A Statistical Model for Lung Function Trajectory and
Mortality in Patients with Fibrotic ILD
Journal:	American Journal of Respiratory and Critical Care Medicine	
Manuscript ID	Blue-202505-1169OC	
Manuscript Type:	OC - Original Contribution	
Date Submitted by the Author:	14-May-2025	
Complete List of Authors:	Wendelberger, Barbara; Berry Consultants, LLC
Jensen, Thomas; Berry Consultants, LLC
Quintana, Melanie; Berry Consultants, LLC
Molyneaux, Philip; Imperial College London, Interstitial Lung Disease
Unit, National Heart and Lung Institute; Royal Brompton & Harefield NHS Foundation Trust
Maher, Toby; University of Southern California Keck School of Medicine, PCCSM; Imperial College London, Interstitial Lung Disease Unit, National Heart and Lung Institute
Oldham, Justin; University of Michigan, Division of Pulmonary and Critical Care Medicine
Johnson, Simon; University of Nottingham, Centre for Respiratory
Research and Biomecal Research Centre; University of Nottingham School of Medicine, Centre and Biodiscovery Institute
Fahy, William; GlaxoSmithKline, Diseases of the Developing World
Khan, Fasihul; University Hospitals of Leicester NHS Trust, Glenfield Hospital
Stewart, Iain; Imperial College London, National Heart & Lung Institute Abdulqawi, Rayid; King Faisal Specialist Hospital and Research Center, Lung Health Center
Allen, Richard; University of Leicester College of Life Sciences, Department of Population Health Sciences
Corte, Tamera J.; Royal Prince Albert Hospital, Department of Respiratory Medicine,; The University of Sydney Sydney Medical School, Central Clinical School
Funke-Chambour, Manuela; Inselspital University Hospital Bern, Department of Pulmonary Medicine;
Glaspole, Ian; Alfred Health, Department of Respiratory and Sleep Medicine
Holland, Anne; Alfred Health, Department of Respiratory Medicine;
Monash University, School of Translational Medicine; Institute for
Breathing and Sleep; Alfred Health, Physiotherapy
Johannson, Kerri; University of Calgary, Department of Medicine;
University of Calgary, Snyder Institute for Chronic Diseases
Kreuter, Michael; University Medical Center of the Johannes Gutenberg University Mainz, Mainz Centre for Pulmonary Medicine, Departments of Pneumology
Kulkarni, Tejaswini; The University of Alabama at Birmingham, Pulmonary, Allergy and Critical Care	

	Martinez, Fernando; University of Massachusetts Chan Medical School, Department of Medicine
Montesi, Syndey; Massachusetts General Hospital, Division of Pulmonary and Critical Care Medicine
Rajan, Sujeet; Bombay Hospital Institute of Medical Sciences Rivera-Ortega, Pilar; Royal Devon and Exeter Hospital, Interstitial Lung Disease Unit
Ryerson, Christopher; The University of British Columbia, Medicine; The University of British Columbia, Centre for Heart Lung Innovation Sakamoto, Koji; Nagoya University, Department of Respiratory Medicine Wain, Louise; University of Leicester, Department of Population Health Sciences; Glenfield Hospital, National Institute for Health Research, Leicester Respiratory Biomedical Research Centre
Wells, Athol; Imperial College London, Interstitial Lung Disease Unit, National Heart and Lung Institute; Royal Brompton & Harefield NHS Foundation Trust
Adams, Wendy; Action for Pulmonary Fibrosis (Formerly)
Kawano-Dourado, Leticia; HCor, Hcor Research Institute; University of Sao Paulo, Pulmonary Division, Heart Institute (InCor); MAGIC Evidence Ecosystem Foundation
Jenkins, R. Gisli; Imperial College London, Margaret Turner Warwick Centre for Fibrosing Lung Disease, National Heart & Lung Institute Lewis, Roger; Berry Consultants, LLC; David Geffen School of Medicine, Department of Emergency Medicine; The Lundquist Institute for Biomedical Innovation, Department of Emergency Medicine	
Subject Category:	9.23 Interstitial Lung Disease < LUNG DISEASES, BIOSTATISTICS	
Keywords:	interstitial pulmonary fibrosis, disease modeling, competing risk models	

A Statistical Model for Lung Function Trajectory and Mortality in Patients with Fibrotic
Interstitial Lung Disease
Barbara Wendelberger, PhD,1 Thomas P. Jensen, MS,1 Melanie Quintana, PhD,1 Philip L. Molyneaux, MD,2,3 Toby M. Maher, MD, PhD2,4 Justin M. Oldham, MD, MS,5 Simon R. Johnson, MD,6,7 William A. Fahy, MD,8 Fasihul Khan, MD,9 Iain Stewart, PhD,10
Rayid Abdulqawi, MD,11 Richard Allen, PhD,12 Tamera J. Corte, MBBS, PhD13,14
Manuela Funke-Chambour, MD,15 Ian Glaspole, MD,16 Anne E. Holland, PhD,17-20
Kerri A. Johannson, MD,21,22 Michael Kreuter, MD,23 Tejaswini Kulkarni, MD,24
Fernando J. Martinez, MD,25 Sydney B. Montesi, MD,26 Sujeet Rajan, MD,27
Pilar Rivera-Ortega, MD,28 Christopher J. Ryerson, MD,29,30 Koji Sakamoto, MD,31
Louise V. Wain, PhD,12,32 Athol U. Wells, MD,2,3 Wendy Adams,33
Leticia Kawano-Dourado, MD,34-36 R. Gisli Jenkins, MD,*37 Roger J. Lewis, MD,
PhD*1,38,39 on behalf of the REMAP-ILD Consortium Investigators
* Denotes equal contributions as senior authors.
1.		Berry Consultants, LLC, Austin, Texas, United States
2.		Interstitial Lung Disease Unit, National Heart and Lung Institute, Imperial College London, United Kingdom
3.		Royal Brompton and Harefield NHS Foundation Trust, Sydney Street, London, United Kingdom
4.		Keck Medicine of University of Southern California, Los Angeles, California, United States
5.		Division of Pulmonary and Critical Care Medicine, University of Michigan, Ann Arbor, Michigan, United States
6.		Centre for Respiratory Research, NIHR Respiratory Biomedical Research Centre and Biodiscovery Institute, Translational Medical Sciences, University of Nottingham, United Kingdom
7.		Centre and Biodiscovery Institute, School of Medicine, University of Nottingham, United Kingdon
8.		Discovery Medicine, GlaxoSmithKline, Stevenage, United Kingdom
9.		Glenfield Hospital, University Hospitals of Leicester NHS Trust, Leicester, United Kingdom
10.		National Heart and Lung Institute, Imperial College London, London, United Kingdom
11.		King Faisal Specialist Hospital and Research Centre, Riyadh, Sauda Arabia
12.		Department of Population Health Sciences, University of Leicester, Leicester, United Kingdom
13.		Department of Respiratory Medicine, Royal Prince Alfred Hospital Camperdown, NSW, Australia
14.		Central Clinical School, Sydney Medical School, University of Sydney, NSW, Australia
15.		Department for Pulmonary Medicine, Allergology and Clinical Immunology, Inselspital, Bern University Hospital, University of Bern, Switzerland
16.		Department of Respiratory and Sleep Medicine, Alfred Health, Melbourne, Australia
17.		Department of Respiratory Medicine, Alfred Health, Melbourne, Australia
18.		School of Translational Medicine, Monash University, Melbourne, Australia
19.		Department of Physiotherapy, Alfred Health, Melbourne, Australia
20.		Institute for Breathing and Sleep, Melbourne, Australia
21.		Department of Medicine, University of Calgary, Calgary, Alberta, Canada
22.		Snyder Institute for Chronic Diseases, University of Calgary, Calgary, Alberta, Canada
23.		Mainz Centre for Pulmonary Medicine, Departments of Pneumology, Mainz University Medical Centre and of Pulmonary, Critical Care & Sleep Medicine, Marienhaus Clinic Mainz, Mainz, Germany
24.		The University of Alabama at Birmingham Heersink School of Medicine, Birmingham, Alabama, United States
25.		University of Massachusetts Chan/UMass Memorial Health, Worcester, MA, United States
26.		Division of Pulmonary and Critical Care Medicine, Massachusetts General Hospital, Boston, Massachusetts, United States
27.		Bombay Hospital Institute of Medical Sciences and Bhatia Hospital, Mumbai, India
28.		Interstitial Lung Disease Unit, Royal Devon and Exeter Hospital, Royal Devon University Healthcare NHS Foundation Trust, Exeter, United Kingdom
29.		Department of Medicine, University of British Columbia, Vancouver, British Columbia, Canada
30.		Centre for Heart Lung Innovation, University of British Columbia, Vancouver, British Columbia, Canada
31.		Department of Respiratory Medicine, Nagoya University, Nagoya, Japan
32.		National Institute for Health Research, Leicester Respiratory Biomedical Research Centre, Glenfield Hospital, Leicester, United Kingdom
33.		Action for Pulmonary Fibrosis Foundation, London, United Kingdom
34.		Hcor Research Institute, Hcor Hospital, Sao Paulo, Brazil
35.		Pulmonary Division, Heart Institute (InCor), University of Sao Paulo, Sao Paulo, Brazil
36.		MAGIC Evidence Ecosystem Foundation, Oslo, Norway
37.		Margaret Turner Warwick Centre for Fibrosing Lung Disease, National Heart and Lung Institute, Imperial College London, London, United Kingdom
38.		Department of Emergency Medicine, David Geffen School of Medicine at UCLA, Los Angeles, California, United States
