Package 'EMGCR'

Liver Cancer Data

Description

A sample of 1,736 patients who have been diagnosed with liver cancer between 2012 and 2016, whose cancer grades are well identified. Available individual-level covariates include age at diagnosis, sex, race, grade of liver cancer, median household income, radiation indicator, and chemotherapy indicator. The grade of disease is categorized into four levels: well-differentiated (Grade I), moderately differentiated (Grade II), poorly differentiated (Grade III), and undifferentiated/anaplastic (Grade IV).

Usage

data(liver)

Format

A data frame with 1,736 observations and 10 variables:

ID

Unique patient identifier

time

Survival time in months

status

Event indicator: death due to liver cancer = 1, censored = 0

sex

Sex of patient: male = 1, female = 0

age

Scaled age of diagnosis

medh

Scaled median household income of the subject

race

Race: white = 1, other = 0

grade

Pathological grade of liver cancer (1 = I, 2 = II, 3 = III, 4 = IV)

chemo

Chemotherapy: yes = 1, no = 0

radio

Radiation: yes = 1, no = 0

Examples

data(liver)
head(liver)

---

Fit a Mixture Cure Rate (MCR) Survival Model

Description

Fits a mixture cure rate model by the Expectation-Maximization (EM) algorithm, with a flexible link function for the incidence (cure) component and a choice of survival distributions for the latency component.

Usage

MCRfit(
  formula,
  data,
  dist = "weibull",
  link = "logit",
  tau = 1,
  maxit = 1000,
  tol = 1e-05
)

Arguments

formula	A two-part formula of the form Surv(time, status) ~ x | w, where x are covariates for the survival part, and w are covariates for the incidence part (the probability of being uncured).
data	A data frame containing the variables in the model.
dist	A character string indicating the baseline distribution. Supported values are "weibull", "exponential", "rayleigh", "lognormal", "loglogistic", and "invgauss".
link	A character string specifying the link function for the probability of being uncured. Options are "logit", "probit", "plogit", "rplogit", and "cauchit".
tau	A numeric value used when link = "plogit" or "rplogit". Defaults to 1.
maxit	Maximum number of iterations for the EM-like algorithm. Defaults to 1000.
tol	Convergence tolerance. Defaults to 1e-5.

Details

In the latency component, the covariates enter through the scale parameter $\lambda_i = \exp(x_i^\top \beta)$, as in an accelerated failure time model (the same convention as survreg). A positive coefficient therefore means longer survival times for uncured individuals. With shape parameter $\alpha$, the survival functions of the uncured are:

• exponential: $S(t) = \exp(-t/\lambda)$;

• Rayleigh: $S(t) = \exp\{-(t/\lambda)^2\}$;

• Weibull: $S(t) = \exp\{-(t/\lambda)^\alpha\}$;

• log-normal: $\log T \sim N(\log \lambda, \alpha^2)$;

• log-logistic: $S(t) = 1/\{1 + (t/\lambda)^\alpha\}$;

• inverse Gaussian: mean $\lambda$ and shape $\alpha$.

Value

An object of class "MCR", which is a list containing:

coefficients	Estimated regression coefficients for the survival part.
coefficients_cure	Estimated coefficients for the incidence part, that is, for the probability of being uncured.
scale	Estimated shape parameter $\alpha$ of the latency distribution (fixed at 1 for the exponential and 2 for the Rayleigh).
loglik	Final log-likelihood value.
n	Number of observations used in the model.
deleted	Number of incomplete cases removed before fitting.
ep	Estimated standard errors.
iter	Number of EM iterations.
convergence	Logical; TRUE if the EM algorithm converged within maxit iterations.
dist	Distribution used.
link	Link function used.
tau	Tau parameter used (if applicable).

Examples

require(EMGCR)

data(liver)
names(liver)
liver$sex <- factor(liver$sex)
liver$grade <- factor(liver$grade)
liver$radio <- factor(liver$radio)
liver$chemo <- factor(liver$chemo)
str(liver)
model <- MCRfit(
  survival::Surv(time, status) ~ age + sex + grade + radio + chemo |
    age + medh + grade + radio + chemo,
  dist = "loglogistic",
  link = "plogit",
  tau = 0.15,
  data = liver
)
summary(model)

---

Compare fitted MCR models with the Kaplan-Meier curve

Description

Plots the Kaplan-Meier estimate of the survival function together with the population survival function implied by one or more fitted mixture cure rate models.

Usage

## S3 method for class 'MCR'
plot(...)

Arguments

...	One or more fitted MCR objects from MCRfit().

Details

For each model, the fitted population survival function is averaged over the observations,

$\hat S_{pop}(t) = \frac{1}{n} \sum_{i=1}^n \{1 - \hat\theta_i + \hat\theta_i \hat S(t \mid x_i)\},$

where $\hat\theta_i$ is the estimated probability of being uncured and $\hat S(t \mid x_i)$ is the estimated survival function of the uncured. The Kaplan-Meier curve (dashed) is computed from the data of the first model, so all models should be fitted to the same data. Curves are coloured by latency distribution.

Value

A ggplot object.

---

QQ-Plot of Residuals for MCR Model

Description

Produces a Q-Q plot of residuals from a Mixture Cure Rate (MCR) model fitted via MCRfit. Optionally, a simulation envelope can be included for Cox-Snell residuals.

