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 2

Model results by cause of failure and USP types − with random effects (formula (7)).

Cause of failure c₁	Estimate	SE	p-value	HR
β_1(a)	0.0000	–	–	–
β_1(b)	–3.7449	1.3078	0.0042	exp(–3.7449) = 0.024
γ_1(a)	–0.8905	0.2520	0.0004	exp(–0.8905) = 0.411
γ_1(b)	–0.3330	0.8330	0.6893	exp(–0.3330) = 0.719
δ_1(a)	1.4895	0.2996	0.0000	exp(1.4895) = 4.435
δ_1(b)	0.6290	1.1210	0.5747	exp(0.6290) = 1.876
$σ_{1}^{2}$	0.8325	–	–	–
$τ_{1}^{2}$	1.7653	–	–	–

Cause of failure c₂	Estimate	SE	p-value	HR
β_2(a)	0.0000	–	–	–
β_2(b)	–0.5812	0.1977	0.0033	exp(–0.5812) = 0.559
γ_2(a)	–0.1960	0.1162	0.0917	exp(–0.1960) = 0.822
γ_2(b)	–0.0469	0.1433	0.7436	exp(–0.0469) = 0.954
δ_2(a)	0.3976	0.1350	0.0032	exp(0.3976) = 1.488
δ_2(b)	0.0222	0.1795	0.9014	exp(0.0222) = 1.022
$σ_{2}^{2}$	0.0359	–	–	–
$τ_{2}^{2}$	0.0996	–	–	–

Cause of failure c₃	Estimate	SE	p-value	HR
β_3(a)	0.0000	–	–	–
β_3(b)	–0.9730	0.8306	0.2414	exp(–0.9730) = 0.378
γ_3(a)	–1.8625	0.5933	0.0017	exp(–1.8625) = 0.155
γ_3(b)	–3.1684	1.7152	0.0647	exp(–3.1684) = 0.042
δ_3(a)	1.3990	0.4829	0.0038	exp(1.3990) = 4.051
δ_3(b)	1.2654	0.7607	0.0962	exp(1.2654) = 3.545
$σ_{3}^{2}$	11.9089	–	–	–
$τ_{3}^{2}$	2.0580	–	–	–

Cause of failure c₄	Estimate	SE	p-value	HR
β_4(a)	0.0000	–	–	–
β_4(b)	–0.1523	0.3437	0.6577	exp(–0.1523) = 0.859
γ_4(a)	–0.4134	0.2056	0.0444	exp(–0.413) = 0.661
γ_4(b)	–0.4003	0.2489	0.1077	exp(–0.4003) = 0.670
δ_4(a)	1.0576	0.2473	0.0000	exp(1.0576) = 2.880
δ_4(b)	0.6880	0.2861	0.0162	exp(0.6880) = 1.990
$σ_{4}^{2}$	0.2195	–	–	–
$τ_{4}^{2}$	0.8522	–	–	–

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