累積和（CUSUM）管制圖

資料

（使用前 20 筆資料計算流程平均值及標準差）

完成表 2 的步驟

一旦流程已經被校正回到目標， `C_i^+` 及 `C_i^-` 的值應該被設定為 0。然而，有一個方稱為快速起始反應（Fast Initial Response, FIR），利用它，值不用設回 0，但會被設為一個特定值，通常是 2 倍的標準差。這提供一個快速回應到偏離目標的作業。你也可以利用FIR開始畫累積和圖。圖 4 顯示當使用FIR的方法時，圖 3 使用 2 個標準差而不是 0。意即，表 2，會有以下改變：

你可以看到圖 2 `C_i^+` 及 `C_i^-` 距離 0 更遠。如果流程偏離目標，累積和管制圖將能較快偵測到。如果在管制中，累積和值的趨勢將回到 0，如第 7 個資料點。

The tabular cusum used in the basic example above has a single measurement in each sample (subgroups n = 1). Cusum can also be used for the averages of rational subgroups (n > 1), for example, the averages of likert scores on a satisfaction survey each day at an outpatient clinic. In this case, replace σ by `σ_x = σ ÷ sqrt(n)`. With Shewhart charts, the use of averages of rational subgroups substantially improves control chart performance.

However, this does not always happen with the cusum. If, for example, you have a choice of taking a sample of size n = 1 every half-hour or a rational subgroup sample of size n = 5 every 2.5 hours (note that both choices have the same sampling intensity), the cusum will often work best with the choice of n = 1 every half-hour. Only if there is some significant economy of scale or some other valid reason for taking samples of size greater than one should subgroups of size greater than one be used with the cusum.

Example: Satisfaction survey

This section uses satisfaction surveys as the example, based on a 5-point Likert scale for answering a question about the effects of a quality improvement strategy.
The scenario was an outpatient clinic for a single doctor. The quality improvement team were using PDSA methodology, and wanted to test the effectiveness of a strategy that they designed to resolve a problem that patients complained of in the free-text section of satisfaction questionnaires.
Each patient, as they left the clinic after seeing the doctor, was asked to rate the result using a 5-point Likert scale (1=very unhappy, 5 extremely satisfied). Table a1 shows the Likert score given by each patient (a~j). Rows indicate "day at clinic", for example, all the survey results for day 1 are in B2 to K2 {1, 2, 3, 4, 1, 2, 3, 4, 2, 3}.
Baseline data for one week (7 days), shown by a thick black line between day 7 and day 8.
From day 8, the new method was continued for another two weeks.

The result of interest was improvement, so a one-sided CUSUM `C^+` was used. Data was analysed in the following ways:

1. Each day, the average satisfaction was calculated and used as a single data value (n=1) to calculate (Table a2) and graph (Figure A2) the result.
2. Each day, the average satisfaction was calculated and used as a single data set (n>1) containing multiple values (different number of patients each day). The calculations were adjusted to reflect this (Table a3) and graph (Figure A3) by replacing σ with `sigma_x = sigma ÷ sqrt(n)` (Figure A1a)
3. Each patient value (n=1) was used to calculate the CUSUM values (Table a4) and graph (Figrue A4). There were 62 patient surveys collecting during the baseline period.

Figure A4. CUSUM graph for Table a4.

1. Table a2 first detects the change (`Y^+`= "Y") at E18, which is day 16, 9 days after the improvement strategy was begun. Using `N^+` to backtrack, we can see that the change started (`N^+` = "1") on day 8.
   The result is easily seen in Figure A2, where the first positive bar column appears at day 8, and the decision interval (H=5.018) is breached at day 16.
2. Table a3 first detects the change (`Y^+`= "Y") at E10, which is day 8, 1 day after the improvement strategy was begun.
   The result is seen in Figure A3, where the first positive bar column appears at day 8, and immediately breaches the decision interval (H=0.637).
3. Table a4 first detects the change (`Y^+`= "Y") at E71, which is patient 69, the fifth patient on day 8, the day that the improvement strategy was begun.
   The result is easily seen in Figure A4, where the consistently positive bar columns appear at patient 63, and reaches the decision interval (H=5.018) by patient 69.