JDS
HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS
QUICK SEARCH:   [advanced]


     

This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Similar articles in this journal
Right arrow Similar articles in PubMed
Right arrow Alert me to new issues of the journal
Right arrow Download to citation manager
Right arrow reprints & permissions
Citing Articles
Right arrow Citing Articles via HighWire
Right arrow Citing Articles via Google Scholar
Google Scholar
Right arrow Articles by St-Pierre, N. R.
Right arrow Search for Related Content
PubMed
Right arrow PubMed Citation
Right arrow Articles by St-Pierre, N. R.
J. Dairy Sci. 86:344-350
© American Dairy Science Association, 2003.

Reassessment of Biases in Predicted Nitrogen Flows to the Duodenum by NRC 2001

N. R. St-Pierre

Department of Animal Sciences The Ohio State University, Columbus 432101

Corresponding author:
N. R. St-Pierre; e-mail:
st-pierre.8{at}osu.edu.


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
The appearance of numerous plots in recent literature from which the residuals are plotted against observed values (Y) to assess a model’s potential bias raises this question: should residuals be regressed against Y or against predicted values (Y)? The answer requires knowing the expected relationship under the assumption of an unbiased model. The objectives of this paper are: 1) to derive the expected relationship between residuals, Y, and Y; 2) to determine whether Y or Y should be used for the assessment of bias; and 3) to reassess the extent of mean and linear bias in the prediction of N flows to the duodenum by the NRC (2001). In the simplest case, we can assume a true model of the form Y = Xß + {varepsilon}. This model is estimated by Y = Xb + e, and Y = Xb. The correlation between the residual vector e and the vector of observations Y can easily be derived. The numerator of the correlation coefficient is shown to be equal to e'e, the residual sum of squares. The denominator of this correlation is equal to the square root of e'e multiplied by the total sum of squares. Algebraic simplifications show that the correlation between e and Y is equal to the square root of (1-R2). That is, under the assumption of an unbiased model, the residuals are correlated with the observed values and the slope of e regressed on Y is equal to (1-R2). Thus, a graph of e versus Y will show a positive slope between e and Y unless the model is a perfect predictor (i.e., R2 is equal to 1.0). Significant slopes linking e to Y have been erroneously interpreted as evidence of biased models in the NRC (2001). Conversely, the slope of e regressed on Y is expected to be zero under the assumption of an unbiased model. Therefore, residuals should be regressed against Y and not Y. When Y, as opposed to Y, was used to assess biases in the prediction of flows to the duodenum of microbial N, nonammonia-nonmicrobial N and nonammonia N in NRC (2001), mean biases became nonsignificant and linear biases over the range of predicted values are of the same magnitude or smaller than the standard errors of measurements reported in literature. Thus, although N flow predictions from NRC (2001) may not be precise, they appear to have insignificant and inconsequential biases.

Key Words: prediction bias • model evaluation • National Research Council • accuracy

Abbreviation key: MN = microbial nitrogen, MSS = model sum of squares, NANMN = nonammonia-nonmicrobial nitrogen, RSS = residual sum of squares, TSS = total sum of squares


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
The evaluation and validation of models, whether empirical or mechanical, require an assessment of prediction bias generally presented in the form of residual plots. Numerous residual plots have appeared in recent literature (NRC, 2001; Ruiz et al., 2001; Haig et al., 2002) from which residuals are plotted or regressed against observed values (Y) to assess linear bias even though statisticians (Draper and Smith, 1988) state that residuals should be plotted against predicted values (Y). It has long been known that residuals are correlated to observed values of the dependent variable in linear models. This is clearly stated in numerous statistical textbooks (e.g., Draper and Smith, 1988), although the proof of this relationship is not provided.

