# Sensitivity analysis of the INRA 2018 feeding system for ruminants by hybrid local and global approaches: Comparing the contribution of dietary input variables to multiple response prediction in dairy cattle

Seoyoung Jeon, Tristan Senga Kiessé, Sophie Lemosquet, Pierre Nozière

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Seoyoung Jeon, Tristan Senga Kiessé, Sophie Lemosquet, Pierre Nozière. Sensitivity analysis of the INRA 2018 feeding system for ruminants by hybrid local and global approaches: Comparing the contribution of dietary input variables to multiple response prediction in dairy cattle. Journal of Dairy Science, 2025, 108 (1), pp.527-537. 10.3168/jds.2024-25297. hal-04874121

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# Sensitivity analysis of the INRA 2018 feeding system for ruminants by hybrid local and global approaches: Comparing the contribution of dietary input variables to multiple response prediction in dairy cattle

Seoyoung Jeon,<fnref n="1" /> ◐ Tristan Senga Kiessé,<fnref n="2" /> ◐ Sophie Lemosquet,<fnref n="3" /> ◐ and Pierre Nozière<fnref n="1" />* ◐

## ABSTRACT

We conducted sensitivity analysis (SA) of the French National Institute of Agricultural Research (Institut national de la recherche agronomique; INRA) 2018 feeding system for ruminants applied to dairy cows. We evaluated which dietary input variables contribute most to changes in each output variable, considering the potential interactions presence among input variables. We applied a quantitative analysis (one-at-a-time analysis, OAT; i.e., local SA) and a relative comparative analysis (global SA; GSA) through variance-based SA considering potential interactions and nonmonotonicity. The 5 likely influential dietary input variables were selected: CP, gross energy (GE), OM apparent digestibility (OMd), effective degradability of nitrogen assuming a passage rate of 6%/h, and true intestinal digestibility of nitrogen. The sensitivity of 5 selected animal responses (output variables) to input variables was analyzed: DMI, milk protein yield (MPY), energy in methane (ECH4), nitrogen utilization efficiency (NUE), and the ratio between urine and total N excretion (UN/TN). Six diets for dairy cattle, reflecting the diversity of diets commonly used in practice, were formulated to meet 95% of the potential milk production (37.5 kg/d) of a multiparous dairy cow at wk 14 of lactation. For each diet, the 5 input variables were randomly sampled around the INRA 2018 feed table values (reference point), and the animal responses around this reference situation were calculated using the rationing software INRAtion V5. In OAT, the sensitivity of animal responses was quantified by calculating the normalized tangent value at the reference point, and in GSA, the Sobol indices were calculated for relative influence of each input and their interaction. The influence of the 5 key input variables on the 5 main animal

responses predicted from the INRA feeding system was consistent across both SA approaches. With the 6 diets, GE and Omd appeared as the main contributors to changes in DMI, MPY, ECH4, and NUE. Crude protein was the main contributor to changes in UN/TN and another major contributor to changes in NUE. When considering OAT, the sensitivity of outputs showed differences depending on diet, more particularly for DMI and MPY. With grass hay-based diets (GH), DMI was less sensitive and MPY was more sensitive to variations in input variables than other diets. When considering GSA, interactions between input variables were also noticeable for DMI and MPY; the interactions were high with the GH diets for DMI, and with fresh ryegrass and grass silage diets for MPY. On the other hand, for MPY, the non-GH diets were less sensitive to variations in input variables, and the interaction between inputs was higher than with GH diets. Our results support the hypothesis that MPY, unlike DMI, is more responsive to energy-related factors at a high true intestinal digestible protein (PDI) to net energy for lactation (UFL) ratio (e.g., GH diets >117 g PDI/UFL), than at a lower PDI/UFL ratio. Hence, hybridizing the SA methods can help to interpret the system and facilitate a more precise evaluation thereof, especially GSA, which is amenable to nonmonotonic models such as those characterizing complex feeding systems integrating multiple nutritional and animal factors.

**Key words:** INRAtion V5, feeding system, dairy cattle, sensitivity analysis

## INTRODUCTION

The French national feeding system for ruminants, developed by the French National Institute of Agricultural Research (INRA) in 1978, is widely used in France and in other European countries. The system has been continuously updated since 1978, and the most recent version was released in 2018 (INRA, 2018) with the updated rationing software (i.e., INRAtion V5). For dairy cows, the

<footnote n="1">¹UMR Herbivores, INRAE, VetAgro Sup, 63122 Saint-Genès-Chamanelle, France</footnote>

