The interpretation of exp(Assoct) in JM

Dear Professor Rizopoulos, I have some doubts regarding the interpretation of exp(Assoct) in joint model. In your book "Joint models for longitudinal and time-to-event data _ with applications in R", section 4.1.1 The Survival Submodel mentions:

h0(t) exp{γ⊤wi + αmi(t)}

exp(α) denotes the relative increase in the risk for an event at time t that results from one unit increase in yi(t) at the same time point. `However, here mi(t) is defined as:

yi(t) = mi(t) + εi(t), mi(t) = x ⊤ i (t)β + z ⊤ i (t)bi

the mi(t) was linear.

Because my y is non-linear, I used bs(time,3) to fit the nonlinear process of y. Below, I've included the code and results：

fitLME <- lme(y~ bs(chartday,3), random = ~ 1+chartday|subject_id,
data = data_for_lme)
fitSURV <- coxph(Surv(time, status) ~ apsiii+charlson_pasthistory_charlson_comorbidity_index,
data = data_for_cox, x = TRUE , model = TRUE)
fitJOINT <- jointModel(fitLME, fitSURV, timeVar = "chartday")
summary(fitJOINT)

Call:
jointModel(lmeObject = fitLME, survObject = fitSURV, timeVar = "chartday")

Data Descriptives:
Longitudinal Process Event Process
Number of Observations: 74816 Number of Events: 1391 (20.7%)
Number of Groups: 6713

Joint Model Summary:
Longitudinal Process: Linear mixed-effects model
Event Process: Weibull relative risk model
Parameterization: Time-dependent

log.Lik AIC BIC
-37348.72 74723.43 74811.99

Variance Components:
StdDev Corr
(Intercept) 0.6976 (Intr)
chartday 0.0807 -0.2029
Residual 0.2904

Coefficients:
Longitudinal Process
                   Value Std.Err  z-value p-value
(Intercept)       5.0622  0.0088 572.5697 <0.0001
bs(chartday, 3)1 -0.6039  0.0163 -36.9510 <0.0001
bs(chartday, 3)2  1.5291  0.0348  43.9721 <0.0001
bs(chartday, 3)3  0.1008  0.0443   2.2755  0.0229

Event Process
                                                  Value Std.Err  z-value p-value
(Intercept)                                     -5.1218  0.1958 -26.1581 <0.0001
apsiii                                           0.0210  0.0009  24.0735 <0.0001
charlson_pasthistory_charlson_comorbidity_index  0.1153  0.0089  12.9207 <0.0001
Assoct                                          -0.3082  0.0285 -10.8297 <0.0001
log(shape)                                       0.0302  0.0234   1.2906  0.1968

Scale: 1.0307

Integration:
method: (pseudo) adaptive Gauss-Hermite
quadrature points: 3

Optimization:
Convergence: 0

Can I still interpret exp(Assoct) as denoting the relative increase in the risk for an event at time t that results from one unit increase in yi(t) at the same time point at this point? Can exp(Assoct) be interpreted as this no matter how mi(t) is fitted (bs(), ns(), poly()...)? Thank you very much!

Best, Qian

drizopoulos / JM

The interpretation of exp(Assoct) in JM #50