Stage 1 fits the random-intercept linear mixed model
y = X beta + b_i + e by EM; stage 2 fits a Cox
proportional-hazards model for the event time with the empirical
Bayes random effects as an extra covariate, so its coefficient is the
association parameter eta. eta = 0 means the longitudinal trajectory
carries no information about the hazard. The Cox model uses the
Breslow handling of ties and Newton-Raphson. Both stages are
deterministic with a fixed iteration count. The two-stage estimator
is biased towards zero relative to the full joint likelihood; that is
a documented property of the method, not an artefact here.
Arguments
- long_y
Longitudinal measurements, one per record.
- time
Event or censoring time (constant within subject).
- event
Event indicator, 0 or 1 (constant within subject).
- X
Fixed-effect design for the longitudinal model; an intercept is prepended. May be
NULL.- Z
Subject-level covariates for the survival model; one row per subject or per record. May be
NULL.- cluster
Subject label of each record.
Value
List with estimate (eta), eta, se,
gamma, gamma_se, beta, tau2,
sigma2, icc, b, partial_loglik,
n_subjects, n_events, n_covariates, n,
method.
Details
Formula: y_ij = x_ij' beta + b_i + e_ij; lambda_i(t) = lambda_0(t) exp(gamma' W_i + eta b_i).
References
Henderson, Diggle and Dobson (2000), Joint modelling of longitudinal measurements and event time data, Biostatistics 1(4):465-480, doi:10.1093/biostatistics/1.4.465 ; Rizopoulos (2012), Joint Models for Longitudinal and Time-to-Event Data, Chapman and Hall/CRC, chapter 4. doi:10.1201/b12208
Examples
Jntlmm(long_y = c(1, 2, 3, 4, 5, 6, 7, 8), time = c(1, 2, 3, 4, 5, 6, 7, 8),
event = c(0, 1, 0, 1, 1, 0, 1, 0), X = c(1, 2, 3, 4, 5, 6, 7, 8),
Z = c(1, 2, 3, 4, 5, 6, 7, 8), cluster = c(1, 2, 3, 4, 5, 6, 7, 8))
#> $estimate
#> [1] 9.08111e-07
#>
#> $eta
#> [1] 9.08111e-07
#>
#> $se
#> [1] 1e+05
#>
#> $gamma
#> [1] -2.757274e+01 9.081110e-07
#>
#> $gamma_se
#> [1] 97857.53 100000.00
#>
#> $beta
#> [1] 1.660216e-11 1.000000e+00
#>
#> $tau2
#> [1] 2.534046e-23
#>
#> $sigma2
#> [1] 4.612197e-23
#>
#> $icc
#> [1] 0.3545984
#>
#> $b
#> [1] -4.578852e-12 -3.270563e-12 -1.962433e-12 -6.541442e-13 6.541442e-13
#> [6] 1.962433e-12 3.270721e-12 4.579009e-12
#>
#> $partial_loglik
#> [1] -4.433787e-12
#>
#> $n_subjects
#> [1] 8
#>
#> $n_events
#> [1] 4
#>
#> $n_covariates
#> [1] 2
#>
#> $n
#> [1] 8
#>
#> $method
#> [1] "stage 1 random-intercept LMM by EM, stage 2 Cox on the empirical Bayes b_i, Henderson et al (2000)"
#>
#> attr(,"class")
#> [1] "morie_rich_result" "list"