39.		The Lundquist Institute for Biomedical Innovation, Torrance, California, United States
Revision Date: May 13, 2025
Word count:
Abstract: 248
Body: 3817
References: 31
Tables: 3
Figures: 4
Appendices: 4
Address for Correspondence:
Roger J. Lewis, MD, PhD
Berry Consultants, LLC
3345 Bee Caves Rd, Suite 201
Austin, Texas 78746
Email: roger@berryconsultants.net
Mobile: +1 310-720-1661
Abstract
Background. Fibrotic interstitial lung diseases (ILDs) cause loss of forced vital capacity (FVC) and increased risk of death over time. Most clinical trials aim to slow FVC decline and reduce mortality. However, the association of lower FVC with higher mortality will bias estimates of differences in FVC progression between groups. Therefore, both the time-dependent decline in FVC and increase in mortality should be jointly modelled. Methods. We developed a Bayesian, joint random-effects disease progression model (DPM), using minimally informative prior distributions, for FVC trajectory and hazard for mortality over time. This model minimizes bias due to mortality in estimating differences in the rate of FVC decline and is suitable for use when characterizing populations or in estimating a treatment effect in a clinical trial. The DPM was applied to individual patient data from prospective cohort studies of fibrotic ILD (PROFILE and INJUSTIS).
Results. The DPM yields a higher estimated rate of FVC decline (6.0 vs 4.7%/year), a more precise fit than a mixed linear model of FVC alone, and replicates the non-linear pattern in observed data. By modelling the full FVC trajectory rather than only the change from baseline at a given time point, the approach increases the information obtained from each patient and reduces both the time to information and the effect of variability in baseline FVC measurements on the estimation of treatment effects.
Conclusions. The joint DPM provides an integrated approach to minimizing bias in the estimation of treatment effects in clinical trials in fibrotic ILDs.
Introduction
Fibrotic interstitial lung disease (ILD) is a heterogeneous group of disorders characterized by inflammation and fibrosis of the pulmonary parenchyma.1 Fibrotic ILD is typically progressive, with high symptom burden, morbidity, and risk of mortality.1 Recent data suggest that both the incidence and prevalence of ILD are increasing globally.2-5
There is no cure for fibrotic ILD and few effective therapies, with the sentinel clinical trials of antifibrotics published approximately a decade ago.6,7 Several potentially effective drugs are in various stages of clinical investigation,8,9 and, while identifying safe and effective therapeutics is a key priority in the field, progress has been slow. This is due to multiple factors, including the failure of promising drugs to demonstrate efficacy, as well as the cost and time burdens associated with testing therapies using standard trial designs that rely on long follow-up to assess treatment response. There remains an urgent need to accelerate the identification of effective therapies for patients with fibrotic ILD to halt further disease progression or even reverse the fibrotic process.
The optimal endpoint in clinical trials of fibrotic ILD—and the optimal analysis strategy for the endpoint—have been the subject of considerable debate.10,11 These choices influence the required size and duration of clinical trials, the likelihood of success given an effective treatment, regulatory approval, and clinical impact. While mortality is objective and clearly important to patients and their families, the value of mortality as a primary endpoint is limited by the time required for mortality events to accrue, the large sample sizes required for adequate power, and the failure of mortality to capture symptom burden prior to death.10,11
Change in Forced Vital Capacity (FVC) is currently considered the preferred endpoint in fibrotic ILD by many trialists and regulatory bodies, providing a general measure of overall disease progression and acting as a surrogate for mortality.12 Regulatory approvals of antifibrotics have been based on treatment effects quantified as a difference in the change from baseline FVC.6,7,12
However, quantifying the outcome in fibrotic ILD as the change from baseline requires a strong assumption about the relationship of baseline FVC and treatment effect.13,14 This is commonly addressed by including the observed baseline FVC as a covariate in the primary analysis model. Furthermore, the high variability in FVC measurements at any one time reduces the precision of the measurement of the difference. The signal-to- noise ratio in the assessment of each patient’s course can be improved by evaluating the disease trajectory over time and taking advantage of the information in a greater number of FVC assessments at intermediate time points, mirroring an approach used successfully in other settings.15-20
Lastly, analyzing the change in FVC is subject to bias from informative censoring due to death. Informative censoring exists when there is an association between the probability of having missing outcome data—being censored—and the value of the outcome data that would have been obtained had it been measured. Here, the fact that a patient dies contains information about the likely FVC trajectory for the patient around the time of 
death. The association between lower FVC and more rapid rates of decline with an increased risk of mortality is likely to lead to a progressive depletion of the sickest individuals from a clinical trial cohort over time. As patients with the lowest FVC measurements, or the steepest FVC declines, are lost to follow up, this results in an upward bias in the estimation of the population FVC trajectory, and an unpredictable bias in the estimation of treatment effects in a clinical trial. This last challenge can only be directly addressed by explicitly considering mortality in the primary endpoint analysis strategy.
This study presents a Bayesian joint model for the FVC trajectory and hazard for mortality over time for patients with fibrotic ILD that addresses the limitations noted above and is suitable for use in characterizing or comparing populations or in estimating treatment effects in clinical trials.
Methods
Overall Approach and Data Sources
We propose a Bayesian joint competing-risks21 model of FVC progression and survival times (i.e., time to death), termed the “disease progression model” (DPM), as a primary model for analyses of outcomes of patients with fibrotic ILD. The DPM is intended both for the analysis and comparison of patient populations and for the estimation of treatment effects in clinical trials. The DPM estimates FVC progression using a linear mixed-effects model and estimates survival time using a survival regression model with time-varying covariates and a Weibull hazard rate function.22 The FVC trajectory and survival models are linked, creating a joint model, by allowing the hazard for death to depend on the FVC and, separately, on the rate of FVC decline. The Bayesian 
framework of the proposed DPM uses minimally informative prior distributions and is suitable for use in adaptive, platform, or other innovative clinical trial designs.23 The joint DPM is structured to address the potential bias that occurs when analyzing FVC alone, without accounting for the informative nature of the missing data due to death. The DPM accounts for death as a censoring event of special interest due to its association with FVC.
We illustrate the use of the DPM using data from two independent cohorts: the PROFILE24 and INJUSTIS25 studies, which include patients with both idiopathic pulmonary fibrosis (IPF) and non-IPF fibrotic ILD, or simply non-IPF for brevity. These data include measurements of FVC and mortality events or patient withdrawals including those who underwent lung transplantation.
Censoring events within cohort and clinical trial data in fibrotic ILD are generally not limited to death, and may include withdrawal of consent, other loss to follow up, and lung transplantation. An assumption of the DPM is that these non-mortality censoring events are unrelated to the FVC-based outcome and therefore do not need special consideration in the analysis approach. The PROFILE and INJUSTIS cohort datasets capture the reason for censoring events, making them ideal for illustrating the DPM. Other observational cohort studies may not capture the nature of censoring events, limiting the ability to apply a joint model. This is less likely to be a limitation in the setting of a prospective clinical trial.
