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Table 4 Posterior estimates under two-stage model and joint model for lung function decline and pulmonary exacerbation onset

From: Flexible semiparametric joint modeling: an application to estimate individual lung function decline and risk of pulmonary exacerbations in cystic fibrosis

Parameter Two-stage model Joint model (Model III)
Posterior mean 95% HPD CI Posterior mean 95% HPD CI
Semiparametric longitudinal submodel for FEV 1
Curve
 \(\beta_{0}\), intercept 18.15 [15.54, 20.95] 16.89 [14.63, 20.39]
 \(\beta_{1}\), age 2.86 [2.477, 3.189] 3.121 [2.74, 3.382]
 \(\beta_{2}\), age2 − 0.1306 [− 0.1418, − 0.117] − 0.1367 [− 0.1481, − 0.124]
 \(\beta_{3}\), age3 − 0.0011 [− 0.0013, − 0.0009] − 0.0016 [− 0.0017, − 0.0014]
 \(b_{1}\) 0.0051 [0.0039, 0.0061] 0.0073 [0.0058, 0.0084]
 \(b_{2}\) 0.0108 [0.0097, 0.0119] 0.0103 [0.0095, 0.0111]
 \(b_{3}\) − 0.0177 [− 0.0226, − 0.0133] − 0.0206 [− 0.0248, − 0.0167]
 \(b_{4}\) − 0.0001 [− 0.0044, 0.0065] 0.0029 [− 0.001, 0.0085]
 \(b_{5}\) 0.0033 [− 0.0024, 0.0083] 0.0015 [− 0.0033, 0.007]
 \(b_{6}\) − 0.0009 [− 0.0114, 0.0095] 0.0008 [− 0.0095, 0.0101]
 Baseline FEV1 0.686 [0.6728, 0.6978] 0.6834 [0.6676, 0.6945]
 Male − 0.15 [− 0.582, 0.2736] − 0.1650 [− 0.6054, 0.2971]
 Birth cohort     
  < 1981 11.43 [9.837, 13.29] 12.39 [10.89, 13.95]
  1981–1988 4.0 [2.992, 4.978] 4.936 [3.995, 5.916]
  1989–1994 − 0.1924 [− 0.7038, 0.3305] 0.0608 [− 0.4059, 0.5908]
  > 1994 (reference)     
 Lower SES − 0.623 [− 1.091, − 0.175] − 0.5984 [− 1.062, − 0.1119]
 CFRD − 0.4613 [− 0.7895, − 0.1506] − 0.3695 [− 0.6731, − 0.0633]
 MRSA − 0.816 [− 1.026, − 0.6022] − 0.8114 [− 1.016, − 0.6134]
 B. cepacia − 0.7964 [− 1.465, − 0.1291] − 0.7705 [− 1.441, − 0.1022]
 Pa − 0.5643 [− 0.6953, − 0.4345] − 0.5673 [− 0.6971, − 0.4373]
Sources of variation
 \(\sigma^{2}\) (measurement error) 59.29 [58.83, 59.78] 59.24 [58.77, 59.73]
 \(\sigma_{0}^{2}\) (between patients, intercept) 324.1 [307.6, 341.3] 326.8 [311, 343.8]
 \(\sigma_{1}^{2}\) (between patients, slope) 2.564 [2.443, 2.69] 2.602 [2.484, 2.727]
 \(\sigma_{01}\) (covariance, intercept and slope) − 25.46 [− 26.81, − 24.18] − 25.76 [− 27.11, − 24.49]
Weibull event submodel for pulmonary exacerbation
\(k\) 2.524 [2.456, 2.601] 2.564 [2.487, 2.634]
intercept − 3.509 [− 3.802, − 3.26] − 3.568 [− 3.804, − 3.296]
\(\theta_{0}\) (random intercept) − 0.0098 [− 0.0149, − 0.0048] − 0.0086 [− 0.0137, − 0.0036]
\(\theta_{1}\) (random slope) − 0.3273 [− 0.3825, − 0.2728] − 0.3532 [− 0.4104, − 0.2979]
Baseline age 0.0002 [− 0.0136, 0.0161] 0.0023 [− 0.0131, 0.0176]
Baseline FEV1 − 0.0193 [− 0.0211, − 0.0175] − 0.0198 [− 0.0218, − 0.0179]
Male − 0.2919 [− 0.36, − 0.2194] − 0.2940 [− 0.3606, − 0.2267]
Birth cohort
 < 1981 − 0.5046 [− 0.8512, − 0.1807] − 0.5348 [− 0.9056, − 0.1726]
 1981–1988 − 0.2297 [− 0.4382, − 0.0415] − 0.2348 [− 0.4405, − 0.0339]
 1989–1994 0.1171 [0.0086, 0.2197] 0.1261 [0.0178, 0.2335]
 >1994 (reference)
Lower SES 0.1091 [0.0369, 0.1799] 0.1059 [0.0382, 0.1804]