bernoulli_cusum {success}

Risk-adjusted Bernoulli CUSUM

Description

This function can be used to construct a risk-adjusted Bernoulli CUSUM chart for survival data. It requires the specification of one of the following combinations of parameters as arguments to the function:

• glmmod & theta

• p0 & theta

• p0 & p1

Usage

bernoulli_cusum(data, followup, glmmod, theta, p0, p1, h, stoptime, assist,
  twosided = FALSE)

Arguments

data

A data.frame containing the following named columns for each subject:

entrytime:

time of entry into study (numeric);

survtime:

time from entry until event (numeric);

censorid:

censoring indicator (0 = right censored, 1 = observed) (integer);

and optionally additional covariates used for risk-adjustment.

followup

The value of the follow-up time to be used to determine event time. Event time will be equal to entrytime + followup for each subject.

glmmod

Generalized linear regression model used for risk-adjustment as produced by the function glm(). Suggested:
glm(as.formula("(survtime <= followup) & (censorid == 1) ~ covariates"), data = data).
Alternatively, a list containing the following elements:

formula:

a formula() in the form ~ covariates;

coefficients:

a named vector specifying risk adjustment coefficients for covariates. Names must be the same as in formula and colnames of data.

theta

The \theta value used to specify the odds ratio e^\theta under the alternative hypothesis. If \theta >= 0, the chart will try to detect an increase in hazard ratio (upper one-sided). If \theta < 0, the chart will look for a decrease in hazard ratio (lower one-sided). Note that

p_1 = \frac{p_0 e^\theta}{1-p_0 +p_0 e^\theta}.

p0

The baseline failure probability at entrytime + followup for individuals.

p1

The alternative hypothesis failure probability at entrytime + followup for individuals.

h

(optional): Control limit to be used for the procedure.

stoptime

(optional): Time after which the value of the chart should no longer be determined.

assist

(optional): Output of the function parameter_assist()

twosided

(optional): Should a two-sided Bernoulli CUSUM be constructed? Default is FALSE.

Details

The Bernoulli CUSUM chart is given by

S_n = \max(0, S_{n-1} + W_n),

where

W_n = X_n \ln \left( \frac{p_1 (1-p_0)}{p_0(1-p_1)} \right) + \ln \left( \frac{1-p_1}{1-p_0} \right)

and X_n is the outcome of the n-th (chronological) subject in the data. In terms of the Odds Ratio:

W_n = X_n \ln \left( e^\theta \right) + \ln \left( \frac{1}{1-p_0 + e^\theta p_0} \right)

For a risk-adjusted procedure (when glmmod is specified), a patient specific baseline failure probability p_{0i} is modelled using logistic regression first. Instead of the standard practice of displaying patient numbering on the x-axis, the time of outcome is displayed.

Value

An object of class bercusum containing:

• CUSUM: A data.frame containing the following named columns:

  time:

  times at which chart is constructed;

  value:

  value of the chart at corresponding times;

  numobs:

  number of observations at corresponding times.

• call: the call used to obtain output;

• glmmod: coefficients of the glm() used for risk-adjustment, if specified;

• stopind: indicator for whether the chart was stopped by the control limit.

Author(s)

Daniel Gomon

See Also

plot.bercusum, runlength.bercusum

Examples

#We consider patient outcomes 100 days after their entry into the study.
followup <- 100
#Determine a risk-adjustment model using a generalized linear model.
#Outcome (failure within 100 days) is regressed on the available covariates:
exprfitber <- as.formula("(survtime <= followup) & (censorid == 1)~ age + sex + BMI")
glmmodber <- glm(exprfitber, data = surgerydat, family = binomial(link = "logit"))
#Construct the Bernoulli CUSUM on the 1st hospital in the data set.
bercus <- bernoulli_cusum(data = subset(surgerydat, unit == 1), glmmod = glmmodber,
 followup = followup, theta = log(2))
#Plot the Bernoulli CUSUM
plot(bercus)