Table 2 Summary table of joint modelling articles applied to localised prostate cancer and clinical failure, in chronological order

1) Pauler & Finkelstein, 2002 [37]

Bayesian change-point SPJM.

N = 676, E = 176

PSA data during the first two years was dropped from analysis due to rapid drops of PSA post-EBRT & HT. The random effects include the intercept and the slopes (before & after the change-point). The change-point indicator predicts recurrence.

Logged-PSA is modelled with covariates age, presenting PSA, T-stage, with change-point indicator.

Change-point occurring within 10 years. Relapse landmark by four years with a prediction horizon of 10 years.

None performed.

C routine dfpmin, and S-PLUS surv.fit function.

2) Law et al., 2002 [38]

Frequentist cure SPJM.

N = 458, E = 92

Two models are fitted, joint-cure and logistic-Cox (no longitudinal PSA consideration).

Nonlinear exponential- decay & growth modelled longitudinal logged-PSAs using presenting PSA, T-stage, and Gleason.

Not specified, estimated probabilities of recurrence are given for each patient at some time in the future.

Simulation study performed showing that joint-cure model has better sensitivity and discrimination compared to logistic-Cox model.

MATLAB

3) Yu et al., 2004 [39]

Cure SPJM (comparing Bayesian and Frequentist).

N = 458, E = 92

Modelled current PSA value and the PSA gradient trajectory. Random effects are modelled parametrically by exponential- decay & growth models adjusting for presenting PSA, T-stage, and Gleason.

Not specified, estimated probabilities of recurrence are given for each patient at some time in the future.

Not done – comparisons are made between the two estimation methods and are shown to be similar to one another.

MATLAB & C++

4) Taylor et al. 2005 [40]

Bayesian cure SPJM.

N = 934, E = 140

PSA value & slope and time-dependent hormone therapy commencement indicator is considered, adjusting for baseline covariates: presenting PSA, T-stage, Gleason, age, total dose (Gy), and treatment duration.

Landmarks from last contact, with a prediction window of four years.

Validation performed on data of the same patients used for development, but with further follow-up. The model is shown to be well calibrated and accurately predict new PSA values and recurrence risk.

C++

5) Yu et al., 2008 [41]

Bayesian cure SPJM.

N = 928, E = 146

PSA value & slope and time-dependent hormone therapy commencement indicator is considered, adjusting for baseline covariates: presenting PSA, T-stage, Gleason, age, total dose (Gy), and treatment duration.

Landmarks from last contact, with a prediction window of four years.

Validation performed on data of the same patients used for development, but with further follow-up. The model is shown to be well calibrated and accurately predict new PSA values and recurrence risk. Kaplan-Meier plot shows the higher predicted risks go on to have more recurrences indicating its validity.

C++

6) Proust-Lima & Taylor, 2009 [42]

Frequentist JLCM.

Model development and validation:

N = 2386, E = 317

Baseline covariates included: presenting PSA, T-stage, and Gleason. The main- and random effects are of the biphasic initial decline and long-term rise. Five latent classes were identified.

Landmarks taken at every six months from 1—3½ years, with a prediction window of three years.

External validation of prediction is performed on two external cohorts. A range of models are explored, the 5-JLCM shows consistently lower absolute- and weighted prediction errors in both cohorts, using prediction windows of 1 and three years.

Not stated but presumably R using the lcmm package.

7) Jacqmin-Gadda et al., 2010 [43]

Frequentist JLCM.

N = 459, E = 74

Similar to [42] with biphasic longitudinal components for the logged-PSA, considering presenting PSA, T-stage, and Gleason. Four latent classes were identified to be best fitting where the proposed score test did not reject the null of conditional independence.

Only mean evolutions for each of the four classes are given with predicted recurrence-free survival. No windows are specified.

Simulation study performed to appraise score test, where baseline hazard function was misspecified. This methodology was applied to prostate cancer cohort.

Not stated but presumably R using the lcmm package.

8) Taylor et al. 2013 [44]

Bayesian SPJM.

