Non-linear mixed models in the study of growth of naturalized chickens

Author Contributions

Conceptualization: V. Ibiapina Neto, F.J.V. Barbosa, J.E.G. Campelo and J.L.R. Sarmento. Data curation: V. Ibiapina Neto. Formal analysis: V. Ibiapina Neto. Funding acquisition: F.J.V. Barbosa. Investigation: V. Ibiapina Neto, F.J.V. Barbosa, J.E.G. Campelo and J.L.R. Sarmento. Methodology: V. Ibiapina Neto, F.J.V. Barbosa, J.E.G. Campelo and J.L.R. Sarmento. Project administration: V. Ibiapina Neto, F.J.V. Barbosa and J.E.G. Campelo. Resources: V. Ibiapina Neto and F.J.V. Barbosa. Software: V. Ibiapina Neto. Supervision: F.J.V. Barbosa, J.E.G. Campelo and J.L.R. Sarmento. Validation: F.J.V. Barbosa, J.E.G. Campelo and J.L.R. Sarmento. Visualization: F.J.V. Barbosa, J.E.G. Campelo and J.L.R. Sarmento. Writing-original draft: V. Ibiapina Neto. Writing-review & editing: F.J.V. Barbosa, J.E.G. Campelo and J.L.R. Sarmento.

Model	Random effect	Formula
von Bertalanffy	Absent	$y = A {(1 - Bexp (- kt))}^{3}$
	A	$y = (A + u 1) {(1 - Bexp (- kt))}^{3}$
	B	$y = A {(1 - (B + u 2) \exp (- kt))}^{3}$
	k	$y = A {(1 - Bexp (- (k + u 3) t))}^{3}$
	A and B	$y = (A + u 1) {(1 - (B + u 2) \exp (- kt))}^{3}$
	A and k	$y = (A + u 1) {(1 - Bexp (- (k + u 3) t))}^{3}$
	B and k	$y = A {(1 - (B + u 2) \exp (- (k + u 3) t))}^{3}$
	A, B, and k	$y = (A + u 1) {(1 - (B + u 2) \exp (- (k + u 3) t)]}^{3}$
Gompertz	Absent	$y = Aexp (- Bexp (- kt))$
	A	$y = (A + u 1) \exp (- Bexp (- kt))$
	B	$y = Aexp (- (B + u 2) \exp (- kt))$
	k	$y = Aexp (- Bexp (- (k + u 3) t))$
	A and B	$y = (A + u 1) \exp (- (B + u 2) \exp (- kt))$
	A and k	$y = (A + u 1) \exp (- Bexp (- (k + u 3) t))$
	B and k	$y = Aexp (- (B + u 2) \exp (- (k + u 3) t))$
	A, B, and k	$y = (A + u1) exp (- (B + u 2) exp (- (k + u 3) t))$
Logistic	Absent	$y = A / (1 + Bexp (- kt))$
	A	$y = (A + u 1) / (1 + Bexp (- kt))$
	B	$y = A / (1 + (B + u 2) \exp (- kt))$
	k	$y = A / (1 + Bexp (- (k + u 3) t))$
	A and B	$y = (A + u 1) / (1 + (B + u 2) \exp (- kt))$
	A and k	$y = (A + u 1) / (1 + Bexp (- (k + u 3) t))$
	B and k	$y = A / (1 + (B + u 2) \exp (- (k + u 3) t))$
	A, B, and k	$y = (A + u 1) / (1 + (B + u 2) \exp (- (k + u 3) t))$

