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)
Log-Normal Regression for Duration Dependent Variables with lognorm
The log-normal model describes an event’s duration, the dependent
variable, as a function of a set of explanatory variables. The
log-normal model may take time censored dependent variables, and allows
the hazard rate to increase and decrease.
Syntax
z.out <- zelig(Surv(Y, C) ~ X, model = "lognorm", weights = w, data = mydata)
x.out <- setx(z.out)
s.out <- sim(z.out, x = x.out)
Log-normal 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) of each subject, 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 lognormal 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
rm(list=ls(pattern="\\.out"))
suppressWarnings(suppressMessages(library(Zelig)))
set.seed(1234)
Attach the sample data:
data(coalition)
Estimate the model:
z.out <- zelig(Surv(duration, ciep12) ~ fract + numst2, model ="lognorm", 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 (qi$ev) and first differences (qi$fd):
s.out <- sim(z.out, x = x.low, x1 = x.high)
summary(s.out)
plot(s.out)
Model
Let $Y_i^*$ be the survival time for observation $i$ with
the density function $f(y)$ and the corresponding distribution
function $F(t)=\int_{0}^t f(y) dy$. 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, $Y^$. For the lognormal model,
there are two equivalent representations:
$$
\begin{aligned}
Y_i^ \; \sim \; \textrm{LogNormal}(\mu_i, \sigma^2) & \textrm{ or
} & \log(Y_i^) \; \sim \; \textrm{Normal}(\mu_i, \sigma^2)\end{aligned}
$$
where the parameters $\mu_i$ and $\sigma^2$ are the mean
and variance of the Normal distribution. (Note that the output from
zelig() parameterizes scale\ $ = \sigma$.)
In addition, survival models like the lognormal 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 lognormal
model,
$$
\begin{aligned}
h(t) &=& \frac{1}{\sqrt{2 \pi} \, \sigma t \, S(t)}
\exp\left{-\frac{1}{2 \sigma^2} (\log \lambda t)^2\right} \
S(t) &=& 1 - \Phi\left(\frac{1}{\sigma} \log \lambda t\right) \
H(t) &=& -\log \left{ 1 - \Phi\left(\frac{1}{\sigma} \log \lambda t\right) \right}\end{aligned}
$$
where $\Phi(\cdot)$ is the cumulative density function for the
Normal distribution.
- The systematic component is described as:
$$
\mu_i = x_i \beta .
$$
Quantities of Interest
- The expected values (qi$ev) for the lognormal model are simulations
of the expected duration:
$$
E(Y) = \exp\left(\mu_i + \frac{1}{2}\sigma^2 \right),
$$
given draws of $\beta$ and $\sigma$ from their sampling
distributions.
-
The predicted value is a draw from the log-normal distribution given
simulations of the parameters $(\lambda_i, \sigma)$.
-
The first difference (qi$fd) 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
$$
\frac{1}{\sum_{i=1}^n t_i}\sum_{i:t_i=1}^n { Y_i(t_i=1) - E[Y_i(t_i=0)] },
$$
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
$$
\frac{1}{\sum_{i=1}^n t_i} \sum_{i:t_i=1}^n { Y_i(t_i=1) -
\widehat{Y_i(t_i=0)} },
$$
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 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$get_predict()
provides predicted values for all observations in the dataset from the analysis.
See also
The exponential function is part of the survival package by Terry
Therneau, ported to R by Thomas Lumley. Advanced users may wish to refer
to help(survfit) in the survival package
z5 <- zlognorm$new()
z5$references()
IQSS/Zelig documentation built on Dec. 11, 2023, 1:51 a.m.
R Package Documentation
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Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Built using Zelig version r packageVersion("Zelig")
knitr::opts_knit$set( stop_on_error = 2L ) knitr::opts_chunk$set( fig.height = 11, fig.width = 7 ) options(cite = FALSE)
Log-Normal Regression for Duration Dependent Variables with lognorm
The log-normal model describes an event’s duration, the dependent variable, as a function of a set of explanatory variables. The log-normal model may take time censored dependent variables, and allows the hazard rate to increase and decrease.
Syntax
z.out <- zelig(Surv(Y, C) ~ X, model = "lognorm", weights = w, data = mydata) x.out <- setx(z.out) s.out <- sim(z.out, x = x.out)
Log-normal 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) of each subject, 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 lognormal 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
rm(list=ls(pattern="\\.out")) suppressWarnings(suppressMessages(library(Zelig))) set.seed(1234)
Attach the sample data:
data(coalition)
Estimate the model:
z.out <- zelig(Surv(duration, ciep12) ~ fract + numst2, model ="lognorm", 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 (qi$ev) and first differences (qi$fd):
s.out <- sim(z.out, x = x.low, x1 = x.high)
summary(s.out)
plot(s.out)
Model
Let $Y_i^*$ be the survival time for observation $i$ with the density function $f(y)$ and the corresponding distribution function $F(t)=\int_{0}^t f(y) dy$. 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, $Y^$. For the lognormal model, there are two equivalent representations:
$$ \begin{aligned} Y_i^ \; \sim \; \textrm{LogNormal}(\mu_i, \sigma^2) & \textrm{ or } & \log(Y_i^) \; \sim \; \textrm{Normal}(\mu_i, \sigma^2)\end{aligned} $$ where the parameters $\mu_i$ and $\sigma^2$ are the mean and variance of the Normal distribution. (Note that the output from zelig() parameterizes scale\ $ = \sigma$.)
In addition, survival models like the lognormal 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 lognormal model,
$$ \begin{aligned} h(t) &=& \frac{1}{\sqrt{2 \pi} \, \sigma t \, S(t)} \exp\left{-\frac{1}{2 \sigma^2} (\log \lambda t)^2\right} \ S(t) &=& 1 - \Phi\left(\frac{1}{\sigma} \log \lambda t\right) \ H(t) &=& -\log \left{ 1 - \Phi\left(\frac{1}{\sigma} \log \lambda t\right) \right}\end{aligned} $$ where $\Phi(\cdot)$ is the cumulative density function for the Normal distribution.
- The systematic component is described as:
$$ \mu_i = x_i \beta . $$
Quantities of Interest
- The expected values (qi$ev) for the lognormal model are simulations of the expected duration:
$$ E(Y) = \exp\left(\mu_i + \frac{1}{2}\sigma^2 \right), $$
given draws of $\beta$ and $\sigma$ from their sampling distributions.
-
The predicted value is a draw from the log-normal distribution given simulations of the parameters $(\lambda_i, \sigma)$.
-
The first difference (qi$fd) 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
$$ \frac{1}{\sum_{i=1}^n t_i}\sum_{i:t_i=1}^n { Y_i(t_i=1) - E[Y_i(t_i=0)] }, $$
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
$$ \frac{1}{\sum_{i=1}^n t_i} \sum_{i:t_i=1}^n { Y_i(t_i=1) - \widehat{Y_i(t_i=0)} }, $$
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 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$get_predict()
provides predicted values for all observations in the dataset from the analysis.
See also
The exponential function is part of the survival package by Terry
Therneau, ported to R by Thomas Lumley. Advanced users may wish to refer
to help(survfit) in the survival package
z5 <- zlognorm$new() z5$references()
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