diff --git a/.gitignore b/.gitignore index a8f8b7b..6b69397 100644 --- a/.gitignore +++ b/.gitignore @@ -7,4 +7,6 @@ Manifest.toml .DS_Store .vscode + +!example/Manifest.toml !docs/quarto_old/pages/Manifest.toml diff --git a/docs/src/FirstExample.md b/docs/src/FirstExample.md index aa6988d..ecef95f 100644 --- a/docs/src/FirstExample.md +++ b/docs/src/FirstExample.md @@ -157,7 +157,7 @@ Now a two step joint model, where we use the posterior mean of the population an end end # using previously sampled posterior for longitudinal process -two_step_chn = sample(example_survival(Y, t_m, T, Δ, a_hat, b_hat), NUTS(), 100) +two_step_chn = sample(example_two_step(Y, t_m, T, Δ, a_hat, b_hat), NUTS(), 100) ``` Finally a joint model where the posterior of the longitudinal and joint survival model are sampled simultaneously. @@ -195,6 +195,28 @@ end joint_model_chn = sample(example_joint_model(Y, t_m, T, Δ), NUTS(), 100) ``` +## Maximum Likelihood +This implementation of joint models can also be used for maximum likelihood optimizations. The `Optim.jl` package contains many different optimizers that can be used to find parameters of your model. Here is an example of finding the parameters of the baseline hazard and link coefficient, while keeping the longitudinal parameters fixed: +```julia +using Optim + +function loglikelihood(args) + result = 0 + m(i) = t -> parametric_m_i(t, i, a, b) + h_0(t) = parametric_h_0(t, args[1], args[2]) + joint_models = [JointSurvivalModel(h_0, args[3], m(i)) for i in 1:n] + for i in 1:length(T) + result += logpdf(censored(joint_models[i], upper = 50 + Δ[i]), T[i]) + end + return result +end +minprob(args) = - loglikelihood(args) + +res = optimize(minprob, [1,100,0.0]) # res.minimizer = 0.5616, 67.0875, 0.0513 +``` + +There is also an interface for MLE and MAP estimations using `Turing.jl` which can be found [here](https://turing.ml/dev/docs/using-turing/guide#maximum-likelihood-and-maximum-a-posterior-estimates). In this specific example the Turing interface does not work, since the Weibull density is not defined for theta equals 0. + ## Link functions In the example above the identity link was used for simplicity. The julia ecosystem contains suitable options to explore link functions. 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"1270edf5-f2f9-52d2-97e9-ab00b5d0237a" +version = "2021.5.5+0" + +[[deps.x265_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9" +uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" +version = "3.5.0+0" + +[[deps.xkbcommon_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"] +git-tree-sha1 = "9c304562909ab2bab0262639bd4f444d7bc2be37" +uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" +version = "1.4.1+1" diff --git a/example/case_study.jl b/example/case_study.jl index fa975af..2a692b2 100644 --- a/example/case_study.jl +++ b/example/case_study.jl @@ -2,6 +2,7 @@ using JointSurvivalModels using Turing, Distributions, StatsPlots, CSV, DataFrames, Survival using ReverseDiff using LogExpFunctions: logit +using Survival 1 # -------------------------------------------------------------- # data processing @@ -36,14 +37,21 @@ function sld(t, Ψ, tx = 0.0) end end -h_0(t, κ, λ) = κ/λ * (t/λ)^(κ - 1) +h_0(t, λ) = 1/λ # -------------------------------------------------------------- # bayesian model using Turing.jl @model # -------------------------------------------------------------- -@model function identity_link(longit_ids, longit_times, longit_measurements, surv_ids, surv_times, surv_event) +@model function identity_link( + longit_ids, + longit_times, + longit_measurements, + surv_ids, + surv_times, + surv_event, +) # treatment at study star tx = 0.0 # number of longitudinal and survival measurements @@ -53,53 +61,52 @@ h_0(t, κ, λ) = κ/λ * (t/λ)^(κ - 1) # ---------------- Priors ----------------- ## priors longitudinal # μ priors, population parameters - μ_BSLD ~ LogNormal(3.5,1) - μ_d ~ Beta(1,100) - μ_g ~ Beta(1,100) - μ_φ ~ Beta(2,4) + μ_BSLD ~ LogNormal(3.5, 1) + μ_d ~ Beta(1, 100) + μ_g ~ Beta(1, 100) + μ_φ ~ Beta(2, 4) μ = (BSLD = μ_BSLD, d = μ_d, g = μ_g, φ = μ_φ) # ω priors, mixed/individual effects - ω_BSLD ~ LogNormal(0,1) - ω_d ~ LogNormal(0,1) - ω_g ~ LogNormal(0,1) - ω_φ ~ LogNormal(0,1) + ω_BSLD ~ LogNormal(0, 1) + ω_d ~ LogNormal(0, 1) + ω_g ~ LogNormal(0, 1) + ω_φ ~ LogNormal(0, 1) Ω = (BSLD = ω_BSLD^2, d = ω_d^2, g = ω_g^2, φ = ω_φ^2) # multiplicative error - σ ~ LogNormal(0,1) - ## priors survival - κ ~ truncated(LogNormal(0,1),0.8,1.2) - λ ~ LogNormal(5,1) - + σ ~ LogNormal(0, 1) + ## prior survival + λ ~ LogNormal(5, 1) + ## prior joint model - β ~ truncated(Normal(0,0.5),-0.1,0.1) - + γ ~ truncated(Normal(0, 0.5), -0.1, 0.1) + ## η describing the mixed effects of the population - η_BSLD ~ filldist(Normal(0,Ω.BSLD),n) - η_d ~ filldist(Normal(0,Ω.d),n) - η_g ~ filldist(Normal(0,Ω.g),n) - η_φ ~ filldist(Normal(0,Ω.