Timetoevent data, including both survival and censoring times, are created using functions defSurv
and genSurv
. The survival data definitions require a variable name as well as a specification of a scale value, which determines the mean survival time at a baseline level of covariates (i.e. all covariates set to 0). The Weibull distribution is used to generate these survival times. In addition, covariates (which have been defined previously) that influence survival time can be included in the formula
field. Positive coefficients are associated with longer survival times (and lower hazard rates). Finally, the shape of the distribution can be specified. A shape
value of 1 reflects the exponential distribution.
The density, mean, and variance of the Weibull distribution that is used in the data generation process are defined by the parameters \(\lambda\) (scale) and \(\nu\) (shape) as shown below.
\[\begin{aligned} f(t) &= \frac{t^{\frac{1}{\nu}1}}{\lambda \nu} exp\left(\frac{t^\frac{1}{\nu}}{\lambda}\right) \\ E(T) &= \lambda ^ \nu \Gamma(\nu + 1) \\ Var(T) &= (\lambda^2)^\nu \left( \Gamma(2 \nu + 1)  \Gamma^2(\nu + 1) \right) \\ \end{aligned}\]The survival time \(T\) data are generated based on this formula:
\[ T = \left( \frac{log(U) \lambda}{exp(\beta ^ \prime x)} \right)^\nu, \]
where \(U\) is a uniform random variable between 0 and 1, \(\beta\) is a vector of parameters in a Cox proportional hazard model, and \(x\) is a vector of covariates that impact survival time. \(\lambda\) and \(\nu\) can also vary by covariates.
Here is an example showing how to generate data with covariates. In this case the scale and shape parameters will vary by group membership.
# Baseline data definitions
< defData(varname = "x1", formula = 0.5, dist = "binary")
def < defData(def, varname = "x2", formula = 0.5, dist = "binary")
def < defData(def, varname = "grp", formula = 0.5, dist = "binary")
def
# Survival data definitions
set.seed(282716)
< defSurv(varname = "survTime", formula = "1.5*x1", scale = "grp*50 + (1grp)*25",
sdef shape = "grp*1 + (1grp)*1.5")
< defSurv(sdef, varname = "censorTime", scale = 80, shape = 1)
sdef
sdef
## varname formula scale shape
## 1: survTime 1.5*x1 grp*50 + (1grp)*25 grp*1 + (1grp)*1.5
## 2: censorTime 0 80 1
The data are generated with calls to genData
and genSurv
:
# Baseline data definitions
< genData(300, def)
dtSurv < genSurv(dtSurv, sdef)
dtSurv
head(dtSurv)
## id x1 x2 grp survTime censorTime
## 1: 1 0 0 1 9.21 96.0
## 2: 2 0 1 0 25.52 46.8
## 3: 3 0 1 0 604.20 31.6
## 4: 4 1 1 0 23.63 338.4
## 5: 5 1 0 0 108.28 287.6
## 6: 6 0 1 1 8.12 53.4
# A comparison of survival by group and x1
round(mean(survTime), 1), keyby = .(grp, x1)] dtSurv[,
## grp x1 V1
## 1: 0 0 156.2
## 2: 0 1 19.0
## 3: 1 0 43.3
## 4: 1 1 14.1
Observed survival times and censoring indicators can be generated by defining new fields:
< defDataAdd(varname = "obsTime", formula = "pmin(survTime, censorTime)", dist = "nonrandom")
cdef < defDataAdd(cdef, varname = "status", formula = "I(survTime <= censorTime)",
cdef dist = "nonrandom")
< addColumns(cdef, dtSurv)
dtSurv
head(dtSurv)
## id x1 x2 grp survTime censorTime obsTime status
## 1: 1 0 0 1 9.21 96.0 9.21 TRUE
## 2: 2 0 1 0 25.52 46.8 25.52 TRUE
## 3: 3 0 1 0 604.20 31.6 31.62 FALSE
## 4: 4 1 1 0 23.63 338.4 23.63 TRUE
## 5: 5 1 0 0 108.28 287.6 108.28 TRUE
## 6: 6 0 1 1 8.12 53.4 8.12 TRUE
# estimate proportion of censoring by x1 and group
round(1  mean(status), 2), keyby = .(grp, x1)] dtSurv[,
## grp x1 V1
## 1: 0 0 0.51
## 2: 0 1 0.13
## 3: 1 0 0.37
## 4: 1 1 0.17
Here is a KaplanMeier plot of the data by the four groups:
Here is a survival analysis (using a Cox proportional hazard model) of a slightly simplified data set:
# Baseline data definitions
< defData(varname = "x1", formula = 0.5, dist = "binary")
def < defData(def, varname = "x2", formula = 0.5, dist = "binary")
def
# Survival data definitions
< defSurv(varname = "survTime", formula = "1.5*x1  .8*x2", scale = 50, shape = 1/2)
sdef < defSurv(sdef, varname = "censorTime", scale = 80, shape = 1)
sdef
< genData(300, def)
dtSurv < genSurv(dtSurv, sdef)
dtSurv
< defDataAdd(varname = "obsTime", formula = "pmin(survTime, censorTime)", dist = "nonrandom")
cdef < defDataAdd(cdef, varname = "status", formula = "I(survTime <= censorTime)",
cdef dist = "nonrandom")
< addColumns(cdef, dtSurv)
dtSurv < survival::coxph(Surv(obsTime, status) ~ x1 + x2, data = dtSurv) coxfit
The 95% confidence intervals of the parameter estimates include the values used to generate the data:
Characteristic  log(HR)^{1}  95% CI^{1}  pvalue 

x1  1.5  1.2, 1.8  <0.001 
x2  1.0  1.2, 0.70  <0.001 
^{
1
}
HR = Hazard Ratio, CI = Confidence Interval
