Bivariate Copula-based Linear Mixed-effects Models: An Application to Longitudinal Child Growth Data

FERREIRA, P.H.; FIACCONE, R.L.; LORDELO, J.S.; SENA, S.O.L.; DURAN, V.R.

doi:10.5540/tema.2019.020.01.037

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Articles • TEMA (São Carlos) 20 (1) • Jan-Apr 2019 • https://doi.org/10.5540/tema.2019.020.01.037 copy

Bivariate Copula-based Linear Mixed-effects Models: An Application to Longitudinal Child Growth Data

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

Multiple longitudinal outcomes are common in public health research and adequate methods are required when there is interest in the joint evolution of response variables over time. However, the main drawback of joint modeling procedures is the requirement to specify the joint density of all outcomes and their correlation structure, as well as numerical difficulties in statistical inference, when the dimension of these outcomes increases. To overcome such difficulty, we present two procedures to deal with multivariate longitudinal data. We first present an univariate approach, for which linear mixed-effects models are considered for each response variable separately. Then, a novel copula-based modeling is presented, in order to characterize relationships among the response variables. Both methodologies are applied to a real Brazilian data set on child growth.

Keywords:
bivariate copula; linear mixed-effects model; longitudinal growth data; time-varying dependence

Copula	C(u, v)	c(u, v)	θ ∊
Gaussian	$\int_{- \infty}^{Φ^{- 1} (u)} \int_{- \infty}^{Φ^{- 1} (v)} \frac{1}{2 π {(1 - θ^{2})}^{\frac{1}{2}}} \times \exp \{- \frac{s^{2} - 2 θ s t + t^{2}}{2 (1 - θ^{2})}\} d t d s$	$\sqrt{\frac{1}{1 - θ^{2}}} \exp \{- \frac{ζ_{1}^{2} - 2 ζ_{1} ζ_{2} + ζ_{2}^{2}}{2 (1 - θ^{2})}\} \exp \{\frac{1}{2} (ζ_{1}^{2} + ζ_{2}^{2})\}$ where $ζ_{1} = Φ^{- 1} (u)$ and $ζ_{2} = Φ^{- 1} (v)$	[-1, 1]

Student’s t _v	$\int_{- \infty}^{T_{v}^{- 1} (u)} \int_{- \infty}^{T_{v}^{- 1} (v)} \frac{1}{2 π {(1 - θ^{2})}^{\frac{1}{2}}} \times {(1 + \frac{s^{2} - 2 θ s t + t^{2}}{v (1 - θ^{2})})}^{- \frac{v + 2}{2}} d t d s$	$\frac{1}{2} (\frac{Γ (\frac{v}{2})}{Γ (\frac{v + 1}{2})}) v {(1 - θ^{2})}^{- \frac{1}{2}} \times {[(1 + \frac{{[T_{v}^{- 1} (u)]}^{2}}{v}) (1 + \frac{{[T_{v}^{- 1} (v)]}^{2}}{v})]}^{\frac{v + 1}{2}} \times {(1 + \frac{{[T_{v}^{- 1} (u)]}^{2} - 2 θ T_{v}^{- 1} (u) T_{v}^{- 1} (v) + {[T_{v}^{- 1} (v)]}^{2}}{(1 - θ^{2}) v})}^{- 1 - \frac{v}{2}}$	[-1, 1]

Clayton	${(u^{- θ} + v^{- θ} - 1)}^{- \frac{1}{θ}}$	$(1 + θ) {(u v)}^{- 1 - θ} {(u^{- θ} + v^{- θ} - 1)}^{- \frac{1}{θ} - θ}$	(0, ∞)

Frank	$- \frac{1}{θ} \log (1 + \frac{[\exp \{- θ u\} - 1] [\exp \{- θ v\} - 1]}{\exp \{- θ\} - 1})$	$\frac{θ [1 - \exp \{- θ\}] \exp \{- θ (u + v)\}}{1 - \exp \{- θ\} - (1 - \exp \{- θ u\}) - (1 - \exp \{- θ v\})}$	$ℝ - \{0\}$

