Construct CUSUM charts for hospital performance

Overview

This vignette describes CUSUM charts based on a simulated false-signal-probability for hospital performance data in the R package cusum. This is a practical guide to constructing and evaluating non-risk-adjusted and risk-adjusted CUSUM charts following Steiner et al. (Biostatistics 1.4 (2000), pp. 441-52).

The cusum packages takes different factors into account that influence the signal rate of CUSUM charts. Some are given by the process to be monitored; these factors are:

• number of patients: How many observations do we expect in a monitoring period (e.g. a month/a year)?
• risk-adjustment: Is risk adjustment available? Can we allocate different risks to different observations?
  • if yes:
    • patient risks: What are these risks and how are they distributed?
  • if no:
    • accepted failure probability: What kind of failure rate do we expect on average?

The primary control input when constructing a CUSUM chart is the control limit. The control limit signals performance deterioration once crossed by the cumulated sum.

The control limit depends on a number of variables:

• the desired target odds multiplier associated to an out-of-control process
• the false signal probability accepted
• number of simulations

Motivating example

To illustrate how cusum can be used for monitoring, we employ a simple and artificial data set generated to closely follow the performance data of German hospitals for one non-risk-adjusted performance indicator and one risk-adjusted performance indicator in 2016 and 2017.

Non-risk-adjusted performance indicator

data("cusum_example_data", package = "cusum")

head(cusum_example_data)
#>   t     y year
#> 1 1 FALSE 2016
#> 2 2 FALSE 2016
#> 3 3 FALSE 2016
#> 4 4 FALSE 2016
#> 5 5 FALSE 2016
#> 6 6 FALSE 2016

Risk-adjusted performance indicator

data("racusum_example_data", package = "cusum")

head(racusum_example_data)
#>   t     y   score year
#> 1 1 FALSE 0.00237 2016
#> 2 2 FALSE 0.00237 2016
#> 3 3 FALSE 0.02412 2016
#> 4 4 FALSE 0.01893 2016
#> 5 5 FALSE 0.00725 2016
#> 6 6 FALSE 0.00810 2016

First, CUSUM charts are constructed on performance data from 2016 (Phase I), and then applied and evaluated on performance data from 2017 (Phase II).

cusum_example_p1 <- cusum_example_data[cusum_example_data$year == 2016, ]
cusum_example_p2 <- cusum_example_data[cusum_example_data$year == 2017, ]

racusum_example_p1 <- racusum_example_data[racusum_example_data$year == 2016, ]
racusum_example_p2 <- racusum_example_data[racusum_example_data$year == 2017, ]

Non-risk-adjusted CUSUM chart

failure_probability <- mean(cusum_example_p1$y)

n_patients <- nrow(cusum_example_p1)

cusum_limit <- cusum_limit_sim(failure_probability,
                               n_patients,
                               odds_multiplier = 2, 
                               n_simulation = 1000,
                               alpha = 0.05,
                               seed = 2046)


print(cusum_limit)
#> [1] 6.498476

patient_outcomes <- cusum_example_p2$y


cusum_cs <- cusum(failure_probability,
                  patient_outcomes,
                  limit = cusum_limit,
                  odds_multiplier = 2,
                  reset = FALSE)

head(cusum_cs)
#>   t failure_probability ct signal    limit
#> 1 1               0.216  0      0 6.498476
#> 2 2               0.216  0      0 6.498476
#> 3 3               0.216  0      0 6.498476
#> 4 4               0.216  0      0 6.498476
#> 5 5               0.216  0      0 6.498476
#> 6 6               0.216  0      0 6.498476
plot(cusum_cs)

cusum_cs <- cusum(failure_probability,
                  patient_outcomes,
                  limit = cusum_limit,
                  odds_multiplier = 2,
                  reset = TRUE)

plot(cusum_cs)

n_patients <- nrow(cusum_example_p2)

cusum_alpha <- cusum_alpha_sim(failure_probability,
                               n_patients,
                               odds_multiplier = 2,
                               n_simulation = 1000,
                               limit = cusum_limit,
                               seed = 2046)

print(cusum_alpha)
#> [1] 0.05

Risk-adjusted CUSUM chart

patient_risks <- racusum_example_p1$score

racusum_limit <- racusum_limit_sim(patient_risks,
                                   odds_multiplier = 2,
                                   n_simulation = 1000,
                                   alpha = 0.05,
                                   seed = 2046)

print(racusum_limit)
#> [1] 3.742861

patient_risks <- racusum_example_p2$score

patient_outcomes <- racusum_example_p2$y

racusum_cs <- racusum(patient_risks,
                      patient_outcomes,
                      limit = racusum_limit,
                      odds_multiplier = 2,
                      reset = FALSE)
plot(racusum_cs)

racusum_cs <- racusum(patient_risks,
                      patient_outcomes,
                      limit = racusum_limit,
                      odds_multiplier = 2,
                      reset = TRUE)

plot(racusum_cs)

racusum_alpha <- racusum_alpha_sim(patient_risks,
                                   odds_multiplier = 2,
                                   n_simulation = 1000,
                                   limit = racusum_limit,
                                   seed = 2046)

print(racusum_alpha)
#> [1] 0.058

CUSUM Chart for process improvement

CUSUM charts for detecting process improvements can be constructed similarly, but the CUSUM statistic is restricted to non-positive values.

cusum_limit_improve <- cusum_limit_sim(failure_probability,
                                       n_patients,
                                       odds_multiplier = .5,
                                       n_simulation = 1000,
                                       alpha = 0.5,seed = 2046)

cusum_cs_improve <- cusum(failure_probability,
                          patient_outcomes = cusum_example_p2$y,
                          limit = cusum_limit_improve,
                          odds_multiplier = .5)

plot(cusum_cs_improve)

cusum_alpha_sim(failure_probability,
                n_patients,
                odds_multiplier = 0.5,
                n_simulation = 1000,
                limit = cusum_limit_improve,
                seed = 2046)
#> [1] 0.505