<footnote n="2">²UMR SAS, INRAE, Institut Agro, 35000 Rennes, France</footnote>

<footnote n="3">³UMR PEGASE, INRAE, Institut Agro, 35590 Saint-Gilles, France</footnote>
<table><caption>**Table 1.** Ingredients and nutritional composition of reference diets for multiparous cows<fnref n="1" />,<fnref n="2" /></caption><thead><tr><th rowspan="2">Item</th><th colspan="6">Diet<fnref n="3" /></th></tr><tr><th>RF</th><th>GH1</th><th>GH2</th><th>GH3⁴</th><th>GS</th><th>CS</th></tr></thead><tbody><tr><td colspan="7">Diet ingredient (% DM)</td></tr><tr><td>Fresh perennial ryegrass</td><td>72.9</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Permanent grassland hay (first growth)</td><td>—</td><td>54.2</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Permanent grassland hay (second growth)</td><td>—</td><td>—</td><td>84.2</td><td>52.9</td><td>—</td><td>—</td></tr><tr><td>Permanent grassland silage</td><td>—</td><td>—</td><td>—</td><td>—</td><td>62.1</td><td>—</td></tr><tr><td>Corn silage</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>75.6</td></tr><tr><td>Barley</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>10.8</td></tr><tr><td>Corn</td><td>23.6</td><td>24.7</td><td>15.8</td><td>28.6</td><td>33.7</td><td>—</td></tr><tr><td>Soybean meal</td><td>3.6</td><td>—</td><td>—</td><td>—</td><td>4.2</td><td>13.6</td></tr><tr><td>Rapeseed meal</td><td>—</td><td>21.1</td><td>—</td><td>18.6</td><td>—</td><td>—</td></tr><tr><td>Predicted DMI (kg/d)</td><td>21.4</td><td>25.4</td><td>22.0</td><td>24.9</td><td>22.6</td><td>22.6</td></tr><tr><td>Predicted MPY<fnref n="5" /> (g/d)</td><td>1,067</td><td>1,073</td><td>1,054</td><td>1,072</td><td>1,053</td><td>1,045</td></tr><tr><td colspan="7">Nutritive values<fnref n="6" /></td></tr><tr><td>OM (g/kg DM)</td><td>926</td><td>937</td><td>922</td><td>939</td><td>928</td><td>955</td></tr><tr><td>CP (g/kg DM)</td><td>143</td><td>150</td><td>183</td><td>150</td><td>140</td><td>141</td></tr><tr><td>NDF (g/kg DM)</td><td>414</td><td>443</td><td>486</td><td>446</td><td>389</td><td>345</td></tr><tr><td>ADF (g/kg DM)</td><td>199</td><td>239</td><td>245</td><td>237</td><td>207</td><td>175</td></tr><tr><td>EE (g/kg DM)</td><td>36.8</td><td>25.4</td><td>30.7</td><td>26.1</td><td>33.8</td><td>27.9</td></tr><tr><td>GE (kcal/kg DM)</td><td>4,339</td><td>4,448</td><td>4,527</td><td>4,456</td><td>4,481</td><td>4,495</td></tr><tr><td>UFL (/kg DM)</td><td>0.98</td><td>0.81</td><td>0.93</td><td>0.83</td><td>0.94</td><td>0.93</td></tr><tr><td>OMd (%)</td><td>73.9</td><td>61.3</td><td>70.2</td><td>62.6</td><td>69.8</td><td>68.3</td></tr><tr><td>ED6_N (%)</td><td>66.3</td><td>58.2</td><td>63.1</td><td>58.7</td><td>64.5</td><td>62.5</td></tr><tr><td>dr_N (%)</td><td>86.0</td><td>79.3</td><td>86.9</td><td>80.6</td><td>82.7</td><td>83.5</td></tr><tr><td>PDI (g/kg DM)</td><td>94.5</td><td>97.6</td><td>109</td><td>98.0</td><td>89.3</td><td>92.6</td></tr><tr><td>PDI/UFL (g PDI/UFL)</td><td>96.4</td><td>121</td><td>117</td><td>119</td><td>94.6</td><td>99.2</td></tr><tr><td>LysDI (% PDI)</td><td>6.68</td><td>6.55</td><td>6.58</td><td>6.54</td><td>6.49</td><td>6.76</td></tr><tr><td>MetDI (% PDI)</td><td>2.01</td><td>2.07</td><td>1.86</td><td>2.04</td><td>2.03</td><td>1.87</td></tr><tr><td colspan="7">Nutritional balance</td></tr><tr><td>UFL balance (UFL/d)</td><td>0.01</td><td>-0.58</td><td>-0.70</td><td>-0.56</td><td>0.45</td><td>0.15</td></tr><tr><td>PDI efficiency (g/g)</td><td>0.73</td><td>0.65</td><td>0.61</td><td>0.65</td><td>0.75</td><td>0.72</td></tr></tbody></table>

system is based on net energy for lactation (i.e., “Unité Fourragère Lait” [UFL]; 1 UFL = 1.76 Mcal of NEL), MP (protein truly digestible in the intestine; PDI) including effects of amino acids digestible in the intestine, and fill units (i.e., “Unité d’Encombrement Lait” [UEL]). These units describe both feed characteristics and the animal’s requirements and intake capacity, and are used to predict animal’s responses to diets. Regarding feed characterization, the INRA 2018 feeding system requires information on the chemical composition, rumen degradability, and total-tract apparent and intestinal true digestibilities. In previous work (Jeon et al., 2024), we quantified how intake as well as the digestive, productive, and environmental responses of dairy cows (output variables) predicted by the INRA 2018 feeding system are affected

by variation in feed characteristics (input variables). In the present study, we evaluated which input variables in feedstuffs contribute most to changes in each output variable, considering the potential presence of interactions among input variables.

For such a sensitivity analysis (SA), 2 main complementary methods could be adopted. The one-at-a-time (OAT) method consists in changing only one target input variable at a time around a reference value (as known as local analysis), in order to quickly identify groups of influential and noninfluential input variables, considered as independent. Likewise, global SA (GSA) consists in changing input variables in a variation range to quantify their relative contributions to output variation, with the possibility of taking into account interactions among

<footnote n="1">¹Second parity; BW 608 kg; 50 mo of age; wk 14 of lactation; 10,000 kg of milk yield per year.</footnote>

<footnote n="2">²This table is reproduced from Jeon et al. (2024; creative commons BY license).</footnote>

<footnote n="3">³RF = fresh perennial ryegrass-based diet; GH1 = first growth permanent grassland hay-based diet; GH2 = second growth permanent grassland hay-based diet; GH3 = first growth and second growth permanent grassland hay-based diet; GS = permanent grassland silage-based diet; CS = corn silage-based diet.</footnote>

<footnote n="4">⁴GH3 diet contained a fixed amount (3 kg) of first growth permanent grassland hay.</footnote>

<footnote n="5">⁵MPY = milk protein yield.</footnote>

<footnote n="6">⁶EE = ether extract; GE = gross energy; UFL = net energy for lactation ("Unité fourragère Lait"; 1 UFL = 1.76 Mcal of NEL); OMd = OM apparent digestibility; ED6_N = effective degradability of nitrogen assuming a passage rate of 6%/h; dr_N = true intestinal digestibility ("digestibilité réelle") of nitrogen; PDI = protein truly digestible in the intestine; LysDI = lysine truly digestible in the intestine; MetDI = methionine truly digestible in the intestine.</footnote>
inputs. In the animal nutrition field, the OAT has been applied to several feeding systems for ruminants, such as Norfor (Volden, 2011) and INRA 2018 (Jeon et al., 2024). A linear model-based GSA has been performed on *Nutrient Requirements of Beef Cattle* (NASEM, 2016) and *Nutrient Requirements of Dairy Cattle* (NASEM, 2021), as well as the Cornell Net Carbohydrate and Protein System (Fox et al., 2004). A variance-based GSA method (Sobol analysis) has been applied by Cadero et al. (2018), Garcia-Launay et al. (2018), and Lautrou et al. (2020) to evaluate a system in swine nutrition. Because of their respective advantages and disadvantages, in other fields (e.g., environmental engineering), some attempts have also been made to combine both OAT and GSA approaches (Haaker and Verheijen, 2004; Chakraborty and Chowdhury, 2017). However, there have been few such attempts, especially in the field of ruminant nutrition. This study thus aims to evaluate, by applying both OAT and GSA methods, the respective impact of feed input variables on the output variables (i.e., the animal's responses) predicted by the INRA 2018 feeding system for dairy cattle.

