Overview
In the Guo-Carlin joint model, the longitudinal CD4 process and the
survival process are linked through subject-specific random effects.
The longitudinal random-effects component is
\[
W_{1i}(t) = U_{1i} + U_{2i}t
\]
and the survival random-effects component is
\[
W_{2i}(t)
=
r_1U_{1i}
+
r_2U_{2i}
+
r_3(U_{1i}+U_{2i}t)
+
U_{3i}.
\]
The shared random effects \(U_{1i}\)
and \(U_{2i}\) create the statistical
association between the repeated CD4 measurements and survival.
1. Longitudinal Model
The longitudinal submodel is
\[
y_{ij}
=
\beta_{11}
+\beta_{12}s_{ij}
+\beta_{13}s_{ij}Drug_i
+\beta_{14}Gender_i
+\beta_{15}PrevOI_i
+\beta_{16}Stratum_i
+U_{1i}
+U_{2i}s_{ij}
+\epsilon_{ij}.
\]
The fixed-effects part describes the average CD4 trajectory in the
population.
The random-effects part is
\[
W_{1i}(s_{ij})
=
U_{1i}
+
U_{2i}s_{ij}.
\]
1.1 Random intercept: \(U_{1i}\)
\(U_{1i}\) is the
subject-specific random intercept.
It represents how much subject \(i\)’s underlying CD4 level differs from the
population-average CD4 level.
If
\[
U_{1i} > 0,
\]
the subject tends to have a higher CD4 trajectory than the population
average.
If
\[
U_{1i} < 0,
\]
the subject tends to have a lower CD4 trajectory than the population
average.
At baseline, the subject-specific longitudinal level is
approximately
\[
\text{population baseline}
+
U_{1i}.
\]
Therefore, \(U_{1i}\) allows
different patients to start at different CD4 levels.
1.2 Random time slope: \(U_{2i}\)
\(U_{2i}\) is the
subject-specific random slope for time.
It describes how subject \(i\)’s CD4
rate of change differs from the population-average time slope.
For the drug group coded \(Drug_i=0\), the subject-specific slope
is
\[
\beta_{12}+U_{2i}.
\]
For the drug group coded \(Drug_i=1\), the subject-specific slope
is
\[
\beta_{12}+\beta_{13}+U_{2i}.
\]
If
\[
U_{2i}>0,
\]
the subject’s CD4 trajectory is more positive, or less negative, than
the population-average trajectory.
If
\[
U_{2i}<0,
\]
the subject’s CD4 declines faster than the population-average
trajectory.
1.3 Why the term is \(U_{2i}s_{ij}\)
The random slope contribution is
\[
U_{2i}s_{ij}.
\]
Its effect increases with time.
At time \(s_{ij}=0\),
\[
U_{2i}s_{ij}=0.
\]
At later times, the contribution of the subject-specific slope
becomes larger.
Therefore,
\[
U_{1i}
\]
controls the vertical position of the subject’s trajectory,
whereas
\[
U_{2i}
\]
controls the direction and rate of change of the trajectory.
1.4 Name of the longitudinal model
Because the model contains both
\[
U_{1i}
\]
and
\[
U_{2i}s_{ij},
\]
it is called a
linear mixed-effects model with a random intercept and random
slope for time.
Equivalently, it can be called a
random-intercept and random-slope linear mixed
model.
The two random effects are usually modeled jointly as
\[
\begin{pmatrix}
U_{1i}\\
U_{2i}
\end{pmatrix}
\sim
N_2
\left(
\begin{pmatrix}
0\\
0
\end{pmatrix},
\Sigma
\right).
\]
The covariance matrix
\[
\Sigma
=
\begin{pmatrix}
Var(U_1) & Cov(U_1,U_2)\\
Cov(U_1,U_2) & Var(U_2)
\end{pmatrix}
\]
allows the random intercept and random time slope to be
correlated.
2. Why Time Can Have a Random Slope
A subject-level random slope is most natural for a predictor that
varies within a subject.
Time varies repeatedly within the same patient, for example,
Patient 1: time = 0, 2, 6, 12, 18
Patient 2: time = 0, 2, 6, 12, 18
Therefore, each patient can have a different time trajectory.
This is why a random time slope is meaningful and identifiable.
By contrast, baseline variables such as gender, PrevOI, treatment
group, and AZT stratum do not vary within the same patient.
For example,
Patient 1: PrevOI = AIDS, AIDS, AIDS, AIDS
Patient 2: PrevOI = No AIDS, No AIDS, No AIDS, No AIDS
Therefore, a patient-level random slope for these baseline variables
is generally not meaningful because it cannot be separated cleanly from
the subject-specific random intercept.
However, interactions such as
\[
Time \times Gender
\]
or
\[
Time \times PrevOI
\]
can be included as fixed effects if the scientific
question is whether average CD4 trajectories differ between groups.
3. Survival Random-Effects Model
Guo-Carlin write the survival random-effects component as
\[
W_{2i}(t)
=
r_1U_{1i}
+
r_2U_{2i}
+
r_3(U_{1i}+U_{2i}t)
+
U_{3i}.
\]
A survival model can therefore be written as
\[
\log \lambda_i(t)
=
X_i^T\beta_2
+
W_{2i}(t),
\]
where \(X_i^T\beta_2\) contains the
observed survival covariates.
The important point is that the survival model uses the same \(U_{1i}\) and \(U_{2i}\) that appear in the longitudinal
CD4 model.
This is what links the two outcomes.
4. Interpretation of \(r_1U_{1i}\)
The term
\[
r_1U_{1i}
\]
represents the association between the subject-specific longitudinal
intercept and the survival hazard.