The NRC (2001) presents important plots of residuals against Y (Figures 5-6, 5-7, and 5-8) for the prediction of flows of three N pools to the duodenum. In most instances, there are clear negative slopes in these graphics, leading many to conclude that the NRC predictions have linear biases. The NRC (2001) stated that "the degree of the negative slope-bias that is evident in the residual plots are of concern" (NRC, 2001 p. 65). The report concluded that "errors in the structure of the model are probably major contributors to the negative slope biases" (NRC, 2001 p. 65). The objectives of this paper are 1) to show that a negative slope is expected when negative residuals are plotted against observed values if the predictions are truly unbiased, and 2) that the NRC predictions of N flows to the duodenum have small biases relative to measurement errors.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
The Expected Relationship Between Residuals and Observed Values
Let Yi be the ith observed value and Yi be the ith predicted value. Residuals ei are defined as ei = Yi - Yi, that is the difference between the observed and the predicted value. The NRC (2001) and others defined the residuals = Yi - Yi. Thus and the rest of this derivation could be expressed in terms of the. The traditional expression of residuals in use in the field of statistics was chosen here. The expected relationship between ei and Yi under the conditions of an unbiased model must be established prior to making judgment that a certain pattern of residuals is symptomatic of biased predictions. That is, we must establish the expectation for the slope of the regression of ei on Yi, and determine whether this expectation is equal to zero under unbiasedness. Likewise, we need to establish the expected value of reY, the simple correlation coefficient between residuals and observed values, when the model is unbiased. As the following mathematical derivation will show, even in the simplest case, that of a simple linear model, the slope of the regression of ei on Yi and the correlation coefficient reY are not equal to zero. Similar conclusions can be reached for more complex model structures although the algebra gets very complex.

The correlation between ei and Yi is derived as follows. In the general case, the Pearson product-moment correlation r between two random variables U and W is given by:


([1])

where and are the means of the two random variables. Thus, the correlation between ei and Yi is given by:


([2])

where and are the means of residuals and of observed values respectively. A linear model is expressed as:


([3])

where Y is a column vector of observed values, X is the matrix of independent variables, b is the vector of parameter estimates, and e is the column vector of residuals. The predicted values, Y, are calculated as:


([4])

The numerator in [2]Go can be simplified as follows:


([5])

because = 0 under the assumption of an unbiased model. That is, if the model is unbiased, then the mean residual (error) must be equal to zero. As we proceed with the multiplication of the factors in [5]Go, the {Sigma}ei part reduces to:


([6])

where n is the number of observations. Because = {Sigma}ei/n and = 0 under the assumption of an unbiased model, then {Sigma}ei must also be equal to zero. Thus,


([7])

Consequently, [5]Go reduces to


([8])

In matrix notation,


([9])

where e' is the transpose (row vector) of the column vector e defined in [3]Go. Rearranging [3]Go,


([10])

The least squares estimates of b are unbiased and are found as:


([11])

where X' is the transpose of X, the design matrix defined in [3]Go and (X'X)-1 is the inverse of the X'X matrix. We define


([12])

noting that H is symmetric and idempotent (i.e., H'H = H). Using [11]Go and [12]Go, equation [10]Go becomes:


([13])


([14])


([15])

Factoring Y:


([16])

where I is the identity matrix, a symmetric and idempotent matrix.

Taking the transpose of [16]Go


([17])

Using [16]Go and [17]Go ,


([18])

Because both I and H are idempotent, I - H is also idempotent and, by definition of an idempotent matrix,


([19])

Thus, [18]Go simplifies to


([20])

and using [16]Go


([21])

and


([22])

due to the commutative property of vector multiplication. Combining [5]Go, [8]Go, [9]Go, and [22]Go:


([23])


([24])


([25])


([26])

ande'e is known as the residual sum of squares (RSS). The denominator in [2]Go can be greatly simplified. The first part of the denominator reduces to:

([27])

because = 0 under the assumption that the model is unbiased. Using matrix notation.


([28])

where e'e is again the RSS. The second half of the denominator in [2]Go, {Sigma}(Yi - )2 is, by definition, the total sum of squares (TSS). Finally, by definition, the model sum of squares (MSS) = TSS - RSS. Using [26]Go, [27]Go and the definition of MSS, RSS, and TSS, [2]Go can be expressed as:


([29])


([30])


([31])


([32])


([33])

Because R2 = MSS/TSS, [33]Go becomes:


([34])


([35])

Thus, under the assumption that the model is unbiased, a correlation is expected between the residuals and the observed values. This correlation is always positive and approaches 1 as R2 approaches 0. Even R2 considered to be high would lead to significant r eY. For example, reY = 0.32 when R2 = 0.9.

