===== Cumulative Sum at Case =====

The **cumulative sum at case** (CUSUM value) represents the running total of deviations from the expected performance level up to a specific case number.

In a binary outcome model (e.g., success = 0, complication = 1), the CUSUM at each case is calculated using the formula:

''Cₙ = max(0, Cₙ₋₁ + (Xₙ - k))''

Where:
  * ''Cₙ'' = cumulative sum at case ''n''
  * ''Cₙ₋₁'' = cumulative sum at the previous case
  * ''Xₙ'' = actual outcome of the current case (0 or 1)
  * ''k'' = target complication rate (e.g., 0.1 for 10%)

==== Interpretation of the CUSUM Value ====

  * A **CUSUM of 0** means performance is matching or better than expected.
  * A **rising CUSUM** indicates more complications than expected.
  * A **declining or flat CUSUM** suggests improving or consistent performance.

==== Case-by-Case Example ====

If the target complication rate ''k'' is 0.1 and the outcome sequence is:

  * Case 1: success → ''C₁ = max(0, 0 + (0 - 0.1)) = 0''
  * Case 2: complication → ''C₂ = max(0, 0 + (1 - 0.1)) = 0.9''
  * Case 3: success → ''C₃ = max(0, 0.9 + (0 - 0.1)) = 0.8''

The curve visually builds a story of performance across cases, and the **CUSUM value at each case becomes a snapshot of deviation** from target performance.

==== Why It Matters ====

Tracking the cumulative sum at each case:
  * Enables real-time feedback
  * Helps detect early deviations from the expected outcome rate
  * Facilitates monitoring during training and protocol changes
  * Supports objective decision-making in quality assurance