A user-defined hazard model: fractional polynomials

This tutorial will illustrate some of the more advanced capabilities of merlin when modelling survival data, but hopefully with a rather simple example. In some previous work, Paul Lambert and I developed stgenreg for modelling survival data with a general user-specified hazard function, which is then estimated using numerical integration. Before continuing with this example, please look through the accompanying material provided with stgenreg; the methods paper can be found here and a software tutorial can be found here.

Consider our standard proportional hazards model,

$$ h(t) = h_{0}(t) \exp (X \beta)$$

Within a general hazard model, we can essentially specify any function for our baseline hazard function, subject to the constraint that $h_{0}(t)>0$ for all $t>0$. The easiest way to do this is to model on the log hazard scale. Let’s model our baseline log hazard function with fractional polynomials, such as,

$$ \log h_{0}(t) = \gamma_{0} + \gamma_{1} t + \gamma_{2} \log(t)$$

This model can be fitted using stgenreg, but with the introduction of merlin, we can do the same as stgenreg, and a whole lot more. I’ll use the catheter dataset to fit some models.

. . webuse catheter, clear
(Kidney data, McGilchrist and Aisbett, Biometrics, 1991)

Our dataset consists of the following,

. . list patient time infect age female in 1/6, noobs

  +----------------------------------------+
  | patient   time   infect   age   female |
  |----------------------------------------|
  |       1     16        1    28        0 |
  |       1      8        1    28        0 |
  |       2     13        0    48        1 |
  |       2     23        1    48        1 |
  |       3     22        1    32        0 |
  |----------------------------------------|
  |       3     28        1    32        0 |
  +----------------------------------------+

with patient our individual patient identifier, time is our time of infection at the catheter insertion point, infect is our event indicator with an event being an infection, age is patient age at baseline and female a binary indicator variable. We immediately see that patients can experience multiple infections, and so we have events nested within patients. For now, I will ignore this clustering.

To fit our model with fractional polynomials for our baseline log hazard function, we need to write a little Mata function which calculates and returns our hazard function. This is really easy to do:

. mata:
------------------------------------------------- mata (type end to exit) ---------------
: real matrix userhaz(transmorphic gml, real colvector t)
> {
>         real matrix linpred
>         real colvector gammas
>         
>         linpred = merlin_util_xzb(gml)
>         gammas = merlin_util_ap(gml,1)\merlin_util_ap(gml,2)
>         return(exp(linpred :+ merlin_fp(t,(0,1)) * gammas))
> }

: end
-----------------------------------------------------------------------------------------

Let’s go through it line by line. First thing to note is that we declare a chunk of Mata code within

mata:

end

We need to define a function, called whatever we like, in this case I’ll call it userhaz(), which returns a real matrix, and it’s going to have two inputs. The first is a transmorphic object called gml. This is the internal struct which contains all the information needed by merlin in the background. You shouldn’t attempt to alter its contents. The second argument is a real colvector which I’m calling t. This represents the vector of time points that we wish to calculate our hazard function at. Internally, our function will be called by merlin at both our core time variable, and our quadrature points needed to calculate the cumulative hazard, and therefore this second input is needed.

Next we declare smoe intermediate vectors/matrices that we’ll need. Explicit declaration of each object’s type is good programming practice.

	real matrix linpred
	real colvector gammas

Now we call our first utility function merlin_util_xzb(), passing it the gml structure and also our time vector. This returns our main complex linear predictor, it’s as simple as that.

	linpred = merlin_util_xzb(gml,t)

We then have two other ancillary parameters to handle, i.e. the coefficients of the fractional polynomial terms, which we extract using merlin_util_ap(gml,i) where i is the ancillary parameter number. In this case we have two extra parameters to estimate, so we build a column vector called gammas as follows,

	gammas = merlin_util_ap(gml,1)\merlin_util_ap(gml,2)

Finally we need to return our hazard function, which is done very simply,

	return(exp(linpred :+ merlin_fp(t,(1,0)) * gammas))

This makes use of the internal merlin_fp() function, which returns fractional polynomials, in this case an FP2 function with powers 1 and 0. In just a few lines of code we have defined our model framework, which can now be used with anything specified in the linear predictor when we fit our merlin models. This provides a very powerful modelling framework.