Patient Cohorts
Data for demonstration and evaluation of the DPM was obtained from investigators for the PROFILE24 and INJUSTIS25 observational cohort studies. Data from the two studies were combined and patients were classified as having IPF or non-IPF (see details in Appendix 1), to yield a final patient cohort of 892 patients, 621 with IPF and 271 with non-IPF. Baseline was defined as the first date for which percent-predicted FVC (ppFVC) measurement was available. The patient data were truncated to include FVC measurements and survival status up to 910 days (2.5 years) following the baseline visit. Only death was labelled as an informative censoring event in the analysis, whereas early discontinuation for consent withdrawal, loss to follow up, lung transplantation, lack of complete follow up, and other reasons were treated as noninformative censoring events.
Joint Disease Progression Model (DPM)
The DPM is composed of two component sub-models: one for the “underlying” FVC progression and one for survival. In this context, the underlying FVC progression rate denotes the FVC progression rate in the population of interest had there been no censoring. The FVC progression sub-model is a Bayesian linear mixed effects model with fixed effects including an overall baseline intercept (i.e., the average baseline FVC measurement for the population) and an overall linear effect of time (i.e., the average slope or rate of loss of FVC). Random effects include a patient-specific intercept offset and a patient-specific linear effect of time, i.e., a difference in the rate of each patient’s FVC decline compared to the overall population. FVC is modelled as the percent predicted (ppFVC) on an absolute scale as opposed to a relative change from baseline. The complete structure of the DPM is provided in Table 1.
The survival sub-model is a Bayesian time-to-event model with a Weibull baseline hazard rate, scaled by a time-varying hazard ratio.22 The time varying hazard ratio for death for each patient is a function of the individual patient’s FVC and, separately, their rate of FVC decline, as estimated by the FVC sub-model. This links the two sub-models together via shared parameters at the level of the individual patient. Specifically, the survival sub-model hazard ratio includes a term capturing the effect of the estimated FVC ( 1in Table 1), and a second term for capturing the effect of the estimated rate of FVC decline ( 2in Table 1), on the hazard for death for each patient. This allows the model to minimize the bias in estimating FVC decline due to informative censoring and to directly estimate the effects of both FVC and the rate of FVC decline on mortality.
Similar to a traditional survival analysis model, such as a Cox model, the survival submodel naturally accommodates both informative and non-informative censoring mechanisms using an indicator to specify which censoring events were not random (e.g., due to death) and which were random (e.g., loss to follow up unrelated to disease progression or death). Because the hazard ratio that measures the difference between individual survival functions and the overall population-averaged survival curve depends on time, via the FVC-related terms, a proportional hazards assumption does not hold for the joint model.
When performing comparisons of subgroups or populations of interest (e.g., patients in two arms of a clinical trial), two additional fixed effects are included in the FVC progression sub-model, a group-specific intercept offset and group-specific slope offset ( and in Table 1). We also include a term quantifying a fixed effect on survival, as a HR, to account for differing mortality rates across groups that are independent of FVC and the rate of FVC decline ( in Table 1). In a clinical trial setting, the terms for the group-specific slope and the adjustment to the hazard ratio reflect the direct treatment effect on FVC decline and on survival, respectively. There is also an indirect treatment effect on survival that is mediated by the effect of treatment on FVC trajectory. Because the proportional hazards assumption does not hold for this model, the hazard ratio for survival differs over the duration of follow up, and it is necessary to select one or more specific time points of interest at which survival is to be compared between populations.
To illustrate the bias that occurs when simple linear models ignoring informative censoring are fit to the FVC data, we also fit a Bayesian linear mixed effects model to the FVC data without regard to censoring. This model included random intercepts and slopes for each patient, as would typically be done in this setting.
Using a Bayesian framework for estimating these models naturally yields measures of precision (e.g., 95% Bayesian credible intervals [CIs] around the population-averaged underlying FVC trajectory, population-averaged survival curves, or population differences or treatment effects). The model is fit using a Markov-chain Monte Carlo (MCMC) approach to numerical integration.26 Each iteration of the MCMC process yields a set of estimated FVC and survival curves that includes an FVC curve and survival curve for each patient due to the random-effects modelling, which can be averaged across population subgroups. The distribution of averaged, population subgroup curves across each iteration of the MCMC algorithm then allows for the quantification of uncertainty in the estimated average curves (e.g., the generation of 95% Bayesian CIs).
Two analyses of the combined cohort populations were conducted: one characterizing the combined cohort with no distinction between IPF and non-IPF and one comparing the IPF and non-IPF subgroups. The latter analysis--with the DPM modified to enable comparisons as above--mirrors the analysis that would be used in evaluating a treatment effect in a clinical trial, i.e., comparing outcomes in two populations defined by two different treatment arms. The complete mathematical structure for the DPM is included in Appendix 2 and code for model fitting (using R version 4.3.3)27 is provided in Appendix 3.
Results
Initially the DPM was fit to the combined patient cohorts, without regard to IPF and non- IPF status, to yield estimated underlying FVC trajectory and survival curves, as well as model predictions for the observed FVC progression rates. The FVC linear sub-model is displayed for the underlying absolute ppFVC (Figure 1a) and for the change from baseline of ppFVC (See Figure 1b), displaying the underlying loss of lung function had censoring due to death not occurred. The survival sub-model (Figure 1c) shows the decline in the probability of survival, which accelerates over time. The corresponding quantitative results are presented in Table 2a. Figure 1d illustrates the DPM-estimated 
Joint Modelling of FVC and Mortality in ILD population-averaged FVC trajectories, obtained by combining the underlying linear FVC loss and informative censoring at the patient-level through weighting of FVC progression by the probability of survival. The DPM’s predictions for the population- averaged, observed FVC progression rate in the presence of censoring due to mortality can be compared visually or quantitatively with the observed data. The DPM reproduces the non-linearity observed in the population averages for both absolute ppFVC (Figure 1d) and change from baseline in ppFVC (Figure 1e).
For comparison, the Bayesian linear mixed model was fit to the combined cohort population data, with results shown for the absolute ppFVC (Figure 2a) and for the change from baseline in ppFVC (Figure 2b). The linear mixed model yields a 22% lower estimate for the underlying annual rate of FVC decline than the DPM (-4.7% [95% CI - 5.3%, -4.1%] versus -6.0% [95% CI -6.7%, -5.3%]), because the model fails to account for the depletion of the patients with more severe disease over time (Table 2a). Since it is a linear model, the linear mixed model is unable to reproduce the observed nonlinearity in the population-averaged FVC progression. The DPM results in a better fit to the observed data, as quantified by a lower root-mean-square error (Appendix 4).
Next, the DPM was fit to the combined cohort data distinguishing the IPF and non-IPF disease categories, with IPF as the reference category, to compare lung function trajectories across groups (Figure 3). This analysis estimates distinct lung function and survival probabilities for the IPF and non-IPF subgroups. The DPM estimates for the underlying FVC progression for the IPF and non-IPF disease categories suggest that the non-IPF subpopulation has both higher ppFVC at baseline (88% in non-IPF vs 80% in IPF; Table 2b and Figure 3a) and less rapid annual decline in ppFVC (-5.0%/year non-IPF vs -6.4%/year in IPF; Table 2b and Figure 3b). The DPM-based estimate for the difference in the annual rate of FVC decline in non-IPF vs IPF is 1.4%/year (corresponding to the term in Table 2b and indicating a slower rate of decline in non- IPF) with a 95% CI of 0.1%/year to 2.7%/year.
Similarly, there is a lower risk of mortality across time among non-IPF patients compared with IPF patients (Figure 3c). The model estimates for the hazard ratio for non-IPF relative to IPF varies with time but averages 0.43 with a 95% CI of 0.27 to 0.64 over the first 2.5 years (Table 3) (the hazard ratio less than one indicates lower mortality in patients with non-IPF). The component of the HR associated with disease category alone, independent of FVC or the rate of decline of FVC, is obtained by exponentiating the subgroup hazard effect ( in Table 2b), resulting in an estimated HR of 0.80 (95% CI 0.53, 1.18). This is consistent with the mortality associated with non-IPF being lower than with IPF, even for patients with the same FVC and rate of FVC decline. The terms
1		and 2, representing the effects of FVC and of the rate of decline of FVC on mortality are both non-zero (Table 2b), demonstrating the links between both the FVC and the rate of FVC decline and the mortality rate and, thus, the existence of informative censoring.