φ),n) - η = [(BSLD=η_BSLD[i], d=η_d[i], g=η_g[i], φ=η_φ[i]) for i in 1:n] + η_BSLD ~ filldist(Normal(0, Ω.BSLD), n) + η_d ~ filldist(Normal(0, Ω.d), n) + η_g ~ filldist(Normal(0, Ω.g), n) + η_φ ~ filldist(Normal(0, Ω.φ), n) + η = [(BSLD = η_BSLD[i], d = η_d[i], g = η_g[i], φ = η_φ[i]) for i = 1:n] # Transforms - Ψ = [[μ_BSLD * exp(η_BSLD[i]), - μ_g * exp(η_g[i]), - μ_d * exp(η_d[i]), - inverse(logit)(logit(μ_φ) + (η_φ[i]))] - for i in 1:n] + Ψ = [ [μ_BSLD * exp(η_BSLD[i]), + μ_g * exp(η_g[i]), + μ_d * exp(η_d[i]), + inverse(logit)(logit(μ_φ) + (η_φ[i])),] + for i = 1:n] # ---------------- Likelihoods ----------------- # add the likelihood of the longitudinal process - for measurement in 1:m - id = Int(longit_ids[measurement]) - meas_time = longit_times[measurement] + for ij in 1:m + id = Int(longit_ids[ij]) + meas_time = longit_times[ij] sld_prediction = sld(meas_time, Ψ[id], tx) if isnan(sld_prediction) || sld_prediction < 0 sld_prediction = 0 end - longit_measurements[measurement] ~ Normal(sld_prediction, sld_prediction * σ) + longit_measurements[ij] ~ Normal(sld_prediction, sld_prediction * σ) end # add the likelihood of the survival model with link - baseline_hazard(t) = h_0(t, κ, λ) - for i in 1:n + baseline_hazard(t) = h_0(t, λ) + for i = 1:n id = Int(surv_ids[i]) id_link(t) = sld(t, Ψ[id], tx) censoring = Bool(surv_event[id]) ? Inf : surv_times[id] @@ -116,27 +123,32 @@ end # sample posterior with a prior # -------------------------------------------------------------- # for faster convergence use reverse diffrerentation -Turing.setadbackend(:reversediff) -Turing.setrdcache(true) n = 100 init_params = ( μ_BSLD=60,μ_d=0.0055, μ_g=0.0015, μ_ϕ=0.2, ω_BSLD=0.7 ,ω_d=1.0, ω_g=1.0 , ω_ϕ=1.5, σ=0.18, - κ = 1, λ=1450, β = 0, + λ=1450, β = 0, η_BSLD=zeros(n) ,η_d=zeros(n), η_g=zeros(n) , η_ϕ=zeros(n), ) identity_link_model = identity_link(longit_id, longit_time, longit_sld, surv_id, surv_time, surv_indicator) -identity_link_chn = sample(identity_link_model, NUTS(100, 0.9, max_depth = 8), 200, init_params=init_params) - -# [κ, λ, β] = [1.1500626704863541, 3.9408207824225534, 0.026148519352791527] +identity_link_chain = sample(identity_link_model, NUTS(50, 0.9, max_depth = 8; adtype=AutoReverseDiff(false)), 100, init_params=init_params) +# more samples +# identity_link_chain = sample(identity_link_model, NUTS(1000, 0.9, max_depth = 8; adtype=AutoReverseDiff(false)), 2000, init_params=init_params) # -------------------------------------------------------------- # diagnostics # -------------------------------------------------------------- -using Survival + +# posterior +param_names = [:μ_BSLD, :μ_d, :μ_g, :μ_φ, :σ, :κ, :λ, :β] +display(identity_link_chain[param_names]) +plot(identity_link_chain[param_names]) + + +# individual predictions r = range(0,700,2000) p_survival_all = plot(title = "Overall Survival", ylabel = "Survival percentage", xlabel = "Years") function (km::KaplanMeier)(t::Real) @@ -185,7 +197,7 @@ pred_model = identity_link(sld_id, sld_timepoints, sld_pred, s_id, s_time, s_eve # Predict using zero $\sigma$ -prediction_chn = predict(pred_model, identity_link_chn) +prediction_chn = predict(pred_model, identity_link_chain) #Plotting longitudinal PPC chn_longit = group(prediction_chn, :longit_measurements) diff --git a/paper/paper.bib b/paper/paper.bib index 8c304b9..d0b68a4 100644 --- a/paper/paper.bib +++ b/paper/paper.bib @@ -15,6 +15,17 @@ @article{Wu2007 pages = {308–320} } +@book{Rizopoulos2012, + title = {Joint Models for Longitudinal and Time-to-Event Data}, + ISBN = {9781439872871}, + url = {http://dx.doi.org/10.1201/b12208}, + DOI = {10.1201/b12208}, + publisher = {Chapman and Hall/CRC}, + author = {Rizopoulos, Dimitris}, + year = {2012}, + month = jun +} + @article{Khnel2021, title = {Simultaneous modeling of Alzheimer’s disease progression via multiple cognitive scales}, volume = {40}, diff --git a/paper/paper.md b/paper/paper.md index b0acb3b..860ee40 100644 --- a/paper/paper.md +++ b/paper/paper.md @@ -1,63 +1,54 @@ --- -title: 'JointSurvivalModels.jl: Numeric approach to joint models' +title: 'JointSurvivalModels.jl: A Julia package for Bayesian joint modeling of survival and longitudinal data' tags: - - Julia - - Joint Model - - biostatistics - Survival analysis - - Bayesian statistics + - Nonlinear + - Biostatistics + - Time-to-event + - Mixed-effects model authors: - name: Yannik Ammann + orcid: 0009-0009-3296-3577 affiliation: "1, 2" - - name: Adrian M. Price-Whelan - orcid: 0000-0000-0000-0000 - equal-contrib: true - affiliation: "1, 2" # (Multiple affiliations must be quoted) - - name: Author Without ORCID - equal-contrib: true # (This is how you can denote equal contributions between multiple authors) + - name: Daniel Sabanés Bové affiliation: 2 - - name: Author with no affiliation - corresponding: true # (This is how to denote the corresponding author) + - name: Francois Mercier + orcid: 0000-0002-5685-1408 affiliation: 2 - - given-names: Ludwig - dropping-particle: van - surname: Beethoven + - name: Douglas O. Kelkhoff + orcid: 0009-0003-7845-4061 affiliation: 2 + affiliations: - name: ETH Zürich, Switzerland index: 1 - name: Hoffmann-La Roche Ltd. index: 2 -date: 02 February 2022 +date: 02 February 2024 bibliography: paper.bib - -# Optional fields if submitting to a AAS journal too, see this blog post: -# https://blog.joss.theoj.org/2018/12/a-new-collaboration-with-aas-publishing -aas-doi: 10.3847/xxxxx <- update this with the DOI from AAS once you know it. -aas-journal: Astrophysical Journal <- The name of the AAS journal. --- # Summary -This software implements a numerical approach to define a distribution based on the description of the hazard function from survival analysis. In particular this allows to define joint models of time-to-even data and longitudinal measurements. Using numerical integration, the likelihood of events can be calculated, allowing Bayesian inference frameworks to sample the posterior distribution of the model's parameters. Additionally, this implementation is able to generate samples of joint models. This allows its use in simulations, which are common in Bayesian workflows [@BayesianWorkflow]. +`JointSurvivalModels.jl` implements a numerical approach to define a distribution based on the hazard function. In particular, this provides a mechanism for defining joint models of time-to-event data and longitudinal measurements. Using numerical integration, the likelihood of events can be calculated. This allows Bayesian inference frameworks to sample the posterior distribution of the model's parameters or optimization frameworks to find parameters that maximize the likelihood. Additionally, this implementation can generate samples of joint models. This allows its use in simulations and predictions, which are common in Bayesian workflows [@BayesianWorkflow]. # Statement of need -Over the last decade, there has been a growing interest in joint models for longitudinal and time-to-event outcome data. This is the case in clinical research where biomarkers are often measured repeatedly over time, with the expectation that they may provide insights into the likelihood of a long-term clinical outcome (such as death). As a consequence, joint models have been proposed to leverage this data and used for the prediction of individual risks, the evaluation of surrogacy of a biomarker for a clinical outcome, and for making inferences about treatment in various therapeutic areas. +Over the last decade, there has been a growing interest in joint models for longitudinal and time-to-event outcome data. This is the case in clinical research where biomarkers are often measured repeatedly over time, with the expectation that they may provide insights into the likelihood of a long-term clinical outcome such as disease progression or adverse events. As a consequence, joint models have been proposed to leverage this data and be used for the prediction of individual risks, the evaluation of a biomarker for a clinical outcome, and for