\(U_{1i}\) measures whether the
patient’s underlying CD4 level is above or below the population
average.
Therefore, \(r_1\) asks:
Is the patient’s underlying CD4 level associated with mortality
risk?
For example, if
\[
r_1<0,
\]
then patients with larger \(U_{1i}\), meaning higher underlying CD4
levels, tend to have lower mortality hazards.
The hazard ratio associated with a one-unit increase in \(U_{1i}\), ignoring the \(r_3\) contribution temporarily, is
\[
\exp(r_1).
\]
5. Interpretation of \(r_2U_{2i}\)
The term
\[
r_2U_{2i}
\]
represents the association between the subject-specific CD4 slope and
mortality risk.
\(U_{2i}\) measures whether a
patient’s CD4 trajectory declines faster or slower than the
population-average trajectory.
Therefore, \(r_2\) asks:
Is the patient’s individual CD4 rate of change associated with
mortality?
For example, if the overall CD4 trajectory is declining and
\[
U_{2i}<0,
\]
the patient has a faster-than-average decline.
If
\[
r_2<0,
\]
then
\[
r_2U_{2i}>0,
\]
which increases the log hazard.
This corresponds to faster CD4 decline being associated with higher
mortality.
6. Interpretation of \(r_3(U_{1i}+U_{2i}t)\)
The term
\[
r_3(U_{1i}+U_{2i}t)
\]
can be written as
\[
r_3W_{1i}(t).
\]
Because
\[
W_{1i}(t)=U_{1i}+U_{2i}t,
\]
this term represents the association between survival and the
subject-specific longitudinal random-effects trajectory evaluated at
time \(t\).
It asks:
At time \(t\), is the patient’s
deviation from the average CD4 trajectory associated with mortality
risk?
This term is time-dependent because it contains
\[
U_{2i}t.
\]
As time increases, the contribution of the patient’s individual CD4
slope to mortality risk can become stronger or weaker.
It is important to note that
\[
U_{1i}+U_{2i}t
\]
is not the patient’s complete predicted CD4 value.
It is only the random-effects component of the
longitudinal trajectory.
The full predicted CD4 trajectory also contains all fixed
effects.
7. Interpretation of \(U_{3i}\)
The term
\[
U_{3i}
\]
is a survival-specific random effect, often called a
frailty.
It represents unobserved subject-level mortality risk that is not
explained by
\[
U_{1i},
\]
\[
U_{2i},
\]
or the observed survival covariates.
Two patients could have similar measured covariates and similar CD4
trajectories but still have different mortality risks because of
unmeasured characteristics.
That additional heterogeneity is represented by
\[
U_{3i}.
\]
A positive \(U_{3i}\) increases the
subject’s log hazard, whereas a negative \(U_{3i}\) decreases it.
8. Expanding the Survival Random-Effects Term
Starting from
\[
W_{2i}(t)
=
r_1U_{1i}
+
r_2U_{2i}
+
r_3(U_{1i}+U_{2i}t)
+
U_{3i},
\]
we can expand it:
\[
W_{2i}(t)
=
r_1U_{1i}
+
r_2U_{2i}
+
r_3U_{1i}
+
r_3U_{2i}t
+
U_{3i}.
\]
Collecting terms gives
\[
W_{2i}(t)
=
(r_1+r_3)U_{1i}
+
(r_2+r_3t)U_{2i}
+
U_{3i}.
\]
This representation is useful because it shows that the effective
association between the random time slope and survival is
time-dependent.
The effective coefficient for \(U_{1i}\) is
\[
r_1+r_3.
\]
The effective coefficient for \(U_{2i}\) at time \(t\) is
\[
r_2+r_3t.
\]
Therefore, the influence of the patient’s CD4 slope on mortality can
change over follow-up.
9. How the Random Effects Link the Two Models
The longitudinal model contains
\[
U_{1i}+U_{2i}t.
\]
The survival model contains
\[
r_1U_{1i}
+
r_2U_{2i}
+
r_3(U_{1i}+U_{2i}t).
\]
Therefore, the same latent subject-specific quantities influence both
outcomes.
Conceptually,
U1i
random longitudinal intercept
/ \
/ \
v v
Longitudinal Survival
CD4 hazard
^ ^
\ /
\ /
U2i
random time slope
This shared-random-effects structure is the central mechanism of the
joint model.
10. Summary Table
| \(U_{1i}\) |
Subject-specific random intercept for CD4 |
| \(U_{2i}\) |
Subject-specific random time slope for CD4 |
| \(U_{1i}+U_{2i}t\) |
Subject-specific deviation from the average CD4 trajectory at time
\(t\) |
| \(r_1\) |
Association between longitudinal random intercept and survival |
| \(r_2\) |
Association between longitudinal random slope and survival |
| \(r_3\) |
Time-dependent association through the current random-effects
trajectory |
| \(U_{3i}\) |
Survival-specific frailty or unexplained subject-level survival
heterogeneity |
11. Main Interpretation
The longitudinal model allows each patient to have:
- their own baseline CD4 level through \(U_{1i}\), and
- their own CD4 rate of change through \(U_{2i}\).
The survival model then uses these same subject-specific
characteristics to explain mortality risk.
Therefore, Guo-Carlin’s joint model asks whether mortality is
associated with:
- the patient’s underlying CD4 level,
- the patient’s CD4 rate of change,
- the patient’s time-dependent deviation from the population CD4
trajectory, and
- additional survival-specific unobserved heterogeneity.
This is why the model is called a shared-random-effects joint
model for longitudinal and survival data.