Parametric survival models

• Our example from last time compared the drop-out rate between black and white high school students over a two-year window.
• This is an example of event history analysis. We are essentially asking how the drop-out rate is affected by a covariate (in this case race).
• We could express our relationship as follows:
  $$\lambda_{i}(t)=\beta_{0}+\beta_{1}x_{i}$$
  $x_{i}$ is a dummy indicating that the student is black. This model says that the individual hazard rate is a function of race. White students will have a drop-out rate of $\beta_{0}$ and black students will have a drop-out rate of $\beta_{0}+\beta_{1}$. Since, we are assuming that the harzard rate is constant across time:
  $$\lambda_{i}=\beta_{0}+\beta_{1}x_{i}$$

• The hazard rate will always be positive. Therefore, we will generally prefer a model that is linear in the log of the hazard rate:
  $$\log(\lambda_{i})=\beta_{0}+\beta_{1}x_{i}$$
  In this format, the drop-out rate for white students is $e^{\beta_{0}}$ and the relative risk for black students is $e^{\beta_{1}}$. We calculated these values by hand last session.
  $$\beta_{0}=\log($$white student drop-out rate$$)=\log(0.1563)=-1.86$$
  $$\beta_{1}=\log($$relative risk for black students$$)=\log(2.17)=0.77$$
  So,
  $$\log(\lambda_{i})=-1.86+0.77x_{i}$$

• From this model, we can develop a general model for the hazard rate:
  $$\log(\lambda_{i})=\beta_{0}+\beta_{1}x_{i1}+\beta_{2}x_{i2}+\ldots+\beta_{p}x_{ip}$$
  For categorical variables, the exponential of the coefficient gives the relative risk for the groups. For continuous variables, the exponential of the coefficient gives the relative risk between two individuals who only differ by one unit on the independent variable. $\beta_{0}$ gives the hazard rate for the baseline group.
• How do we estimate such models?
  • The secret lies in breaking $\lambda_{i}$ into it's component pieces:
    $$\lambda_{i}=\frac{\mu_{i}}{t_{i}}$$
    Where $\mu_{i}$ is the expected number of events for observation $i$ and $t_{i}$ is the length of time observation $i$ was observed.
  • Plugging, this back into our equation:
    $$\log(\frac{\mu_{i}}{t_{i}})=\beta_{0}+\beta_{1}x_{i1}+\beta_{2}x_{i2}+\ldots+\beta_{p}x_{ip}$$

  • By algebraic rules:
    $$\log(\mu_{i})-\log(t_{i})=\beta_{0}+\beta_{1}x_{i1}+\beta_{2}x_{i2}+\ldots+\beta_{p}x_{ip}$$
    $$\log(\mu_{i})=\log(t_{i})+\beta_{0}+\beta_{1}x_{i1}+\beta_{2}x_{i2}+\ldots+\beta_{p}x_{ip}$$

  • We are trying to estimate the log of the mean number of events for the $i$th individual as a linear function of the independent variables. The only wrinkle is the $\log(t_{i})$ on the right hand side. This is referred to as an offset variable - i.e. a variable whose coefficient we are assuming equals one.
  • The only remaining trick is how to model $\mu_{i}$. This is an expected number of events (i.e. counts). That's right, this is a straightforward GLM with a poisson error, log link, and an offset of the log of exposure time.
  • So in order to model this we treat the number of events for each individual (in most cases with individual data, it will be one or zero) as the dependent variable and relate the independent variables to it using a log link, a poisson error distribution, and log-exposure as an offset. If we apply this to our school data, we get:

    Variable	Coefficient (SE)
    Intercept	-1.86 (1.00)
    Black	0.77 (1.22)

    This is precisely what we derived by hand earlier.
• Parameteric models
  • Our example treats the hazard rate as if it was constant over the duration time.
    $$\lambda(t)=\lambda$$
    This assumption leads to a an exponential survival curve:
    $$S(t)=e^{-\lambda t}$$