The slope (b1) of the regression of ei on Yi is easily found:


([36])

Using [26]Go and the definitions of MSS, RSS and TSS, equation [36]Go becomes:


([37])

([38])


([39])

This implies that for R2 < 1, there is a positive slope between ei and Yi; and the lower the R2, the greater the slope.

In [39]Go, the slope is for the regression of ei on Yi. Remember that NRC (2001) defined the residuals as -ei. Thus, under the assumption of unbiased prediction, a negative slope would be expected between the NRC residuals and the observed values. The magnitude of this slope would be equal to -(1 - R2) = R2 - 1. Clearly, biases of models predicting flows of nitrogenous compounds to the duodenum were not assessed properly. In nonlinear models with nonoptimized parameter estimates, the regression of ei on Yi can take a multitude of forms depending on the nature of the bias, if bias is present. The point is that the method of plotting (or regressing) residuals on observed values fails to properly identify biases with the simplest, most basic model (simple linear regression).

The Expected Relationship Between Residuals and Predicted Values
What can be said of the correlation of ei with the predicted values Yi? The numerator of the correlation reiY is:


([40])


([41])

because = 0 under the assumption of an unbiased model. Factoring,


([42])

([43])

because {Sigma}ei = 0 under the assumption of an unbiased model. This is because of = 0 and that = {Sigma}ei ÷ n. In matrix notation,


([44])

We know from [4]Go that


([45])

And from [11]Go that


([46])

Thus:


([47])


([48])

Also, from [16]Go


([49])

and from [17]Go


([50])

So, using [48]Go and [50]Go,


([51])


([52])

Factoring,


([53])

Because H is idempotent andI is the identity matrix,


Which results in:


([54])

Thus, the numerator of reiY is zero and the correlation between e i and Yi is zero. That is, the residuals are not correlated with the predictions, and the slope of ei regressed on Yi is zero if the model is unbiased. A positive or negative slope of ei on Yi is a test of biased prediction. This is why we should plot the residuals versus the predicted values and not the observed values.

Reevaluation of Biases in NRC Predictions of N Flows
Data in figures 5-6, 5-7 and 5-8 of NRC (2001) were digitized using a GTCO CalComp AccuTabII digitizer (Columbia, MD) set at a resolution of 0.05 mm, to recalculate observed, predicted and residual values for microbial nitrogen (MN), nonammonia-nonmicrobial nitrogen (NANMN), and nonammonia nitrogen (NAN) flows to the duodenum. For all three N flow variables, residuals were regressed against their predicted values according to the following model (expressed here for MN):


([55])

where


In [55]Go, the regressor is shifted and centralized to its mean value. This method of shifting the independent variable was extensively used by Harvey (1977; 1982) in the software LSML76 to test the interaction of a discrete (class) variable by a continuous variable. By centering the independent variable, the intercept term of the linear model is estimated at the mean value of the independent variable as opposed to a value of zero. The intercept term at the mean value of the regressor measures the overall prediction bias, also known as mean prediction bias. A t-test on the estimate of the intercept determines the statistical significance of this bias. The slope of the regression is an estimate of the linear prediction bias and a t-test is used to assess its significance. In instances where the slope of the regression of residuals on predicted values is significant, the magnitude of the bias within the range of the predicted values must be quantified. This is done by calculating the bias at the minimum and maximum levels of the predicted values. Equation [55]Go is used with the minimum and maximum predicted values as inputs. Then, the bias at minimum and maximum predictions is judged relative to the size of the standard error ({sigma}êi), or compared to the 95% confidence intervals of measurements reported in the literature.


   RESULTS AND DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
Table 1Go reports summary statistics regarding observed, predicted and residual values for MN, NANMN and NAN that are presented in figures 5-6, 5-7 and 5-8 of NRC (2001).


