*λ*

_{d}may have limited practical and biological meaning.

*k*

_{p}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

*k*

_{p}and because published equations predict

*k*

_{p}with low precision [the best equation for predicting

*k*

_{p}of forages in

^{−1}with R

^{2}= 0.39], 95% confidence limits would be much greater in more practical cases (e.g., in which

*k*

_{p}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

*λ*

_{d}to calculate digestibility.

### Analysis Where a and b Are Known with Certainty

*a*and

*b*may often be measured or estimated readily by using solubility and chemical assays, such as buffer-soluble N (

*a*for CP, and lignin, which is used to predict

*b*for NDF (

*a*and

*b*are measured, the 95% confidence limits were recalculated, assuming that

*a*and

*b*were known with certainty and that

*λ*

_{d}alone had uncertainty (Table 5).

*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.

*k*

_{p}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

*k*

_{p}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 (

*λ*

_{d}contributes appreciable uncertainty in calculated digestibility, techniques to measure or effectively estimate

*λ*

_{d}efficiently should continue to be developed. One promising approach is that proposed by

## Appendix

*a*

_{i,j}and

*b*

_{i,j}that is digested in the rumen:

where ${a}_{i,j}\prime $ and ${b}_{i,j}\prime $ are

*a*

_{i,j}and

*b*

_{i,j}that is digestible (g·g

^{−1}). The terms ${a}_{i,j}\prime $ and ${b}_{i,j}\prime $ are explicitly defined as

and

where ${p}_{{a}_{i,j}}$ and ${p}_{{b}_{i,j}}$ are the fractions (g·g

^{−1};

*a*

_{i,j}or

*b*

_{i,j}) of

*a*

_{i,j}and

*b*

_{i,j}that are digestible. It is assumed that the

*a*

_{i,j}fraction is completely digestible (i.e., ${p}_{{a}_{i,j}}=1$ ), following its definition that it is instantly degraded. For the G2 model used in this report, ${p}_{{b}_{i,j}}$ is equal to

*b*

_{i,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

*r*

_{i,j}(

*t*), the rate function for digestion of

*b*

_{i,j}(h

^{−1}), multiplied by

*B*

_{i,j}(

*t*), the amount of

*b*

_{i,j}remaining over time (g·g

^{−1};

*b*

_{i,j}):

*r*

_{i,j}(

*t*) is

(

*B*

_{i,j}(

*t*),

is the solution to the differential equation that describes the change of

*B*

_{i,j}(

*t*) over time by digestion and passage:

## References

- Modeling Ruminant Digestion and Metabolism.Chapman and Hall, London, UK1995
- Influence of in situ bag rinsing technique on determination of dry matter disappearance.
*J. Dairy Sci.*1990; 73: 391-397 - Methodology for estimating digestion and passage kinetics of forages.in: Fahey Jr., G.C. Collins M. Mertens D.R. Moser L.E. Forage Quality, Evaluation, and Utilization. Am. Soc. Agron. Crop Sci. Soc. Am., Soil Sci. Soc. Am., Madison, WI1994: 682-753
- Comparing relative feed value with degradation parameters of grass and legume forages.
*J. Anim. Sci.*2008; 86: 2344-2356 - Error Propagation in Environmental Modelling with GIS.Taylor and Francis, Bristo, PA1998
- In situ techniques for the estimation of protein degradability and postrumen availability.in: Givens D.I Owen E. Axford R.F.E. Omed H.M. Forage Evaluation in Ruminant Nutrition. CABI, Wallingford, UK2000: 233-258
- 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.
*Br. J. Nutr.*1983; 50: 555-568 - A study of the artificial fibre bag technique for determining the digestibility of feeds in the rumen.
*J. Agric. Sci.*1977; 88: 645-650 Mertens, D. R. 1973. Application of theoretical mathematical models to cell wall digestion and forage intake in ruminants. PhD Diss. Cornell Univ., Ithaca, NY.

- Forage in Ruminant Nutrition.Academic Press, New York1990
- In situ and other methods to estimate ruminal protein and energy digestibility: A review.
*J. Dairy Sci.*1988; 71: 2051-2069 - Nutrient Requirements of Beef Cattle.7th updated ed. Natl. Acad. Press, Washington, DC2000
- Nutrient Requirements of Dairy Cattle.7th rev. ed. Natl. Acad. Press, Washington, DC2001
- The estimation of protein degradability in the rumen from incubation measurements weighted according to rates of passage.
*J. Agric. Sci.*1979; 92: 499-503 - Compartment models for estimating attributes of digesta flow in cattle.
*Br. J. Nutr.*1988; 60: 571-595 - Development and evaluation of empirical equations to predict feed passage rate in cattle.
*Anim. Feed Sci. Technol.*2006; 128: 67-83 - Relationships of forage compositions with rates of cell wall digestion and indigestibility of cell walls.
*J. Dairy Sci.*1972; 55: 1140-1147 - A net carbohydrate and protein system for evaluating cattle diets: II. Carbohydrate and protein availability.
*J. Anim. Sci.*1992; 70: 3562-3577 - An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements.2nd ed. University Science Books, Sausalito, CA1997
- Predicting forage indigestible NDF from lignin concentration.
*J. Anim. Sci.*1998; 76: 1469-1480 - Rumen balance and rates of fiber digestion.in: Proc. Cornell Nutr. Conf. Feed Manuf, Cornell University, Ithaca, NY2000: 150-166
- Standardization of in situ techniques for ruminant feedstuff evaluation.
*J. Anim. Sci.*1998; 76: 2717-2729 - Degradability characteristics of dry matter and crude protein of forages in ruminants.
*Anim. Feed Sci. Technol.*1996; 57: 291-311 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.

## Article info

### Publication history

### Identification

### Copyright

### User license

Elsevier user license |## Permitted

### For non-commercial purposes:

- Read, print & download
- Text & data mine
- Translate the article

## Not Permitted

- Reuse portions or extracts from the article in other works
- Redistribute or republish the final article
- Sell or re-use for commercial purposes

Elsevier's open access license policy