## MATERIALS AND METHODS

In the present work, an SA was conducted to analyze which dietary input variables mainly affect an animal's multiple responses (intake, production, efficiency, and waste), by successively implementing OAT and GSA methods. The following input variables selected by Jeon et al. (2024) as being key variables at the feedstuff level were considered: CP (g/kg DM), gross energy (GE, kcal/kg DM), OM apparent digestibility (OMd, %), effective degradability of nitrogen assuming a passage rate of 6%/h (ED6_N, %), and true intestinal digestibility of nitrogen (dr_N, i.e., “digestibilité réelle,” %). The following key output variables from INRAtion V5 were considered: DMI (kg/d), milk protein yield (MPY, g/d), energy in methane (ECH4, kcal/kg DM), nitrogen utilization efficiency (NUE, N in milk as proportion of intake N, g/g), and ratio between urine and total N excretion (UN/TN, g/g). The OAT analysis, changing one input variable at a time, was first performed to evaluate the absolute quantitative changes, including direction (increase or decrease) in the outputs as a function of changes in inputs around the reference situation. A GSA method (Sobol, 1993), changing all input variables simultaneously, was then applied to measure the relative contribution of each key input variable (and its interactions) to the animal's multiple responses considered. Thus, both methods are expected to provide complementary information (quantitative and relative) on the effects of dietary input variables on the animal's multiple responses.

## Reference Diets for Sensitivity Analysis

Diets designed to reach 95% of the potential milk yield (37.5 kg/d of milk yield and 1,121 g/d of MPY) of a reference animal (multiparous cow at 14 wk of lactation) are called reference diets. Six reference diets, to cover a wide range of diets commonly used on dairy farms in France, were designed using INRAtion V5, with forages (fresh perennial ryegrass, first and second growth permanent grassland hay, permanent grassland silage, and corn silage), energy concentrates (barley and corn), with or without protein concentrates (soybean meal and rapeseed meal), as described by Jeon et al. (2024). The composition of the 6 reference diets is shown in Table 1: diets based on fresh perennial ryegrass (RF); first-growth (GH1), second-growth (GH2), and first and second-growth (GH3) permanent grassland hay; permanent grassland silage (GS); corn silage (CS). The proportion of each feed ingredient in the reference diet was maintained throughout the SA.

## One-at-a-Time Sensitivity Analysis: Normalized Tangent at the Reference Situation

The feed table for OAT analysis with INRAtion V5 was constructed as follows. For each feedstuff, random values for the 5 key input variables (i.e., CP, GE, OMd, ED6_N, and dr_N) were sampled assuming 10% variation (or actual observed variation, when given by BIPEA, 2022) around the reference value (feed table from INRA, 2018) through Latin hypercube sampling. For each key input variable, 50 random samples were created assuming a normal distribution (n = 50, respectively), the values of the other 4 key input variables being fixed to the tabulated reference values (INRA, 2018). The values of each input variable of the feeds composing the diet were sorted in ascending order, so that all the feedstuffs with the lowest (highest) value composed the diet with the lowest (highest) value of the target input variable. All other variables describing the feed value (e.g., UFL, PDI, UEL) were then calculated using PrevAlim (the application for prediction of feed value included in INRAtion V5 software).

To absolutely quantify the influence of each input variable (i) on each output variable (j), the normalized tangent (NTij) was used (Hughes et al., 2013), as follows:

$$NT_{ij}(\%) = \frac{\Delta y_j}{\Delta x_i} \times \frac{Avg(x_{1i}, x_{2i})}{Avg(y_{1j}, y_{2j})} \times 100 = \frac{y_{2j} - y_{1j}}{x_{2i} - x_{1i}} \times \frac{Avg(x_{1i}, x_{2i})}{Avg(y_{1j}, y_{2j})} \times 100,$$
where $x_{2i}$ represents the closest value larger than the reference value $x$ and $y_{2j}$ is the corresponding response, $x_{1i}$ represents the closest value smaller than the value $x$, and $y_{1j}$ is the corresponding response. The $NT_{ij}$ provides information on the sensitivity of the model output to the variation of an input variable around the reference point $(x_i, y_j)$, which was conducted by the OAT method. For a given $NT_{ij}$, the sign may vary depending on the diet; thus, its absolute value ($|NT_{ij}|$) was considered to assess the magnitude (mean) and the variability (SE) of this indicator. For example, an $NT_{CP,DMI}$ of -10% for a diet at 150 g CP/kg DM and 20 kg DMI/d at the reference point can be interpreted as follows: as CP increases by 1% (+1.50 g/kg DM), DMI would decrease by 10% (-2.0 kg/d).

All statistical analyses were conducted using R software (version 4.1.3; R Core Team, 2022). The sensobol package of R was used to calculate Sobol indices (Puy et al., 2021). The Sobol matrix was created in 'QRN' type (n = 10,000) so that the same matrix was generated every run. Under the assumption that the input variables are independent (Saltelli, 2002), random numbers were generated based on the mean value ± 5.5% (average variation for feedstuffs used in this study) in the normal distribution of each input variable and assigned to the Sobol matrix. However, to stay in the validity range of the previous model of fill value of corn silage, a 3.0% variation was applied only to the CS diet. The R scripts and example files for SA used in this study can be found in the supplemental materials (see Notes).