View this table:
[in this window]
[in a new window]
  		Table 1. Descriptive statistics of observed, predicted and residuals values of microbial N, nonammonia-nonmicrobial N (NANMN), and nonammonia N (NAN) flows to the duodenum.1
 
Microbial N Flow to the Duodenum
Figure 1Go shows the relationship between residuals MN (i.e., observed MN minus predicted MN) and observed MN. The slope of the regression of residuals on observed MN is 0.66 (±0.04). From [39]Go, one would expect this slope to equal (1-R2) if the predictions are unbiased. The actual R2 (the square of the simple Pearson correlation coefficient between observed and predicted values) is 0.28. Thus the slope of the regression of residuals on observed values would be 1 - 0.28 = 0.72 under the assumption of an unbiased model. The difference between 0.66 and 0.72 is due to (1) the nonlinear prediction model used by NRC, (2) random variance in estimation, and, (3) because the model is not optimized with respect to the residual values (i.e., the parameters used in the NRC model are not optimized with respect to the residual values—they are not least-squares estimates). The similarity of the slope and (1-R2) should be taken as evidence of an unbiased model.



View larger version (15K):
[in this window]
[in a new window]
  		Figure 1. Plot of observed minus predicted microbial N flow to the duodenum (MN) vs. observed MN. Data are from NRC (2001) figure 5Go–6Go. Both the intercept and the slope are significantly different from 0 (P < 0.01), leading to the erroneous conclusion that the model generating the prediction is biased.

 
Figure 2Go shows the relationship between residual MN and predicted MN, with the predicted values shifted so as to be centered at the mean predicted MN of 246 g/d. That is, the independent variable is predicted MN minus the mean predicted MN. As explained previously, this procedure results in an exact test for the mean bias. By shifting the independent variable, the two parameter estimates (slope and intercept) are orthogonal and, consequently, independent of each other. Thus, a test on the location of the intercept is independent from the slope estimate. Results showed an absence of mean bias (P = 0.73), but a significant linear bias (P = 0.02). The magnitude of this linear bias translates, however, to values of less than 27 g/d at the minimum (119 g/d) and 22 g/d at the maximum (352 g/d) predicted MN values. These are less than the standard error (66 g/d), and well within the 95% confidence interval of least-square means reported in the literature for MN measurements (Aldrich et al., 1993; Cunningham et al., 1994; Beauchemin et al., 1999).



View larger version (17K):
[in this window]
[in a new window]
  		Figure 2. Plot of observed minus predicted microbial N flow to the duodenum (MN) vs. predicted MN. Data are from NRC (2001) figure 5Go–6Go. The independent variable predicted MN was centered around the mean predicted value before the residuals were regressed on the predicted values. The mean bias (-1.4 g/d) is not significant. The linear bias (0.197), although significant statistically (P = 0.02) results in a maximum bias of less than 27 g/d over the full range of predicted values.

 
Nonammonia-Nonmicrobial N Flow to the Duodenum
The relationship between residual NANMN and observed NANMN is shown in Figure 3Go. The slope of the regression of residuals on observed NANMN is 0.45 (± 0.04). Again, this value is close to (1 - R2) where R2 equals 0.48. The regression of residuals on predicted NANMN values is shown in Figure 4Go. The mean bias is clearly nonsignificant (intercept = 0.2 ± 3.9, P = 0.97) whereas the linear bias (slope = -0.131 ± 0.05, P = 0.02) although significant, implies a bias of less than 20 g/d at the minimum (58.1 g/d) and of less than 21 g/d at the maximum (373.7 g/d) predicted NANMN values. These biases are less than the standard error (65 g/d) and within the 95% confidence range of measurements found in the literature (McCarthy et al., 1989; Cunningham et al., 1994).



View larger version (16K):
[in this window]
[in a new window]
  		Figure 3. Plot of observed minus predicted nonammonia-nonmicrobial N flow to the duodenum (NANMN) vs. observed NANMN. Data are from NRC (2001) figure 5-7. Both the intercept and the slope are significantly different from 0 (P < 0.01), leading to the erroneous conclusion that the model generating the prediction is biased.