Gumbel	$\exp \{- {[{(- \log u)}^{θ} + {(- \log v)}^{θ}]}^{\frac{1}{θ}}\}$	$\frac{{(- \log u)}^{θ - 1} {(- \log v)}^{θ - 1}}{u v} \times \exp \{- {[{(- \log u)}^{θ} + {(- \log v)}^{θ}]}^{\frac{1}{θ}}\} \times ({[{(- \log u)}^{θ} + {(- \log v)}^{θ}]}^{{(\frac{1 - θ}{θ})}^{2}}) + ([θ - 1] {[{(- \log u)}^{θ} + {(- \log v)}^{θ}]}^{(\frac{1 - 2 θ}{θ})})$	[1, ∞)

Copula	τ _K	ρ _S	λ _L	λ _U
Gaussian	$\frac{\arcsin (θ)}{π / 2}$	$\frac{\arcsin (θ / 2)}{π / 6}$	0	0
Student’s t _v	$\frac{\arcsin (θ)}{π / 2}$	$\frac{\arcsin (θ / 2)}{π / 6}$	$2 T_{ν + 1} (- \sqrt{\frac{(ν + 1) (1 - θ)}{1 + θ}})$ ‡	$2 T_{ν + 1} (- \sqrt{\frac{(ν + 1) (1 - θ)}{1 + θ}})$
Clayton	$\frac{θ}{θ + 2}$	No closed-form	$2^{\frac{1}{θ}}$	0
Frank	$1 - \frac{4 [D_{1} (θ) - 1]}{θ}$ †	$1 - \frac{12 [4 D_{1} (θ) - 2 D_{2} (θ)]}{θ}$	0	0
Gumbel	$\frac{θ - 1}{θ}$	Complicated	0	$2 - 2^{\frac{1}{θ}}$

	Model (2.2): Height			Model (2.3): Weight
Fixed-Effect Parameter ‡	Value	SE	p-value	Value	SE	p-value
Intercept	55.097	0.614	<0.0001	4495.871	156.308	<0.0001
Gender (Female)	-1.439	0.307	<0.0001	-286.155	77.529	0.0003
TDB	0.114	0.097	0.2415	59.229	24.142	0.0153
Age	2.612	0.035	<0.0001	697.081	14.011	<0.0001
Hemoglobin Level	-0.395	0.023	<0.0001	-90.555	6.413	<0.0001
Log-likelihood	-1454.434	—	—	-6061.202	—	—
AIC	2924.868	—	—	12140.400	—	—
BIC	2962.385	—	—	12182.610	—	—

Copula	$\hat{θ}$	SE	2.5%	97.5%	${\hat{τ}}_{K}$	${\hat{ρ}}_{S}$	${\hat{λ}}_{L}$	${\hat{λ}}_{U}$	LL	AIC	BIC
Gaussian	0.920	0.044	0.784	0.950	0.743	0.912	0.000	0.000	-7198.01	14432.02	14516.54
Student’s t ₂	0.885	0.063	0.686	0.932	0.691	0.875	0.647	0.647	-7208.43	14452.86	14537.39
Clayton	2.634	0.668	1.222	3.851	0.568	0.755	0.768	0.000	-7214.14	14464.28	14548.81
Frank	15.046	2.768	8.219	18.584	0.763	0.929	0.000	0.000	-7197.90	14431.80	14516.32
Gumbel	4.747	0.891	2.620	6.118	0.789	0.937	0.000	0.842	-7196.68	14429.37	14513.89