As expected, by accounting for informative censoring due to mortality through patientlevel weighting of FVC progression by probability of survival, the DPM-estimated population-averaged FVC trajectories for IPF and non-IPF, respectively, reproduce the
Joint Modelling of FVC and Mortality in ILD non-linearity observed in the respective subpopulations for both absolute ppFVC (Figure 3d) and change from baseline in ppFVC (Figure 3e).
Using the linear mixed model to fit the data and similarly distinguishing IPF patients from non-IPF patients also shows the decline in both absolute ppFVC (Figure 4a) and change from baseline in ppFVC (figure 4b). However, the linear mixed model yields a 22% lower point estimate for the underlying annual rate of FVC decline for IPF (-5.0% [95% CI -5.7%, -4.2%] for the linear model compared to -6.4% [95% CI -7.2%, -5.6%] for the DPM) and an 18% lower point estimate for the underlying annual rate of FVC decline for non-IPF (-4.1% [95% CI -5.1%, -3.0%] for the linear mixed model compared with -5.0% [95% CI -6.1%, -3.8%] for the DPM) (see Table 2b). The mixed linear model produces an estimate of 0.9% (95% CI -0.4, 2.2) for the difference in annual FVC decline between IPF and non-IPF disease categories, compared with the estimate of 1.4% (95% CI 0.1, 2.7) from the DPM (Table 2b). In this example, only the DPM demonstrates the difference in FVC decay rates between the disease categories.
The DPM, modified to estimate differences in the two patient populations, results in a better fit to the observed data than the corresponding Bayesian mixed linear model, as quantified by a lower root-mean-square error (Appendix 4).
Discussion
Missing FVC data in longitudinal studies of fibrotic ILD are often due to mortality during the follow up period. The fundamental linkage between decline in FVC and mortality in patients with fibrotic ILD complicates the analysis of outcome data, even when the goal is only to evaluate patterns in FVC decline or the effects of therapeutic strategies on 
FVC decline. The DPM presented here allows accurate and less biased estimation of the underlying FVC trajectory. This cannot be achieved without simultaneously modelling mortality and the relationship between FVC and mortality. Furthermore, the DPM allows the untangling of effects on FVC and mortality, which may also be important in the assessment of safety.
Although shorter trials analysing FVC data at 6 months, or even 3 months, reduce the impact of mortality on the primary endpoint of change in FVC,28 the shorter duration of follow-up provides less time to characterize each patient’s decline in FVC, reducing the information obtained from each trial participant and thus increasing the required sample size. Furthermore, longer trials of 12-18 months are required to ensure sufficient time to evaluate adverse effects of novel therapies. Thus, there is a need for effective statistical approaches that can characterize FVC trajectories accurately over a sufficient period of time, in the context of informative censoring events.
The progressive removal of the sickest patients from the study population over time can result in non-linear FVC progression patterns in the population, with the predicted effect of a steeper initial decline in FVC followed by a plateau at later time points, even if each individual patient’s FVC trajectory is linear over the period of observation. While a number of models have been used to address this missingness through a variety of imputation strategies,29 none have accurately accounted for loss of FVC data due to mortality. The joint DPM fits the observed, non-linear FVC progression rate seen in populations with fibrotic ILD and progressive informative censoring, utilizing a linear FVC sub-model for the underlying FVC progression rate. The joint model provides estimates that are relatively unbiased by the censoring mechanism because the DPM directly associates FVC progression with mortality.
The challenges posed by informative censoring are further compounded when comparing two or more study groups. For example, groups with high rates of early mortality may appear to experience less progression of FVC as compared with groups whose mortality occurs at later times. Early mortality is likely to disproportionally affect the segment of the population that was likely to experience faster decline in FVC, so the surviving population is enriched for those with more slowly progressing disease. A simple analysis of FVC in this scenario may fail to properly handle the partial competing risks of FVC progression and mortality, potentially leading to faulty conclusions about the observed differences between groups of interest. Specifically, without a joint DPM, this could paradoxically make a treatment whose only effect is to increase mortality among the patients with more advanced disease appear to slow FVC decline.
It is important to use a model structure that reflects what is known about the disease being studied, even in settings and with data for which the lack of fit of a simpler modelling approach (e.g., linear mixed models, perhaps with time varying covariates) is not obvious and cannot be demonstrated with regression diagnostics. Specifically, the known association between lower FVC and the higher risk for death or being lost to follow up is an important piece of information that should be utilized in modelling and data analysis to improve precision and reduce bias in analyses. This is analogous to the principle that the covariates included in a multivariable model should be chosen primarily based on what are believed to be important predictors of outcome, rather than based on the presence of statistical significance of the associated regression coefficients.30 Using data from two prior independent cohort studies, the proposed DPM demonstrates that survival is impacted by both the patient’s current FVC and the rate of FVC decline, both of which may change over time during follow up. Thus, for a given current FVC, the patient who is declining more rapidly is at increased risk of death.
Ideally, effective therapies for fibrotic ILD will both slow decline in lung function and reduce mortality. The DPM allows the simultaneous estimation and evaluation of treatment effects on FVC decline and mortality, furthermore the DPM partitions any effect on mortality into components mediated through an effect on FVC decline and through a separate direct effect on mortality. For example, a therapy that does not slow FVC decline but instead reduces the frequency of acute exacerbations might exhibit a beneficial effect on survival that is not mediated by an effect on FVC decline. The DPM is capable of properly capturing and characterizing this treatment effect pattern.
The DPM has limitations: first, the model does not address the need to assess the safety of novel treatments. Second, in its current form it cannot address important outcomes not manifested by either FVC or mortality such as patient reported outcomes or quality of life, though it could be extended to account for additional outcomes.31 Finally, the benefits of unbiased estimation using the DPM are greater as mortality events increase and thus the DPM may not be helpful in trials with shorter follow up, e.g., 3 months.
Conclusion
The Bayesian joint DPM provides an integrated approach to reducing bias in the estimation of treatment effects on both FVC decline and survival in trials of therapeutics for fibrotic ILD. The approach to analysis of longitudinal outcomes is well suited for both observational studies and for implementation in clinical trials.
