BAYESIAN ESTIMATION FOR THE STABLE DISTRIBUTIONS IN THE PRESENCE OF COVARIATES WITH APPLICATIONS IN CLINICAL ISSUES

Achcar, Jorge Alberto; Souza, Roberto Molina de; Bussola, Daiane; Moala, Fernando A.

doi:10.1590/0101-7438.2022.042.00254533

Jorge Alberto Achcar

University of São Paulo, Department of Public Health, Av. Bandeirantes, 3900, Monte Alegre, 14049-900 Ribeirão Preto, SP, Brazil - E-mail: achcar@fmrp.usp.br

http://orcid.org/0000-0002-9868-9453

Roberto Molina de Souza

Federal Technological University of Paraná, Department of Mathematics, Av. Alberto Carazzai, 1640, Centro, 86300000 Cornélio Procópio, PR, Brazil - E-mail: rmolinasouza@utfpr.edu.br

http://orcid.org/0000-0002-0638-3650

Daiane Bussola

University of São Paulo, Department of Public Health, Av. Bandeirantes, 3900, Monte Alegre, 14049-900 Ribeirão Preto, SP, Brazil - E-mail: dpsampaiob@gmail.com

http://orcid.org/0000-0002-0362-2871

Fernando A. Moala ^**Corresponding author

São Paulo State University, Department of Statistics, R. Roberto Símonsen, 305, Centro Educacional, 19060-900 Presidente Prudente, SP, Brazil - E-mail: f.moala@unesp.br

http://orcid.org/0000-0002-2445-0407

*Corresponding author

	LSE	SE	T-Value	P-Value
Constant	-1.9414	0.0790	-24.58	< 0.001
Age	0.0236	0.0033	7.04	< 0.001
gendercoded	0.0288	0.0118	2.45	0.015
smokercoded	-0.0471	0.0210	-2.25	0.025
Height	0.0168	0.0006	25.34	< 0.001

	mean	s.d.	Lower 95%	Upper 95%
α	1.912	0.0467	1.83	1.993
β	-0.0755	0.4961	-0.9024	0.9151
µ	1.997	0.0024	1.991	2.0
σ	0.7389	0.0261	0.6908	0.7926

	Mean	s.d.	Lower 95%	Upper 95%
α	1.975	0.0215	1.923	1.999
β	-0.0138	0.2831	-0.4783	0.4713
d₀	2.015	0.0157	2.0	2.061
d₁	-0.0119	3,11E-01	-0.0124	-0.0112
d₂	0.2394	0.0049	0.2292	0.2479
d₃	0.2355	0.0426	0.1491	0.3178
d₄	0.0499	0.1060	-0.1615	0.2431
e₀	-0.8975	0.1053	-0.9978	-0.613
e₁	-0.0066	0.0013	-0.0095	-0.0044
e₂	0.0854	0.0156	0.0549	0.1167
e₃	0.3526	0.0581	0.2427	0.4694
e₄	0.1683	0.1060	-0.0281	0.3937

	Coef	SE	T-value	P-Value
Constant	12.37	2.39	5.18	< 0.001
Age	0.0188	0.0207	0.91	0.366
Educ years	0.0010	0.0459	0.02	0.983
Durationdiagnosis in months	0.0022	0.0014	1.52	0.131
Medication compliance	-0.0263	0.0127	-2.07	0.040
Barrier score	-0.0100	0.0166	-0.60	0.549
Dietary knowledge skill score	-0.1267	0.0446	-2.84	0.005
BMI	0.0581	0.0289	2.01	0.046
Gender	0.057	0.387	0.15	0.883
total HEI score	-0.0221	0.0179	-1.23	0.220

	mean	s.d.	Lower 95%	Upper 95%
α	1.65	0.1424	1.389	1.909
β	-0.2276	0.7033	-0.9878	0.975
µ	8.791	0.1734	8.484	9.153
σ	1.393	0.103	1.208	1.616

	mean	s.d.	Lower 95%	Upper 95%
α	1.82	0.1479	1.47	1.994
β	0.0196	0.6218	-0.9489	0.9678
d₀	8.92	0.2138	8.506	9.3380
d₁	0.0267	0.0186	-0.0098	0.0629
d₂	-4,75E-01	0.0413	-0.0804	0.0879
d₃	0.0021	0.0014	4.09E-4	0.0059
d₄	-0.0164	0.0129	-0.0404	0.0121
d₅	-0.0428	0.0228	-0.0897	0.0014
d₆	-0.1468	0.0404	-0.2229	-0.0646
d₇	0.0734	0.0309	0.0100	0.1313
d₈	-0.0251	0.0179	-0.0603	0.0093
d₉	0.2887	0.3430	-0.4444	0.91
e₀	0.192	0.0995	0.0026	0.3804
e₁	0.0157	0.0080	0.0031	0.0316
e₂	-0.0213	0.0193	-0.0592	0.0163
e₃	-6,85E-01	0.0010	-0.0027	0.0012
e₄	0.0014	0.0058	-0.0113	0.0117
e₅	-0.0116	0.0067	-0.0246	0.0013
e₆	-0.0384	0.0206	-0.0813	0.0020
e₇	0.0155	0.0126	-0.0105	0.0393
e₈	-0.0079	0.0081	-0.0233	0.0098
e₉	0.2315	0.1399	-0.0732	0.4706