Random effect in:	Gompertz model
	A		B		k		Age		Weight
	M	F	M	F	M	F	M	F	M	F
Absent	1,438a	1,261b	6.52a	5.70a	0.0214a	0.0197b	38	38	529	464
A	1,482a	1,269b	6.07a	5.25b	0.0201a	0.0186b	39	39	545	467
B	1,437a	1,261b	6.51a	5.96b	0.0214a	0.0197b	38	39	528	464
k	1,809a	1,532b	4.54a	4.29b	0.0147a	0.0141b	45	45	665	563
A and B	1,482a	1,269b	6.07a	5.25b	0.0201a	0.0186b	39	39	545	467
A and k	1,546a	1,532a	5.58a	4.29b	0.0186a	0.0141b	40	45	568	563
B and k	1,864a	1,261b	4.46b	5.69a	0.0141b	0.0197a	46	38	685	464
A, B, and k	1,808a	1,385b	4.54b	4.85a	0.0147b	0.0167a	45	41	665	509
Logistic model
Absent	1,310a	1,129b	40.48a	35.47b	0.0367a	0.0354b	101	101	655	565
A	1,335a	1,126b	36.19a	30.59b	0.0349a	0.0335b	103	102	668	563
B	1,310a	1,128b	40.47a	35.46b	0.0367a	0.0354b	101	101	655	564
k	1,465a	1,226b	27.42a	25.00b	0.0299a	0.0294a	111	109	733	613
A and B	1,336a	1,226b	36.18a	30.60b	0.0349b	0.0355a	103	96	668	613
A and k	1,362a	1,226b	39.78a	25.01b	0.0356a	0.0294b	103	109	681	613
B and k	1,465a	1,226b	27.43a	25.02b	0.0299a	0.0294a	111	110	733	613
A, B, and k	1,364a	1,152b	38.75a	29.25b	0.0352a	0.0322b	104	105	682	576
von Bertalanffy model
Absent	1,570a	1,365b	1.16a	1.09b	0.0155a	0.0147b	98	97	465	344
A	1,569a	1,365b	1.16a	1.09b	0.0155a	0.0147b	98	97	465	344
B	1,569a	1,365b	1.16a	1.09b	0.0155a	0.0147b	98	97	465	344
k	2,430a	2,071b	0.84a	0.81b	0.0084a	0.0081b	130	130	720	249
A and B	1,569a	1,413b	1.16a	1.01b	0.0155a	0.0133b	98	99	465	344
A and k	2,680a	2,071b	1.08a	0.81b	0.0139a	0.0081b	101	130	794	320
B and k	2,430a	1,365b	0.84b	1.09a	0.0084b	0.0147a	130	97	720	249
A, B, and k	1,670a	1,356b	1.08a	1.05b	0.0139b	0.0143a	101	96	495	320

Random effect in:	Gompertz model
	Females					Males
	MSE	AAD	R²	AIC	BIC	MSE	AAD	R²	AIC	BIC
Absent	12653	83.42	0.956	7354	7371	21028	106.9	0.945	6749	6766
A	6220	58.41	0.975	7170	7192	8408	69.57	0.983	6514	6536
B	12653	83.42	0.956	7356	7378	21028	106.9	0.945	6751	6772
k	6214	55.71	0.956	7181	7203	8774	67.23	0.959	6540	6561
A and B	6220	58.41	0.975	7174	7204	8409	69.57	0.982	6518	6548
A and k	6215	55.71	0.956	7185	7216	8200	67.79	0.981	6510	6540
B and k	12653	83.42	0.956	7360	7390	7635	63.19	0.954	6541	6571
A, B, and k	3339	42.88	0.988	7069	7113	8771	67.22	0.959	6550	6592
Logistic model
Absent	12105	78.16	0.923	7327	7345	20226	100.3	0.910	6729	6746
A	6096	55.41	0.941	7152	7174	8012	64.42	0.946	6490	6512
B	12105	78.16	0.923	7329	7351	20226	100.3	0.910	6731	6752
k	6138	57.13	0.930	7167	7189	8448	68.44	0.931	6519	6540
A and B	6096	55.41	0.941	7156	7187	8011	64.42	0.946	6494	6524
A and k	6138	57.12	0.930	7171	7202	4940	49.72	0.969	6428	6458
B and k	6137	57.12	0.930	7171	7202	8447	68.43	0.931	6523	6553
A, B, and k	2972	40.09	0.947	7024	7067	4916	49.55	0.967	6433	6476
von Bertalanffy model
Absent	13060	86.34	0.963	7372	7390	21623	109.7	0.944	6764	6781
A	13060	86.34	0.963	7374	7396	21623	109.7	0.944	6766	6787
B	13060	86.34	0.963	7374	7396	21623	109.7	0.944	6766	6787
k	6746	59.59	0.957	7228	7250	9662	71.76	0.959	6587	6608
A and B	6380	60.70	0.987	7190	7221	21623	109.7	0.944	6770	6800
A and k	6747	59.59	0.957	7232	7263	8589	70.60	0.991	6532	6562
B and k	13060	86.34	0.963	7378	7409	9663	71.76	0.959	6591	6621
A, B, and k	6486	61.34	0.988	7193	7215	8587	70.60	0.991	6538	6580