REMAP-ILD Consortium Investigators
1.		Amanda Bravery, Imperial College Clinical Trials Unit, Imperial College London, London, United Kingdom
2.		Amanda Goodwin, University of Nottingham, Nottingham, United Kingdom
3.		Ana Etges, Federal University of Rio Grande do Sul, Farroupilha, Porto Alegre - RS, Brazil
4.		Ana Boshoff, Imperial College Clinical Trials Unit, Imperial College London, London, United Kingdom
5.		Andreas Günther, Justus-Liebig-University of Giessen, Goethestrasse 58, Giessen, Germany
6.		Andrew Briggs, London School of Hygiene and Tropical Medicine, London United Kingdom
7.		Andrew Palmer, University of Tasmania, Australia
8.		Andrew Wilson, University of East Anglia, Norwich, United Kingdom
9.		Anjali Crawshaw, Department of Respiratory Medicine, University Hospitals Birmingham NHS Foundation Trust, Birmingham, United Kingdom
10.		Anna-Maria Hoffmann-Vold, Department of Rheumatology, Oslo University Hospital, Norway; Department of Rheumatology, University Hospital Zurich, University of Zurich, Switzerland
11.		Anne Bergeron, University Hospitals Geneva, Geneva, Switzerland
12.		Anthony C. Gordon, Division of Anaesthetics, Pain Medicine and Intensive Care, Imperial College London, London, United Kingdom
13.		Antje Prasse, Hannover Medical School, Hannover, Germany
14.		Argyris Tzouvelekis, Department of Respiratory Medicine, University of Patras, Greece
15.		Athina Trachalaki, Imperial College London, South Kensington, London, United Kingdom
16.		Avinash Anil Nair, Christian Medical College, Vellore, India
17.		Ayodeji Adegunsoye, University of Chicago, Chicago Illinois, USA
18.		Ben Hope-Gill, Cardiff and Vale University Hospital, Wales, United Kingdom
19.		Bhavika Kaul, U.S. Department of Veterans Affairs Center for Innovation in Quality, Effectiveness, and Safety; Baylor College of Medicine and University of California San Francisco
20.		Bibek Gooptu, University of Leicester, University Road, Leicester, United Kingdom
21.		Bruno Guedes Baldi, Pulmonary Division, Heart Institute (InCor), University of Sao Paulo Medical School, Sao Paulo, Brazil
22.		Bruno Crestani, Hôpital Bichat, Assistance Publique Hôpitaux de Paris, Paris, France
23.		Carisi Anne Polanczyk, Federal University of Rio Grande do Sul, Sao Paulo, Brazil
24.		Carlo Vancheri, University – Hospital Policlinico “G. Rodolico – San Marco” University of Catania, Catania, Italy
25.		Carlos Crobal, European Respiratory Society, Sheffield, United Kingdom
26.		Charlotte Summers, University of Cambridge, Cambridge, United Kingdom
27.		Chris Grainge, John Hunter Hospital, Newcastle, Australia and University of Newcastle, Newcastle, Australia
28.		Christophe Von Garnier, Centre Hospitalier Universitaire Vaudois, Lausanne, Switzerland
29.		Christopher Huntley, University Hospitals Birmingham, Birmingham, United Kingdom
30.		Claudia Ravaglia, University of Bologna, Bologna, Spain
31.		Claudia Valenzuela, Hospital Universitario de La Princesa, Madrid, Spain
32.		Conal Hayton, Wythenshawe Hospital, Manchester University NHS Foundation Trust, Manchester, United Kingdom
33.		Cormac McCarthy, University College Dublin, Belfield, Dublin, Ireland
34.		Daniel Chambers, The University of Queensland, Brisbane, Australia
35.		Daphne Babalis, Imperial College Clinical Trials Unit, Imperial College London, London, United Kingdom
36.		David Thicket, University of Birmingham, Birmingham, United Kingdom
37.		David Turner, University of East Anglia, Norwich, United Kingdom
38.		Deepak Talwar, Metro Respiratory Centre Pulmonology & Sleep Medicine, Noida, Uttar Pradesh, India
39.		Devaraj Anand, Royal Brompton Hospital, London, United Kingdom
40.		Devesh Dhasmana, University of St. Andrews, St. Andrews, Scotland, United Kingdom
41.		Dhruv Parek, Birmingham University, Birmingham, United Kingdom
42.		Diane Griffiths, University Hospitals Birmingham, Birmingham, United Kingdom
43.		Diego Castillo Villegas, Hospital de la Santa Creu I Sant Pau. Barcelona, Spain
44.		Duncan Richards, Nuffield Department of Orthopaedics, Rheumatology and Musculoskeletal Sciences, University of Oxford, United Kingdom
45.		Eliana Santucci, Hcor Research Institute, Hcor Hospital, Sao Paulo, Brazil
46.		Elisabeth Bendstrup, Aarhus University, Aarhus C, Denmark
47.		Elisabetta Balestro, University of Padua, Padova PD, Italy
48.		Eliza Tsitoura, University of Crete, Heraklion, Greece
49.		Emanuela Falaschetti, Imperial College Clinical Trials Unit, Imperial College London, London, United Kingdom
50.		Ena Gupta, University of Vermont Health Network, Burlington, Vermont, USA
51.		Erica Farrand, University of California, San Francisco, USA
52.		Felix Chua, Royal Brompton Hospital, London, United Kingdom
53.		Francesco Bonella, Essen University Hospital, Essen, Germany
54.		Francesco Lombardi, Division of Pulmonary Medicine, Fondazione Policlinico Universitario Agostino Gemelli IRCCS, Rome, Italy
55.		Gary M. Hunninghake, Brigham and Women's Hospital, Massachusetts, USA
56.		Gauri Saini, Nottingham University Hospital, Nottingham, United Kingdom
57.		George Chalmers, Glasgow Royal Infirmary, Glasgow, Scotland, United Kingdom
58.		Gunnar Gudmundsson, University of Iceland, Reykjavik, Iceland
59.		Harold Collard, University of California, California, USA
60.		Haruyuki Ishii, Department of Respiratory Medicine, Kyorin University, Mitaka, Japan
61.		Helen Parfrey, Royal Papworth Hospital NHS Foundation Trust, Cambridge, United Kingdom
62.		Helmut Prosch, Dept. of Biomedical Imaging and Image-guided Therapy, Medical University of Vienna, Vienna, Austria
64.		Huzaifa Adamali, North Bristol NHS Trust, Bristol, United Kingdom
65.		Ian Forrest, Newcastle Hospitals NHS Foundation Trust, Newcastle, United Kingdom
66.		Iazsmin Bauer-Ventura, University of Chicago, Chicago, Illinois, USA
67.		Imre Noth, University of Virginia, Charlottsville, Vermont, USA
68.		Ingrid Cox, University of Tasmania, Australia
69.		Irina Strambu, Institutul de Pneumoftiziologie "Marius Nasta", Bucharest, Romania
70.		Jacobo Sellares, Grup de Treball de Malalties Pulmonars Intersticials. Pneumology Service. Hospital Clinic. Barcelona, Spain
71.		James Eaden, Sheffield Teaching Hospitals NHS FT, Sheffield, United Kingdom
72.		Janet Johnston, Wythenshawe Hospital, Manchester University NHS Foundation Trust, Manchester, United Kingdom
73.		Jeff Swigris, National Jewish Health, Denver, Colorado, USA
74.		Joel Rodrigo Francesqui Candela, Hospital Clinic de Barcelona, Barcelona, Spain
75.		John Blaikley, Manchester University, Manchester, United Kingdom
76.		John S Kim, University of Virginia, Charlottsville, Vermont, USA
77.		Jonas Schupp, Section of Pulmonary, Critical Care and Sleep Medicine, Yale School of Medicine, New Haven, CT United States
78.		Jonathan Chung, University of Chicago, Illinois, USA
79.		Joseph A. Lasky, Tulane & Pulmonary Fibrosis Foundation, Louisiana, USA
80.		Joseph Jacob, Satsuma Lab, Hawkes Institute, University College London; UCL Respiratory, University College London, United Kingdom
81.		Joyce Lee, University of Colorado, Colorado, USA
82.		Juan Ignacio Enghelmayer, José de San Martin Clinical Hospital, Bueno Aires, Argentina
83.		Juergen Behr, LMU University Hospital, LMU Munich, Germany
84.		Karin Storrer, Federal University of Parana, Curitiba, Brazil
85.		Katarzyna B. Lewandowska, National Research Institute of Tuberculosis & Lung Disease, Warsaw, Poland
86.		Kate Johnson, The University of British Colombia, Vancouver, Canada
87.		Katerina Antoniou, Dept of Respiratory Medicine, School of Medicine, University of Crete, Greece
88.		Katrin Hostettler, University Hospital Basel, Basel, Switzerland
89.		Kensuke Kataoka, Department of Respiratory Medicine and Allergy, Tosei General Hospital, Tosei, Japan
90.		Killian Hurley, Royal College of Surgeons in Ireland, Dublin, Ireland
91.		Kirsty Hett, Cardiff and Vale University Health Board, Wales, United Kingdom
92.		Larissa Schwarzkopf, The Institute for Therapy Research, Munich, Germany
93.		Laura Fabbri, Margaret Turner Warwick Centre for Fibrosing Lung Diseases, NHLI, Imperial College, London, UK; Interstitial Lung Disease Unit, Royal Brompton and Harefield Clinical Group, Guy's and St Thomas' NHS Foundation Trust, London, United Kingdom
