Rizopoulos 2016 - Association Structures and Transformation Functions

Summary

The function $f[H_i(t),{\mathbf{b}}_i,\mathbf{\alpha }]$ decides which feature of the longitudinal profile drives risk: current value, slope, area under the curve, weighted history, or the random effects themselves. On PBC, current level and slope are both strongly associated with risk; the estimated weight function shows only the last 0.3 years matter; the shared-random-effects parametrisation dominates the model-averaging weights for Patient 2.

Main Content

Association structures

Written for the default param, with extraForm giving the fixed/random formulas and the index vectors indFixed, indRandom of the original mixed model used by the extra term.

1. Current value ("td-value"): $h_i(t)=h_0(t)\exp \{{\gamma }^{\top }{w}_i+\alpha {\eta }_i(t)\}$.
2. Value + slope (Eq. 8, "td-both"): $h_i(t)=h_0(t)\exp \{{\gamma }^{\top }{w}_i+{\alpha }_1{\eta }_i(t)+{\alpha }_2{\eta }_i^{\prime }(t)\}$, ${\eta }_i^{\prime }(t)=d\{{\mathbf{x}}_i^{\top }\mathbf{\beta }+{\mathbf{z}}_i^{\top }{\mathbf{b}}_i\}/dt$, computed with dns() (central difference of the natural spline).
3. Cumulative effect ("td-extra"): $h_i(t)=h_0(t)\exp \{{\gamma }^{\top }{w}_i+\alpha {\int }_0^t{\eta }_i(s)ds\}$ via ins() (Gauss-Kronrod integral of ns()).
4. Weighted cumulative effect: $h_i(t)=h_0(t)\exp \{{\gamma }^{\top }{w}_i+\alpha {\int }_0^t\varpi (t-s){\eta }_i(s)ds\}$ with $\varpi$ a user density (normal, Student-$t$, logistic) with up to three shape parameters, normalised over $[0,t_{\max }]$ (estimateWeightFun=TRUE).
5. Shared random effects (Eq. 9, "shared-RE"): $h_i(t)=h_0(t)\exp ({\gamma }^{\top }{w}_i+{\mathbf{\alpha }}^{\top }{\mathbf{b}}_i)$; with random intercept and slope, $\alpha$ components say whether baseline level or trajectory steepness drives risk.

PBC estimates by association structure

Fit	Terms	Estimates (SD)	Time
pbc1 value	Assoct	1.4099 (0.0935)	3.8 min
pbc12 value + slope	Assoct; AssoctE	1.3530 (0.1010); 2.0905 (0.5425)	4.4 min
pbc13 cumulative	AssoctE	0.2198 (0.0197)	3.8 min
pbc13w weighted	Assoct; shape (SD of normal weight)	1.3734 (0.0998); 0.0898 (0.0371)	59.3 min
pbc14 shared RE	intercept; ns1; ns2	1.3655 (0.1512); 0.5381 (0.0629); 0.1545 (0.1010)	5.5 min
pbc15 transformed	Assoct; Assoct:${}^2$; AssoctE; AssoctE:D-penicil	0.8572 (0.3179); 0.1457 (0.0949); 2.4226 (0.7515); 0.1818 (0.9590)	4.6 min

Readings: (i) hazard ratio for a 50% rise within a year at the same level $=\exp (0.4054\times 2.0905)=2.33$ (95% CI 1.5-3.6). (ii) Cumulative term: unit increase of the area multiplies risk by 1.2 (CI 1.2-1.3). (iii) Weight-function SD of 0.09 means only measurements within about 0.3 years (3 SD) before $t$ matter, so the weighted model reproduces the current-value $\alpha$. (iv) No evidence of non-linearity in the current value or of treatment-specific slope effects.

Transformation functions ( transFun, Sec. 4.4)

A function (or list with components value and extra) applied to the current value or the extra term before it enters the linear predictor, vectorised in x, with access to data. Example: tf1 <- function(x, data) cbind(x, "^2" = x*x) and tf2 <- function(x, data) cbind(x, "D-penicil" = x*(data$drug=="D-penicil")) fit ${\alpha }_1\eta +{\alpha }_2{\eta }^2+{\alpha }_3{\eta }^{\prime }+{\alpha }_4{\eta }^{\prime }\times \text{drug}$.

Choosing a structure. Hard in general: more than one can hold (not available together in JMbayes). For inference treat as model selection by DIC; for prediction use model averaging (Rizopoulos 2016 - Dynamic Predictions and Bayesian Model Averaging). Supporting functions (4.5): coef, fixef, ranef, vcov, fitted, residuals (martingale residuals for survival), anova (DIC, pD, LPML), plot, predict, survfitJM, logLik (with Laplace-approximate marginal likelihood), xtable.

Connections

Model Comparison (DIC/LPML), Hierarchical Models, Survival Analysis.

See Also

Rizopoulos 2016 - JMbayes Joint Models Overview