Control Charts and Run Charts

Definitions

• Common cause variation – random variation or “noise” that is inherent to the process itself. If variation is common cause alone, the process is considered stable and predictable.
• Special cause variation – when one or more data points vary in an unpredictable manner due to some factor(s) that is not inherent to the process. If a special cause is present, it signals that the process has been changed.

Control, aka Shewhart, charts help to distinguish common cause vs special cause variation.

• Trend – a long series of consecutive increases/decreases in the data, i.e. at least 6 or 7 points are all going up or all going down.[^carey]
• Run – series of points in a row on one side of the median, i.e. crossing the median line counts as a new run.

Generate Data

I’ll try to find a nice open dataset to play with this concept later, but for now I’ll just simulate some data to work with.

library(qicharts)

set.seed(1)

x <- rnorm(24)
y <- c(rnorm(mean = 5, 12), rnorm(mean = 4, 12))

Run Chart

Run charts are very simple and don’t require any sort of statistical analysis. You also can construct a run chart using different types of data, such as counts, percentages, ratios, etc.

#qic(x) #generates a run chart of x
qic(y) #generates a run chart of y

Ideally, you should have at least 16 points (excluding those on the median) to have enough statistical power to identify a special cause.[^carey] Perla, et al. puts this number at 10.

Rules to Interpret a Run Chart:[^perla] 1. Shift – 6 or more points either all above or all below the median. Values on the median do not count towards this tally. 2. Trend – at least 5 points that are all going up or all going down. If 2 or more values are the same, only count the first point and exclude the repeating values. Of note, Carey puts the number of points at 6 or 7 to qualify as a “trend.” 3. Runs – series of points in a rown on one side of the median. Trick to count the number of runs: tally the number of times the line connecting the points crosses the median and add 1. 4. Astronomical Point – the blatant outlier

Figure 3 of Perla’s 2011 paper, The run chart: a simple analytical tool for learning from variation in healthcare processes, does an excellent job of visually illustrating these principles.

Control Charts

You’ll notice that an obvious difference between control charts and run charts is the addition of 3 lines – the median and the upper and lower control limits. The upper and lower control limits are NOT the confidence interval. What’s the difference? The confidence interval describes the variability of a distribution of data, whereas the control limits describe the variability of a process. This is a subtle point, but an important one to know. Furthermore, a point that falls beyond the control limit is considered a special cause, rather than an outlier with confidence intervals. Additionally, the formulas used to calculate the standard deviation for confidence intervals and control limits is different.

Rules to Detect Special Cause:[^carey] 1. At least 1 point beyond the control limit (UCL or LCL), i.e. greater than 3 standard deviations (σ) 2. 2 out of 3 consecutive values are both: a) on the same side of the mean, and b) more than 2 σ away from the mean. 3. 8 or more successive values lie on the same side of the centerline (mean). 4. Trend of 6 or more successive values steadily increasing/decreasing.

Refer to Figure 2.2 and pgs 16-17 of Carey’s Improving Healthcare with Control Charts for more.

Choosing the “right” control chart

There are a few control charts to choose from, and while it seems daunting at first, all you need to really do is understand your dataset and reference a decision tree algorithm.

1. Are you working with continuous or discrete variables?
2. If you’re working with discrete data, are you working with binary, e.g. 0 or 1, data – examples include death (yes/no), infection (yes/no)?

Although not shown in the figure above, if working with a continuous variable and n=1, you’ll want to use an i-chart (aka XMR chart).

library(rvest)
library(qicharts)
library(qicharts2)
library(qcc)
library(dplyr)

# Scrape the table from the site
surg_table <- read_html("https://academic.oup.com/intqhc/article/22/5/402/1786749/The-p-control-chart-a-tool-for-care-improvement")

surgcomp <- surg_table %>% html_table()

# Clean up
surgcomp <- surgcomp[1]
surgcomp <- as.data.frame(surgcomp)
colnames(surgcomp)

## [1] "Month"                      "No..of.complications"      
## [3] "No..of.surgical.procedures"

surgcomp <- surgcomp[1:30,]

# rename the columns for simplicity
surgcomp <- rename(surgcomp, 
                   month = Month, 
                   complications = No..of.complications, 
                   surgeries = No..of.surgical.procedures)

surgcomp$month <- as.integer(surgcomp$month)
glimpse(surgcomp)

## Observations: 30
## Variables: 3
## $ month         <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1...
## $ complications <int> 14, 12, 10, 12, 9, 7, 9, 11, 9, 12, 10, 7, 12, 9...
## $ surgeries     <int> 105, 97, 115, 100, 95, 111, 68, 47, 83, 108, 115...

# with qicharts
qicharts::qic(complications, 
              n = surgeries, 
              x = month, 
              data = surgcomp, 
              chart = "p", 
              main = "P-chart for Surgical Complications", 
              ylab = "Proportion of patients", 
              xlab = "Month")

# with qicharts2
qicharts2::qic(complications, 
               n = surgeries, 
               x = month, 
               data = surgcomp, 
               chart = "p", 
               title = "P-chart for Surgical Complications", 
               ylab = "Proportion of patients", 
               xlab = "Month")

# with qcc

qcc(data = surgcomp$complications, 
    type = "p", 
    sizes = surgcomp$surgeries, 
    title = "P-chart for Surgical Complications")

## List of 11
##  $ call      : language qcc(data = surgcomp$complications, type = "p", sizes = surgcomp$surgeries,      title = "P-chart for Surgical Complications")
##  $ type      : chr "p"
##  $ data.name : chr "surgcomp$complications"
##  $ data      : int [1:30, 1] 14 12 10 12 9 7 9 11 9 12 ...
##   ..- attr(*, "dimnames")=List of 2
##  $ statistics: Named num [1:30] 0.1333 0.1237 0.087 0.12 0.0947 ...
##   ..- attr(*, "names")= chr [1:30] "1" "2" "3" "4" ...
##  $ sizes     : int [1:30] 105 97 115 100 95 111 68 47 83 108 ...
##  $ center    : num 0.1
##  $ std.dev   : num 0.3
##  $ nsigmas   : num 3
##  $ limits    : num [1:30, 1:2] 0.01219 0.00864 0.0161 0.01002 0.00768 ...
##   ..- attr(*, "dimnames")=List of 2
##  $ violations:List of 2
##  - attr(*, "class")= chr "qcc"