94.		Laura Price, Royal Brompton Hospital, London, United Kingdom
95.		Laurence Pearmain, Manchester University, Manchester, United Kingdom
96.		Liam Galvin, European Pulmonary Fibrosis Federation, Brussels, Belgium
97.		Lisa G. Spencer, Liverpool Interstitial Lung Disease Service, Aintree Hospital, Liverpool University Hospitals NHS Foundation Trust, Liverpool, United Kingdom
98.		Lisa Watson, Sheffield Teaching Hospitals NHS FT, Sheffield, United Kingdom
99.		Louise Crowley, Queen Elizabeth Hospital, University Hospitals Birmingham, Birmingham, United Kingdom
100.		Luca Richeldi, Agostino Gemelli IRCCS University Hospital Foundation, Rome, Italy
101.		Lucilla Piccari, Department of Pulmonary Medicine, Hospital del Mar, Barcelona, Spain
102.		Makoto Ishii, Division of Respirology, Nagoya University, Nagoya, Japan
103.		Maria Molina-Molina, ILD Unit, Respiratory Department, University Hospital of Bellvitge, IDIBELL, Barcelona, Spain
104.		Maria Otaola, Institute of Psychophysical Rehabilitation, Buenos Aires, Argentina
105.		Mark Jones, Southampton University, Southampton, United Kingdom
106.		Mark Spears, University of Dundee, Scotland, United Kingdom
107.		Mark Toshner, University of Cambridge, Cambridge, United Kingdom
108.		Marlies Wijsenbeek-Lourens, Erasmus University Medical Hospital, Rotterdam, Netherlands
109.		Martin Brutsche, Kantonsspital St.Gallen, Sankt Gallen, Switzerland
110.		Martina Vasakova, Thomayer University Hospital, Prague, Czech Republic
111.		Masashi Bando, Division of Pulmonary Medicine, Department of Medicine, Jichi Medical University, Shimotsuke, Japan
112.		Michael Gibbons, University of Exeter, Exeter, United Kingdom
113.		Michael Henry, Cork University Hospital, Cork, Ireland
114.		Michael P. Keane, University College Dublin, Ireland
115.		Milena Man, Iuliu	University of Medicine and Pharmacy, Cluj-Napoca, Romania
116.		Mohsen Sadatsafavi, The University of British Colombia, Vancouver, Canada
117.		Naftali Kaminski, Pulmonary, Critical Care and Sleep Medicine, Yale School of Medicine, Connecticut, USA
118.		Nazia Chaudhuri, Department of health and life sciences, School of medicine, University of ulster, North Ireland
119.		Nick Weatherley, Sheffield University Hospitals, Sheffield, United Kingdom
120.		Nik Hirani, The University of Edinburgh, Scotland, United Kingdom
121.		Ovidiu Fira Mladinescu, Victor University of Medicine and Pharmacy, Timisoara, Romania
122.		Paolo Spagnolo, University of Padua, Padova, Italy
123.		Paul Beirne, Leeds Teaching Hospitals NHS Foundation Trust, Leeds, United Kingdom
124.		Peter Bryce, The Pulmonary Fibrosis Trust, Lichfield, United Kingdom
125.		Peter George, Royal Brompton Hospital, London, United Kingdom
126.		Radu Crisan-Dabija, University of Medicine and Pharmacy "Grigore T. Popa" Iasi, Romania
127.		Rahul Maida, University of Birmingham, Birmingham, United Kingdom
128.		Raphael Borie, Hôpital Bichat, Assistance Publique Hôpitaux de Paris, Paris, France
129.		Reoto Takei, Osaka Prefectural Medical Centre for Respiratory and Allergic Diseases, Japan
130.		Rui Rolo, Braga Hospital, Braga, Portugal
131.		Sabina Guler, University Hospital of Bern, Bern, Switzerland
132.		Sabrina Paganoni, Massachusetts General Hospital, Boston, USA
133.		Sally Singh, University of Leicester, Leicester, United Kingdom
134.		Sara Freitas, University Hospital Coimbra, Coimbra, Portugal
135.		Sara Piciucchi, Department of Radiology, GB Morgagni Hospital; Azienda USL Romagna, Italy
136.		Shama Malik, Action for Pulmonary Fibrosis, United Kingdom
137.		Shaney Barratt, North Bristol NHS Trust, Bristol, Unted Kingdom
138.		Simon Hart, University of Hull, Hull, United Kingdom
139.		Simone Dal Corso, Universidade Nove de Julho, Graduate Program in Rehabilitation Sciences, Sao Paulo, Brazild; Central Clinical School, Monash University, Melbourne, Australia
140.		Sophie V. Fletcher, University Hospital Southampton NHS Foundation Trust, NIHR Southampton Respiratory Biomedical Research Centre; University of Southampton, School of Clinical and Experimental Sciences, Faulty of Medicine, Southampton, United Kingdom
141.		Stefan Stanel, Wythenshawe Hospital, Manchester University NHS Foundation Trust, Manchester, United Kingdom
142.		Stephen Bianchi, Sheffield Teaching Hospitals NHS Foundation Trust, Sheffield, United Kingdom
143.		Steven Jones, Action for Pulmonary Fibrosis, United Kingdom
144.		Steven Nathan, Inova Medical Group, Virginia, USA
146.		Takafumi Suda, Second Division, Department of Internal Medicine, Hamamatsu University School of Medicine, Hamamatsu, Japan
147.		Tanzira Zaman, Cedars Sinai Medical Center, Los Angeles, California, USA
148.		Timothy Gatheral, University Hospitals of Morecambe Bay NHS Foundation Trust, Cumbria, United Kingdom
149.		Tom McMillan, Pulmonary Fibrosis NI, Ballycastle, United Kingdom
150.		Valerie Quinn, Margaret Turner Warwick Centre for Fibrosing Lung Disease, National Heart and Lung Institute, Imperial College London, United Kingdom
151.		Venerino Poletti, University of Bologna, Bologna, Italy
152.		Victoria Cornelius, Imperial College Clinical Trials Unit, Imperial College London, London, United Kingdom
153.		Vincent Cottin, Louis Pradel Hospital/Claude Bernard University, Lyon, France
154.		Wim Wuyts, Leuven University Hospital, Leuven, Belgium
155.		Yasuhiro Kondoh, Department Respiratory Medicine and Allergy, Tosei General Hospital, Aichi, Japan
156.		Yasunari Miyazaki, Department Respiratory Medicine and Allergy, Tokyo Medical and Dental University, Tokyo, Japan
157.		Yet Hong Khor, Respiratory Research@Alfred, Central Clinical School, Monash University, Melbourne, Victoria, Australia; Department of Respiratory and Sleep Medicine, Austin Health, Heidelberg, Victoria, Australia
158.		Yussef Haider, The London School of Hygiene and Tropical Medicine, London, United Kingdom
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Table 1.
Table 1: Bayesian Joint Disease Progression Model + Prior Distributions NI = Non-informative; SD = Standard Deviation;	
Parameter	Notation	Prior Distribution	Prior Justification	
FVC Submodel
=	+	;
=	+	( )= +	+	+	( )= +
()
is the percent predicted FVC score measured at time	during visit j for participant i,
(i): subgroup of participant i; 0 control subgroup, 1 for comparator subgroup	
Intercepts	Intercept for referent/control group		(	)	NI centered on anticipated mean baseline FVC value	
	Fixed additive effect to intercept for comparator subgroup		()	NI centered on 0	
Slopes	Slope for referent/control group		()	NI centered on 0	
	Subgroup effect on FVC: Fixed additive effect to slope for comparator subgroup		()	NI centered on 0	
Residual standard error		(	)+	NI with a wide tail over positive values only	
Participantlevel Random effects	Random effect on intercept (grouping 1) for participant i		[,]	()
=	NI centered on 0	
	Random effect on slope (grouping 2) for participant i				
	Variance of grouping k		(	)+	NI with a wide tail over positive values only	
	Correlation between groupings			NI with prior mode on the identity matrix	
Survival Submodel
( ) =	{	}	()= +	( )+	( )
( )is the hazard function for participant i across continuous survival times
: subgroup of participant i; 0 control subgroup, 1 for comparator subgroup
( ): mean of the FVC submodel for participant i across continuous survival times
( ): first derivative of the mean ( ) with respect to time	
Shape parameter for Weibull hazard		(	)+	NI with a wide tail over positive values only	
Log scale parameter for Weibull hazard		()	NI centered on 0	
Subgroup effect on survival independent of FVC: multiplicative adjustment to HR		(	)	NI centered on 0	
Effect of FVC value on survival (value association): multiplicative adjustment to HR		()	NI centered on 0	
Effect of rate of decline in FVC value on survival (slope association): multiplicative adjustment to HR		()	NI centered on 0	

Tables 2a and 2b.
Table 2a: DPM and linear model parameter estimates from fit to data without distinguishing non- IPF and IPF disease categories. Estimates list posterior medians and 95% credible intervals.
