Mixed nonlinear models in ruminal <i>in situ</i> degradability trials

Medeiros, Simone Daniela Sartorio de; Lima, César Gonçalves de; Savian, Taciana Villela; Malheiros, Euclides Braga; Werner, Simone Silmara

doi:10.1590/1809-6891v21e-57596

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Zootecnia • Ciênc. anim. bras. 21 • 2020 • https://doi.org/10.1590/1809-6891v21e-57596 copy

Mixed nonlinear models in ruminal in situ degradability trials

Modelos não lineares mistos em ensaios de degradabilidade ruminal in situ

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

Classical methods of analysis of nonlinear models are widely used in studies of ruminal degradation kinetics. As this type of study involves repeated measurements in the same experimental unit, the use of mixed nonlinear models (MNLM) is proposed, in order to solve problems of heterogeneity of variances of the responses, correlation among repeated measurements and consequent lack of sphericity in the covariance matrix. The aims of this work are to present an evaluation of the applicability of MNLM in the estimation of parameters to describe the in situ ruminal degradation kinetics of the dry matter of Tifton 85 hay and to compare the results with those obtained from the usual analysis in two-phases. The steers used in the trial were fed diets composed of three different combinations of roughage and concentrate and two hays with different nutritional qualities. The proposed approach was proven as effective as the traditional one for estimating model parameters. However, it adequately models the correlation among the longitudinal data, which can affect the estimates obtained, the standard error associated with them and potentially change the results of the inferences. It is quite attractive when the research seeks to understand the behavior of the process of food degradation throughout the incubation times.

Keywords:
Ruminal degradation kinetics; Longitudinal data; Covariance matrix; Random effects; Dry matter

Treatment		Time (h)
Treatment		3	6	12	24	48	60	72	84	96	120
70:30 TIF4	Mean	23.2	28.4	33.4	48.8	61.4	66.5	68.7	70.5	71.4	73.6
70:30 TIF4	s.d.	1.24	1.75	2.79	5.32	4.17	1.66	1.14	1.08	0.67	0.77
70:30 TIF10	Mean	23.3	28.8	35.4	50.7	61.9	65.5	68.6	70.2	71.4	72.7
70:30 TIF10	s.d.	1.52	0.80	3.25	4.78	3.49	3.00	1.67	0.91	0.86	0.67
50:50 TIF4	Mean	23.1	27.3	34.9	49.3	58.9	64.1	67.3	68.3	69.8	72.1
50:50 TIF4	s.d.	1.53	2.87	1.52	2.87	4.71	2.03	2.36	2.87	3.02	2.8
50:50 TIF10	Mean	23.3	28.5	33.9	49.8	61.1	64.9	67.3	68.6	70.5	72.S
50:50 TIF10	s.d.	1.91	1.19	3.36	3.85	5.41	2.79	2.06	2.69	2.03	1.57
30:70 TIF4	Mean	23.5	28.1	31.6	45.0	56.5	62.1	63.3	66.2	68.5	71.1
30:70 TIF4	s.d.	1.88	1.52	1.91	3.17	2.35	2.74	1.66	1.73	1.41	1.37
30:70_TIF10	Mean	23.3	27.6	32.0	47.8	56.2	61.5	65.3	66.1	68.9	71.7
30:70_TIF10	s.d.	1.62	1.56	2.21	3.15	3.10	3.03	2.58	2.83	1.64	1.56

R:C	Parameter
	β₁	β₂	β₃
	Mean	Mean	Mean
70:30	17.56 a^* * Means of the same parameter followed by equal letters do not differ between Tukey's test (5%)	56.72 a	0.0334 a
50:50	18.03 ab	55.09 ab	0.0326 a
30:70	19.40 b	53.98 b	0.0260 b
CV(%)	11.15	5.98	21.34
s²	2.11	6.01	1.27×10^s

Model	Random effect in^#	Expression	AIC	logLik
MFO^* * MF: fixed effects model: & MR: model with random effects;	-	$β_{1 k} + β_{2 k} [1 - \exp (- β_{3 k} x_{i})]$	1700.33	^ Ordinary Least Squares Method; #The seven random effects models were adjusted considering the matrix D structure with pdSymm
MR1 ^&^#	β ₁	$(β_{1 k} + b_{1}) + β_{2 k} [1 - \exp (- β_{3 k} x_{i})]$	1683.65	-821.82
MR2	β ₂	$β_{1 k} + (β_{2 k} + b_{2}) [1 - \exp (- β_{3 k} x_{i})]$	1680.87	-820.44
MRS	β ₃	$β_{1 k} + β_{2 k} [1 - \exp (- (β_{3 k} + b_{3}) x_{i})]$	1670.90	-815.45
MR4	β₁ e β₂	$(β_{1 k} + b_{1}) + (β_{2 k} + b_{2}) [1 - \exp (- β_{3 k} x_{i})]$	1681.61	-818.80
MR5	β₁ e β₃	$(β_{1 k} + b_{1}) + β_{2 k} [1 - \exp (- (β_{3 k} + b_{3}) x_{i})]$	1675.06	-815.53
MR6	β₂ e β₃	$β_{1 k} + (β_{2 k} + b_{2}) [1 - \exp (- (β_{3 k} + b_{3}) x_{i})]$	1661.57	-808.79
MR7	β₁, β₂ e β₃	$(β_{1 k} + b 1) + (β_{2 k} + b_{2}) [1 - \exp (- (β_{3 k} + b_{3}) x_{i})]$	1667.47	-808.74