 


View larger version (17K):
[in this window]
[in a new window]
  		Figure 4. Plot of observed minus predicted nonammonia-nonmicrobial N flow to the duodenum (NANMN) vs. predicted NANMN. Data are from NRC (2001) figure 5-7. The independent variable predicted NANMNMN was centered around the mean predicted value before the residuals were regressed on the predicted values. The mean bias (-0.2 g/d) is not significant. The linear bias (-0.13), although significant statistically (P = 0.02) results in a maximum bias of less than 21 g/d over the full range of predicted values.

 
Nonammonia N Flow to the Duodenum
Figure 5Go shows the relationship between residual NAN and observed NAN. The slope of the regression of residuals on observed NAN is 0.30 (± 0.04) compared to (1 - R2) = 0.41. The regression of residual NAN on predicted NAN is shown in Figure 6Go. The mean bias is nonsignificant (intercept = -8.9 ± 5.2, P = 0.09) whereas the linear bias (slope = -0.161 ± 0.044, P < 0.01) implies a bias of less than 44 g/d at the minimum (136.4 g/d) and of less than 49 g/d at the maximum (713.6 g/d) predicted NAN values. These biases are smaller than the standard error (83.7 g/d) and within the 95% confidence range of the measured values reported in the literature (McCarthy et al., 1989; Aldrich et al., 1993; Cunningham et al., 1994).



View larger version (16K):
[in this window]
[in a new window]
  		Figure 5. Plot of observed minus predicted nonammonia N flow to the duodenum (NAN) vs. observed NAN. Data are from NRC (2001) figure 5-8. Both the intercept and the slope are significantly different from 0 (P < 0.01), leading to the erroneous conclusion that the model generating the prediction is biased.

 


View larger version (18K):
[in this window]
[in a new window]
  		Figure 6. Plot of observed minus predicted nonammonia N flow to the duodenum (NAN) vs. predicted NAN. Data are from NRC (2001) figure 5-8. The independent variable predicted MN was centered around the mean predicted value before the residuals were regressed on the predicted values. The mean bias (-9 g/d) is not significant. The linear bias (-0.16), although significant statistically (P < 0.01) results in a maximum bias of less than 49 g/d over the full range of predicted values.

 

   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
Residual plots for the detection of mean and linear model biases should be based on predicted values for the X-axis. Under the assumption of an unbiased model, the slope of the regression between the residuals and the observed values is expected to be equal to 1-R2. A proper analysis of the residual values reported by NRC (2001) for MN, NANMN and NAN failed to detect any mean bias of prediction for the NRC model. Linear biases were in all instances small and of lesser magnitude over the full range of predicted values than reported errors of measurements. Thus, predictions of N flows to the duodenum using the NRC (2001) model appear to have small biases relative to the error of measurements. It is likely that future improvements in prediction properties will only be gained if methods of measurements and experimental designs are improved to yield much smaller errors of measurements.


   FOOTNOTES
 
1 Salaries and research support were provided by state and federal funds appropriated to the Ohio Agricultural Research and Development Center, The Ohio State University. Manuscript No. 21-02AS. Back

Received for publication June 17, 2002. Accepted for publication August 14, 2002.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 


Aldrich, J. M., L. D. Muller, G. A. Varga, and L. C. Griel Jr. 1993. Nonstructural carbohydrate and protein effects on rumen fermentation, nutrient flow, and performance of dairy cows. J. Dairy Sci. 76:1091–1105.[Abstract/Free Full Text]

Beauchemin, K. A., W. Z. Yang, and L. M. Rode. 1999. Effects of grain source and enzyme additive on site and extent of nutrient digestion in dairy cows. J. Dairy Sci. 82:378–390.[Abstract]

Cunningham, K. D., M. J. Cecava, and T. R. Johnson. 1994. Flow of nitrogen and amino acids in dairy cows fed diets containing supplemental feather meal and blood meal. J. Dairy Sci. 77:3666–3675.[Abstract]

Draper, N. R., and H. Smith. 1988. Pages 63–64 in Applied Regression Analysis. 3rd ed. John Wiley and Sons, Inc., New York, NY.