Submodel	Parameter	DPM	Linear Model	
FVC	Intercept	0	82.5 (81.2, 83.8)	82.4 (81.0, 83.7)	
	Slope	0	-6.0 (-6.7, -5.3)	-4.7 (-5.3, -4.1)	
	Residual standard error		6.5 (6.2, 6.7)	6.5 (6.2, 6.8)	
	Random intercept SD	11	19.6 (18.6, 20.6)	19.6 (18.7, 20.6)	
	Random slope SD	22	5.7 (5.0, 6.5)	5.1 (4.4, 5.8)	
	Random effects correlation	12	0.19 (0.07, 0.30)	0.08 (-0.04, 0.21)	
Survival	Weibull shape		1.4 (1.2, 1.7)		
	Log Weibull scale	0	-0.2 (-1.4, 0.8)	--	
	Value association	1	-0.04 (-0.05, -0.03)	-	
	Slope association	2	-0.14 (-0.22, -0.08)	-	

Table 2b: DPM and linear model parameter estimates from fit to data, comparing non-IPF and IPF disease categories with IPF as the reference group. Estimates list posterior medians and 95% credible intervals.
Submodel	Parameter	DPM	Linear Model	
FVC	IPF group intercept	0	80.1 (78.5, 81.7)	79.9 (78.4, 81.5)	
	Non-IPF group intercept	0+	87.9 (85.5, 90.3)	87.8 (85.4, 90.1)	
	Intercept effect estimate (non-IPF vs IPF)		7.8 (4.9, 10.6)	7.9 (5.0, 10.6)	
	IPF group slope	0	-6.4 (-7.2, -5.6)	-5.0 (-5.7, -4.2)	
	Non-IPF group slope	0+	-5.0 (-6.1, -3.8)	-4.1 (-5.1, -3.0)	
	Slope effect estimate (non-IPF vs IPF)		1.4 (0.1, 2.7)	0.9 (-0.4, 2.2)	
	Residual standard error		6.5 (6.2, 6.7)	6.5 (6.3, 6.8)	
	Random intercept SD	11	19.2 (18.3, 20.2)	19.3 (18.3, 20.3)	
	Random slope SD	22	5.7 (5.0, 6.4)	5.1 (4.4, 5.7)	
	Random effects correlation	12	0.17 (0.06, 0.28)	0.07 (-0.05, 0.20)	
	Disease rate ratio (proportional slowing of progression)	( 0+	)
0	0.78 (0.59, 0.99)	0.82 (0.59, 1.08)	
Survival	Weibull shape		1.4 (1.2, 1.7)	-	
	Log Weibull scale	0	-0.2 (-1.4, 0.7)	-	
	Subgroup hazard effect		-0.2 (-0.6, 0.2)	-	
	Value association	1	-0.04 (-0.05, -0.03)	-	
	Slope association	2	-0.14 (-0.22, -0.08)	-	

Table 3.
Table 3: Hazard ratio estimates from the DPM fit to data, comparing non-IPF and IPF disease types, with IPF as the reference group. Estimates list posterior medians and 95% credible intervals and are interpreted as the “effect” of non-IPF relative to IPF.
Hazard ratio by component	Proportional Effect?	Time (years)	Estimate	
Full hazard ratio including effect due to both FVC and survival independent of FVC. (Denoted
HR in Appendix 2)	No	0.5	0.45 (0.29, 0.66)	
		1.0	0.44 (0.28, 0.65)	
		1.5	0.42 (0.27, 0.63)	
		2.0	0.41 (0.26, 0.62)	
		2.5	0.40 (0.25, 0.61)	
		Average over all time	0.43 (0.27, 0.64)	
Hazard ratio for effect on survival independent of FVC. (Denoted HRS in Appendix 2)	Yes	-	0.80 (0.53, 1.18)	
Full FVC hazard ratio including effect due to difference in baseline values and slopes. (Denoted HRF in Appendix 2)	No	0.5	0.56 (0.41, 0.73)	
		1.0	0.55 (0.39, 0.72)	
		1.5	0.53 (0.37, 0.72)	
		2.0	0.51 (0.35, 0.72)	
		2.5	0.50 (0.33, 0.71)	
		Average over all time	0.54 (0.38, 0.72)	
Hazard ratio for effect of difference in baseline values alone. (Denoted HRF:BLN in Appendix 2)	Yes	-	0.71 (0.61, 0.82)	
Hazard ratio for effect of difference in slopes alone. (Denoted HRF:SLP in Appendix 2)	No	0.5	0.80 (0.60, 0.99)	
		1.0	0.77 (0.57, 0.98)	
		1.5	0.75 (0.54, 0.98)	
		2.0	0.72 (0.50, 0.98)	
		2.5	0.70 (0.48, 0.98)	
		Average over all time	0.76 (0.55, 0.98)	

Figure Captions
Figure 1. Application of the disease progression model (DPM) to the combined patient cohorts, without separation of data from patients with IPF and non-IPF. The top panels illustrate the FVC sub-model estimate and uncertainty, both as an absolute FVC measurement (1a as percent predicted) and as a change from the model-estimated baseline (1b). The middle panel (1c) illustrates the survival sub-model estimate and uncertainty for the survival curve. The bottom panels (1d and 1e) illustrate the modelgenerated prediction for the observed FVC results with the informative censoring (solid curve) and the uncertainty. The data values are shown by the whisker plots (mean ± 1 standard error).
Figure 2. The results of fitting a Bayesian linear mixed-effects models to the data shown in Figure 1. Panel 2a illustrates the model fit on the ppFVC scale and panel 2b as a change from model-estimated baseline.
Figure 3. Application of the disease progression model (DPM) to the combined patient cohort, comparing data from patients with IPF (reference group) to those with non-IPF. The top panels (3a and 3b) illustrate the FVC sub-model estimates and uncertainty, as in Figure 1, for each of the subpopulations. The middle panel (3c) illustrates the survival sub-model estimates and uncertainty for the survival curves for both IPF and non-IPF patients. The bottom panels (3d and 3e) illustrate the model-generated prediction for the observed FVC results with the informative censoring (solid curve) and the uncertainty, for each subpopulation. The data values are shown by the whisker plots (mean ± 1 standard error).
Figure 4. The results of fitting a Bayesian linear mixed-effects model to the data shown in Figure 3. Panel 4a illustrates the model fits on the ppFVC scale and panel 4b as a change from model-estimated baselines.

Figure 1

Figure 3.

Figure 4

Appendix 1. Patient Cohorts and Data Preparation.
Data for demonstration and evaluation of the DPM was obtained from investigators for the PROFILE24 and INJUSTIS25 observational cohort studies. The PROFILE study was a multicenter, prospective, observational cohort study of 632 individuals either with incident IPF (n=563) or idiopathic non-specific interstitial pneumonia (iNSIP)(n=69) in the United Kingdom, conducted between May 2010 and Sept 2017.24 Patients aged 1885 years were enrolled through a coordinating centre (Nottingham University Hospitals, Nottingham, UK; ClinicalTrials.gov identifier NCT01134822), as well as the Royal Brompton Hospital (London, UK; NCT01110694). All participants with IPF or iNSIP were eligible for inclusion and exclusion criteria only included individuals diagnosed more than 6 months prior to baseline visit and those not providing consent. Participants were longitudinally assessed for 36 months.
The INJUSTIS study enrolled 272 participants aged 18 or older with confirmed diagnoses of fibrotic ILD, recruited between November 2018 and May 2023. Participants were distributed across five diagnostic subtypes: IPF (n=67), asbestosis (n=54), fibrotic hypersensitivity pneumonitis (fHP, n=54), rheumatoid arthritis-associated ILD (RA-ILD) (n=47), and unclassifiable ILD (uILD) (n=50).25 We classify all but the first group as non-IPF (n=205). Recruitment occurred at outpatient clinics across 24 UK sites. All participants with fibrotic ILD were eligible for inclusion, irrespective of their progression status. Exclusions included individuals with inflammatory radiological changes lacking fibrosis, or those diagnosed more than 18 months before study commencement. Participants were longitudinally assessed for 24 months. The study was registered on clinicaltrials.gov (NCT03670576). Data were still premature for roughly 25% of the INJUSTIS population at the time of this writing, and as such the INJUSTIS data presented here should not be considered final.
Of the 632 patients in PROFILE, 7 patients were removed for missing all post-baseline FVC or survival status data and of the 272 patients in INJUSTIS, 5 patients were removed for similar reasons (Figure). Follow up was still ongoing for the INJUSTIS study at the time of this analysis; approximately 50 patients had yet to complete follow up.
During the analysis, after the transfer of data, it became apparent that some patient records were internally inconsistent. After consultation with the investigators providing the data, the following changes were made to the data that had been transferred.
