vlad (version 0.2.0)

racusum_arl_mc: Compute ARLs of RA-CUSUM control charts using Markov chain approximation

Description

Computes the Average Run Length of a risk-adjusted cumulative sum control chart using Markov chain approximation.

Usage

racusum_arl_mc(pmix, RA, RQ, h, scaling = 600, rounding = "p",
  method = "Toep")

Arguments

pmix

Numeric Matrix. A three column matrix. First column is the risk score distribution. Second column are the predicted probabilities from the risk model. Third column can be either the predicted probabilities from the risk model or average outcome per risk score, see examples.

RA

Double. Odds ratio of death under the alternative hypotheses. Detecting deterioration in performance with increased mortality risk by doubling the odds Ratio RA = 2. Detecting improvement in performance with decreased mortality risk by halving the odds ratio of death RA = 1/2. Odds ratio of death under the null hypotheses is 1.

RQ

Double. Defines the true performance of a surgeon with the odds ratio ratio of death RQ. Use RQ = 1 to compute the in-control ARL and other values to compute the out-of-control ARL.

h

Double. h is the control limit (>0).

scaling

Double. The scaling parameter controls the quality of the approximation, larger values achieve higher accuracy but increase the computation burden (larger transition probability matrix).

rounding

Character. If rounding = "p" a paired rounding implementation similar to Webster and Pettitt (2007) is used, if rounding = "s" a simple rounding method of Steiner et al. (2000) is used.

method

Character. If method = "Toep" a combination of Sequential Probability Ratio Test and Toeplitz matrix structure is used to calculate the ARL. "ToepInv" computes the inverted matrix using Toeplitz matrix structure. "BE" solves a linear equation system using the classical approach of Brook and Evans (1972) to calculate the ARL.

Value

Returns a single value which is the Run Length.

References

Steiner SH, Cook RJ, Farewell VT and Treasure T (2000). Monitoring surgical performance using risk-adjusted cumulative sum charts. Biostatistics, 1(4), pp. 441--452.

Brook D and Evans DA (1972) An approach to the probability distribution of CUSUM run length. Biometrika, 59(3), pp. 539--549

Webster RA and Pettitt AN (2007) Stability of approximations of average run length of risk-adjusted CUSUM schemes using the Markov approach: comparing two methods of calculating transition probabilities. Communications in Statistics - Simulation and Computation 36(3), pp. 471--482

Examples

# NOT RUN {
library(vlad)
library(dplyr)
data("cardiacsurgery", package = "spcadjust")

## preprocess data to 30 day mortality and subset phase I (In-control) of surgeons 2
SALLI <- cardiacsurgery %>% rename(s = Parsonnet) %>%
  mutate(y = ifelse(status == 1 & time <= 30, 1, 0),
        phase = factor(ifelse(date < 2*365, "I", "II"))) %>%
  filter(phase == "I") %>% select(s, y)

## estimate risk model, get relative frequences and probabilities
mod1 <- glm(y ~ s, data = SALLI, family = "binomial")
fi  <- as.numeric(table(SALLI$s) / length(SALLI$s))
usi <- sort(unique(SALLI$s))
pi1 <- predict(mod1, newdata = data.frame(s = usi), type = "response")
pi2 <- tapply(SALLI$y, SALLI$s, mean)

## set up patient mix (risk model)
pmix1  <- data.frame(fi, pi1, pi1)

## Average Run Length for detecting deterioration RA = 2:
racusum_arl_mc(pmix = pmix1, RA = 2, RQ = 1, h = 4.5)

## Average Run Length for detecting improvement RA = 1/2:
racusum_arl_mc(pmix = pmix1, RA = 1/2, RQ = 1, h = 4)

## set up patient mix (model free)
pmix2  <- data.frame(fi, pi1, pi2)

## Average Run Length for detecting deterioration RA = 2:
racusum_arl_mc(pmix = pmix2, RA = 2, RQ = 1, h = 4.5)

## Average Run Length for detecting improvement RA = 1/2:
racusum_arl_mc(pmix = pmix2, RA = 1/2, RQ = 1, h = 4)

## compare results with R-code function 'findarl()' from Steiner et al. (2000)
source("https://bit.ly/2KC0SYD")
all.equal(findarl(pmix = pmix1, R1 = 2, R = 1, CL = 4.5, scaling = 600),
         racusum_arl_mc(pmix = pmix1, RA = 2, RQ = 1, h = 4.5, scaling = 600, rounding = "s"))
# }