In IQSS/Zelig: Everyone's Statistical Software
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).
Syntax
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.
Input Values
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.
Example
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)
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 exponential distribution whose density function is given
by
$$
f(y_i^\mid \lambda_i) = \frac{1}{\lambda_i} \exp\left(-\frac{y_i^}{\lambda_i}\right)
$$
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).
- 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 exponential model are simulations
of the expected duration given $x_i$ and draws of $\beta$
from its posterior,
$$
E(Y) = \lambda_i^{-1} = 1/\exp(x_i \beta).
$$
-
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.
- 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 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
$t_i=0$.
- 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 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 $t_i=0$.
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$get_predict() provides predicted values for all observations in the dataset from the analysis.
See also
The exponential function is part of the survival package by Terry
Therneau, ported to R by Thomas Lumley. Advanced users may wish to refer
to help(survfit) in the survival package
z5 <- zexp$new()
z5$references()
IQSS/Zelig documentation built on Dec. 11, 2023, 1:51 a.m.
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Built using Zelig version r packageVersion("Zelig")
knitr::opts_knit$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).
Syntax
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.
Input Values
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.
Example
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)
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 exponential distribution whose density function is given by
$$ f(y_i^\mid \lambda_i) = \frac{1}{\lambda_i} \exp\left(-\frac{y_i^}{\lambda_i}\right) $$
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).
- 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 exponential model are simulations of the expected duration given $x_i$ and draws of $\beta$ from its posterior,
$$ E(Y) = \lambda_i^{-1} = 1/\exp(x_i \beta). $$
-
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.
- 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 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 $t_i=0$.
- 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 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 $t_i=0$.
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$get_predict() provides predicted values for all observations in the dataset from the analysis.
See also
The exponential function is part of the survival package by Terry
Therneau, ported to R by Thomas Lumley. Advanced users may wish to refer
to help(survfit) in the survival package
z5 <- zexp$new() z5$references()
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