  • This is one form of a parameteric survival model. We refer to it as parametric because we are assuming some underlying shape to the survival curve. The exponential model is the simplest form of parametric survival model.
  • Most parametric survival models assume proportional hazards. That is they assume that there is an underlying hazard rate over time, and differences in the covariates simly lead to differences in the relative hazard rate at a point in time. in other words, they assume no interaction between time and covariates. Mathematically,
    $$\lambda_{i}(t)=h_{0}(t)e^{\beta_{1}x_{i1}+\beta_{2}x_{i2}+\ldots+\beta_{p}x_{ip}}$$
    Where $h_{0}(t)$ is some time function of the hazard rate. Because the hazard rate depends on the length of times spent in the state, it is called duration dependent.
  • Different specifications of this $h_{0}(t)$ lead to different parametric survival models.
    • The Weibull model:
      $$h_{0}(t)=\lambda p(\lambda t)^{p-1}$$
      The Weibull distribution leads to hazard rates which either increase or decrease monotonically over time. If $p$ is greater than one, the hazard rate is increasing, if $p$ is less than one, the hazard rate is decreasing, and if $p=1$, we have a constant hazard rate.
    • The Gompertz model:
      $$h_{0t}=\lambda e^{rt}$$
      The hazard rate increases or decreases at an exponential rate. This is a useful model for mortality at ages past about 30, because mortality rates tend to follow this pattern.
  • These models are all more complex to estimate than the exponential model because, they require modeling the parameters of the hazard function as well as the covariates, but intuitively they are similar. Choosing between them often requires a diagnostic analysis such as the Kaplan-Meier curve. (what do we think about our high school students).

    Let's try fitting these parametric survival models to our data:

    Model	log(RR)	RR
    Exponential	0.77	2.17
    Weibull	0.79	2.21
    Gompertz	0.78	2.18

• Adding time-varying covariates
  • One of the great advantages of longitudinal analysis is that we can observe not only changes over time in the state of interest, but also change over time in the characteristics which might predict the state
  • In the drop out example, what are some covariates that might affect drop-out rates and change over time.
    • Grades last term
    • Disciplinary problems
    • Grade in school
    • Romantic relationships
  • We would like to be able to incorporate these changes into our estimates of the effect of the covariates. There is a straightforward technique for doing this. We have to change the format of our dataset into a split-episode event data set.
    1. Break the overall interval for individual $i$ into sub-intervals broken by each point where one of individual $i$'s covariates changed.
    2. record the value of each observed covariate at the start of each sub-interval.
    3. Treat each sub-interval as an observation in itself. If there are lots of changes, each individual will contribute many observations to the data. This will not artificially inflate the size of your sample.
  • Let's take grades in math as an example for our high school students. Since grades are recorded each semester, we will get 2 observations per school year for each individual.

    Student	Race	Previous Grade	Semester	Exposure time	Dropped out
    1	White	A	Fall	.5	N
    1	White	A	Spring	.5	N
    2	White	B	Fall	.5	N
    2	White	A	Spring	.5	N
    2	White	B	Fall	.5	N
    2	White	C	Spring	.5	N
    3	White	B	Fall	.5	N
    3	White	B	Spring	.5	N
    3	White	A	Fall	.5	N
    3	White	B	Spring	.5	N
    4	White	D	Fall	.5	N
    4	White	C	Spring	.5	N
    4	White	D	Fall	.4	Y
    5	Black	B	Fall	.5	N
    5	Black	A	Spring	.5	N
    5	Black	B	Fall	.5	N
    5	Black	C	Spring	.5	N
    6	Black	D	Fall	.5	N
    6	Black	F	Spring	.2	Y
    7	Black	B	Fall	.5	N
    7	Black	B	Spring	.5	N
    7	Black	C	Fall	.5	N
    7	Black	B	Spring	.5	N
    8	Black	C	Fall	.5	N
    8	Black	C	Spring	.5	N
    8	Black	D	Fall	.2	Y

  • Now I can run a model which includes both a grade variable and a race variable. I could also include a semester variable to see whether drop-out rates are higher in the fall or spring. Let's include a quantitative grade variable which goes from 0 (F) to 4 (A).

    Variable	Coefficient (SE)
    Intercept	4.49 (2.29)
    Black	-0.32 (1.41)
    Letter Grade	-2.26 (0.95)

    Once grades are controlled for, the higher dropout rate of black students goes away (of course this data is made up).
  • In many cases, covariates will be recorded at regular intervals such as this, meaning that splitting the data will be easier.