Random regression models to estimate genetic parameters for milk production of Guzerat cows using orthogonal Legendre polynomials

Peixoto, Maria Gabriela Campolina Diniz; Santos, Daniel Jordan de Abreu; Borquis, Rusbel Raul Aspilcueta; Bruneli, Frank Ângelo Tomita; Panetto, João Cláudio do Carmo; Tonhati, Humberto

doi:10.1590/S0100-204X2014000500007

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GENETICS • Pesq. agropec. bras. 49 (5) • May 2014 • https://doi.org/10.1590/S0100-204X2014000500007 copy

Random regression models to estimate genetic parameters for milk production of Guzerat cows using orthogonal Legendre polynomials

Modelos de regressão aleatória para estimação de parâmetros genéticos para produção de leite da raça Guzerá com uso de polinômios ortogonais de Legendre

Authorship SCIMAGO INSTITUTIONS RANKINGS

The objective of this work was to compare random regression models for the estimation of genetic parameters for Guzerat milk production, using orthogonal Legendre polynomials. Records (20,524) of test-day milk yield (TDMY) from 2,816 first-lactation Guzerat cows were used. TDMY grouped into 10-monthly classes were analyzed for additive genetic effect and for environmental and residual permanent effects (random effects), whereas the contemporary group, calving age (linear and quadratic effects) and mean lactation curve were analized as fixed effects. Trajectories for the additive genetic and permanent environmental effects were modeled by means of a covariance function employing orthogonal Legendre polynomials ranging from the second to the fifth order. Residual variances were considered in one, four, six, or ten variance classes. The best model had six residual variance classes. The heritability estimates for the TDMY records varied from 0.19 to 0.32. The random regression model that used a second-order Legendre polynomial for the additive genetic effect, and a fifth-order polynomial for the permanent environmental effect is adequate for comparison by the main employed criteria. The model with a second-order Legendre polynomial for the additive genetic effect, and that with a fourth-order for the permanent environmental effect could also be employed in these analyses.

Bos indicus ; covariance functions; lactation curve; test-day model

Trait	N	Days of lactation		Milk yield
Trait	N	Mean	SD	Mean	SD
Month 1	1,815	19.33	6.81	8.13	2.94
Month 2	2,441	45.42	8.70	8.39	2.85
Month 3	2,452	75.96	8.55	7.93	2.68
Month 4	2,464	106.59	8.80	7.43	2.63
Month 5	2,385	136.96	8.44	6.94	2.51
Month 6	2,288	167.19	8.66	6.48	2.36
Month 7	2,089	197.93	8.60	6.12	2.25
Month 8	1,901	228.27	8.60	5.69	2.21
Month 9	1,508	258.67	8.35	5.37	2.08
Month 10	1,181	288.18	8.51	5.12	2.04

Number	RRM	p	log L	AIC	BIC	LRT
1	Leg.3.3.1	13	-16256.805	32539.610	32642.430	(2-1) 238
2	Leg.3.3.4	16	-16137.754	32307.508	32434.058	(3-2) 78^ Significant at 1% probability.
3	Leg.3.3.6	18	-16098.332	32232.664	32375.032	(4-3) 4^ns ns Nonsignificant.
4	Leg.3.3.10	22	-16096.008	32236.016	32410.022	-
5	Leg.3.4.6	22	-16003.640	32051.280	32225.284	(6-5)122^ Significant at 1% probability.
6	Leg.3.5.6	27	-15942.464	31938.928	32152.480⁽¹⁾ (1) Values indicate the best models based on log L, AIC, and BIC.	(7-6) 64^ Significant at 1% probability.
7	Leg.3.6.6	33	-15910.165	31886.330	32147.338⁽¹⁾ (1) Values indicate the best models based on log L, AIC, and BIC.	-
8	Leg.4.3.6	22	-16022.206	32088.412	32262.416	(8-7)56^ Significant at 1% probability.
9	Leg.4.4.6	26	-15993.842	32039.684	32245.326	(9-8) 122^ Significant at 1% probability.
10	Leg.4.5.6	31	-15932.546	31927.092	32172.282	(10-9) 66^ Significant at 1% probability.
11	Leg.4.6.6	37	-15899.519	31873.038	32165.684	-
12	Leg.5.3.6	27	-15981.911	32017.822	32231.374	(13-12) 63^ Significant at 1% probability.
13	Leg.5.4.6	31	-15950.057	31962.114	32207.304	(14-13) 45^ Significant at 1% probability.
14	Leg.5.5.6	36	-15927.227	31926.454	32211.19	(15-14) 66^ Significant at 1% probability.
15	Leg.5.6.6	42	-15893.881	31871.762	32203.952	-
16	Leg.6.3.6	33	-15947.777	31961.554	32222.562	(17-16) 68^ Significant at 1% probability.
17	Leg.6.4.6	37	-15913.543	31901.086	32193.732	(18-17) 51^ Significant at 1% probability.
18	Leg.6.5.6	42	-15887.997	31859.994	32192.186	(19-18)26^ Significant at 1% probability.
19	Leg.6.6.6	48	-15874.503⁽¹⁾ (1) Values indicate the best models based on log L, AIC, and BIC.	31845.006⁽¹⁾ (1) Values indicate the best models based on log L, AIC, and BIC.	32224.654	-