making inferences about treatment. -In oncology, it is well known that the individual risk of death depends on the treatment-induced changes in tumor size over time [@Tardivon2019]. Because pharmacologic effects are typically transient, nonlinear mixed-effect models are required to capture central tendency and inter-individual variability in tumor size, while parametric models can be used for analysing time to death. Joint models have been used in other therapeutic areas such as neurology [Khnel2021], cardiovascular disease [KassahunYimer2020], or infection diseases [@Wu2007]. +In oncology, it is well known that the individual risk of death correlates with the treatment-induced changes in tumor size over time [@Tardivon2019]. Because pharmacologic effects are typically transient, nonlinear mixed-effect models are required to capture central tendency and inter-individual variability in tumor size, while parametric models can be used for analyzing time to death. Joint models have been used in other therapeutic areas such as neurology [@Khnel2021], cardiovascular disease [@KassahunYimer2020], or infection diseases [@Wu2007]. -The current landscape of open-source software allowing end-users to easily fit and use joint models consists primarily of R packages such as JMbayes [@JMbayes], rstanarm [@rstanarm], joineR [@joineR], or JM [@JM]. These packages typically limit the longitudinal model to linear forms (for the parameters) preventing users from fitting joint models with saturating biological processes resulting in nonlinear profiles for the biomarker. -In contrast, the present software supports fitting any longitudinal and survival model, provided the joint model fulfills continuity and certain smoothness characteristics for numerical procedures. Such flexibility is crucial for studying complex biological processes. +The current landscape of open-source software allowing end-users to easily fit and use joint models consists primarily of R packages such as JMbayes [@JMbayes], rstanarm [@rstanarm], joineR [@joineR], and JM [@JM]. These packages typically limit the longitudinal model to linear forms (for the parameters) preventing users from fitting joint models with saturating biological processes resulting in nonlinear profiles for the biomarker. +In contrast, the present software supports fitting any longitudinal and survival model, provided the joint model fulfills continuity and certain smoothness characteristics required for numerical procedures. Such flexibility is crucial for studying complex biological processes. # Formulation -To build a joint model, we augment the survival analysis hazard function $h(t) = \lim_{\delta \to 0} P(t\leq T\leq t+\delta | T \geq t)/\delta$ by incorporating a link $l$ to a longitudinal process. The longitudinal process is modeled by a function $m:\mathbb{R} \to \mathbb{R}$, for example a non-linear mixed effects model [@Kerioui2020]. Let the function $h_0:\mathbb{R} \to \mathbb{R}$ describe a baseline hazard and $\gamma\in\mathbb{R}$ be a coefficient of the link contribution. The hazard of the joint model is +To build a joint model, we augment the survival analysis hazard function in time $h(t) = \lim_{\delta \to 0} P(t\leq T\leq t+\delta | T \geq t)/\delta$ for a positive random variable $T$, by incorporating a link $l$ to a longitudinal process. The longitudinal process is modeled by a function $m:\mathbb{R} \to \mathbb{R}$, for example a non-linear mixed effects model [@Wu2007]. Let the function $h_0:\mathbb{R} \to \mathbb{R}_{\geq 0}$ describe a baseline hazard and $\gamma\in\mathbb{R}$ be a coefficient of the link contribution. The hazard of the joint model is -$$ h(t) = h_0(t) \exp(\gamma\cdot l(m(t))).$$ +$$ h(t) = h_0(t) \exp(\gamma\cdot l(m(t))),$$ -The link $l$ is not a function over the reals. In general, the link is an operator on the longitudinal models. Some examples of links such as $l(m(t)) = (l \circ m)(t)$ are given in [@Kerioui2020] such as the derivative $\frac{d}{dt} m(t)$ or integral $\int_0^t m(u) \, du$ operators. +where $l$ is the link to the longitudinal model. Some examples of links $l(m(t)) = (l \circ m)(t)$ are given in @Rizopoulos2012 such as the derivative $d/dt \; m(t)$ or integral $\int_0^t m(u) \, du$ operators. Now we extend this idea to multiple longitudinal models. Suppose that we have $k\in \mathbb{N}$ longitudinal models $\{m_{1},\dots, m_{k}\}$ as well as $k$ link functions $\{l_{1},\dots, l_{k}\}$. Let $M: \mathbb{R} \to \mathbb{R}^k$ and $L:\mathbb{R}^k \to \mathbb{R}^k$ be the vector of functions @@ -77,31 +68,34 @@ $$L(M(t)) =\begin{pmatrix} l_1(m_{1}(t)) \\ l_2(m_{2}(t)) \\ \dots \\ l_k(m_{k}(t)) \end{pmatrix}.