ϕ (t; α, β, μ, σ) = E [e^{i t X}] = [\begin{matrix} e x p \{i μ t - {|σ t|}^{α} (1 - i β (s i g n t) \tan (\frac{r α}{2})\}, α \neq 1 \\ e x p \{i μ t - α |t| (1 + i β \frac{2}{π} (s i g n t) \ln |t|)\}, α = 1 \end{matrix}] .

π (α, β, μ, σ | x) \propto \in {(\frac{α}{|α - 1| σ})}^{n} e x p \{- \sum_{i = 1}^{n} {|\frac{z_{i}}{t_{α, β} (y)}|}^{θ}\} \prod_{i = 1}^{n} {|\frac{z_{i}}{t_{α, β} (y)}|}^{θ} \frac{1}{|z_{i}|} π_{0} (α, β, μ, σ) d y .

π (α, β, μ, σ, y | x) \propto {(\frac{α}{|α - 1| σ})}^{n} e x p \{- \sum_{i = 1}^{n} {|\frac{z_{i}}{t_{α, β} (y_{i})}|}^{θ}\} \prod_{i = 1}^{n} {|\frac{z_{i}}{t_{α, β} (y_{i})}|}^{θ} \frac{1}{|z_{i}|} \prod_{i = 1}^{n} h (y_{i}) π_{0} (α, β, μ, σ) .

π (α, β, d, e | x) \propto \int {(\frac{α}{|α - 1| σ_{i}})}^{n} e x p \{\sum_{i = 1}^{n} {|\frac{z_{i}}{t_{α, β} (Y_{i})}|}^{θ}\} \prod_{i = 1}^{n} |\frac{z_{i}}{t_{α, β} (y_{i})}| \frac{1}{|z_{i}|} π_{0} (α, β, d, e) d y,

π (α, β, d, e, y | x) \propto {(\frac{α}{|α - 1| σ_{i}})}^{n} e x p \{\sum_{i = 1}^{n} |\frac{z_{i}}{t_{α, β} (y_{i})}|\} \prod_{i = 1}^{n} {|\frac{z_{i}}{t_{α, β} (y_{i})}|}^{θ} \frac{1}{|z_{i}|} \cdot \prod_{i = 1}^{n} h (y_{i}) π_{0} (α, β, d, e),

μ_{i} = d_{0} + d_{1} h e i g h t_{i} + d_{2} a g e_{i} + d_{3} g e n d e r_{i} + d_{4} s m o k e r_{i} σ_{i} = e x p (e_{0} + e_{1} h e i g h t_{i} + e_{2} a g e_{i} + e_{3} g e n d e r_{i} + e_{4} s m o k e r_{i}) .

H B A 1 c . a v e r a g e = 12.37 + 0.0188 a g e + 0.0010 e d u c . y e a r s + 0.00221 d u r a t i o n . d i a g n o s i s - 0.0263 m e d i c a t i o n . c o m p l i a n c e - 0.0100 b a r r i e r . s c o r e - 0.1267 d i e t a r y . k n o w l e d g e . s c o r e + 0.0581 B M I + 0.057 g e n d e r - 0.0221 t o t a l . H E I . s c o r e

μ_{i} = d_{0} + d_{1} a g e_{i} + d_{2} e d u c . y e a r s_{i} + d_{3} d u r a t i o n . d i a g n o s i s_{i} + d_{4} m e d i c a t i o n . c o m p l i a n c e_{i} + d_{5} b a r r i e r . s c o r e_{i} + d_{6} d i e t a r y . k n o w l w d g e . s c o r e_{i} + d_{7} B M I_{i} + d_{8} t o t a l . H E I . s c o r e_{i} + d_{9} g e n d e r_{i} σ_{i} = e x p (e_{0} + e_{1} a g e_{i} + e_{2} e d u c . y e a r s_{i} + e_{3} d u r a t i o n . d i a g n o s i s_{i} + e_{4} m e d i c a t i o n . c o m p l i a n c e_{i} + e_{5} b a r r i e r . s c o r e_{i} + e_{6} d i e t a r y . k n o w l e d g e . s c o r e_{i} + e_{7} B M I_{i} + e_{8} t o t a l . H E I . s c o r e_{i} + e_{9} g e n d e r_{i})

[1] *Corresponding author

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BAYESIAN ESTIMATION FOR THE STABLE DISTRIBUTIONS IN THE PRESENCE OF COVARIATES WITH APPLICATIONS IN CLINICAL ISSUES

ABSTRACT