• To identify biomarker relevant to a disease

• To identify the effect of a treatment on a disease
  (independently of the association between the disease and the biomarker)

For example:

• Biomarker: CD4 counts
• Event: death

(Extended) Cox model: \[\begin{align} \lambda_i(t)=\lambda_0(t) \exp(\beta Y_i(t) + \alpha Z_i + \eta W_i) \end{align}\]

Where:

• \(\lambda_i(t)\) is the hazard function at time \(t\) for individual \(i\);
• \(\lambda_0(t)\) is the unspecified baseline hazard function;
• \(\alpha\) measures the effect of \(Z_i\) on the hazard function;
• \(\beta\) measures the association between the trajectory function \(Y_i(t)\) and the hazard function.

coxph(Surv(Time, death) ~ drug, data = aids.id)

coxph(Surv(Time, death) ~ drug, data = aids.id)

#> Call:
#> coxph(formula = Surv(Time, death) ~ drug, data = aids.id, x = TRUE)
#> 
#>   n= 467, number of events= 188 
#> 
#>           coef exp(coef) se(coef)     z Pr(>|z|)
#> drugddI 0.2102    1.2339   0.1462 1.437    0.151
#> 
#>         exp(coef) exp(-coef) lower .95 upper .95
#> drugddI     1.234     0.8104    0.9264     1.643
#> 
#> Concordance= 0.531  (se = 0.019 )
#> Rsquare= 0.004   (max possible= 0.99 )
#> Likelihood ratio test= 2.07  on 1 df,   p=0.2
#> Wald test            = 2.07  on 1 df,   p=0.2
#> Score (logrank) test = 2.07  on 1 df,   p=0.1

1. are measured at determined time points (\(t_{ij}\))
2. can have missing values over time
   • => Imputation?
   • \(\Rightarrow\) Bias introduction
3. are measured with some degree of error
   • \(\Rightarrow\) Noise in the biomarker trajectory (\(Y_i(t_{ij}) \neq X_i(t_{ij})\))
4. can be endogenous
   • \(\Rightarrow\) Trajectory can change when the event occurs
   • \(\Rightarrow\) Bias introduction

(Generalised) linear mixed effect model: \[Y_{i}(t_{ij})=X_{i}(t_{ij})+\epsilon_{i}(t_{ij})\]

where:

• \(Y_{i}(t_{ij})\) is the observed value
• \(X_{i}(t_{ij})\) is the true (unobserved) value of the longitudinal measurement at time \(t_{ij}\) for individual \(i\).
• \(\epsilon_{i}(t_{ij})\) is a random error term, usually:
  \[\epsilon_{i}(t_{ij})\sim \mathcal{N}(0,\sigma^2)\]

lme(sqrt(CD4) ~ obstime*drug - drug, random = ~ 1|patient, data = aids)

lme(sqrt(CD4) ~ obstime*drug - drug, random = ~ 1|patient, data = aids)

#> Linear mixed-effects model fit by REML
#>  Data: aids 
#>        AIC      BIC    logLik
#>   2730.729 2756.957 -1360.364
#> 
#> Random effects:
#>  Formula: ~1 | patient
#>         (Intercept)  Residual
#> StdDev:   0.8719493 0.4106749
#> 
#> Fixed effects: sqrt(CD4) ~ obstime * drug - drug 
#>                      Value  Std.Error  DF  t-value p-value
#> (Intercept)      2.5087153 0.04306984 936 58.24761  0.0000
#> obstime         -0.0338783 0.00352643 936 -9.60697  0.0000
#> obstime:drugddI  0.0055813 0.00498557 936  1.11949  0.2632
#>  Correlation: 
#>                 (Intr) obstim
#> obstime         -0.153       
#> obstime:drugddI  0.003 -0.691
#> 
#> Standardized Within-Group Residuals:
#>         Min          Q1         Med          Q3         Max 
#> -5.09514877 -0.46243667  0.01324625  0.48993786  5.17139372 
#> 
#> Number of Observations: 1405
#> Number of Groups: 467

fitCOX <- coxph(Surv(Time, death) ~ drug, data = aids.id, x = TRUE)
fitLME <- lme(sqrt(CD4) ~ obstime*drug - drug, random = ~ 1 | patient, data = aids)
fitJOINT <- jointModel(fitLME, fitCOX, timeVar = "obstime", method = "piecewise-PH-aGH")

#> Variance Components:
#>                StdDev
#> (Intercept) 0.8793445
#> Residual    0.4093577
#>
#> Coefficients:
#> Longitudinal Process
#>                   Value Std.Err  z-value p-value
#> (Intercept)      2.5100  0.0434  57.8778 <0.0001
#> obstime         -0.0362  0.0035 -10.3474 <0.0001
#> obstime:drugddI  0.0045  0.0049   0.9173  0.3590
#>
#> Event Process
#>             Value Std.Err z-value p-value                                  
#> drugddI    0.3458  0.1523  2.2715  0.0231                                  
#> Assoct    -1.0866  0.1184 -9.1786 <0.0001  
#> log(xi.1) -1.6560  0.2529 -6.5475                                          
#> log(xi.*)  ......  .....   ......

Test simultaneously an effect on:

• a biomarker (\(\gamma\));
• an event (\(\alpha\));
• a biomarker and an event (\(\beta\gamma+\alpha\)).