Group	Model	Random effect	Formula
1	Gompertz	Absent	$y = Aexp (- Bexp (- kt))$
	Gompertz	B	$y = Aexp (- (B + u 2) \exp (- kt))$
	Gompertz	B and k	$y = Aexp (- (B + u 2) \exp (- (k + u 3) t))$
	Logistic	Absent	$y = A / (1 + Bexp (- kt))$
	Logistic	B	$y = A / (1 + (B + u 2) \exp (- kt))$
	Bertalanffy	Absent	$y = A {(1 - Bexp (- kt))}^{3}$
	Bertalanffy	A	$y = (A + u 1) {(1 - Bexp (- kt))}^{3}$
	Bertalanffy	B	$y = A {(1 - (B + u 2) \exp (- kt))}^{3}$
	Bertalanffy	B and k	$y = A {(1 - (B + u 2) \exp (- (k + u 3) t))}^{3}$
2	Gompertz	A	$y = (A + u 1) \exp (- Bexp (- kt))$
	Gompertz	k	$y = Aexp (- Bexp (- (k + u 3) t))$
	Gompertz	A and B	$y = (A + u 1) \exp (- (B + u 2) \exp (- kt))$
	Gompertz	A and k	$y = (A + u 1) \exp (- Bexp (- (k + u 3) t))$
	Logistic	A	$y = (A + u 1) / (1 + Bexp (- kt))$
	Logistic	k	$y = A / (1 + Bexp (- (k + u 3) t))$
	Logistic	A and B	$y = (A + u 1) / (1 + (B + u 2) \exp (- kt))$
	Logistic	A and k	$y = (A + u 1) / (1 + Bexp (- (k + u 3) t))$
	Logistic	B and k	$y = A / (1 + (B + u 2) \exp (- (k + u 3) t))$
	Bertalanffy	k	$y = A {(1 - Bexp (- (k + u 3) t))}^{3}$
	Bertalanffy	A and B	$y = (A + u 1) {(1 - (B + u 2) \exp (- kt))}^{3}$
	Bertalanffy	A and k	$y = (A + u 1) {(1 - Bexp (- (k + u 3) t))}^{3}$
	Bertalanffy	A, B, and k	$y = (A + u 1) {(1 - (B + u 2) \exp (- (k + u 3) t))}^{3}$
3	Gompertz	A, B, and k	$y = (A + u 1) \exp (- (B + u 2) \exp (- (k + u 3) t))$
3	Logistic	A, B, and k	$y = (A + u 1) / (1 + (B + u 2) \exp (- (k + u 3) t))$

Group	Model	Random effect in:	Formula
1	Gompertz	Absent	$y = Aexp (- Bexp (- kt))$
	Gompertz	B	$y=Aexp(-(B+u2)exp(-kt))$
	Bertalanffy	Absent	$y = A {(1 - Bexp (- kt))}^{3}$
	Bertalanffy	A	$y = (A + u 1) {(1 - Bexp (- kt))}^{3}$
	Bertalanffy	B	$y = A {(1 - (B + u 2) \exp (- kt))}^{3}$
	Bertalanffy	A and B	$y = (A + u 1) {(1 - (B + u 2) \exp (- kt))}^{3}$
2	Logistic	Absent	$y = A / (1 + Bexp (- kt))$
2	Logistic	B	$y = A / (1 + (B + u 2) \exp (- kt))$
3	Logistic	k	$y = A / (1 + Bexp (- (k + u 3) t))$
3	Logistic	B and k	$y = A / (1 + (B + u 2) \exp (- (k + u 3) t))$
4	Logistic	A	$y = (A + u 1) / (1 + Bexp (- kt))$
4	Logistic	A and B	$y = (A + u 1) / (1 + (B + u 2) \exp (- kt))$
5	Gompertz	A	$y = (A + u 1) \exp (- Bexp (- kt))$
	Gompertz	A and B	$y = (A + u 1) \exp (- (B + u 2) \exp (- kt))$
	Gompertz	A and k	$y = (A + u1) exp (- Bexp (- (k + u 3) t))$
6	Bertalanffy	k	$y = A {(1 - Bexp (- (k + u 3) t))}^{3}$
6	Bertalanffy	B and k	$y = A {(1 - (B + u 2) \exp (- (k + u 3) t))}^{3}$
7	Bertalanffy	A and k	$y = (A + u 1) {(1 - Bexp (- (k + u 3) t))}^{3}$
7	Bertalanffy	A, B, and k	$y = (A + u 1) {(1 - (B + u 2) exp (- (k + u 3) t))}^{3}$
8	Logistic	A and k	$y = (A + u 1) / (1 + Bexp (- (k + u 3) t))$
8	Logistic	A, B, and k	$y = (A + u 1) / (1 + (B + u 2) \exp (- (k + u 3) t))$
9	Gompertz	k	$y = Aexp (- Bexp (- (k + u 3) t))$
9	Gompertz	A, B, and k	$y = (A + u 1) \exp (- (B + u 2) \exp (- (k + u 3) t))$
10	Gompertz	B and k	$y = Aexp (- (B + u 2) \exp (- (k + u 3) t))$