## Global Sensitivity Analysis: Sobol Indices

Sobol indices (Sobol, 1993) were used to quantify the relative influence of each key input variable (i.e. CP, GE, OMd, ED6_N, and dr_N) on the outputs (i.e. DMI, MPY, ECH4, NUE, and UN/TN) of the INRAtion V5, considering all the variation range of the response law (not only close to the reference situation). The individual effect an input variable (i) has on an output variable (j) was measured by the first-order index ($S_i^{[j]}$) given by

$$S_i^{[j]} = \frac{V_i}{Var(y_j)} \times 100,$$

where $V_i$ is the variation of $x_i$ and $Var(y_j)$ is the variation of output $y_j$.

The total contribution of $x_i$ to $y_j$, when including its interactions with other inputs $x_{k \neq i}$, was also measured by the total Sobol index, $ST_i^{[j]}$. For instance, when considering inputs $x_1, x_2, x_3$, the total contribution of $x_1$ to $y_j$ was given by

$$ST_1^{[j]} = S_1^{[j]} + S_{1,2}^{[j]} + S_{1,3}^{[j]} + S_{1,2,3}^{[j]},$$

where $S_{1,2}^{[j]}, S_{1,3}^{[j]}$, and $S_{1,2,3}^{[j]}$ are indices that measure the contribution of interactions between the pairs $(x_1, x_2)$ and $(x_1, x_3)$ and the triplet $(x_1, x_2, x_3)$, respectively, to the variation of $y_j$. For example, if the $ST_{OMd}^{[MPY]}$ is 90% in a diet where MPY varies by 100 g/d, this means that variation in OMd contributed to a variation of 90 g MPY/d. The difference between the sum of total indices and the sum of first-order indices ($\sum ST_i^{[j]} - \sum S_i^{[j]}$) reflects the contribution of interactions among the input variables to output variation.

## RESULTS

### Dry Matter Intake

**OAT.** The sum of $|NT_{i,DMI}|$ (i.e., the degree to which DMI depends on the variability of all input variables $i$, based on OAT) was much lower for grass hay (GH)-based diets (GH1, GH2, and GH3) than for the other diets (i.e., RF, GS, and CS; Figure 1a). For these 3 diets, the input variable that had the most influence on DMI was GE, followed by OMd and ED6_N. For all diets, increases in CP and dr_N increased DMI, and increases in GE and ED6_N decreased DMI. The increase in OMd acted as a decreasing factor for RF and GS, but as an increasing factor for the other 4 diets.

**GSA.** The $ST_i^{[DMI]}$ (i.e., the degree to which each input variable $i$ contributes to the variation of DMI when changing all input variables simultaneously while considering the interaction between input variables) revealed that GE and OMd had a major influence on DMI (Figure 1b). The influence of CP was low, averaging less than 6% in all diets except GH2 ($ST_{CP}^{[DMI]}$ = 33% for GH2). For all diets, the relative influence of ED6_N and dr_N appeared low. The GH diets had higher interactions between input variables than the other 3 diets, which was mainly the result of GE and OMd interacting with other input variables (Table 2).

### Milk Protein Yield

**OAT.** The relative change in MPY according to the variation of the 5 input variables was the highest with the GH diets ($\sum |NT_{i,MPY}| > 230\%$, Figure 2a). The input variable with the highest influence on MPY on average was OMd (average $NT_{OMd,MPY}$ = 98%), but in the GS diet, it was ED6_N. The other most influential input variables in GH diets were GE and CP, but it was not consistent in non-GH diets. In all diets, as CP, OMd, and dr_N in-
creased, MPY also increased, and as ED6_N increased, MPY also decreased. In addition, MPY increased as GE increased with all diets except GS.

**GSA.** When the 5 input variable values were changed simultaneously, most of the variation in MPY (73% on average for the 6 diets, up to 89% with the CS diet) occurred mainly due to variations in OMd (Figure 2b). The second input variable with the highest contribution varied depending on the diet. In the GH diets, it was GE (average of ST<sub>GE</sub><sup>[MPY]</sup> = 24.4%), the contribution of the other input variables being very low (less than 2% each). In contrast, with other diets, the contributions of ED6_N, CP, and GE were non-negligible. The interaction between input variables occurred mainly with RF and GS, and it primarily resulted from GE and OMd interacting with other input variables (Table 2).

## Energy in Methane

**OAT.** The relative change in ECH4 according to the variation of the 5 input variables was slightly lower with GH diets (Σ|NT<sub>i,ECH4</sub>| = 119 ± 6.2%) than with non-GH diets (Σ|NT<sub>i,ECH4</sub>| = 161 ± 10.8%, Figure 3a). In all diets, OMd was the main input variable involved in ECH4 variation (average of NT<sub>OMd,ECH4</sub> = 103 ± 3.8%). With the non-GH diets, GE was the second highest contributor (NT<sub>GE, ECH4</sub> >27.6 ± 2.81%). On average, the influence of CP, ED6_N, and dr_N on variations in ECH4 was low (average of |NT<sub>CP,ECH4</sub>| = 7.4 ± 3.27%, NT<sub>ED6_N,ECH4</sub> = 9.5 ± 2.49%, and NT<sub>dr_N,ECH4</sub> = -3.4 ± 1.13%), but NT<sub>CP,ECH4</sub> was relatively high with the GH2 diet (22%). With all diets, increase in OMd, GE, and ED6_N increased ECH4, whereas increase in dr_N decreased ECH4. The effect of CP on ECH4 was positive with all diets except for GH1 and GS.

**GSA.** When the 5 input variables were changed simultaneously, OMd was the main variable involved in variations in ECH4 (ST<sub>OMd</sub><sup>[ECH4]</sup> >90% in all diets, Figure 3b), the ST<sub>i</sub><sup>[ECH4]</sup> value of the other 4 input variables being less than 10%. We observed no significant interaction between input variables in variations in ECH4, irrespective of the diet.