A gain in statistical power to detect those effects

• if \(\beta\neq0\), to detect a joint effect of \(Z\): \(\beta\gamma+\alpha\neq0\)
  (from Chen, Ibrahim, & Chu (2011));
• compared to the extended Cox model.

The standard (joint likelihood) formulation involves two components:

• a longitudinal component
• a time-to-event (survival) component.

With:

• \(n\), the sample size;
  
• \(i\), an individual (\(i=1,\cdots,n\));
• \(m_i\), the number of measurements on individual \(i\);
• \(t_{ij}\), a time points (\(j=1,\cdots,m_i\)).

(Generalised) linear mixed effect model: \[Y_{i}(t_{ij})=X_{i}(t_{ij})+\epsilon_{i}(t_{ij})\]

\(X_{ij}\) is the trajectory function, and could be defined: \[\begin{gather}X_{i}(t_{ij})=\theta_{0i} + \theta_{1i}t_{ij} + \cdots + \theta_{pi}t_{ij}^p &, & \boldsymbol\theta_p \sim \mathcal{N}(\boldsymbol\mu_, \boldsymbol\Sigma)\end{gather}\]

For simplicity here, we assume linearity over time (\(\theta_{0i}+\theta_{1i}t_{ij}\)): \[\begin{gather}Y_{i}(t_{ij})=\theta_{0i}+\theta_{1i}t_{ij}+\gamma Z_i+\delta W_i+\epsilon_{ij} &, & \boldsymbol{\theta} \sim \mathcal{N}_2 (\boldsymbol{\mu},\boldsymbol{\Sigma})\end{gather}\]

(Generalised) linear mixed effect model: \[Y_{i}(t_{ij})=\theta_{0i}+\theta_{1i}t_{ij}+\gamma Z_i+\delta W_i+\epsilon_{ij}\]

With:

• \(Y_{i}(t_{ij})\), the observed value;
• \(X_{i}(t_{ij})\), the true (unobserved) value of the longitudinal measurement at time \(t_{ij}\) for individual \(i\);
• \(\epsilon_{ij}\),a random error term, usually: \(\epsilon_{ij}\sim \mathcal{N}(0,\sigma^2)\);
• \(Z_i\), a vector denoting the genotype of individual \(i\);
• \(W_i\), a set of adjusting covariates.

(Extended) Cox model (proportional hazards): \[\begin{align} \lambda_i(t)&=\lim_{dt \to 0} \frac{P\{t\leq T_i<t+dt|T_i\geq t, \bar{Y_i}(t), Z_i, W_i\}}{dt}\\ &=\lambda_0(t) \exp\{\beta X_{i}(t) + \alpha Z_i + \eta W_i\} \end{align}\]

With:

• \(\bar{Y_i}(t)=\{Y_i(u),0 \leq u \leq t\}\), the history of the trajectory;
• \(T_i\), the event time for individual \(i\);
• \(C_i\), the right censoring time (end of the follow-up);
• \(\Delta_i\), an event indicator: \(\begin{cases} \Delta_i=0, & \text{if }\ T_i>C_i. \\ \Delta_i=1, & \text{if }\ T_i <= C_i. \end{cases}\)

• Are Joint Model estimators good?
  • Bias, variance and RMSE (Root-Mean Square Error)
    \[\begin{align} \operatorname{MSE}(\hat\phi)&= \operatorname{Bias}(\hat\phi)^2 + \operatorname{Var}(\hat\phi)\\ \operatorname{RMSE}(\hat{\phi})&=\sqrt{\operatorname{MSE}(\hat\phi)}\\ &=\sqrt{E\{(\hat{\phi}-\phi)^2\}} \end{align}\] \[\phi=(\beta, \gamma, \alpha)\]
• Can we do a whole genome analysis…
  … in a reasonable time frame?

What if, we split the job in two?

\(\Rightarrow\) "Two-Step"? (Tsiatis, DeGruttola, & Wulfsohn, 1995)

1. (Generalised) linear mixed effect model
   \[\begin{align} Y_{i}(t)&=X_i(t)+ \epsilon_{i}(t)\\ X^*_{i}(t)&=E\{X_{i}(t)|\bar{Y_i}(t), T_i\geq t\} \end{align}\]

2. (Extended) Cox model (proportional hazards)
   \[h_i(t)=h_0(t) \exp\{\beta X^*_{i}(t)\}\]

Let's keep it simple, i.e., without covariates:

• the trajectory: \(Y_{i}(t)=\theta_{0i} + \theta_{1i}t + \gamma Z_i + \epsilon_{i}(t)\)

• the event: \(\lambda_i(t)=\lambda_0(t) \exp\{\beta X_{i}(t) + \alpha Z_i\}\)

• the time of event, e.g., the exponential distribution (Austin, 2012):
  \[\begin{gather} H_i(T_i)=\int_0^{T_i}\lambda_0(t) \exp(\beta X_i(t)+\alpha Z_i)dt , & \lambda_0(t)=\lambda\\ F_i(T_i)=1-exp(-H_i(T_i))=u , & u\sim\mathcal{U}(0, 1) \end{gather}\] \[T_i=\frac{1}{\beta\theta_{1i}}\log\left(1-\frac{\beta\theta_{1i}\times \log(1-u)}{\lambda \exp(\beta\theta_{0i}+(\beta\gamma+\alpha)Z_i)}\right)\]