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Table 1 Posterior estimates for lung function decline and pulmonary exacerbation onset based on separate and joint models

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

Parameter Separate model (Model I) Joint model (Model III)
Posterior mean 95% HPD CI Posterior mean 95% HPD CI
Semiparametric longitudinal submodel for FEV 1
Curve
 \(\beta_{0}\), intercept 18.57 [16.33, 20.68] 16.89 [14.63, 20.39]
 \(\beta_{1}\), age 2.846 [2.545, 3.143] 3.121 [2.74, 3.382]
 \(\beta_{2}\), \({\text{age}}^{2}\) − 0.1295 [− 0.1393, − 0.1197] − 0.1367 [− 0.1481, − 0.124]
 \(\beta_{3}\), \({\text{age}}^{3}\) − 0.0012 [− 0.0014, − 0.0009] − 0.0016 [− 0.0017, − 0.0014]
 \(b_{1}\) 0.0058 [0.0044, 0.0077] 0.0073 [0.0058, 0.0084]
 \(b_{2}\) 0.0083 [0.0076, 0.0089] 0.0103 [0.0095, 0.0111]
 \(b_{3}\) − 0.0143 [− 0.0187, − 0.0114] − 0.0206 [− 0.0248, − 0.0167]
 \(b_{4}\) − 0.0023 [− 0.0056, 0.0027] 0.0029 [− 0.001, 0.0085]
 \(b_{5}\) 0.0041 [− 0.0015, 0.0101] 0.0015 [− 0.0033, 0.007]
 \(b_{6}\) − 0.0011 [− 0.012, 0.0102] 0.0008 [− 0.0095, 0.0101]
 Baseline FEV1 0.6828 [0.6722, 0.6933] 0.6834 [0.6676, 0.6945]
 Male − 0.1665 [− 0.6403, 0.3137] − 0.1650 [− 0.6054, 0.2971]
 Birth cohort     
  < 1981 11.37 [9.651, 12.97] 12.39 [10.89, 13.95]
  1981–1988 3.766 [2.749, 4.839] 4.936 [3.995, 5.916]
  1989–1994 − 0.1912 [− 0.698, 0.3584] 0.0608 [− 0.4059, 0.5908]
  > 1994 (reference)     
 Lower SES − 0.6318 [− 1.075, − 0.1686] − 0.5984 [− 1.062, − 0.1119]
 CFRD − 0.4528 [− 0.765, − 0.1465] − 0.3695 [− 0.6731, − 0.0633]
 MRSA − 0.8209 [− 1.033, − 0.6121] − 0.8114 [− 1.016, − 0.6134]
 B. cepacia − 0.8318 [− 1.531, − 0.1367] − 0.7705 [− 1.441, − 0.1022]
 Pa − 0.5661 [− 0.6963, − 0.4314] − 0.5673 [− 0.6971, − 0.4373]
Sources of variation
 \(\sigma^{2}\) (measurement error) 59.29 [58.83, 59.76] 59.24 [58.77, 59.73]
 \(\sigma_{0}^{2}\) (between patients, intercept) 324.9 [307.6, 342.2] 326.8 [311, 343.8]
 \(\sigma_{1}^{2}\) (between patients, slope) 2.566 [2.441, 2.694] 2.602 [2.484, 2.727]
 \(\sigma_{01}\) (covariance, intercept and slope) − 25.51 [− 26.82, − 24.16] − 25.76 [− 27.11, − 24.49]
Weibull event submodel for pulmonary exacerbation
\(k\) 2.467 [2.395, 2.537] 2.564 [2.487, 2.634]
Intercept − 3.705 [− 3.988, − 3.416] − 3.568 [− 3.804, − 3.296]
\(\theta_{0}\) (random intercept) − 0.0086 [− 0.0137, − 0.0036]
\(\theta_{1}\) (random slope) − 0.3532 [− 0.4104, − 0.2979]
Baseline age − 0.0021 [− 0.0187, 0.0147] 0.0023 [− 0.0131, 0.0176]
Baseline FEV1 − 0.0158 [− 0.0178, − 0.0138] − 0.0198 [− 0.0218, − 0.0179]
Male − 0.2973 [− 0.3618, − 0.2314] − 0.2940 [− 0.3606, − 0.2267]
Birth cohort     
 < 1981 − 0.3785 [− 0.7538, − 0.0028] − 0.5348 [− 0.9056, − 0.1726]
 1981–1988 − 0.1805 [− 0.3947, 0.0182] − 0.2348 [− 0.4405, − 0.0339]
 1989–1994 0.1431 [0.031, 0.2544] 0.1261 [0.0178, 0.2335]
 > 1994 (reference)     
Lower SES 0.1173 [0.049, 0.1872] 0.1059 [0.0382, 0.1804]
  1. * In the Bayesian sense, a 95% CI that excludes zero indicates statistical significance at the 0.05 level. Parameters \(b_{1} - b_{6}\) are regression coefficients of the cubic truncated power functions defined in “Appendix 1