The standard joint model was described and illustrated with merlin here.
In this example we consider time-dependent effects in the survival submodel. Once again we have,
$$h(t) = h_{0}(t) \exp(X_{1ij}\mathbf{\beta_{1}} + \alpha m_{i}(t))$$
and
$$y_{}(t) = m_{i}(t) + \epsilon_{ij}$$
where
$$m_{i}(t) = X_{2ij}(t)\mathbf{\beta}_{2} + Z_{ij}(t)b_{i}$$
and
$$b_{i} \sim N(0,\Sigma)$$
Given the survival submodel is a proportional hazards model, we are making the assumption that both our hazard ratios $\exp (\beta_{1})$ for the covariates $X_{1ij}$, and the hazard ratio for the effect of our biomarker, $\exp(\alpha)$, are constant across follow-up time. We can relax these assumptions to allow for non-proportional hazards i.e. by modelling time-dependent effects. First we consider the following model,
$$h(t) = h_{0}(t) \exp(X_{1ij}\beta_{1}(t) + \alpha m_{i}(t))$$
where if we assume $X_{1ij}$ is a binary treatment variable, we can allow its associated hazard ratio $\beta_{1}(t)$ to be time-dependent, by making it a function of $\log (t)$
$$h(t) = h_{0}(t) \exp(X_{1ij} \beta_{1} + X_{1ij} \times \beta_{2} \times \log (t) + \alpha m_{i}(t))$$
We can fit this model with merlin as follows
. use https://www.mjcrowther.co.uk/data/jm_example.dta, clear
(Example dataset for joint modelling, Michael Crowther 2011)
. merlin (stime trt trt#fp(stime,pow(0)) EV[logb], family(weibull, failure(died)) timevar(stime)) ///
> (logb fp(time,pow(1)) fp(time,pow(1))#M2[id]@1 M1[id]@1, family(gaussian) timevar(time))
variables created for model 1, component 2: _cmp_1_2_1 to _cmp_1_2_1
variables created for model 2, component 1: _cmp_2_1_1 to _cmp_2_1_1
variables created for model 2, component 2: _cmp_2_2_1 to _cmp_2_2_1
Fitting fixed effects model:
Fitting full model:
Iteration 0: log likelihood = -3328.5706
Iteration 1: log likelihood = -2539.587
Iteration 2: log likelihood = -2107.9207
Iteration 3: log likelihood = -1962.2205
Iteration 4: log likelihood = -1933.5759
Iteration 5: log likelihood = -1932.1907
Iteration 6: log likelihood = -1932.1887
Iteration 7: log likelihood = -1932.1887
Mixed effects regression model Number of obs = 1,945
Log likelihood = -1932.1887
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
stime: |
trt | .1236867 .2526515 0.49 0.624 -.3715012 .6188746
trt#fp() | -.0929622 .1662182 -0.56 0.576 -.4187439 .2328196
EV[] | 1.261099 .0955683 13.20 0.000 1.073788 1.448409
_cons | -4.508608 .3321893 -13.57 0.000 -5.159688 -3.857529
log(gamma) | .0631748 .1101585 0.57 0.566 -.152732 .2790815
-------------+----------------------------------------------------------------
logb: |
fp() | .1690606 .013109 12.90 0.000 .1433674 .1947537
fp()#M2[id] | 1 . . . . .
M1[id] | 1 . . . . .
_cons | .4993858 .0595809 8.38 0.000 .3826093 .6161623
sd(resid.) | .345686 .006619 .3329534 .3589055
-------------+----------------------------------------------------------------
id: |
sd(M1) | 1.026386 .0433006 .9449324 1.114861
sd(M2) | .1805098 .0123637 .1578334 .2064441
------------------------------------------------------------------------------
We can also relax the proportional hazards assumption on our association parameter, as follows,
$$h(t) = h_{0}(t) \exp(X_{1ij}\mathbf{\beta_{1}} + \alpha_{1} m_{i}(t) + \alpha_{2} \times \log (t) \times m_{i}(t) )$$
which can be fitted using,
. merlin (stime trt EV[logb] EV[logb]#fp(stime,pow(0)) ///
> , family(weibull, failure(died)) timevar(stime)) ///
> (logb fp(time,pow(1)) fp(time,pow(1))#M2[id]@1 M1[id]@1 ///
> , family(gaussian) timevar(time))
variables created for model 1, component 3: _cmp_1_3_1 to _cmp_1_3_1
variables created for model 2, component 1: _cmp_2_1_1 to _cmp_2_1_1
variables created for model 2, component 2: _cmp_2_2_1 to _cmp_2_2_1
Fitting fixed effects model:
Fitting full model:
Iteration 0: log likelihood = -3328.8883 (not concave)
Iteration 1: log likelihood = -2385.0711
Iteration 2: log likelihood = -1965.2019
Iteration 3: log likelihood = -1933.4852
Iteration 4: log likelihood = -1929.9781
Iteration 5: log likelihood = -1929.8715
Iteration 6: log likelihood = -1929.8674
Iteration 7: log likelihood = -1929.8674
Mixed effects regression model Number of obs = 1,945
Log likelihood = -1929.8674
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
stime: |
trt | .0760776 .1805074 0.42 0.673 -.2777104 .4298656
EV[] | 1.518905 .1637618 9.28 0.000 1.197938 1.839872
EV[]#fp() | -.2095982 .0977551 -2.14 0.032 -.4011948 -.0180016
_cons | -5.362485 .5439844 -9.86 0.000 -6.428675 -4.296295
log(gamma) | .3840727 .1604287 2.39 0.017 .0696382 .6985072
-------------+----------------------------------------------------------------
logb: |
fp() | .1698028 .0132098 12.85 0.000 .1439119 .1956936
fp()#M2[id] | 1 . . . . .