## Nitrogen Utilization Efficiency

**OAT.** Variations in NUE according to the variation of the 5 input variables were comparable among diets (Σ|NT<sub>i,NUE</sub>| = 290 ± 9.4%) and were mainly influenced by OMd, GE, and CP, irrespective of the diet (average of NT<sub>OMd,NUE</sub> = 108 ± 6.4%, NT<sub>GE,NUE</sub> = 86 ± 7.4%, and NT<sub>CP,NUE</sub> = -84 ± 4.5%; Figure 4a). With all diets, when CP content increased, NUE decreased. Conversely, when GE or OMd increased, NUE also increased. Values for NT<sub>ED6_N,NUE</sub> and NT<sub>dr_N,NUE</sub> were negative and positive,

<figure><img src="image_3.png" /></figure>

**Figure 1.** Results of one-at-a-time analysis (a; normalized tangent, %) and global sensitivity analysis (b) for DMI (kg/d). In the graph (b), the left axis represents total Sobol indices (bar chart; ST<sub>i</sub><sup>[j]</sup>), and the sum of first-order indices (black dots, ●; ΣS<sub>i</sub><sup>[j]</sup>). RF = fresh perennial ryegrass-based diet; GH1 = first growth permanent grassland hay-based diet; GH2 = second growth permanent grassland hay-based diet; GH3 = first growth and second growth permanent grassland hay-based diet; GS = permanent grassland silage-based diet; CS = corn silage-based diet. CP (blue filled square) = crude protein; GE (right diagonal line) = gross energy; OMd (left diagonal line) = organic matter digestibility; ED6_N (horizontal stripes) = effective degradability of nitrogen assuming a passage rate of 6%/h; dr_N (dotted) = true intestinal digestibility of nitrogen.

respectively, but were notably low, particularly in the GH diets.

**GSA.** When all input variables were changed simultaneously, OMd, CP, and GE were also the main contributors to variations in NUE with an average contribution of 47% for ST<sub>OMD</sub><sup>[NUE]</sup>, 28% for ST<sub>CP</sub><sup>[NUE]</sup>, and 24% for ST<sub>GE</sub><sup>[NUE]</sup> (Figure 4b). Irrespective of the diet, we found no difference in the Sobol index and no remarkable interaction between input variables (average interaction = 1.5%).

## Urinary Nitrogen Excretion per Total Nitrogen Excretion

**OAT.** The proportions of N excreted in urine among total N excretion was less affected by variation in input variables with the GH2 diet than with the other 5
<table><caption>**Table 2.** The effect of interactions among the input variables calculated through global analysis on dry matter intake and milk protein yield; values are the difference between the sum of total indices and the sum of first-order indices ($\sum ST_i^{[j]} - \sum S_i^{[j]}$)</caption><thead><tr><th rowspan="2">Item<sup>2</sup></th><th colspan="7">Diet<sup>1</sup></th></tr><tr><th>RF</th><th>GH1</th><th>GH2</th><th>GH3</th><th>GS</th><th>CS</th><th>Average</th><th>SD</th></tr></thead><tbody><tr><td>DMI</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Interaction (%)</td><td>8.6</td><td>61.8</td><td>46.3</td><td>57.4</td><td>4.5</td><td>8.9</td><td>31.2</td><td>26.74</td></tr><tr><td>Contribution (%) to interaction</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>CP</td><td>1.4</td><td>2.6</td><td>1.1</td><td>2.1</td><td>4.3</td><td>0.0</td><td>1.9</td><td>1.45</td></tr><tr><td>GE</td><td>42.1</td><td>40.8</td><td>41.5</td><td>41.2</td><td>31.6</td><td>43.4</td><td>40.1</td><td>4.27</td></tr><tr><td>OMd</td><td>51.4</td><td>39.1</td><td>36.3</td><td>38.8</td><td>57.7</td><td>31.4</td><td>42.4</td><td>9.97</td></tr><tr><td>ED6_N</td><td>5.1</td><td>12.3</td><td>14.5</td><td>12.9</td><td>6.5</td><td>23.3</td><td>12.4</td><td>6.50</td></tr><tr><td>dr_N</td><td>0.0</td><td>5.3</td><td>6.6</td><td>5.0</td><td>0.0</td><td>1.9</td><td>3.1</td><td>2.87</td></tr><tr><td>Milk protein yield</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Interaction (%)</td><td>20.9</td><td>8.5</td><td>4.1</td><td>8.6</td><td>20.7</td><td>4.9</td><td>11.3</td><td>7.57</td></tr><tr><td>Contribution (%) to interaction</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>CP</td><td>0.0</td><td>3.8</td><td>5.5</td><td>3.7</td><td>0.0</td><td>0.0</td><td>2.2</td><td>2.47</td></tr><tr><td>GE</td><td>47.6</td><td>41.6</td><td>41.0</td><td>41.4</td><td>44.8</td><td>46.4</td><td>43.8</td><td>2.85</td></tr><tr><td>OMd</td><td>44.1</td><td>34.6</td><td>30.6</td><td>34.5</td><td>45.5</td><td>29.9</td><td>36.5</td><td>6.73</td></tr><tr><td>ED6_N</td><td>8.2</td><td>14.2</td><td>15.8</td><td>14.4</td><td>9.6</td><td>20.2</td><td>13.7</td><td>4.34</td></tr><tr><td>dr_N</td><td>0.1</td><td>5.8</td><td>7.1</td><td>5.9</td><td>0.0</td><td>3.6</td><td>3.7</td><td>3.08</td></tr></tbody></table>

diets (Figure 5a). Irrespective of the diet, UN/TN variations were largely influenced by all 4 input variables except ED6_N. Among them, CP contributed the most (NT<sub>CP,UN/TN</sub> = 131 ± 31.0%), followed by dr_N, GE, and OMd (NT<sub>dr_N,UN/TN</sub> = 54 ± 10.3%; NT<sub>GE,UN/TN</sub> = -52 ± 12.2%; NT<sub>OMd,UN/TN</sub> = -40 ± 10.5%). Increases in CP and dr_N increased UN/TN, and increases in GE and OMd decreased UN/TN. The effect of ED6_N differed depending on the diet. In the RF and GS diets, an increase in ED6_N only slightly affected UN/TN, whereas in other diets, it increased UN/TN.

**GSA.** When all input variables were changed simultaneously, irrespective of the diet, CP was the main contributor to UN/TN variations, followed by dr_N, GE, and OMd (average of ST<sub>i</sub><sup>[UN/TN]</sup> was 69%, 12%, 11%, and 8%, respectively; Figure 5b). The contribution of ED6_N was markedly minimal (average of ST<sub>ED6_N</sub><sup>[UN/TN]</sup> = 1.1%). Additionally, little interaction between input variables was observed (average of interaction = 0.6%).

