success (version 1.0.1)

bernoulli_cusum: 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)

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.

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.

Author

Daniel Gomon

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.

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)