$$ -For the link vector $L(M(t))$ we consider the coefficient vector $\gamma \in \mathbb{R}^k$. Then we formulate the hazard as follows. +For the link vector $L(M(t))$ we consider the coefficient vector $\gamma \in \mathbb{R}^k$ and formulate the hazard as follows -$$h(t) = h_0(t) \exp\left(\sum_{j\in [k]}\gamma_{j} l_j(m_{j}(t)) \right) = h_0(t) \exp(\gamma^\top \cdot L(M(t))).$$ +$$h(t) = h_0(t) \exp\left(\sum_{j = 1}^{k}\gamma_{j} l_j(m_{j}(t)) \right) = h_0(t) \exp(\gamma^\top \cdot L(M(t))).$$ -In addition, we consider covariates $x\in \mathbb{R}^l, l\in\mathbb{N}$ and coefficients $\beta \in \mathbb{R}^l$. This results in the hazard +In addition, we consider covariates $x\in \mathbb{R}^p, p\in\mathbb{N}$ and coefficients $\beta \in \mathbb{R}^p$. This results in the hazard $$h(t) = h_0(t) \exp\left(\gamma^\top \cdot L(M(t)) + \beta^\top \cdot x \right).$$ -The probability density function in survival analysis can be described by +The probability density function in survival analysis can be described using the hazard function $$f(t) = h(t) \exp\left(-\int_0^t h(u) \, du\right).$$ -Note for nonlinear longitudinal models $\int_0^t h(u) \, du$ generally does not have a analytic form, thus numerical integration is required. +Note for joint models with links to nonlinear longitudinal models $\int_0^t h(u) \, du$ generally does not have a closed form, thus numerical integration is required. ## Likelihood calculations -Suppose that we have $n\in \mathbb{N}$ individuals. For each individual $i\in [n]=\{1,\dots, n\}$ we observe $n_i \in \mathbb{N}$ different longitudinal measurements $\{y_{i1}, \dots, y_{in_i}\}\subseteq \mathbb{R}$ at associated time points $\{t_{i1}, \dots, t_{in_i}\}\subseteq \mathbb{R}$ at which the measurements were recorded. In addition, we measure an event time $\tau_i \in \mathbb{R}$ and an event indicator $\delta_i \in \{0,1\}$. Without loss of generality we will consider right-censored data; this can be adapted to other censoring processes. Let $Y_i := (\tau_i,\delta_i,(y_{i1}, \dots, y_{in_i}),(t_{i1}, \dots, t_{in_i}))$ be the measurements associated with individual $i\in [n]$ and $Y = \{Y_1, \dots, Y_n\}$ all observations. +Suppose that we have $n\in \mathbb{N}$ individuals. For each individual $i\in \{1,\dots, n\}$ we observe $n_i \in \mathbb{N}$ different longitudinal measurements $\{y_{i1}, \dots, y_{in_i}\}\subseteq \mathbb{R}$ at associated time points $\{t_{i1}, \dots, t_{in_i}\}\subseteq \mathbb{R}$ at which the measurements were recorded. In addition, we measure an event time $\tau_i \in \mathbb{R}$ and an event indicator $\delta_i \in \{0,1\}$. Without loss of generality, we will consider right-censored data; this can be adapted to other censoring processes. Let $Y_i := (\tau_i,\delta_i,(y_{i1}, \dots, y_{in_i}),(t_{i1}, \dots, t_{in_i}))$ be the measurements associated with individual $i$ and $Y = \{Y_1, \dots, Y_n\}$ all observations. + + + -We denote the parameters for the joint model with $\theta =(\theta_H, \theta_J, \theta_L)$, where $\theta_H$ describes the parameters for the baseline hazard, $\theta_J$ for the joint model, and $\theta_L$ for longitudinal models. The likelihood of the generalized joint model is comprised of the likelihood of the survival measurements and the longitudinal measurements. +Let $\theta_H$ describe the parameters for the baseline hazard, $\theta_J$ for the joint model, and $\theta_L$ for longitudinal models. The likelihood of the joint model is comprised of the likelihood of the survival measurements and the longitudinal measurements. -$$\log L(Y | \theta) \propto \sum_{i\in[n]} \log ( L((\tau_i, \delta_i) | \theta)) + \sum_{i\in[n], j\in[n_i]} \log( L(t_{ij},y_{ij} | \theta_L) )$$ +$$\log L(Y | (\theta_H, \theta_J, \theta_L)) = \sum_{i=1}^{n} \log ( L((\tau_i, \delta_i) | (\theta_H, \theta_J, \theta_L))) + \sum_{i = 1}^{n}\sum_{ j=1}^{n_i} \log( L(t_{ij},y_{ij} | \theta_L) )$$ -For