Statistical process control compares observed process outcomes with a stable reference model and control limits reflecting expected sampling variation. An observation outside those limits is a signal to investigate, not proof of a specific cause. Varying sample volumes require varying limits, and multiple testing changes the chance of a chance signal somewhere in the display.
A CUSUM accumulates evidence for a change in a monitored probability distribution. With a log-likelihood ratio increment favoring a specified changed distribution, reset the accumulated score at zero and signal when it exceeds a chosen boundary . For , is the largest nonnegative sum of successive increments ending at . This permits repeated detection of a change with an unknown onset, rather than a single fixed-start test.
For independent Poisson observations, comparing mean with gives the displayed log-likelihood ratio. Updating a CUSUM with these scores targets the specified multiplicative increase. Its expected score is negative under the null and positive under the alternative when , so resetting at zero discards earlier evidence against an increase.
The average run length of a CUSUM is the expectation of its first signalling time under a specified data model and initial chart state. A large in-control average run length discourages false alarms; a small out-of-control value indicates rapid detection. The mean alone does not give a finite-horizon Type I error probability or the complete detection-delay distribution.
A fast initial response CUSUM starts a CUSUM with a positive head start below its signalling boundary. Before its first reset, it accumulates a fixed-start log-likelihood ratio like a sequential probability ratio test; after resetting to zero it behaves as an ordinary CUSUM. It gives faster initial detection when the process may already be out of control, at the cost of changed initial false-alarm behavior.
A hospital mortality funnel plot places each hospital's observed mortality proportion against its case volume. Under an independent binomial distribution model with common target p, the standard error is , so uncertainty bands contract as volume grows. This is a provider-performance plot, distinct in purpose and axes from a meta-analysis funnel plot. A signal requires investigation of data quality and patient case mix before drawing a causal conclusion.
The displayed pointwise control limits follow from the normal approximation to an observed binomial proportion. Clip plotted limits to the probability range. Their nominal coverage is approximate and can be poor for small expected counts; exact binomial tails give a discrete alternative. Pointwise coverage does not provide simultaneous coverage of all hospitals; use Bonferroni correction or another familywise error rate procedure when that is the required guarantee.
Control limits bound the sampling outcomes expected under a specified reference process. An observation outside a limit is a statistical hypothesis testing signal to investigate the reference model. These are limits for observations under that model, rather than a confidence interval constructed around each observed estimate. Their calibration may be pointwise, simultaneous or sequential.
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Statistical Process Control (SPC) is a method used in quality control and management that employs statistical methods to monitor and control a process. The main goal of SPC is to ensure that a process operates efficiently, producing more specification-conforming products with less waste (rework or scrap). Here are some key features and concepts of SPC: 1. **Control Charts**: One of the core tools of SPC is the control chart, which visually represents data over time.