Model Name Abbreviation
Name Specification
Int Linear evolution with random intercept
Int + Slope Linear evolution with random intercept and slope
Spline Non-linear evolution by using spline as both average evolution and random effect
Int + Corr Linear evolution with random intercept and exponential correlation structure
Spline + Corr Non-linear evolution with spline as average evolution, random intercept and exponential correlation structure
Lin_GEE Linear evolution with independence correlation structure
Nonlin_GEE Non-linear evolution by using spline with independence correlation structure
Lin_JM Linear evolution with jointly model FEV and survival
Nonlin_JM Non-linear evolution with jointly model FEV and survival

1 Model Comparison using AIC and BIC

We compare all LME models using AIC and BIC. Note that these criterion cannot be used to evaluate fit for the GEE and joint models.

df AIC BIC
Int 5 9363285 9363345
Int + Slope 7 9045074 9045158
Spline 16 8928750 8928942
Int + Corr 7 8833945 8834029
Spline + Corr 12 8830315 8830459

For the GEE model, the criterion Quasi-AIC, known as QIC, is used to compare GEE type models, since GEE model estimation is based on quasi-likelihood; by contrast, the LME models reported here are estimated based on a specified likelihood. The QIC for linear and nonlinear GEE models are below.

Model QIC
Lin_GEE 7593016
Nonlin_GEE 7542590

For the joint model, the AIC value represents fit for the two process (FEV1 and survival), whereas the AIC for the other models are for only one process, FEV1. Thus, we can compare AIC across the single-process models of FEV1, which are LME models in this application; however, the value is not comparable for the joint model and GEE. The AIC and BIC for linear and nonlinear joint models are below.

df AIC BIC
Lin_JM 12 9054910 9055010
Nonlin_JM 17 9046671 9046813

2 Model Comparison using MAD, RMSE and Correlation

The following model shows Median Absolute Deviation (MAD), Root-mean-square deviation (RMSD) and correlation of observations (FEV1) and fitted value. For MAD and RMSE, the smaller the better model; for Corr (correlation), the larger the better model.

MAD RMSE Corr
Int 5.957 10.612 0.912
Int + Slope 4.722 8.913 0.939
Spline 4.244 8.174 0.949
Int + Corr 4.787 7.799 0.961
Spline + Corr 5.036 8.064 0.959
Lin_JM 4.741 9.030 0.937
Nonlin_JM 4.725 8.998 0.938

3 Plot Observation and Estimated Evolution

Here we recall the estimation of population-level evolution by different models we have presented.

Here we are showing the patient-specific evolution by different models. We randomly select 8 subjects from the sample and show the subject’s FEV1 from observations (black points) with estimation from models (color curves). The following figures for subject 916320770, 923790754, 927070700, 953850736, 962680738, 964930738, 973910716, 976470708.