Semi-parametric models

• The assumption of a parametric form to the survival curve is a serious one that can often be problematic. Two semi-parametric methods have been developed to reduce the importance of this assumption.
• Piecewise constant exponential model
  • This model assumes that the hazard is constant not over the whole range of time, but within certain specified intervals of time. So, the constant $\lambda$ from the exponential model becomes $\lambda_{j}$ where $j$ is some interval of time.
    $$\log(\lambda_{ij})=\alpha_{j}+\beta_{1}x_{i1}+\beta_{2}x_{i2}+\ldots+\beta_{p}x_{ip}$$
    Each $\alpha_{j}$ gives the constant hazard within time interval $j$ for the baseline group.
  • If the time units are small enough or the true hazard rate is not changing too dramatically, then the piecewise constant model often gives very acceptable results.
  • The piecewise constant model approach allows the hazard rate to change in non-parametric ways.
  • Implementing this model is straightforward. The time intervals are just treated as time-varying covariates. The researcher splits each observation into the $J$ time intervals and estimates an exponential models with each time inverval as a dummy variable.
  • In our example, let's assume that each student begins the study in the 10th grade. We have reason to believe that the drop-out rate may differ in the 10th and 11th grade, so we run a piecewise constant model. Our split episode file looks like this:

    Student	Race	Grade	Exposure time	Dropped out
    1	White	10	1	N
    2	White	10	1	N
    2	White	11	1	N
    3	White	10	1	N
    3	White	11	1	N
    4	White	10	1	N
    4	White	11	.4	Y
    5	Black	10	1	N
    5	Black	11	1	N
    6	Black	10	0.7	Y
    7	Black	10	1	N
    7	Black	11	1	N
    8	Black	11	1	N
    8	Black	11	.2	Y

  • Now we just throw a grade dummy into our poisson GLM as well as a race dummy:

    Coefficient	Constant	Piecewise Constant
    Intercept	-1.86	-2.49
    Grade
    10th (ref)
    11th		1.21
    Race
    White (ref)
    Black	0.77	0.78

    (go through and interpret these results)
• Cox proportional hazards model
  • Probably the most popular model for estimating event history is the Cox proportional hazard model, also called Cox regression.
  • Cox regression is popular because it makes no assumptions about the underlying hazard function. It simply assumes that relative risks are constant across all times (thus "proportional hazards).
    $$\lambda(t)=h_{0}(t)e^{x_{it}^{'}\beta}$$
    Where $h_{0}(t)$ can be any hazard function.
  • The set up for the Cox regression is similar to that for other models. The model requires the value of covariates, the status of the individual at the end of the interval, and the length of the interval. Time-varying covariates can be added in the normal way.
  • Estimation of the Cox regression model is complex and requires a technique called partial likelihood rather than MLE.
  • The basic idea is that Cox regression uses information on the rank-ordering of survival times to look at the likelihood that the $i$th individual will be the next to experience an event.
  • Caution: when there are many tied event times, Cox regression can give biased results.
• Here is a comparison of the relative risk for black students under various models

  Model	log(RR)	RR
  Exponential	0.77	2.17
  Weibull	0.79	2.21
  Gompertz	0.78	2.18
  Piecewise Constant	0.78	2.18
  Cox	0.83	2.30