Copula	${\hat{θ}}_{0}$	${\hat{θ}}_{1}$	${\hat{θ}}_{2}$	${\hat{θ}}_{3}$	${\hat{θ}}_{4}$	${\hat{θ}}_{5}$	${\hat{θ}}_{6}$	LL	AIC	BIC
Gaussian	0.696	0.825	0.759	0.753	0.683	0.646	0.684	-7773.21	15700.42	15911.56
Gaussian	[0.608, 0.774]	[0.767, 0.879]	[0.681, 0.835]	[0.665, 0.830]	[0.578, 0.781]	[0.524, 0.767]	[0.578, 0.773]	-7773.21	15700.42	15911.56
Student’s t ₂	0.620	0.765	0.656	0.692	0.548	0.468	0.550	-7825.55	15805.09	16016.23
Student’s t ₂	[0.473, 0.736]	[0.657, 0.848]	[0.496, 0.774]	[0.534, 0.801]	[0.357, 0.703]	[0.250, 0.656]	[0.366, 0.691]	-7825.55	15805.09	16016.23
Clayton	1.354	2.009	1.490	1.836	1.233	1.018	1.204	-7846.63	15847.26	16058.39
Clayton	[0.932, 1.862]	[1.390, 2.704]	[0.973, 2.138]	[1.238, 2.549]	[0.768, 1.783]	[0.612, 1.587]	[0.785, 1.768]	-7846.63	15847.26	16058.39
Frank	5.606	8.785	6.620	6.751	5.004	4.558	5.205	-7799.50	15753.00	15964.13
Frank	[4.375, 7.002]	[7.138, 10.724]	[5.189, 8.450]	[5.120, 8.717]	[3.612, 6.637]	[3.125, 6.216]	[3.848, 6.782]	-7799.50	15753.00	15964.13
Gumbel	1.850	2.381	2.030	1.996	1.746	1.591	1.725	-7812.39	15778.77	15989.91
Gumbel	[1.577, 2.209]	[2.005, 2.932]	[1.723, 2.495]	[1.651, 2.463]	[1.471, 2.102]	[1.354, 1.969]	[1.473, 2.116]	-7812.39	15778.77	15989.91

		Month (t)
Copula	Measure	0	1	2	3	4	5	6
Gaussian	${\hat{τ}}_{K}$	0.490	0.617	0.548	0.542	0.478	0.447	0.479
	${\hat{ρ}}_{S}$	0.678	0.812	0.743	0.737	0.665	0.628	0.666
	${\hat{λ}}_{L}$	0.000	0.000	0.000	0.000	0.000	0.000	0.000
	${\hat{λ}}_{U}$	0.000	0.000	0.000	0.000	0.000	0.000	0.000
Student’s t ₂	${\hat{τ}}_{K}$	0.425	0.554	0.455	0.486	0.369	0.310	0.370
	${\hat{ρ}}_{S}$	0.601	0.749	0.638	0.674	0.530	0.451	0.532
	${\hat{λ}}_{L}$	0.387	0.505	0.413	0.441	0.340	0.294	0.341
	${\hat{λ}}_{U}$	0.387	0.505	0.413	0.441	0.340	0.294	0.341
Clayton	${\hat{τ}}_{K}$	0.403	0.501	0.426	0.478	0.381	0.337	0.375
	${\hat{ρ}}_{S}$	0.568	0.684	0.596	0.658	0.540	0.483	0.533
	${\hat{λ}}_{L}$	0.599	0.708	0.628	0.685	0.569	0.506	0.562
	${\hat{λ}}_{U}$	0.000	0.000	0.000	0.000	0.000	0.000	0.000
Frank	${\hat{τ}}_{K}$	0.492	0.629	0.544	0.551	0.456	0.427	0.469
	${\hat{ρ}}_{S}$	0.686	0.829	0.744	0.751	0.643	0.607	0.658
	${\hat{λ}}_{L}$	0.000	0.000	0.000	0.000	0.000	0.000	0.000
	${\hat{λ}}_{U}$	0.000	0.000	0.000	0.000	0.000	0.000	0.000
Gumbel	${\hat{τ}}_{K}$	0.459	0.580	0.507	0.498	0.427	0.371	0.420
	${\hat{ρ}}_{S}$	0.635	0.768	0.691	0.681	0.595	0.525	0.586
	${\hat{λ}}_{L}$	0.000	0.000	0.000	0.000	0.000	0.000	O.000
	${\hat{λ}}_{U}$	0.545	0.662	0.593	0.584	0.512	0.454	0.505

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[1] *Corresponding author: Paulo H. Ferreira - E-mail: paulohenri@ufba.br