DISCUSSION

Hybrid Local and Global Sensitivity Analysis

Both OAT and GSA approaches allow assessment of the respective influence of each input variable on a given output variable. Between them, OAT is used in many fields because it requires a small computation time and the results are easy to interpret. For this reason, many

studies have used SA for various feeding systems by calculating regression analysis or calculating the change in output range for a given input range (Jouven et al., 2008; Huhtanen et al., 2015; Binggeli et al., 2022), or as the tangent at a reference point (Jeon et al., 2024). Among GSA methods that are able to consider interactions between input variables by changing multiple input variables simultaneously, regression-based GSA, which assumes the monotonicity of the model, has been used with the Northern American feeding system (Lanzas et al., 2007a,b; Seo et al., 2009; Higgs et al., 2015; NAS-EM, 2016). On the other hand, variance-based GSA had limited use in animal nutrition due to the greater computational costs and analytical complexity.

To overcome the limitations of OAT and GSA, attempts have been made to combine the 2 methods in the ruminant nutritional system. However, Thompson et al. (2014) applied a GSA based on linear regression, rather than Sobol indices as in the present work. Likewise, Dougherty et al. (2017) used a local analysis as part of pre-processing (i.e., screening) for GSA. In the present study, OAT and GSA were simultaneously applied to the evaluation of the INRA 2018 feeding system in order to take the advantages of both methods and complement their respective disadvantages: OAT (differential-based) can absolutely quantify the contribution of each input, but only around the reference point, whereas GSA (variance-based) considers the whole range of data even when they are not monotonic, but only relative comparison is

<footnote n="1">¹RF = fresh perennial ryegrass-based diet; GH1 = first growth permanent grassland hay-based diet; GH2 = second growth permanent grassland hay-based diet; GH3 = first growth and second growth permanent grassland hay-based diet; GS = permanent grassland silage-based diet; CS = corn silage-based diet.</footnote>

<footnote n="2">²GE = gross energy; OMd = OM apparent digestibility; ED6_n = effective degradability of nitrogen assuming a passage rate of 6%/h; dr_n = true intestinal digestibility ("digestibilité réelle") of nitrogen.</footnote>
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

**Figure 2.** Results of one-at-a-time analysis (a; normalized tangent, %) and global sensitivity analysis (b) for milk protein yield (g/d). In the graph (b), the left axis represents total Sobol indices (bar chart; ST<sub>i</sub><sup>[j]</sup>), and the sum of first-order indices (black dots, •; ΣS<sub>i</sub><sup>[j]</sup>). RF = fresh perennial ryegrass-based diet; GH1 = first growth permanent grassland hay-based diet; GH2 = second growth permanent grassland hay-based diet; GH3 = first growth and second growth permanent grassland hay-based diet; GS = permanent grassland silage-based diet; CS = corn silage-based diet. CP (blue filled square) = crude protein; GE (right diagonal line) = gross energy; OMd (left diagonal line) = organic matter digestibility; ED6_N (horizontal stripes) = effective degradability of nitrogen assuming a passage rate of 6%/h; dr_N (dotted) = true intestinal digestibility of nitrogen.

**Figure 3.** Results of one-at-a-time analysis (a; normalized tangent, %) and global sensitivity analysis (b) for energy in methane (kcal/kg DM). In the graph (b), the left axis represents total Sobol indices (bar chart; ST<sub>i</sub><sup>[j]</sup>), and the sum of first-order indices (black dots, •; ΣS<sub>i</sub><sup>[j]</sup>). RF = fresh perennial ryegrass-based diet; GH1 = first growth permanent grassland hay-based diet; GH2 = second growth permanent grassland hay-based diet; GH3 = first growth and second growth permanent grassland hay-based diet; GS = permanent grassland silage-based diet; CS = corn silage-based diet. CP (blue filled square) = crude protein; GE (right diagonal line) = gross energy; OMd (left diagonal line) = organic matter digestibility; ED6_N (horizontal stripes) = effective degradability of nitrogen assuming a passage rate of 6%/h; dr_N (dotted) = true intestinal digestibility of nitrogen.

available. In most cases, both methods showed consistent results in the present study.

## Dry Matter Intake

The input variables that the most influenced DMI according to both OAT and GSA were GE and OMd. In the INRA 2018 feeding system, DMI is positively influenced by the intake capacity (IC) of the animal and negatively influenced by the fill value (FV) of feedstuffs (Faverdin et al., 2018a). The OMd and CP are the primary factors that have a positive and linear effect on voluntary DMI, and thus a negative and nonlinear effect on the FV (inverse function of voluntary DMI) of forages. Moreover, for a given animal, the IC is also positively modulated by the PDI/UFL dietary ratio, which is

highly and negatively dependent on both GE and OMd and positively dependent on CP. This modulation of IC is not linear, being more sensitive at the lowest PDI/UFL values (Vérité and Delaby, 2000). This supports that the DMI of the GH diets, which had a higher PDI/UFL ratio (>115 g PDI/UFL), was less sensitive (i.e., smaller sum of |NT<sub>ij</sub>|) than the other 3 diets, which had a lower PDI/UFL ratio (<100 g PDI/UFL, Table 1). Additionally, ED6_N is negatively related to the value of PDI, which may explain the non-negligible contribution of ED6_N to these non-GH diets. Also, because the GH2 diet contains a high proportion of high CP forage, it showed a high contribution of CP in GSA.

The level of interaction between input variables was the highest for GH diets, and involved mainly GE and OMd. This means that their respective effects on the variation of DMI (but also on substitution rate [SRg]
and IC, data not shown) depend on each other. In the INRA 2018 feeding system, DMI depends on IC (animal characteristics), FV (feed characteristics), and SRg (interactions between feedstuffs). Both SRg and IC are influenced directly but nonlinearly by the PDI/UFL ratio. The input variables with the greatest impact on the PDI/UFL ratio are GE and OMd ($ST_{OMd}^{[PDI/UFL]}$ = 35% and $ST_{GE}^{[PDI/UFL]}$ = 39%; data not shown), through their contribution to UFL. Therefore, it appears that the highest interaction value observed with the GH diets is related to their highest PDI/UFL ratio. This may explain the highest contribution of ED6_N and dr_N (both involved in the calculation of PDI) to the interaction with these GH diets.

