Reliability with multiple causes of failures: Modeling and practice through a case study on ultrasound probes for medical imaging | International Journal of Metrology and Quality Engineering (IJMQE)

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Table 3

Model results by cause of failure and USP types − with random effects (formula (8)).

Cause of failure c₁	Estimate	SE	p-value	HR
β_1(a)	0.0000	–	–	–
β_1(b)	–3.7301	1.3009	0.0041	exp(–3.7301) = 0.024
γ_1(a)	–0.8712	0.2516	0.0005	exp(–0.8712) = 0.418
γ_1(b)	–0.3216	0.8198	0.6948	exp(–0.3216) = 0.725
δ_1(a)	1.4604	0.2992	0.0000	exp(1.4604) = 4.308
δ_1(b)	0.6131	1.1081	0.5801	exp(0.6131) = 1.846
$σ_{1}^{2}$	0.8014	–	–	–
$τ_{1}^{2}$	1.7108	–	–	–

Cause of failure c₂	Estimate	SE	p-value	HR
β_2(a)	0.0000	–	–	–
β_2(b)	–0.5746	0.1950	0.0032	exp(–0.5746) = 0.563
γ_2(a)	–0.1753	0.1126	0.1195	exp(–0.1753) = 0.839
γ_2(b)	–0.0413	0.1367	0.7627	exp(–0.0413) = 0.960
δ_2(a)	0.3675	0.1332	0.0058	exp(0.3675) = 1.444
δ_2(b)	0.0172	0.1756	0.9219	exp(0.0172) = 1.017
$σ_{2}^{2}$	0.0307	–	–	–
$τ_{2}^{2}$	0.0873	–	–	–

Cause of failure c₃	Estimate	SE	p-value	HR
β_3(a)	0.0000	–	–	–
β_3(b)	–0.9476	0.8244	0.2504	exp(–0.9476) = 0.388
γ_3(a)	–1.8374	0.5932	0.0020	exp(–1.8374) = 0.159
γ_3(b)	–3.1527	1.7064	0.0647	exp(–3.1527) = 0.043
δ_3(a)	1.3520	0.4812	0.0050	exp(1.3520) = 3.865
δ_3(b)	1.2428	0.7531	0.0989	exp(1.2428) = 3.465
$σ_{3}^{2}$	11.7400	–	–	–
$τ_{3}^{2}$	1.9594	–	–	–

Cause of failure c₄	Estimate	SE	p-value	HR
β_4(a)	0.0000	–	–	–
β_4(b)	–0.1374	0.3432	0.6890	exp(–0.1374) = 0.872
γ_4(a)	–0.3930	0.2043	0.0545	exp(–0.3930) = 0.675
γ_4(b)	–0.3897	0.2464	0.1138	exp(–0.3897) = 0.677
δ_4(a)	1.0287	0.2469	0.0000	exp(1.0287) = 2.797
δ_4(b)	0.6833	0.2854	0.0167	exp(0.6833) = 1.980
$σ_{4}^{2}$	0.2059	–	–	–
$τ_{4}^{2}$	0.8185	–	–	–

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