Model	n.p.	AIC	BIC	logLik^ logLik: logarithm of the likelihood maximum function
MR6	22	1661.57	1746.76	-808.79
MR6.5	19	1655.86	1729.43	-808.93
MR6.3	13	1646.46	1696.80	-810.23
MR6.2	10	1644.58	1683.31	-812.29
MR6.1	7	1665.94	1693.05	-825.97

Model	β _1u	β _2u	β _3u	n.p.	AIC	BIC	logLik
MR6.2	differents	differents	differents	10	1644.58	1683.31	-812.29
MR6.2a	β₁₁ = β₁₂	differents	differents	9	1647.69	1682.54	-814.85
MR6.2b	differents	β₂₁ = β₂₂	differents	9	1645.79	1680.63	-813.89
MR6.2c	differents	differents	β₃₁ = β₃₂	9	^* ***** The iterative process of adjustment did not converge. A great sensitivity was noted in the process convergence relative to the initial values provided. In these cases, it is believed that the process did not converge because it was not possible to find adequate initial estimates for the parameter β3, (DM degradability velocity rate) of the two curves, indicating that these parameters of the two curves are really different.	^* ***** The iterative process of adjustment did not converge. A great sensitivity was noted in the process convergence relative to the initial values provided. In these cases, it is believed that the process did not converge because it was not possible to find adequate initial estimates for the parameter β3, (DM degradability velocity rate) of the two curves, indicating that these parameters of the two curves are really different.	^* ***** The iterative process of adjustment did not converge. A great sensitivity was noted in the process convergence relative to the initial values provided. In these cases, it is believed that the process did not converge because it was not possible to find adequate initial estimates for the parameter β3, (DM degradability velocity rate) of the two curves, indicating that these parameters of the two curves are really different.
MR6.2ab	β₁₁ = β₁₂	β₂₁ = β₂₂	differents	8	1647.14	1678.12	-815.57
MR6.2ac	β₁₁ = β₁₂	differents	β₃₁ = β₃₂	8	^* ***** The iterative process of adjustment did not converge. A great sensitivity was noted in the process convergence relative to the initial values provided. In these cases, it is believed that the process did not converge because it was not possible to find adequate initial estimates for the parameter β3, (DM degradability velocity rate) of the two curves, indicating that these parameters of the two curves are really different.	^* ***** The iterative process of adjustment did not converge. A great sensitivity was noted in the process convergence relative to the initial values provided. In these cases, it is believed that the process did not converge because it was not possible to find adequate initial estimates for the parameter β3, (DM degradability velocity rate) of the two curves, indicating that these parameters of the two curves are really different.	^* ***** The iterative process of adjustment did not converge. A great sensitivity was noted in the process convergence relative to the initial values provided. In these cases, it is believed that the process did not converge because it was not possible to find adequate initial estimates for the parameter β3, (DM degradability velocity rate) of the two curves, indicating that these parameters of the two curves are really different.
MR6.2bc	differents	β₂₁ = β₂₂	β₃₁ = β₃₂	8	^* ***** The iterative process of adjustment did not converge. A great sensitivity was noted in the process convergence relative to the initial values provided. In these cases, it is believed that the process did not converge because it was not possible to find adequate initial estimates for the parameter β3, (DM degradability velocity rate) of the two curves, indicating that these parameters of the two curves are really different.	^* ***** The iterative process of adjustment did not converge. A great sensitivity was noted in the process convergence relative to the initial values provided. In these cases, it is believed that the process did not converge because it was not possible to find adequate initial estimates for the parameter β3, (DM degradability velocity rate) of the two curves, indicating that these parameters of the two curves are really different.	^* ***** The iterative process of adjustment did not converge. A great sensitivity was noted in the process convergence relative to the initial values provided. In these cases, it is believed that the process did not converge because it was not possible to find adequate initial estimates for the parameter β3, (DM degradability velocity rate) of the two curves, indicating that these parameters of the two curves are really different.

Approach	R:C	Parameter						ED _5%	PD
Approach	R:C	β ₁	s.e.	β ₂	s.e.	β ₃	s.e.	ED _5%	PD
Mixed*	70:30 and 50:50	18.77	0.32	55.71	0.47	0.0291	0.0008	39.28	74.48
	30:70	20.06	0.52	54.69	0.83	0.0229	0.0013	37.24	74.75

Approach	β ₁	β ₂	β ₃
Two phases	17.56 to 19.40	53.98 to 56.72	0.0260 to 0.0326
MNLM	18.77 to 20.06	54.69 to 55.71	0.0229 to 0.0291

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[1] *Correspondent - sisartorio@gmail.com