Month	1	2	3	4	5	6	7	8	9	10
				Leg.3.5.6
1	-	0.97	0.89	0.77	0.64	0.50	0.37	0.23	0.09	-0.03
2	0.69	-	0.97	0.90	0.80	0.68	0.55	0.40	0.21	0.02
3	0.62	0.79	-	0.98	0.92	0.83	0.71	0.55	0.33	0.08
4	0.55	0.72	0.80	-	0.98	0.93	0.84	0.68	0.45	0.17
5	0.49	0.64	0.74	0.80	-	0.98	0.92	0.79	0.57	0.28
6	0.45	0.58	0.68	0.76	0.81	-	0.98	0.89	0.69	0.42
7	0.41	0.52	0.62	0.70	0.75	0.80	-	0.96	0.82	0.58
8	0.35	0.47	0.54	0.60	0.65	0.71	0.77	-	0.94	0.77
9	0.30	0.42	0.48	0.51	0.55	0.61	0.69	0.76	-	0.94
10	0.24	0.34	0.39	0.41	0.43	0.48	0.55	0.63	0.75	-
				Leg.3.6.6
1	-	0.97	0.89	0.77	0.64	0.51	0.38	0.24	0.09	-0.03
2	0.69	-	0.97	0.90	0.80	0.69	0.56	0.40	0.21	0.01
3	0.62	0.80	-	0.98	0.92	0.83	0.71	0.55	0.33	0.08
4	0.55	0.72	0.80	-	0.98	0.93	0.84	0.68	0.45	0.17
5	0.49	0.64	0.74	0.80	-	0.98	0.92	0.79	0.57	0.27
6	0.44	0.59	0.68	0.75	0.81	-	0.98	0.89	0.69	0.41
7	0.40	0.54	0.62	0.68	0.75	0.82	-	0.96	0.82	0.58
8	0.34	0.47	0.54	0.59	0.64	0.71	0.77	-	0.94	0.77
9	0.29	0.40	0.48	0.52	0.55	0.59	0.66	0.75	-	0.94
10	0.26	0.34	0.39	0.42	0.45	0.49	0.55	0.63	0.76	-
				Leg.6.6.6
1	-	0.83	0.71	0.62	0.49	0.35	0.23	0.11	0.00	-0.06
2	0.68	-	0.97	0.92	0.84	0.75	0.65	0.52	0.28	0.15
3	0.61	0.80	-	0.98	0.93	0.86	0.77	0.63	0.38	0.21
4	0.54	0.72	0.81	-	0.98	0.93	0.84	0.71	0.45	0.28
5	0.48	0.64	0.74	0.80	-	0.98	0.92	0.80	0.55	0.40
6	0.43	0.60	0.69	0.75	0.81	-	0.98	0.89	0.66	0.54
7	0.39	0.55	0.62	0.68	0.75	0.82	-	0.96	0.79	0.69
8	0.34	0.48	0.55	0.59	0.65	0.71	0.77	-	0.92	0.84
9	0.29	0.41	0.48	0.52	0.55	0.59	0.67	0.75	-	0.94
10	0.25	0.34	0.39	0.43	0.45	0.50	0.56	0.63	0.76	-

Month	1	2	3	4	5	6	7	8	9	10
1	-	0.91	0.92	0.89	0.93	0.89	0.79	0.74	0.60	0.59
2	0.70	-	0.95	0.95	0.96	0.93	0.84	0.79	0.67	0.56
3	0.65	0.78	-	0.95	0.94	0.92	0.83	0.79	0.72	0.69
4	0.61	0.70	0.75	-	0.96	0.95	0.90	0.91	0.81	0.75
5	0.61	0.69	0.72	0.77	-	0.99	0.88	0.87	0.72	0.66
6	0.63	0.68	0.71	0.74	0.79	-	0.91	0.91	0.78	0.74
7	0.59	0.68	0.69	0.71	0.73	0.81	-	0.96	0.93	0.85
8	0.55	0.61	0.65	0.62	0.66	0.73	0.78	-	0.94	0.89
9	0.48	0.55	0.56	0.57	0.57	0.63	0.69	0.75	-	0.95
10	0.43	0.51	0.54	0.54	0.55	0.58	0.60	0.65	0.76	-