Usage

qqMCR(
  object,
  type = c("cox-snell", "quantile"),
  envelope = FALSE,
  nsim = 100,
  censor = NULL,
  ...
)

Arguments

object	An object of class MCR, typically returned by MCRfit.
type	Character. Type of residual to use in the QQ-plot. Options are "cox-snell" or "quantile". Defaults to "cox-snell".
envelope	Logical. Whether to add a simulation envelope to the QQ-plot. Default is FALSE.
nsim	Integer. Number of simulations used to construct the envelope. Default is 100.
censor	Logical vector or NULL. Censoring indicator used when simulating data for the envelope. Required only when envelope = TRUE and type = "cox-snell".
...	Additional arguments (currently ignored).

Details

The function generates QQ-plots of either Cox-Snell or quantile residuals. When envelope = TRUE and type = "cox-snell", a simulation envelope is added using Monte Carlo replications.

Value

A QQ-plot is produced as a side effect. Nothing is returned.

See Also

MCRfit, residuals.MCR

Examples

data(liver)
liver$sex <- factor(liver$sex)
liver$grade <- factor(liver$grade)
liver$radio <- factor(liver$radio)
liver$chemo <- factor(liver$chemo)

fit <- MCRfit(
  survival::Surv(time, status) ~ age + sex + grade + radio + chemo |
    age + medh + grade + radio + chemo,
  dist = "loglogistic", link = "plogit", tau = 0.15,
  data = liver
)
qqMCR(fit, type = "quantile", envelope = TRUE, nsim = 50, censor = liver$status)

---

Compute residuals for MCR model

Description

This function computes Global Cox-Snell and randomized quantile residuals for objects of class MCR.

Usage

## S3 method for class 'MCR'
residuals(object, type = c("cox-snell", "quantile"), ...)

Arguments

object	An object of class MCR, typically returned from MCRfit.
type	Type of residual.
...	Additional arguments (not used).

Value

A numeric vector of residuals of class "MCRresiduals". Printing it shows the first six residuals and a summary; use as.vector() to drop the class.

Examples

data(liver)
liver$sex <- factor(liver$sex)
liver$grade <- factor(liver$grade)
liver$radio <- factor(liver$radio)
liver$chemo <- factor(liver$chemo)

model <- MCRfit(
  survival::Surv(time, status) ~ age + sex + grade + radio + chemo |
    age + medh + grade + radio + chemo,
  dist = "loglogistic", link = "plogit", tau = 0.15,
  data = liver
)
residuals(model, type = "quantile")

---

Generate Random Samples for Mixture Cure Rate (MCR) Model

Description

Simulates survival data from a mixture cure rate model with covariates, a chosen link function for the incidence part and a chosen latency distribution. Censoring times are drawn from a uniform distribution on $(0, \code{censor})$.

Usage

rMCM(
  n,
  x,
  w,
  censor,
  alpha,
  beta,
  eta,
  dist = "weibull",
  link = "logit",
  tau = 1
)

Arguments

n	Integer. Number of observations to simulate.
x	Matrix or numeric. Covariate matrix for the latency component. The first column must be the intercept (a column of ones).
w	Matrix or numeric. Covariate matrix for the incidence part (the probability of being uncured). Include a column of ones if an intercept is wanted.
censor	Numeric. Maximum censoring time (uniformly distributed).
alpha	Numeric. Shape parameter $\alpha$ of the latency distribution. Ignored for the exponential and Rayleigh distributions.
beta	Numeric vector. Coefficients for the latency part, which enter through the scale parameter $\lambda = \exp(x^\top \beta)$ (see MCRfit for details).
eta	Numeric vector. Coefficients for the incidence part.
dist	Character. Distribution for the latency part. Options: "weibull", "lognormal", "loglogistic", "invgauss", "exponential", "rayleigh".
link	Character. Link function for the probability of being uncured. Options: "logit", "probit","plogit" ,"rplogit", "cauchit".
tau	A numeric value used when link = "plogit" or "rplogit". Defaults to 1.

Value

A tibble with columns:

time

Observed (possibly censored) survival time.

status

Event indicator (1 = event, 0 = censored).

x1, x2, ...

Columns of x without the intercept.

w1, w2, ...

Columns of w.

It also has two attributes: "pCcensur", the percentage of cured individuals, and "pUCcensur", the percentage of censored observations among the uncured.

Examples

# Example: Simulating survival data using the inverse Gaussian distribution
library(EMGCR)

n <- 500
beta <- c(1, -1, -2)
eta <- c(0.5, -0.5)
alpha <- 1.5

p <- length(beta)
q <- length(eta)

set.seed(10)
X <- matrix(rnorm(n*(p-1),0,1),n,p-1)
X <- cbind(1,X)

set.seed(20)
W <- matrix(runif(n*q,-1,1),n,q)
W <- scale(W)

max_censoring <- 10

set.seed(1234)
sim_data <- rMCM(n=n, x = X, w = W,
                 censor = max_censoring,
                 beta = beta, eta = eta,
                 alpha = alpha,
                 link = "logit", dist = "invgauss", tau = 1)

names(sim_data)
head(sim_data)
attributes(sim_data)
attr(sim_data, "pCcensur")
attr(sim_data, "pUCcensur")

---