individual $i\in[n]$ let $f_i, S_i$ be the joint probability density function and survival function. The likelihood depends on the censoring process, for example for right-censored measurements $(\tau_i, \delta_i)$ is given by +For individual $i$ let $f_i$ be the joint probability density function and $h_i$ the joint hazard function. The likelihood depends on the censoring process, for example for right-censored measurements $(\tau_i, \delta_i)$ is given by -$$\log ( L((\tau_i, \delta_i) | \theta)) = \delta_i \log(f_i(\tau_i)) - (1-\delta_i)\int_0^{\tau_i} h_i(u) du$$ +$$\log ( L((\tau_i, \delta_i) | (\theta_H, \theta_J, \theta_L))) = \delta_i \log(f_i(\tau_i)) - (1-\delta_i)\int_0^{\tau_i} h_i(u) \,du$$ -For the longitudinal model, the likelihood depends on the error process you use. Let $p_{m_i(t_{ij})}$ be the probability density function for measurements for model $m_i$ at time $t_{ij}$ for a given error, for example, the standard error or a multiplicative error. Then the longitudinal likelihood is given by +For the longitudinal model, the likelihood depends on the error process you use. Let $p_{m_i(t_{ij})}$ be the probability density function for measurements for model $m_i$ at time $t_{ij}$ for a given error, for example, the additive error or a multiplicative error. Then the longitudinal likelihood is given by $$\log( L(t_{ij},y_{ij} | \theta_L)) = \log(p_{m_i(t_{ij})}(y_{ij}))$$ @@ -112,7 +106,7 @@ $$\log( L(t_{ij},y_{ij} | \theta_L)) = \log(p_{m_i(t_{ij})}(y_{ij}))$$ # Example -The following example showcases the simplicity and similarity to the mathematical description of the model that is achieved for the modeling of non-linear joint models using `JointSurvivalModels.jl`. It follows the simulation study by [@Kerioui2020]. They specify a longitudinal model for $\Psi = (\text{BSLD}, g, d, \phi) \in \mathbb{R}^4$ as +The following example showcases the simplicity and similarity to the mathematical description of the model that is achieved for the modeling of non-linear joint models using `JointSurvivalModels.jl`. The code can be found in the [example](https://github.com/insightsengineering/JointSurvivalModels.jl/tree/main/example) folder in the project repository. Following @Kerioui2020 a longitudinal model for the sum of longest diameters $\text{SLD}: \mathbb{R} \to \mathbb{R}_{\geq 0}$ is specified with parameters $\Psi = (\text{BSLD}, g, d, \phi)$ where $\text{BSLD}, g, d \in \mathbb{R}_{\geq 0},\; \phi \in [0,1]$ and start of treatment $t_x$ $$\text{SLD}(t,\Psi) = \begin{cases} \text{BSLD}\exp(gt) & t < t_x \\ @@ -133,213 +127,123 @@ function sld(t, Ψ, tx = 0.0) end ``` -They use a baseline Weibull hazard $h_0(t) =\kappa / \lambda * (t/\lambda)^{\kappa-1}$ and the identity function as link. +For the survival distribution, they use an Exponential distribution which has a constant hazard $h_0(t) = 1 / \lambda$ with scale parameter $\lambda \in \mathbb{R}_{\geq 0}$ ```julia -h_0(t, κ, λ) = κ/λ * (t/λ)^(κ - 1) +h_0(t, λ) = 1/λ ``` -The parameters $\Psi$ for the longitudinal model depend on the individual $i$ resulting in the joint hazard +In the mixed effects model every individual $i$ has a different mixed effects parameter denoted by $\Psi_i$ which defines the longitudinal model $M_i(t) = \text{SLD}(t,\Psi_i)$ resulting in the joint hazard for individual $i$ -$$h_i(t) = h_0(t) \exp(b * \text{SLD}(t, \Psi)).$$ +$$h_i(t) = h_0(t) \exp(\gamma \cdot L(M_i(t))) = h_0(t) \exp(\gamma * \text{id}(\text{SLD}(t, \Psi_i))).$$ -In code the distribution of the joint model defined by this hazard is given by: +The identity $\text{id}$ was used as a link. In code, the distribution of the joint model defined by this hazard is given by: ```julia -my_jm(κ, λ, γ, Ψ, tx) = JointSurvivalModel(t -> h_0(t, κ, λ), γ, t -> sld(t, Ψ, tx)) +my_jm(λ, γ, Ψ_i, tx) = JointSurvivalModel(t -> h_0(t, λ), γ, t -> sld(t, Ψ_i, tx)) ``` -The mixed effects model contains population parameters $\mu = (\mu_{\text{BSLD}},\mu_d, \mu_g, \mu_\phi)$ and mixed effects $\eta_i = (\eta_{\text{BSLD},i},\eta_{d,i}, \eta_{g,i}, \eta_{\phi,i})$ which are normally distributed around zero $\eta_i \sim N(0, \Omega), \Omega = (\omega_{\text{BSLD}}^2,\omega_d^2, \omega_g^2, \omega_\phi^2)$. For biological constraints, the parameters were transformed such that $\phi_q(\Psi_{q,i}) = \phi_q\mu_q + \eta_{q,i}$ for $q\in \{\text{BSLD}, d, g, \phi\}$. For $\text{BSLD}, g, d$ a log-normal $(\phi = log)$ transform was assumed and for $\phi$ a logit-normal $(\sigma = \text{logit})$. +The mixed effects model contains population parameters $\mu = (\mu_{\text{BSLD}},\mu_d, \mu_g, \mu_\phi)$ and random effects $\eta_i = (\eta_{\text{BSLD},i},\eta_{d,i}, \eta_{g,i}, \eta_{\phi,i})$ which are normally distributed around zero $\eta_i \sim N(0, \Omega), \Omega = \text{diag}(\omega_{\text{BSLD}}^2,\omega_d^2, \omega_g^2, \omega_\phi^2)$. For $\text{BSLD}, g, d$ a log-normal transform $\log(\Psi_{q,i}) = log (\mu_q) + \eta_{q,i},\, q\in \{\text{BSLD}, g, d\}$ was used while for $\phi$ a logit transform $\text{logit}(\Psi_{\phi,1}) = \text{logit}(\mu_\phi) + \eta_{\phi,1}$ was used. -With this information, a Bayesian model can be specified in Turing.jl [@Turing.jl] by giving prior distributions for the parameters and calculations for the likelihood. To calculate the likelihood of the survival time and event indicator the software is used. This results in a canonical translation of the statistical ideas into code. For longitudinal data, a multiplicative error model is used using $e_{ij} \sim N(0, \sigma^2)$ given by $y_{ij} = \text{SLD}(t_{ij},\Psi_i)(1+e_{ij})$ is used. The model and prior setup from [@Kerioui2020] can be implemented as follows in code: +With this information, a Bayesian model can be specified in `Turing.jl` [@Turing.jl] by giving prior distributions for the parameters and calculations for the likelihood. To calculate the likelihood of the survival time and event indicator the `JointSurvivalModel` is used. This results in a canonical translation of the statistical ideas into code. For longitudinal data, a multiplicative error model is used using $e_{ij} \sim N(0, \sigma^2)$ given by $y_{ij} = \text{SLD}(t_{ij},\Psi_i)(1+e_{ij})$ is used. The model and prior setup from @Kerioui2020 can be implemented as follows in code: ```julia -@model function identity_link(longit_ids, longit_times, longit_measurements, - surv_ids, surv_times, surv_event) - # treatment at study start +@model function identity_link( + longit_ids, + longit_times, + longit_measurements, + surv_ids, + surv_times, + surv_event, +) + # treatment start at study start tx = 0.0 # number of longitudinal and survival measurements n = length(surv_ids) m = length(longit_ids) - # ---------------- Priors ----------------- ## priors longitudinal - # μ priors, population parameters - μ_BSLD ~ LogNormal(3.5,1) - μ_d ~ Beta(1,100) - μ_g ~ Beta(1,100) - μ_φ ~ Beta(2,4) + # population parameters + μ_BSLD ~ LogNormal(3.5, 1) + μ_d ~ Beta(1, 100) + μ_g ~ Beta(1, 100) + μ_φ ~ Beta(2, 4) μ = (BSLD = μ_BSLD, d = μ_d, g = μ_g, φ = μ_φ) - # ω priors, mixed/individual effects - ω_BSLD ~ LogNormal(0,1) - ω_d ~ LogNormal(0,1) - ω_g ~ LogNormal(0,1) - ω_φ ~ LogNormal(0,1) + # standard deviation of random effects + ω_BSLD ~ LogNormal(0, 1) + ω_d ~ LogNormal(0, 1) + ω_g ~ LogNormal(0, 1) + ω_φ ~ LogNormal(0, 1) Ω = (BSLD = ω_BSLD^2, d = ω_d^2, g = ω_g^2, φ = ω_φ^2) + ## η describing the random effects + η_BSLD ~ filldist(Normal(0, Ω.BSLD), n) + η_d ~ filldist(Normal(0, Ω.d), n) + η_g ~ filldist(Normal(0, Ω.g), n) + η_φ ~ filldist(Normal(0, Ω.φ), n) + # Transforms + Ψ = [ [μ_BSLD * exp(η_BSLD[i]), + μ_g * exp(η_g[i]), + μ_d * exp(η_d[i]), + inverse(logit)(logit(μ_φ) + (η_φ[i])),] + for i = 1:n] # multiplicative error - σ ~ LogNormal(0,1) - ## priors survival - κ ~ truncated(LogNormal(0,1),0.8,1.2) - λ ~ LogNormal(5,1) - + σ ~ LogNormal(0, 1) + ## prior survival + λ ~ LogNormal(5, 1) ## prior joint model - γ ~ truncated(Normal(0,0.5),-0.1,0.1) - - ## η describing the mixed effects of the population - η_BSLD ~ filldist(Normal(0,Ω.BSLD),n) - η_d ~ filldist(Normal(0,Ω.d),n) - η_g ~ filldist(Normal(0,Ω.g),n) - η_φ ~ filldist(Normal(0,Ω.