Let’s now fit a model using our userhaz() function. We can call merlin as follows,

. merlin (time age female, family(user, hfunc(userhaz) failure(infect) nap(2)))

Fitting full model:

Iteration 0:   log likelihood = -8198.9889  
Iteration 1:   log likelihood = -339.39401  
Iteration 2:   log likelihood = -334.76772  
Iteration 3:   log likelihood = -334.39315  
Iteration 4:   log likelihood = -334.39242  
Iteration 5:   log likelihood = -334.39242  

Fixed effects regression model                  Number of obs     =         76
Log likelihood = -334.39242
------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
time:        |            
         age |   .0035436   .0092129     0.38   0.701    -.0145134    .0216006
      female |  -.8639105   .2920079    -2.96   0.003    -1.436236   -.2915854
       _cons |  -4.081803   .6379757    -6.40   0.000    -5.332212   -2.831393
        ap:1 |  -.0310416   .1456656    -0.21   0.831    -.3165409    .2544577
        ap:2 |  -.0010321   .0017629    -0.59   0.558    -.0044872    .0024231
------------------------------------------------------------------------------

I’m telling merlin that I want to fit a model with a user defined family, and in particular I provide the name of the Mata function through hazfunction(). The survival time variable and event indicator are declared as normal. I also tell it that there are 2 ancillary parameters to estimate through nap(2). In my linear predictor I’ve adjusted for age and female.

Given that we inevitably have correlation between events suffered by the same patient, we can now add in a random intercept at the patient level to account for this,

. merlin (time age female M1[patient]@1,   ///
>            family(user, hfunc(userhaz) failure(infect) nap(2)))

Fitting fixed effects model:

Fitting full model:

Iteration 0:   log likelihood = -336.03458  
Iteration 1:   log likelihood =  -330.8822  
Iteration 2:   log likelihood = -330.03033  
Iteration 3:   log likelihood = -329.82043  
Iteration 4:   log likelihood = -329.81922  
Iteration 5:   log likelihood = -329.81923  

Mixed effects regression model                  Number of obs     =         76
Log likelihood = -329.81923
------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
time:        |            
         age |   .0077964   .0139772     0.56   0.577    -.0195984    .0351912
      female |  -1.655001      .5304    -3.12   0.002    -2.694566   -.6154359
 M1[patient] |          1          .        .       .            .           .
       _cons |  -4.649168   .9124865    -5.10   0.000    -6.437609   -2.860728
        ap:1 |    .214339   .2014649     1.06   0.287    -.1805251     .609203
        ap:2 |    .000749     .00213     0.35   0.725    -.0034258    .0049238
-------------+----------------------------------------------------------------
patient:     |            
      sd(M1) |     .93747     .28642                      .5151035    1.706162
------------------------------------------------------------------------------

Which gives us a standard deviation for the random intercept of $\sigma = $0.94, indicating substantial heterogeneity between patients.

We can investigate non-proportional hazards, for example in the effect of age as follows, remembering to add the timevar() option,

. merlin (time age age#fp(time, powers(0)) female M1[patient]@1, ///
>            family(user, hfunc(userhaz) failure(infect) nap(2)) timevar(time))
variables created for model 1, component 2: _cmp_1_2_1 to _cmp_1_2_1

Fitting fixed effects model:

Fitting full model:

Iteration 0:   log likelihood = -274.64295  
Iteration 1:   log likelihood = -270.67862  (not concave)
Iteration 2:   log likelihood = -270.42114  
Iteration 3:   log likelihood = -269.13615  
Iteration 4:   log likelihood = -269.09274  
Iteration 5:   log likelihood =  -269.0763  
Iteration 6:   log likelihood = -269.07693  
Iteration 7:   log likelihood = -269.07687  
Iteration 8:   log likelihood = -269.07687  

Mixed effects regression model                  Number of obs     =         76
Log likelihood = -269.07687
------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
time:        |            
         age |   .3845645   .0681329     5.64   0.000     .2510265    .5181025
    age#fp() |  -.0850253   .0148084    -5.74   0.000    -.1140493   -.0560013
      female |  -1.805519   .6942923    -2.60   0.009    -3.166306   -.4447307
 M1[patient] |          1          .        .       .            .           .
       _cons |  -19.06474   3.331987    -5.72   0.000    -25.59532   -12.53417
        ap:1 |   3.816655   .8012333     4.76   0.000     2.246266    5.387043
        ap:2 |  -.0036826   .0028632    -1.29   0.198    -.0092944    .0019292
-------------+----------------------------------------------------------------
patient:     |            
      sd(M1) |   1.289988   .4326368                      .6685124    2.489213
------------------------------------------------------------------------------

I’ve formed an interaction between age and $\log(t)$ by using the # notation, using the fp() element. In this case we find evidence of a time-dependent effect of age.