Smooth operator: Modifying the Anhøj rules to improve runs analysis in statistical process control

INTRODUCTION: The run chart is one form of statistical process control chart that is particularly useful for detecting persistent shifts in data over time. The Anhøj rules test for shifts by looking for unusually long runs (L) of data points on the same side of the process centre (mean or median) and unusually few crossings (C) of the centre depending on the number of available data points (N). Critical values for C and L have mainly been studied in isolation. But what is really of interest is the joint distribution of C and L, which has so far only been studied using simulated data series. We recently released an R package, crossrun that calculates exact values for the joint probabilities of C and L that allowed us to study the diagnostic properties of the Anhøj rules in detail and to suggest minor adjustments to improve their diagnostic value. METHODS: Based on the crossrun R package we calculated exact values for the joint distribution of C and L for N = 10–100. Furthermore, we developed two functions, bestbox() and cutbox() that automatically seek to adjust the critical values for C and L to balance between sensitivity and specificity requirements. RESULTS: Based on exact values for the joint distribution of C and L for N = 10–100 we present measures of the diagnostic value of the Anhøj rules. The best box and cut box procedures improved the diagnostic value of the Anhøj rules by keeping the specificity and sensitivity close to pre-specified target values. CONCLUSIONS: Based on exact values for the joint distribution of longest run and number of crossings in random data series this study demonstrates that it is possible to obtain better diagnostic properties of run charts by making minor adjustment to the critical values for C and L.