Month	2	3	4	5	6	7	8	9	10
Leg.3.5.6
1	0.85	0.73	0.68	0.64	0.62	0.60	0.58	0.54	0.49
2	-	0.97	0.90	0.80	0.73	0.70	0.69	0.67	0.61
3	-	-	0.97	0.90	0.82	0.77	0.73	0.69	0.63
4	-	-	-	0.97	0.92	0.85	0.78	0.70	0.63
5	-	-	-	-	0.98	0.93	0.83	0.72	0.63
6	-	-	-	-	-	0.98	0.89	0.77	0.65
7	-	-	-	-	-	-	0.96	0.86	0.71
8	-	-	-	-	-	-	-	0.96	0.80
9	-	-	-	-	-	-	-	-	0.91
Leg.3.6.6
1	0.86	0.76	0.70	0.65	0.62	0.60	0.59	0.54	0.53
2	-	0.95	0.86	0.78	0.73	0.70	0.68	0.60	0.58
3	-	-	0.96	0.89	0.81	0.75	0.72	0.66	0.61
4	-	-	-	0.97	0.89	0.82	0.76	0.69	0.63
5	-	-	-	-	0.97	0.90	0.82	0.69	0.63
6	-	-	-	-	-	0.97	0.88	0.70	0.64
7	-	-	-	-	-	-	0.95	0.78	0.68
8	-	-	-	-	-	-	-	0.92	0.77
9	-	-	-	-	-	-	-	-	0.88
Leg.6.6.6
1	0.93	0.84	0.75	0.70	0.67	0.65	0.62	0.56	0.55
2	-	0.95	0.85	0.77	0.71	0.68	0.66	0.60	0.56
3	-	-	0.96	0.89	0.80	0.74	0.71	0.67	0.61
4	-	-	-	0.97	0.90	0.82	0.76	0.70	0.62
5	-	-	-	-	0.97	0.91	0.82	0.70	0.61
6	-	-	-	-	-	0.97	0.88	0.72	0.61
7	-	-	-	-	-	-	0.96	0.80	0.65
8	-	-	-	-	-	-	-	0.93	0.75
9	-	-	-	-	-	-	-	-	0.88

α, λ and %	Additive effect						Permanent environmental effect
	α₀	α₁	α₂	α₃	α₄	α₅	α₀	α₁	α₂	α₃	α₄	α₅
	Leg.3.5.6
α₀	1.323	-0.228	-0.114	-	-	-	3.646	-0.172	-0.213	0.103	-0.047	-
α₁	-0.298	0.441	-0.004	-	-	-	-0.143	0.398	0.003	-0.074	0.004	-
α₂	-0.334	-0.022	0.087	-	-	-	-0.232	0.010	0.231	-0.001	-0.048	-
α₃	-	-	-	-	-	-	0.179	-0.392	-0.006	0.091	-0.017	-
α₄	-	-	-	-	-	-	-0.117	0.028	-0.470	-0.263	0.044	-
λ	1.39	0.39	0.08	-	-	-	3.67	0.4	0.23	0.08	0.03	-
%	74.7	21.0	4.3	-	-	-	83.2	9.1	5.2	1.8	0.7	-
	Leg.3.6.6
α₀	1.313	-0.232	-0.111	-	-	-	3.651	-0.190	-0.217	0.092	-0.053	0.033
α₁	-0.306	0.439	-0.004	-	-	-	-0.152	0.425	0.009	-0.059	0.009	-0.029
α₂	-0.3297	-0.019	0.087	-	-	-	-0.231	0.030	0.240	0.011	-0.052	-0.012
α₃	-	-	-	-	-	-	0.144	-0.270	0.072	0.112	-0.012	-0.030
α₄	-	-	-	-	-	-	-0.126	0.061	-0.478	-0.164	0.050	0.003
α₅	-	-	-	-	-	-	0.099	-0.249	-0.141	-0.503	0.081	0.032
λ	1.38	0.38	0.07	-	-	-	3.68	0.43	0.25	0.11	0.03	0.02
%	75.4	20.8	3.8	-	-	-	81.4	9.5	5.5	2.4	0.7	0.4
	Leg.6.6.6
α₀	1.469	-0.310	-0.120	0.070	-0.052	0.069	3.545	-0.121	-0.210	0.035	-0.020	-0.028
α₁	-0.384	0.444	0.006	-0.007	0.017	-0.008	-0.099	0.422	-0.002	-0.055	-0.005	-0.023
α₂	-0.336	0.031	0.087	-0.015	0.005	-0.011	-0.228	-0.006	0.238	0.021	-0.055	-0.003
α₃	0.748	-0.129	-0.638	0.006	-0.005	0.002	0.057	-0.259	0.133	0.106	-0.007	-0.030
α₄	-0.423	0.247	0.150	-0.644	0.010	-0.004	-0.052	-0.041	-0.554	-0.109	0.041	0.007
α₅	0.575	-0.121	-0.365	0.296	-0.347	0.010	-0.100	-0.235	-0.048	-0.627	0.219	0.022
λ	1.57	0.36	0.08	0.01	0.01	0	3.68	0.43	0.25	0.11	0.03	0.02
%	77.3	17.7	3.9	0.5	0.5	0	81.4	9.5	5.6	2.4	0.7	0.4

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[tfn01t02] ⁽¹⁾
Values indicate the best models based on log L, AIC, and BIC.

[tfn02t02] ^ns
Nonsignificant.

[tfn03t02] ^**
Significant at 1% probability.

[4] Notation on RRM: Leg.K_a.K_pe.R, where K_aand K_pe indicate the number of random regression coefficients used to fit the additive genetic effect and the permanent environment effect, respectively, and R indicates the number of categories for the error variances.