φ),n) - η = [(BSLD=η_BSLD[i], d=η_d[i], g=η_g[i], φ=η_φ[i]) for i in 1:n] - # Transforms - Ψ = [[μ_BSLD * exp(η_BSLD[i]), - μ_g * exp(η_g[i]), - μ_d * exp(η_d[i]), - inverse(logit)(logit(μ_φ) + (η_φ[i]))] - for i in 1:n] - + γ ~ truncated(Normal(0, 0.5), -0.1, 0.1) # ---------------- Likelihoods ----------------- # add the likelihood of the longitudinal process - for data in 1:m - id = Int(longit_ids[data]) - meas_time = longit_times[data] + for ij in 1:m + id = Int(longit_ids[ij]) + meas_time = longit_times[ij] sld_prediction = sld(meas_time, Ψ[id], tx) if isnan(sld_prediction) || sld_prediction < 0 sld_prediction = 0 end - longit_measurements[data] ~ Normal(sld_prediction, sld_prediction * σ) + longit_measurements[ij] ~ Normal(sld_prediction, sld_prediction * σ) end # add the likelihood of the survival model with link - baseline_hazard(t) = h_0(t, κ, λ) - for i in 1:n + baseline_hazard(t) = h_0(t, λ) + for i = 1:n id = Int(surv_ids[i]) id_link(t) = sld(t, Ψ[id], tx) censoring = Bool(surv_event[id]) ? Inf : surv_times[id] # here we use the JointSurvivalModel - surv_times[i] ~ censored( - JointSurvivalModel(baseline_hazard, γ, id_link), - upper = censoring - ) + surv_times[i] ~ + censored(JointSurvivalModel(baseline_hazard, γ, id_link), upper = censoring) end end ``` -When sampling the posterior the log probability density function of the joint model is called conditioned on specific parameters. The numerical calculation of the likelihood is then used in the sampling process. Additionally, the implementation of generating random samples of a joint distribution enables Turing.jl [@Turing.jl] to sample a joint distribution. This allows to create posterior predictive checks or simulations, which are a major step in a Bayesian workflow when validating a model [@BayesianWorkflow]. - -![Individual posterior predictions \label{fig:ind_pred}](individual_prediction.svg) - -The figure \autoref{fig:ind_pred} showcases posterior predictions for the mixed effects longitudinal sub model along side the longitudinal observations for a specific individual. The figure also shows the individual survival predictions conditioned on survival until the last measurement based on the joint survival distribution. The light colored ribbons represent the 95% quantile of the posterior predictions while the line represents the median. To create the posterior predictions the above Turing.jl [@Turing.jl] model was used. - - - - - - - - - - - - - - - - - - +When sampling the posterior the log probability density function `logpdf` implemented in `JointSurvivalModels.jl` is called for a distribution given specific parameters. The numerical calculation of the likelihood is then used in the sampling process. Sampling with the `Turing.Inference.NUTS` algorithm 2'000 posterior samples using 1'000 burn-in samples results in posterior statistic: +``` +parameters mean std mcse rhat | ture_parameters + Symbol Float64 Float64 Float64 Float64 | + + μ_BSLD 61.3971 4.4043 0.2737 1.0021 | 60 + μ_d 0.0058 0.0014 0.0001 0.9998 | 0.0055 + μ_g 0.0018 0.0002 0.0000 0.9998 | 0.0015 + μ_φ 0.1729 0.0591 0.0049 1.0001 | 0.2 + σ 0.1767 0.0057 0.0001 1.0008 | 0.18 + λ 1491.2720 307.5059 6.4314 1.0008 | 1450 + γ 0.0103 0.0012 0.0000 1.0005 | 0.01 +``` +Notice that the link coefficient $\gamma$ was sampled around the true value $0.01$ with a small variance. The survival and population parameters are well represented by the posterior samples indicated by the $\hat r$ value close to one and the relative closeness of the mean to the true parameter. +Additionally, `JointSurvivalModels.jl` implements the generation of random samples of a joint distribution. This allows the creation of posterior predictive checks, simulations, or individual predictions \autoref{fig:ind_pred} using `Turing.jl`, which are a major step in a Bayesian workflow when validating a model [@BayesianWorkflow]. +![The figure showcases individual posterior prediction for the mixed effects longitudinal sub-model alongside the longitudinal observations. The figure also shows the individual survival predictions conditioned on survival until the last measurement based on the joint survival distribution. The light-colored ribbons represent the 95% quantile of the posterior predictions while the line represents the median. To create the posterior predictions the above `Turing.jl` model was used. \label{fig:ind_pred}](individual_prediction.svg){ width=80% } +# Acknowledgements +This software was initially developed at Hoffmann-La Roche Ltd. before transitioning into an open source project. Professor Peter Bühlmann from ETH Zurich facilitated the development and open source project. +# References - - -# References \ No newline at end of file