Model development and validation:

N = 3232, E = 458

Covariates include presenting PSA, T-stage, and Gleason grade. Longitudinal parameterisation includes PSA value & slope, and time-dependent HT.

Landmarks are given from most recent PSA values with a prediction window of three years.

External validation is performed on fourth dataset. Simpler visual approaches are undertaken, focusing on estimated risk of recurrence three years after treatment using a three-year prediction window. Patients are assigned to four risk groups, comparing the training and testing Kaplan-Meier plots, treating commencing hormone therapy as either censored and as an event. The model is deemed adequately calibrated with similar patterns being exhibited between training & testing datasets.

C

9) Proust-Lima et al., 2014 [45]

Frequentist JLCM & SPJM.

Model development and validation:

N = 1178, E = 200

Biphasic mixed-effect parameterisation of longitudinal logged-PSA. Baseline covariates: presenting PSA, T-stage and Gleason.

Four latent classes identified for the JLCM, SPJM included PSA value and slope association structure. All other components had the same model structure for direct comparison.

Landmarks taken at every six months from 1—3½ years, with a prediction window of three years.

Internal and external validation is performed. The EPOCE is estimated internally using CVPOL from 1 to 6 years after EBRT. The difference in EPOCE for 4-JLCM and SPJM shows the 4-JLCM to be a better prognostic model in the first four years. External EPOCEs and integrated BS are shown over the follow-up period. The IBSs and EPOCEs show reduced errors for ≥ 3-JLCM and SPJM with little difference between the two approaches.

R: using the lcmm and JM packages – code is available on request from authors.

10) Sène et al., 2016 [46]

Frequentist SPJM.

N = 2386, E = 312

Similar to [42] with biphasic longitudinal components for the logged-PSA, considering presenting PSA, T-stage, Gleason, and corrected total EBRT dose. Several specifications of the time-dependent initiation of salvage HT, and the association structures of the longitudinal value and slope of PSA and random effects.

Landmarks from 1.2, 1.6, 2 and 2.6 years are given with a prediction window of recurrence within the next three years. The predicted recurrence probabilities are given under four scenarios of initiating salvage HT immediately, in 1 or 2 years, or not at all.

Internal validation is performed using cross validation for a prediction window of three years. The CVPOL, CV-BS, and CV-IBS are shown for the six model structures. A simpler random effects joint model is best and chosen for the absence of salvage HT; for immediate HT, the JM that separated the PSA trajectory before and after HT is deemed best.

R: JM package with modifications to source code.

11) Ferrer et al., 2016 [47]

Frequentist multi-state SPJMs

N = 1474; E = 941^*

* sum of all events

Similar to [42] with biphasic longitudinal components for the logged-PSA, considering presenting PSA, T-stage, and Gleason. The longitudinal PSA value and slope was modelled.

For the multi-state process, transition probabilities are given from each transition to any of the other four transitions from the end of treatment throughout follow-up.

A simulation study was undertaken to ensure the estimation process was correct.

Diagnostic plots for the residuals and observed/predicted of the longitudinal model.

R: nlme, survival, mstate and JM packages, with adaptations. Code is readily available at the author’s GitHub account.

12) Ferrer et al., 2018 [48]

Frequentist landmarking and cause-specific SPJMs.

Not explicitly stated but as above.

N = 1474; E = unknown.

Longitudinal logged-PSA modelled similarly as to [47]. Adjusting for: dataset cohort, age, T-stage, Gleason and presenting-PSA.

Predicted recurrence and competing risk of death probabilities for two patients at their landmarks of 1.3 to 2.5 years using a prediction window of 1½ and three years, comparing JM and landmark modelling.

Simulation study using the prostate patient cohorts to generate similar data. Evaluating robustness of JMs and landmark models, under different assumptions. JMs generally more robust to deviations in assumptions than landmark models, other than a strong violation in the longitudinal PSA biomarker specification where the landmark model performs better.

R: nlme, JM, survival, pseudo, and geepack packages. Code is readily available at the author’s GitHub account.