## Milk Protein Yield

The OAT approach revealed that the response of MPY to the variability of input variables was higher with the GH diets than with the other 3 diets. In the INRA 2018 feeding system, the predicted MPY of a given animal (as defined by its genetic merit, i.e., potential), depends on the theoretical energy and protein balances (supplies minus potential requirements), and their interactions as well. This is modeled as a multiphase linear-plateau model derived from Koops and Grossman (1993). In the Koops and Grossman model, when theoretical PDI balance was the same, the marginal response of MPY was greater with a lower theoretical UFL balance (Faverdin et al., 2018b). With this integrated model, the notion of limiting factor (energy versus protein) for MPY may thus be related to the PDI/UFL ratio (Vérité and Delaby, 2000). In our study, GH diets (117–121 g PDI/UFL) had lower PDI efficiency (<0.67), indicating that the PDI supply was relatively excessive and the energy balance was lower (<0 UFL/d) compared with the non-GH diets (94 to 99 g PDI/UFL; PDI efficiency ≥0.72; UFL balance ≥0). Consequently, GH diets were quantitatively and relatively less sensitive to PDI-related input variables (i.e., CP, ED6_N, dr_N) and more sensitive to UFL-related input variables (i.e., GE, OMd). Additionally, the model considers that a part of the NE~L~ intake is used for reconstitution of body reserves (according to parity, milk potential, BCS, and lactation stage). Therefore, MPY appears more significantly regulated by energy-related inputs than by protein-related inputs, with both SA. This suggests that the net energy (UFL) content is more limiting than the PDI content for MPY across the 6 diets, with higher than 94 g PDI per UFL. Furthermore, compared with the other 4 diets, protein-related variables were more involved in variations in MPY, and greater interactions (mainly due to OMd and GE, and to a lesser extent to ED6_N) were observed with the RF and GS diets (Table 2). Simulations were

<figure><img src="image_8.png" /><figcaption>**Figure 4.** Results of one-at-a-time analysis (a; normalized tangent, %) and global sensitivity analysis (b) for nitrogen utilization efficiency (N in milk per N intake, g/g). In the graph (b), the left axis represents total Sobol indices (bar chart; $ST_i^{[j]}$), and the sum of first-order indices (black dots, $\bullet$; $\sum_i S_i^{[j]}$). RF = fresh perennial ryegrass-based diet; GH1 = first growth permanent grassland hay-based diet; GH2 = second growth permanent grassland hay-based diet; GH3 = first growth and second growth permanent grassland hay-based diet; GS = permanent grassland silage-based diet; CS = corn silage-based diet. CP (blue filled square) = crude protein; GE (right diagonal line) = gross energy; OMd (left diagonal line) = organic matter digestibility; ED6_N (horizontal stripes) = effective degradability of nitrogen assuming a passage rate of 6%/h; dr_N (dotted) = true intestinal digestibility of nitrogen.</figcaption></figure>

also performed with DMI fixed at the level of the reference points (data not shown) and revealed no further notable influence of protein-related input variables on Sobol indices and no interactions with the RF and GS diets. This suggests that some part of the contribution of protein-related input variables and interactions observed with the RF and GS diets on MPY are due to variations in DMI. Our results, in which MPY variation was sensitive to energy-related variables, are in line with Binggeli et al. (2022), who conducted an SA on MPY prediction across various feeding systems, including INRA 2018, and reported that among dietary characteristics, MPY was mostly affected by OMd. However, in practice, the influence of variables contributing to PDI estimation on the sensitivity of the MPY response might be greater in diets with CP content <14% and a PDI/UFL ratio <95 g PDI/UFL.
<figure><img src="image_9.png" /><figcaption>**Figure 5.** Results of one-at-a-time analysis (a; normalized tangent, %) and global sensitivity analysis (b) for ratio between urinary N and total excreted N (g/g). In the graph (b), the left axis represents total Sobol indices (bar chart; ST<sub>i</sub><sup>[j]</sup>), and the sum of first-order indices (black dots, •; ΣS<sub>i</sub><sup>[j]</sup>). RF = fresh perennial ryegrass-based diet; GH1 = first growth permanent grassland hay-based diet; GH2 = second growth permanent grassland hay-based diet; GH3 = first growth and second growth permanent grassland hay-based diet; GS = permanent grassland silage-based diet; CS = corn silage-based diet. CP (blue filled square) = crude protein; GE (right diagonal line) = gross energy; OMd (left diagonal line) = organic matter digestibility; ED6_N (horizontal stripes) = effective degradability of nitrogen assuming a passage rate of 6%/h; dr_N (dotted) = true intestinal digestibility of nitrogen.</figcaption></figure>

## Energy in Methane

The input variable that had the greatest impact on ECH4 with both OAT and GSA was OMd, and no interaction between input dietary variables was observed in GSA. In INRA 2018, energy lost as methane is calculated from feeding level (% DMI per BW), proportion of concentrate (PCO), and digestible OM (OM × OMd) content (Sauvant et al., 2011). Because for a given animal (given BW) and diet, PCO was fixed and did not change according to the variability of input variables, ECH4 was expected to vary only according to DMI and OMd. Thus, non-GH diets, in which DMI was more sensitive to dietary inputs, were also more sensitive than GH diets in ECH4. Additionally, with GH diets, the lower contribution of GE to variations in ECH4 was also related to the lower influence of GE on DMI. The marginal response of ECH4 to variations

in dietary OMd, as quantified by NT<sub>OMd,ECH4</sub>, was close between diets, but the difference in response between diets was not null (i.e., CV = 9%), because the influence of OMd on DMI is not linear at diet level. The slight contribution of CP, ED6_N, and dr_N to variations in ECH4 observed with OAT can be partly explained by their combined effect on the rumen protein balance (Supplemental Figure S1, see Notes), which affects OMd. The value of NT<sub>CP,ECH4</sub> is highest with the GH2 diet due to a greater effect of CP on OMd (NT<sub>CP,OMd</sub> = 26% with GH2, vs. 9% on average with the 5 other diets; Supplemental Figure S2, see Notes). However, these indirect effects were small compared with the direct effect of OMd on variations in ECH4, and were thus not observed in GSA.