M1[id] | 1 . . . . .
_cons | .5003171 .0595577 8.40 0.000 .3835862 .6170479
sd(resid.) | .3454356 .0066109 .3327185 .3586388
-------------+----------------------------------------------------------------
id: |
sd(M1) | 1.025622 .0432787 .9442107 1.114053
sd(M2) | .1815506 .0124789 .1586683 .2077327
------------------------------------------------------------------------------
For completeness we can model non-proportional hazards in both our treatment effect and our association parameter as follows,
. merlin (stime trt trt#fp(stime,pow(0)) EV[logb] EV[logb]#fp(stime,pow(0)), ///
> family(weibull, failure(died)) timevar(stime)) ///
> (logb fp(time,pow(1)) fp(time,pow(1))#M2[id]@1 M1[id]@1, ///
> family(gaussian) timevar(time))
variables created for model 1, component 2: _cmp_1_2_1 to _cmp_1_2_1
variables created for model 1, component 4: _cmp_1_4_1 to _cmp_1_4_1
variables created for model 2, component 1: _cmp_2_1_1 to _cmp_2_1_1
variables created for model 2, component 2: _cmp_2_2_1 to _cmp_2_2_1
Fitting fixed effects model:
Fitting full model:
Iteration 0: log likelihood = -3328.5706 (not concave)
Iteration 1: log likelihood = -2384.8956
Iteration 2: log likelihood = -1964.4008
Iteration 3: log likelihood = -1933.1154
Iteration 4: log likelihood = -1929.7005
Iteration 5: log likelihood = -1929.5334
Iteration 6: log likelihood = -1929.5161
Iteration 7: log likelihood = -1929.516
Mixed effects regression model Number of obs = 1,945
Log likelihood = -1929.516
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
stime: |
trt | .2345087 .2600758 0.90 0.367 -.2752304 .7442478
trt#fp() | -.1396825 .1661591 -0.84 0.401 -.4653484 .1859834
EV[] | 1.54404 .1675975 9.21 0.000 1.215555 1.872525
EV[]#fp() | -.220537 .0990953 -2.23 0.026 -.4147602 -.0263138
_cons | -5.544952 .5856712 -9.47 0.000 -6.692847 -4.397058
log(gamma) | .4437689 .1675443 2.65 0.008 .1153881 .7721497
-------------+----------------------------------------------------------------
logb: |
fp() | .1698974 .0132107 12.86 0.000 .1440049 .1957899
fp()#M2[id] | 1 . . . . .
M1[id] | 1 . . . . .
