Weibull

Built using Zelig version 5.1.4.90000

Weibull Regression for Duration Dependent Variables with weibull.

Choose the Weibull regression model if the values in your dependent variable are duration observations. The Weibull model relaxes the exponential model’s assumption of constant hazard, and allows the hazard rate to increase or decrease monotonically with respect to elapsed time.

Syntax

z.out <- zelig(Surv(Y, C) ~ X, model = "weibull", weights = w,
               data = mydata)
x.out <- setx(z.out)
s.out <- sim(z.out, x = x.out)

Weibull models require that the dependent variable be in the form Surv(Y, C), where Y and C are vectors of length $n$. For each observation $i$ in 1, …, $n$, the value $y_i$ is the duration (lifetime, for example), and the associated $c_i$ is a binary variable such that $c_i = 1$ if the duration is not censored (e.g., the subject dies during the study) or $c_i = 0$ if the duration is censored (e.g., the subject is still alive at the end of the study). If $c_i$ is omitted, all Y are assumed to be completed; that is, time defaults to 1 for all observations.

Input Values

In addition to the standard inputs, zelig() takes the following additional options for weibull regression:

robust: defaults to FALSE. If TRUE, zelig() computes robust standard errors based on sandwich estimators (see and ) based on the options in cluster.
cluster: if robust = TRUE, you may select a variable to define groups of correlated observations. Let x3 be a variable that consists of either discrete numeric values, character strings, or factors that define strata. Then

z.out <- zelig(y ~ x1 + x2, robust = TRUE, cluster = "x3",
               model = "exp", data = mydata)

means that the observations can be correlated within the strata defined by the variable x3, and that robust standard errors should be calculated according to those clusters. If $robust = TRUE$ but cluster is not specified, zelig() assumes that each observation falls into its own cluster.

Example

Attach the sample data:

data(coalition)

Estimate the model:

z.out <- zelig(Surv(duration, ciep12) ~ fract + numst2,
               model = "weibull", data = coalition)

## How to cite this model in Zelig:
##   Terry M Therneau, and Thomas Lumley. 2007.
##   weibull: Weibull Regression for Duration Dependent Variables
##   in Christine Choirat, Christopher Gandrud, James Honaker, Kosuke Imai, Gary King, and Olivia Lau,
##   "Zelig: Everyone's Statistical Software," http://zeligproject.org/

View the regression output:

summary(z.out)

## Model: 
## 
## Call:
## z5$zelig(formula = Surv(duration, ciep12) ~ fract + numst2, data = coalition)
##                Value Std. Error      z        p
## (Intercept)  5.49850   0.525646 10.460 1.31e-25
## fract       -0.00384   0.000714 -5.374 7.69e-08
## numst2       0.45232   0.118924  3.803 1.43e-04
## Log(scale)  -0.04435   0.049987 -0.887 3.75e-01
## 
## Scale= 0.957 
## 
## Weibull distribution
## Loglik(model)= -1077   Loglik(intercept only)= -1100.6
##  Chisq= 47.18 on 2 degrees of freedom, p= 5.7e-11 
## Number of Newton-Raphson Iterations: 5 
## n= 314 
## 
## Next step: Use 'setx' method

Set the baseline values (with the ruling coalition in the minority) and the alternative values (with the ruling coalition in the majority) for X:

x.low <- setx(z.out, numst2 = 0)
x.high <- setx(z.out, numst2 = 1)

Simulate expected values and first differences:

s.out <- sim(z.out, x = x.low, x1 = x.high)

summary(s.out)

## 
##  sim x :
##  -----
## ev
##       mean       sd      50%     2.5%    97.5%
## 1 15.27579 1.401267 15.23677 12.75938 18.12979
## pv
##          mean       sd      50%      2.5%    97.5%
## [1,] 15.10834 14.40472 11.24056 0.5767899 54.05405
## 
##  sim x1 :
##  -----
## ev
##       mean       sd      50%     2.5%    97.5%
## 1 23.95177 1.805729 23.93098 20.51706 27.74208
## pv
##          mean       sd      50%     2.5%   97.5%
## [1,] 23.79848 21.39514 17.55209 1.032722 77.9025
## fd
##       mean       sd     50%     2.5%    97.5%
## 1 8.675973 2.265219 8.63897 4.462617 13.20894

plot(s.out)

Graphs of Quantities of Interest for Zelig-weibull

Model

Let $Y_i^*$ be the survival time for observation $i$. This variable might be censored for some observations at a fixed time $y_c$ such that the fully observed dependent variable, $Y_i$, is defined as

\[ Y_i = \left\{ \begin{array}{ll} Y_i^* & \textrm{if }Y_i^* \leq y_c \\ y_c & \textrm{if }Y_i^* > y_c \end{array} \right. \]

The stochastic component is described by the distribution of the partially observed variable $Y^*$. We assume $Y_i^*$ follows the Weibull distribution whose density function is given by

\[ f(y_i^*\mid \lambda_i, \alpha) = \frac{\alpha}{\lambda_i^\alpha} y_i^{* \alpha-1} \exp \left\{ -\left( \frac{y_i^*}{\lambda_i} \right)^{\alpha} \right\} \]

for $y_i^* \ge 0$, the scale parameter $\lambda_i > 0$, and the shape parameter $\alpha > 0$. The mean of this distribution is $\lambda_i \Gamma(1 + 1 / \alpha)$. When $\alpha = 1$, the distribution reduces to the exponential distribution (see Section [exp]). (Note that the output from zelig() parameterizes scale $ = 1 / $.)

In addition, survival models like the Weibull have three additional properties. The hazard function $h(t)$ measures the probability of not surviving past time $t$ given survival up to $t$. In general, the hazard function is equal to $f(t)/S(t)$ where the survival function $S(t) = 1 - \int_{0}^t f(s) ds$ represents the fraction still surviving at time $t$. The cumulative hazard function $H(t)$ describes the probability of dying before time $t$. In general, $H(t)= \int_{0}^{t} h(s) ds = -\log S(t)$. In the case of the Weibull model,

\[ \begin{aligned} h(t) &=& \frac{\alpha}{\lambda_i^{\alpha}} t^{\alpha - 1} \\ S(t) &=& \exp \left\{ -\left( \frac{t}{\lambda_i} \right)^{\alpha} \right\} \\ H(t) &=& \left( \frac{t}{\lambda_i} \right)^{\alpha} \end{aligned} \] For the Weibull model, the hazard function $h(t)$ can increase or decrease monotonically over time.

The systematic component $\lambda_i$ is modeled as

\[ \lambda_i = \exp(x_i \beta), \]

where $x_i$ is the vector of explanatory variables, and $\beta$ is the vector of coefficients.

Quantities of Interest

The expected values (qi$ev) for the Weibull model are simulations of the expected duration:

\[ E(Y) = \lambda_i \, \Gamma (1 + \alpha^{-1}), \]

given draws of $\beta$ and $\alpha$ from their sampling distributions.

The predicted value (qi$pr) is drawn from a distribution defined by $(\lambda_i, \alpha)$.
The first difference (qi$fd) in expected value is

\[ \textrm{FD} = E(Y \mid x_1) - E(Y \mid x). \]

In conditional prediction models, the average expected treatment effect (att.ev) for the treatment group is

\[ \frac{1}{\sum_{i=1}^n t_i}\sum_{i:t_i=1}^n \left\{ Y_i(t_i=1) - E[Y_i(t_i=0)] \right\}, \]

where $t_i$ is a binary explanatory variable defining the treatment ($t_i=1$) and control ($t_i=0$) groups. When $Y_i(t_i=1)$ is censored rather than observed, we replace it with a simulation from the model given available knowledge of the censoring process. Variation in the simulations are due to uncertainty in simulating $E[Y_i(t_i=0)]$, the counterfactual expected value of $Y_i$ for observations in the treatment group, under the assumption that everything stays the same except that the treatment indicator is switched to $t_i=0$.

In conditional prediction models, the average predicted treatment effect (att.pr) for the treatment group is

\[ \frac{1}{\sum_{i=1}^n t_i}\sum_{i:t_i=1}^n \left\{ Y_i(t_i=1) - \widehat{Y_i(t_i=0)} \right\}, \]

where $t_i$ is a binary explanatory variable defining the treatment ($t_i=1$) and control ($t_i=0$) groups. When $Y_i(t_i=1)$ is censored rather than observed, we replace it with a simulation from the model given available knowledge of the censoring process. Variation in the simulations are due to uncertainty in simulating $\widehat{Y_i(t_i=0)}$, the counterfactual predicted value of $Y_i$ for observations in the treatment group, under the assumption that everything stays the same except that the treatment indicator is switched to $t_i=0$.

Output Values

The Zelig object stores fields containing everything needed to rerun the Zelig output, and all the results and simulations as they are generated. In addition to the summary commands demonstrated above, some simply utility functions (known as getters) provide easy access to the raw fields most commonly of use for further investigation.

In the example above z.out$get_coef() returns the estimated coefficients, z.out$get_vcov() returns the estimated covariance matrix, and z.out$get_predict() provides predicted values for all observations in the dataset from the analysis.

2017-10-29