One follow-up date for a month 24 visit for one patient in PROFILE was recorded incorrectly and so was manually changed to reflect exactly 24 months following baseline. The death date of one patient in INJUSTIS was recorded incorrectly as the final visit date and so the death date was manually changed to be the final visit date.
The Figure below illustrates the data sources and population sizes.

Appendix 2. Mathematical representation of the components of the disease progression model.
A mathematical representation of the disease progression model is provided to illustrate each estimated effect and the corresponding interpretation. The FVC and survival submodels are described separately for control and comparator subgroups. The hazard ratio is decomposed to distinct contributing effects which are presented in Table 3. The derivations of this section highlight how to appropriately report effect estimates for the disease progression model.
FVC submodel:
The modelled mixed effects in the FVC submodel for control and comparator subgroups shown separately.
Control group mixed effects model specification:
=( + )+( + ) +
Comparator group mixed effects model specification:
=( + + )+( + + ) +
Survival submodel:
The modelled hazard functions of the control and comparator subgroups shown separately. The implicit hazard ratio of comparator vs control is decomposed into individual components each with distinct meaning.
Control group survival hazard:
( ) =	{ }	{ ((	+ ) + (	+ ) ) +	(	+ )}
Comparator group survival hazard:
()=	{ }	+	(	+	+	)+(	+	+	) +	(	+	+ ) ( ) =	{ }	{ (( + ) + ( + ) ) + ( + )}	+	+	+ ()=	()	+	+	+
Hazard Ratio (HR) for Comparator vs. Control (HR):
Decomposition of the implicit hazard ratio of comparator vs control describes unique component effects contributing to the aggregate hazard ratio. The hazard ratio contains effects for subgroup and differences in intercepts and slopes between subgroups. 
()
=	{ }	+(	+	)
()
1)		Subgroup effect (HR) on survival independent of FVC (HRS):	{ }
2)		Subgroup effect (HR) on survival due to FVC (HRF):
+(	+	)
a.		Component due to differences in baseline FVC values (HRF:BLN): {}
b.		Component due to differences in FVC slopes (HRF:SLP):
{(	+	) }
Appendix 3. R Code for Joint Disease Trajectory and Linear Mixed Effects Models
The Bayesian joint model is fit using the stan_jm function from the rstanarm package
(version 2.26.1) and the Surv function from the survival package (version 3.5.8) in R
(version 4.3.3).
Joint Disease Trajectory Model:
jm_fit <- stan_jm(
formulaLong = response ~ 1 + visit_time + (1 + visit_time | sub), dataLong = dfL,
formulaEvent = Surv(censor_time, mortality_status) ~ 1, dataEvent = dfS,
time_var = "visit_time",
id_var = "sub",
assoc = c("etavalue","etaslope"),
basehaz = "weibull",
priorLong = normal(location = 0, scale = 15, autoscale = FALSE), priorLong_intercept = normal(location = 80, scale = 30, autoscale = FALSE), priorLong_aux = cauchy(location = 0, scale = 15, autoscale = FALSE), priorEvent = normal(location = 0, scale = 15, autoscale = FALSE), priorEvent_intercept = normal(location = 0, scale = 15, autoscale = FALSE), priorEvent_aux = cauchy(location = 0, scale = 10, autoscale = FALSE), priorEvent_assoc = normal(location = 0, scale = 5, autoscale = FALSE), prior_covariance = lkj(regularization = 1, scale = 30, df = 1, autoscale = FALSE) )
Linear Mixed Effects Model:
lme_fit <- stan_mvmer(
formula = response ~ 1 + visit_time + (1 + visit_time | sub), data = df,
prior = normal(location = 0, scale = 15, autoscale = FALSE), prior_intercept = normal(location = 80, scale = 30, autoscale = FALSE), prior_aux = cauchy(location = 0, scale = 15, autoscale = FALSE), prior_covariance = lkj(regularization = 1, scale = 30, df = 1, autoscale = FALSE) )
Where arguments have the following interpretations:
formulaLong (formula): A two-sided linear formula object describing both fixed effects and random effects in the FVC submodel using syntax employed by the R package lme4.
dataLong (data): A long-format data frame containing columns for subject ID, time, grouping (where applicable), and repeated measures of FVC.
formulaEvent: A two-sided formula object describing the event submodel using syntax employed by the R package survival.
dataEvent: A data frame containing columns for subject ID, grouping (where applicable), censoring time, and censoring status (1 indicates an event, 0 indicates censoring).
time_var: A string value of the column name in dataLong associated with time.
id_var: A string value of the column name in dataLong and dataEvent associated with subject ID.
assoc: A string value indicating the modelled association effects of FVC and mortality. A value of “etavalue” indicates an association of estimated FVC value and mortality and “etaslope” indicates an association of estimates FVC rate of decline or slope and mortality.
basehaz: A string value indicating the form of the baseline hazard function (Weibull).
priorLong (prior): Specified prior for the FVC submodel fixed effects, excluding the fixed intercept effect.
priorLong_intercept (prior_intercept): Specified prior for the FVC submodel fixed intercept effect.
priorLong_aux (prior_aux): Specified prior for the measurement error term in the FVC submodel.
priorEvent: Specified prior for the hazard ratio effects in the survival submodel.
priorEvent_intercept: Specified prior for the intercept effect in the Weibull survival submodel.
priorEvent_aux: Specified prior for the shape parameter in the Weibull survival submodel.
priorEvent_assoc: Specified prior for the association effects for FVC and mortality in the Weibull survival submodel.
prior_covariance: Specified prior for the covariance structure of random effects in the FVC submodel.
Appendix 4. Goodness of Fit Statistics
The models’ predictions for the observed FVC progression rate in the presence of censoring due to mortality allow the visual or quantitative comparison with the observed data. To quantify the relative goodness-of-fit, we determined the root mean square (RMS) error as follows.
Methods:
Model fit to the FVC % predicted outcome was evaluated and compared for the DPM vs the linear mixed effects model by calculating a root mean squared error (RMSE) of observed, individual patient data relative to model predictions for the population average. The marginal fit of the linear mixed effects model was calculated by averaging the expected FVC values across individual-level fits at each time t; i.e., [ ( )] = 
( ) where ( ) is the expected FVC value for individual at time . The marginal fit of the joint model was calculated as a weighted average of the individual-level expected FVC values weighted by corresponding expected survival probabilities at time ; i.e., [ ( )] =	( )	( ) /	( ) where ( ) is the same as before and ( ) is the survival probability for individual at time . If the model included a covariate adjustment for fILD subtype, then the marginal fits were calculated for each subtype separately, i.e., only averaged over individuals of a certain subtype. Finally, the error term used for the
RMSE calculation was ( )	[ ( )] where ( ) is the observed FVC value for

individual at time ; i.e.,	=

observations. Since the RMSE can be calculated for each iteration of the MCMC algorithm, the posterior distribution of the RMSE was used to compare the fit of the joint model vs the linear mixed effects model.
Results:
The Figure displays histograms of the posterior RMSE for the joint DPM model fit (orange) vs the Bayesian linear mixed effects model fit (blue). The top and bottom panels show the RMSE when a covariate adjustment for fibrotic ILD (fILD) subtype has and has not been included in the model, respectively. In both cases, the joint DPM model posterior RMSE distribution is less than (to the left of) the linear mixed effects posterior RMSE distribution with no visible overlap between distributions.
The RMSE is also lower, indicating the fit is improved, for both models when a covariate adjustment is included for fILD subtype. The posterior means of each distribution are i) 20.62 (95% CrI: 20.60, 20.66) for the joint DPM model with an adjustment for fILD subtype (top orange), ii) 20.80 (95% CrI: 20.75, 20.86) for the Bayesian linear mixed effects model with an adjustment for fILD subtype (top blue), iii) 20.87 (95% CrI: 20.85, 20.89) for the joint DPM model without an adjustment for fILD subtype (bottom orange), 
and iv) 21.03 (95% CrI: 20.99, 21.08) for the Bayesian linear mixed effects model without an adjustment for fILD subtype (bottom blue).
Figure: Posterior distributions of the RMSE for the joint model (orange) vs a linear mixed effects model (blue) are shown for cases where a covariate adjustment for fILD subtype has (top) and has not (bottom) been included in the model. The RMSE is improved both by using the joint model and by including a covariate adjustment for fILD subtype in the model.