## Nitrogen Utilization Efficiency

The main contributors to NUE variation were CP, GE, and OMd in both SA approaches. Nitrogen utilization efficiency corresponds to the proportion of N intake that is recovered as N in milk. It is thus directly influenced by the CP content of diet, DMI, and MPY. This is consistent with the finding that GE and OMd were the main sources of variation of DMI and MPY in the 6 diets, as discussed in the "Dry Matter Intake" and "Milk Protein Yield" sections. The positive response to CP dietary content is higher for N intake (NT<sub>CP,Nintake</sub> = 116% ± 2.1%) than for MPY (NT<sub>CP,MPY</sub> = 33% ± 5.6%), resulting in a decrease in NUE. In OAT, the relative variation of NUE in response to the variation of input variables (Σ|NT<sub>NUE</sub>|), and their respective contributions, were comparable among diets, although the mechanisms differed between diets: DMI (thus N intake) was more variable with non-GH diets, whereas MPY (thus N in milk) was more variable with GH diets. The results of GSA also revealed no major differences between diets in the relative contributions of input variables, or in the interactions between input variables. In other words, variations in NUE due to variations in input variables were similar among diets and mainly depended on nutrient levels in the diet, especially through OMd and GE (positive effects), and through CP (negative effect).

## Urinary Nitrogen Excretion per Total Nitrogen Excretion

Variability in UN/TN mainly derived from CP, and the influence of GE, OMd, and dr_N was lower but could not be ignored. This result was similar with both the OAT and GSA approaches. Variations in urinary N in a given animal condition from excess ruminally fermented N compared with available energy (N not used for microbial protein synthesis thus absorbed as NH₃, i.e., rumen nitrogen balance), and excess in digestible protein supply (absorbed AA catabolized in the body; Sauvant et
al., 2015). Variations in fecal N flow (g/d) mainly come from variations in DMI (Sauvant et al., 2018; Jeon et al., 2024). In other words, the higher the excess in N intake compared with the microbial requirement for growth and with animal use for production and nonproductive functions, the higher the UN/TN ratio. In agreement with this physiological mechanism represented in the INRA 2018 feeding system, the OAT analysis evidenced that UN/TN increases as the CP level or dr_N increases and energetic parameters (GE and OMd) decrease. Interestingly, ED6_N appears to have a negligible impact on variations in UN/TN due to its concomitant positive effect on urinary N from NH₃ absorption and negative effect on urinary N from unused PDI. The variation of UN/TN was lowest in the GH2 diet, which had the highest CP content; indeed, the positive response of UN/TN to CP content exhibits a curvilinear shape common across diets, with a gradually decreasing slope due to the increase in fecal N flow being higher than the increase in urinary N flow (Jeon et al., 2024). Similarly to NUE, no differences between diets were found in the GSA results, and no interactions between input variables were found. In other words, quantitative variations in the UN/TN ratio due to variations in input variables mainly depend on CP (and to a lesser extent on GE, OMd, dr_N, but not on ED6_N), with the difference being smaller in diets with higher CP content.

## CONCLUSIONS

In conclusion, for the 5 main animal responses predicted from INRAtion V5 reported in this study, the relative importance of input variables was consistent across both SA approaches. With the 6 contrasted diets, GE and OMd were the main contributors to all outputs except UN/TN, and CP was the main contributor to NUE and UN/TN. However, the variability of DMI and MPY and the interaction between input variables differed across diets, and was closely related to the dietary PDI/UFL value. In addition, most of the interactions involved the UFL determinants (i.e., OMd and GE) with these diets >14% CP. These results, together with theoretical knowledge on the model, allowed study of the compartment of the model describing the complex regulatory mechanisms of multiple animal responses. Therefore, the application of both OAT and GSA to complex models such as feeding systems, where various factors interact with each other nonlinearly, allows a thorough understanding and evaluation of such systems.

## NOTES

This research was supported by the INRAE Animal Physiology and Livestock Systems Division (Toulouse,

France). The authors thank the committees in INRAE, Rémy Delagarde, Luc Delaby and Yayu Huang (UMR PEGASE), René Baumont and Gaëlle Maxin (UMR Herbivores), and Patrick Chapoutot (UMR MoSAR) for advice on analysis diet rationing and on explaining the results. Supplemental material for this article is available at https://zenodo.org/records/11445847 (R script for both SA) and https://doi.org/10.57745/9G5XPE (Supplementary Figures S1 and S2). No human or animal subjects were used, so this analysis did not require approval by an Institutional Animal Care and Use Committee or Institutional Review Board. The authors have not stated any conflicts of interest.

**Nonstandard abbreviations used:** CS = corn silage; dr_N = true intestinal digestibility of nitrogen (digestibilité réelle); ECH4 = energy in methane; ED6_N = effective degradability of nitrogen assuming a passage rate of 6%/h; EE = ether extract; FV = fill value; GE = gross energy; GH = grass hay-based diet; GH1 = first growth permanent grassland hay-based diet; GH2 = second growth permanent grassland hay-based diet; GH3 = first and second growth permanent grassland hay-based diet; GS = grass silage; GSA = global sensitivity analysis; IC = intake capacity; INRA = French National Institute of Agricultural Research (Institut national de la recherche agronomique); LysDI = lysine truly digestible in the intestine; MetDI = methionine truly digestible in the intestine; MPY = milk protein yield; NT<sub>ij</sub> = normalized tangent; NUE = nitrogen utilization efficiency; OAT = one-at-a-time analysis; OMd = OM apparent digestibility; PCO = proportion of concentrate; PDI = protein truly digestible in the intestine; RF = fresh perennial ryegrass-based diet; SA = sensitivity analysis; SRg = substitution rate; UEL = fill unit (Unité d'Encombrement Lait); UFL = net energy for lactation (Unité Fourragère Lait); UN/TN = ratio between urinary N and total N excretion.

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## ORCIDS

Seoyoung Jeon, ◎ https://orcid.org/0000-0002-8276-317X

Tristan Senga Kiessé, ◎ https://orcid.org/0000-0003-2710-5825

Sophie Lemosquet, ◎ https://orcid.org/0000-0002-1077-1285

Pierre Nozière ◎ https://orcid.org/0000-0003-1727-8984