_cons | .500362 .0595612 8.40 0.000 .3836241 .6170999
sd(resid.) | .3454272 .00661 .3327118 .3586286
-------------+----------------------------------------------------------------
id: |
sd(M1) | 1.025705 .0432803 .9442896 1.114139
sd(M2) | .1815853 .0124773 .1587055 .2077636
------------------------------------------------------------------------------
In any of the models above, there is no restriction on what functional form we use to model each of the time-dependent
effects. If a more complex form is desired, then we can use the fp() or rcs() syntax to get more flexible time
functions. For example using fractional polynomials,
. merlin (stime trt EV[logb] EV[logb]#fp(stime,pow(1 2)) ///
> , family(weibull, failure(died)) timevar(stime)) ///
> (logb fp(time,pow(1)) fp(time,pow(1))#M2[id]@1 M1[id]@1 ///
> , family(gaussian) timevar(time))
variables created for model 1, component 3: _cmp_1_3_1 to _cmp_1_3_2
variables created for model 2, component 1: _cmp_2_1_1 to _cmp_2_1_1
variables created for model 2, component 2: _cmp_2_2_1 to _cmp_2_2_1
Fitting fixed effects model:
Fitting full model:
Iteration 0: log likelihood = -3328.8883 (not concave)
Iteration 1: log likelihood = -2387.2282
Iteration 2: log likelihood = -2103.1152
Iteration 3: log likelihood = -1961.4764
Iteration 4: log likelihood = -1931.6218
Iteration 5: log likelihood = -1930.0729
Iteration 6: log likelihood = -1930.066
Iteration 7: log likelihood = -1930.066
Mixed effects regression model Number of obs = 1,945
Log likelihood = -1930.066
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
stime: |
trt | .0678679 .1803206 0.38 0.707 -.285554 .4212899
EV[] | 1.489873 .2001533 7.44 0.000 1.09758 1.882166
EV[]#fp():1 | -.0563397 .0658035 -0.86 0.392 -.1853122 .0726328
EV[]#fp():2 | .0012086 .0044446 0.27 0.786 -.0075027 .0099198
_cons | -4.948367 .4383464 -11.29 0.000 -5.80751 -4.089223
log(gamma) | .2286905 .1430941 1.60 0.110 -.0517689 .5091499
-------------+----------------------------------------------------------------
logb: |
fp() | .1689365 .0131413 12.86 0.000 .14318 .194693
fp()#M2[id] | 1 . . . . .
M1[id] | 1 . . . . .
_cons | .5000151 .0596196 8.39 0.000 .3831628 .6168674
sd(resid.) | .3455656 .0066158 .332839 .3587787
-------------+----------------------------------------------------------------
id: |
sd(M1) | 1.026716 .0433295 .9452087 1.115251
sd(M2) | .1805794 .0123842 .1578673 .206559
------------------------------------------------------------------------------
or restricted cubic splines,
. merlin (stime trt EV[logb] EV[logb]#rcs(stime, df(3) log event) ///
> , family(weibull, failure(died)) timevar(stime)) ///
> (logb fp(time,pow(1)) fp(time,pow(1))#M2[id]@1 M1[id]@1 ///
> , family(gaussian) timevar(time))
variables created for model 1, component 3: _cmp_1_3_1 to _cmp_1_3_3
variables created for model 2, component 1: _cmp_2_1_1 to _cmp_2_1_1
variables created for model 2, component 2: _cmp_2_2_1 to _cmp_2_2_1
Fitting fixed effects model:
Fitting full model:
Iteration 0: log likelihood = -3328.8883 (not concave)
Iteration 1: log likelihood = -2385.6777 (not concave)
Iteration 2: log likelihood = -2074.161
Iteration 3: log likelihood = -1952.8568
Iteration 4: log likelihood = -1929.943
Iteration 5: log likelihood = -1929.3383
Iteration 6: log likelihood = -1929.316
Iteration 7: log likelihood = -1929.316
Mixed effects regression model Number of obs = 1,945
Log likelihood = -1929.316
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
stime: |
trt | .0860601 .1812674 0.47 0.635 -.2692175 .4413376
EV[] | 1.574602 .2154813 7.31 0.000 1.152266 1.996937
EV[]#rcs():1 | -.1524369 .1325349 -1.15 0.250 -.4122005 .1073267
EV[]#rcs():2 | -.0205159 .1182128 -0.17 0.862 -.2522088 .211177
EV[]#rcs():3 | .0474122 .1849852 0.26 0.798 -.3151521 .4099765
_cons | -5.395884 .5640056 -9.57 0.000 -6.501314 -4.290453
log(gamma) | .3872765 .1669192 2.32 0.020 .0601209 .7144321
-------------+----------------------------------------------------------------
logb: |
fp() | .1695192 .0132021 12.84 0.000 .1436436 .1953947
fp()#M2[id] | 1 . . . . .
M1[id] | 1 . . . . .
_cons | .5005547 .0596044 8.40 0.000 .3837322 .6173771
sd(resid.) | .3454686 .0066128 .3327479 .3586755
-------------+----------------------------------------------------------------
id: |
sd(M1) | 1.026383 .0433163 .9449015 1.114892
sd(M2) | .1812489 .0124586 .1584039 .2073886
------------------------------------------------------------------------------
I’ve also only shown a time-dependent effect under the current value parameterisation. Any of the other structures shown here can also be made time-dependent in a similar way.