Haig, P. A., T. Mutsvangwa, R. Spratt, and B. W. McBride. 2002. Effects of dietary protein solubility on nitrogen losses from lactating dairy cows and comparison with predictions from the Cornell Net Carbohydrate and Protein System. J. Dairy Sci. 85:1208–1217.[Abstract]

Harvey, W. R. 1976. User’s Guide for LSML76. The Ohio State University, Columbus.

Harvey, W. R. 1982. Mixed model capabilities of LSML76. J. Anim. Sci. 54:1279–1285.[Abstract/Free Full Text]

McCarthy, R. D. Jr., T. H. Klusmeyer, J. L. Vicini, J. H. Clark, and D. R. Nelson. 1989. Effects of source of protein and carbohydrate on ruminal fermentation and passage of nutrients to the small intestine of lactating cows. J. Dairy Sci. 72:2002–2016.

National Research Council. 2001. Pages 65–67 in Nutrient Requirements of Dairy Cattle. 7th rev. ed. Natl. Acad. Sci., Washington, DC.

Ruiz, R., G. L. Albrecht, L. O. Tedeshi, G. Jarvis, J. B. Russell, and D. G. Fox. 2001. Effect of monensin on the performance and nitrogen utilization of lactating dairy cows consuming fresh forage. J. Dairy Sci. 84:1717–1727.[Abstract]


This article has been cited by other articles:


Home page
 J ANIM SCIHome page
J. P. McMeniman, P. J. Defoor, and M. L. Galyean
Evaluation of the National Research Council (1996) dry matter intake prediction equations and relationships between intake and performance by feedlot cattle
J Anim Sci, March 1, 2009; 87(3): 1138 - 1146.
[Abstract] [Full Text] [PDF]

Home page
 J ANIM SCIHome page
E. Kebreab, K. A. Johnson, S. L. Archibeque, D. Pape, and T. Wirth
Model for estimating enteric methane emissions from United States dairy and feedlot cattle
J Anim Sci, October 1, 2008; 86(10): 2738 - 2748.
[Abstract] [Full Text] [PDF]

Home page
 Am. J. Physiol. Regul. Integr. Comp. Physiol.Home page
R. C. Boston and P. J. Moate
A novel minimal model to describe NEFA kinetics following an intravenous glucose challenge
Am J Physiol Regulatory Integrative Comp Physiol, April 1, 2008; 294(4): R1140 - R1147.
[Abstract] [Full Text] [PDF]

Home page
 J DAIRY SCIHome page
P. C. J. van Vliet, J. W. Reijs, J. Bloem, J. Dijkstra, and R. G. M. de Goede
Effects of Cow Diet on the Microbial Community and Organic Matter and Nitrogen Content of Feces
J Dairy Sci, November 1, 2007; 90(11): 5146 - 5158.
[Abstract] [Full Text] [PDF]

Home page
 J DAIRY SCIHome page
S. K. R. Karnati, J. T. Sylvester, S. M. Noftsger, Z. Yu, N. R. St-Pierre, and J. L. Firkins
Assessment of Ruminal Bacterial Populations and Protozoal Generation Time in Cows Fed Different Methionine Sources
J Dairy Sci, February 1, 2007; 90(2): 798 - 809.
[Abstract] [Full Text] [PDF]

Home page
 J DAIRY SCIHome page
S. Seo, C. Lanzas, L. O. Tedeschi, and D. G. Fox
Development of a Mechanistic Model to Represent the Dynamics of Liquid Flow Out of the Rumen and to Predict the Rate of Passage of Liquid in Dairy Cattle
J Dairy Sci, February 1, 2007; 90(2): 840 - 855.
[Abstract] [Full Text] [PDF]