Graúna Dourada ecotype
	Female (y1¹⁾ 1 Gompertz model: y1=(A+u1)exp(−(B+u2)exp(−(k+u3)t)); Logistic model: y2=(A+u1)/(1+(B+u2)exp(−(k+u3)t)); Logistic model: y3=(A+u1)/(1+Bexp(−(k+u3)t)); Logistic model: y4=(A+u1)/(1+(B+u2)exp(−(k+u3)t)).		Female (y2)		Male (y3)		Male (y4)
	Estimate	t-value	Estimate	t-value	Estimate	t-value	Estimate	t-value
A	1,301.44c	30.50² 2 P-value <0.0001.	1,117.50b	40.56² 2 P-value <0.0001.	1,230.70c	36.62² 2 P-value <0.0001.	1,241.03c	42.56² 2 P-value <0.0001.
B	4.59b	26.91² 2 P-value <0.0001.	35.31a	12.79² 2 P-value <0.0001.	55.38a	9.96² 2 P-value <0.0001.	52.86a	10.96² 2 P-value <0.0001.
k	0.0145b	21.04² 2 P-value <0.0001.	0.0332a	36.81² 2 P-value <0.0001.	0.0394a	28.18² 2 P-value <0.0001.	0.0382a	28.63² 2 P-value <0.0001.
MSE	5,004.96		1,797.25		3,058.32		2,923.01
AAD	50.38		30.53		39.72		39.04
R²	0.9719		0.9692		0.9936		0.9904
AIC	4,317.98		4,158.59		3,196.08		3,192.50
BIC	4,357.00		4,197.62		3,221.27		3,228.48
Nordestina ecotype
A	1,467.22b	18.85² 2 P-value <0.0001.	1,159.05b	25.05² 2 P-value <0.0001.	1,530.53b	26.18² 2 P-value <0.0001.	1,517.73b	21.47² 2 P-value <0.0001.
B	5.09a	12.48² 2 P-value <0.0001.	24.31b	7.36² 2 P-value <0.0001.	20.45c	6.34² 2 P-value <0.0001.	24.85b	7.09² 2 P-value <0.0001.
k	0.0203a	14.59² 2 P-value <0.0001.	0.0332a	19.64² 2 P-value <0.0001.	0.0278b	13.66² 2 P-value <0.0001.	0.0306b	14.65² 2 P-value <0.0001.
MSE	3,564.84		3,590.21		9,587.60		6,405.60
AAD	47.25		47.74		78.38		65.12
R²	0.9929		0.9310		0.9177		0.9438
AIC	1,417.45		1,420.36		1,438.13		1,431.72
BIC	1,445.08		1,447.98		1,457.35		1,459.17
Teresina ecotype
A	2,136.56a	7.30² 2 P-value <0.0001.	1,353.36a	20.56² 2 P-value <0.0001.	1,636.01a	28.25² 2 P-value <0.0001.	1,635.92a	28.25² 2 P-value <0.0001.
B	3.75c	18.85² 2 P-value <0.0001.	21.30b	8.08² 2 P-value <0.0001.	27.35b	7.38² 2 P-value <0.0001.	27.36b	7.38² 2 P-value <0.0001.
k	0.0107c	7.65² 2 P-value <0.0001.	0.0281b	19.07² 2 P-value <0.0001.	0.0293b	16.87² 2 P-value <0.0001.	0.0293b	16.87² 2 P-value <0.0001.
MSE	8,265.63		5,572.70		8,199.86		8,199.68
AAD	66.53		53.79		66.65		66.65
R²	0.9105		0.9155		0.9308		0.9308
AIC	1,416.79		1,407.34		1,758.97		1,764.97
BIC	1,444.24		1,434.79		1,779.67		1,794.53

Parameter	Gompertz model - Males
Parameter	A	B	k
A	1	−0.9864	−0.9944
B	6.15 × 10⁻⁶	1	0.9982
k	4.46 × 10⁻⁷	1.4 × 10⁻⁸	1

	Logistic model - Males
A	1	−0.9262	−0.9717
B	0.0009	1	0.9890
k	5.55 × 10⁻⁵	3.28 × 10⁻⁶	1

	von Bertalanffy model - Males
A	1	−0.7439	−0.7340
B	0.0343	1	0.9997
k	0.0381	9.37 × 10⁻¹¹	1

	Gompertz model - Females
A	1	−0.9348	−0.9861
B	0.0006	1	0.9768
k	6.63 × 10⁻⁶	3.06 × 10⁻⁵	1

	Logistic model - Females
A	1	−0.8071	−0.6426
B	0.0154	1	0.9348
k	0.0857	0.0007	1

	von Bertalanffy model - Females
A	1	−0.9817	−0.9933
B	1.51 × 10⁻⁵	1	0.9968
k	7.49 × 10⁻⁷	8.24 × 10⁻⁸	1

[1] *Corresponding author: sarmento@ufpi.edu.br

Brasil

Brasil

Non-linear mixed models in the study of growth of naturalized chickens

ABSTRACT