Built using Zelig version r packageVersion("Zelig")
knitr::opts_knit$set( stop_on_error = 2L ) knitr::opts_chunk$set( fig.height = 11, fig.width = 7 ) options(cite = FALSE)
Exponential Regression for Duration Dependent Variables with exp
.
Use the exponential duration regression model if you have a dependent variable representing a duration (time until an event). The model assumes a constant hazard rate for all events. The dependent variable may be censored (for observations have not yet been completed when data were collected).
z.out <- zelig(Surv(Y, C) ~ X, model = "exp", weights = w, data = mydata) x.out <- setx(z.out) s.out <- sim(z.out, x = x.out)
Exponential 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 and is know to live at least as long as $y_i$).
If $c_i$ is omitted, all Y are assumed to be completed; that is,
time defaults to 1 for all observations.
In addition to the standard inputs, zelig()
takes the following
additional options for exponential regression:
robust: defaults to FALSE. If TRUE, zelig()
computes robust standard
errors based on sandwich estimators and the options
selected 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.
rm(list=ls(pattern="\\.out")) suppressWarnings(suppressMessages(library(Zelig))) set.seed(1234)
Attach the sample data:
data(coalition) library(survival)
Estimate the model:
z.out <- zelig(Surv(duration, ciep12) ~ fract + numst2, model = "exp", data = coalition)
View the regression output:
summary(z.out)
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)
Summarize quantities of interest and produce some plots:
summary(s.out)
plot(s.out)
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
$$ Y_i = \left{ \begin{array}{ll} Y_i^ & \textrm{if }Y_i^ \leq y_c \ y_c & \textrm{if }Y_i^* > y_c \ \end{array} \right. $$
$$ f(y_i^\mid \lambda_i) = \frac{1}{\lambda_i} \exp\left(-\frac{y_i^}{\lambda_i}\right) $$
for $y_i^*\ge 0$ and $\lambda_i>0$. The mean of this distribution is $\lambda_i$.
In addition, survival models like the exponential 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 exponential model,
$$ \begin{aligned} h(t) &=& \frac{1}{\lambda_i} \ S(t) &=& \exp\left( -\frac{t}{\lambda_i} \right) \ H(t) &=& \frac{t}{\lambda_i}\end{aligned} $$
For the exponential model, the hazard function $h(t)$ is constant over time. The Weibull model and lognormal models allow the hazard function to vary as a function of elapsed time (see and respectively).
$$ \lambda_i = \exp(x_i \beta), $$
where $x_i$ is the vector of explanatory variables, and $\beta$ is the vector of coefficients.
$$ E(Y) = \lambda_i^{-1} = 1/\exp(x_i \beta). $$
The predicted values (qi$pr) are draws from the exponential distribution with rate equal to the expected value.
The first difference (or difference in expected values, qi$ev.diff
),
is
$$ \textrm{FD} \; = \; E(Y \mid x_1) - E(Y \mid x), $$
where $x$ and $x_1$ are different vectors of values for the explanatory variables.
$$ \frac{1}{\sum_{i=1}^n t_i}\sum_{i:t_i=1}^n \left{ Y_i(t_i=1) - E[Y_i(t_i=0)] \right}, $$
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 is due to two factors: uncertainty in the imputation process for censored $y_i^*$ and 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 $t_i=0$.
$$ \frac{1}{\sum_{i=1}^n t_i}\sum_{i:t_i=1}^n \left{ Y_i(t_i=1) - \widehat{Y_i(t_i=0)} \right}, $$
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 is due to two factors: uncertainty in the imputation process for censored $y_i^*$ and 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 $t_i=0$.
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$get_predict()
provides predicted values for all observations in the dataset from the analysis.
The exponential function is part of the survival package by Terry
Therneau, ported to R by Thomas Lumley. Advanced users may wish to refer
to help(survfit)
in the survival package
z5 <- zexp$new() z5$references()
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