Home page
 J DAIRY SCIHome page
B. Vlaeminck, V. Fievez, S. Tamminga, R. J. Dewhurst, A. van Vuuren, D. De Brabander, and D. Demeyer
Milk odd- and branched-chain Fatty acids in relation to the rumen fermentation pattern.
J Dairy Sci, October 1, 2006; 89(10): 3954 - 3964.
[Abstract] [Full Text] [PDF]

Home page
 J DAIRY SCIHome page
P. J. Moate, R. C. Boston, I. J. Lean, and W. Chalupa
Short communication: further validation of the fat sub-model in the cornell-penn-miner dairy model.
J Dairy Sci, March 1, 2006; 89(3): 1052 - 1056.
[Abstract] [Full Text] [PDF]

Home page
 J DAIRY SCIHome page
J. L. Firkins, A. N. Hristov, M. B. Hall, G. A. Varga, and N. R. St-Pierre
Integration of Ruminal Metabolism in Dairy Cattle
J Dairy Sci, March 1, 2006; 89(e_suppl_1): E31 - E51.
[Abstract] [Full Text] [PDF]

Home page
 J DAIRY SCIHome page
T. D. Nennich, J. H. Harrison, L. M. VanWieringen, N. R. St-Pierre, R. L. Kincaid, M. A. Wattiaux, D. L. Davidson, and E. Block
Prediction and Evaluation of Urine and Urinary Nitrogen and Mineral Excretion from Dairy Cattle
J Dairy Sci, January 1, 2006; 89(1): 353 - 364.
[Abstract] [Full Text] [PDF]

Home page
 J ANIM SCIHome page
A. P. Schinckel
Critique of "Evaluation of procedures to predict fat-free lean in swine carcasses"
J Anim Sci, December 1, 2005; 83(12): 2719 - 2720.
[Full Text] [PDF]

Home page
 J DAIRY SCIHome page
T. D. Nennich, J. H. Harrison, L. M. VanWieringen, D. Meyer, A. J. Heinrichs, W. P. Weiss, N. R. St-Pierre, R. L. Kincaid, D. L. Davidson, and E. Block
Prediction of Manure and Nutrient Excretion from Dairy Cattle
J Dairy Sci, October 1, 2005; 88(10): 3721 - 3733.
[Abstract] [Full Text] [PDF]

Home page
 J DAIRY SCIHome page
A. N. Hristov, W. J. Price, and B. Shafii
A Meta-Analysis on the Relationship Between Intake of Nutrients and Body Weight with Milk Volume and Milk Protein Yield in Dairy Cows
J Dairy Sci, August 1, 2005; 88(8): 2860 - 2869.
[Abstract] [Full Text] [PDF]

Home page
 J DAIRY SCIHome page
H. G. Bateman II, J. H. Clark, and M. R. Murphy
Development of a System to Predict Feed Protein Flow to the Small Intestine of Cattle
J Dairy Sci, January 1, 2005; 88(1): 282 - 295.
[Abstract] [Full Text] [PDF]

Home page
 J DAIRY SCIHome page
W. P. Weiss and D. J. Wyatt
Macromineral Digestion by Lactating Dairy Cows: Estimating Phosphorus Excretion via Manure
J Dairy Sci, July 1, 2004; 87(7): 2158 - 2166.
[Abstract] [Full Text] [PDF]

Home page
 J DAIRY SCIHome page
A. N. Hristov, W. J. Price, and B. Shafii
A Meta-Analysis Examining the Relationship Among Dietary Factors, Dry Matter Intake, and Milk and Milk Protein Yield in Dairy Cows
J Dairy Sci, July 1, 2004; 87(7): 2184 - 2196.
[Abstract] [Full Text] [PDF]

This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Similar articles in this journal
Right arrow Similar articles in PubMed
Right arrow Alert me to new issues of the journal
Right arrow Download to citation manager
Right arrow reprints & permissions
Citing Articles
Right arrow Citing Articles via HighWire
Right arrow Citing Articles via Google Scholar
Google Scholar
Right arrow Articles by St-Pierre, N. R.
Right arrow Search for Related Content
PubMed
Right arrow PubMed Citation
Right arrow Articles by St-Pierre, N. R.

HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS