Control chart for individual measurement (t-chart)

The T-chart (or time-between chart) is an I-chart used when the incident of interest is relatively rare and a measurement of time between each occurrence of the incident can be obtained. It allows each incident to be evaluated as it occurs rather than having to wait to the end of a standard time period before the data point is plotted.

The times between occurrences follow an exponential distribution, which is highly skewed. This can be transformed to a symmetric Weibull distribution by raising the time measure to the 1/3.6 power [y = t^0.2777].

Collect data by recording the time that the incident occurs and subtract the time that the previous incident occurred. Often the time can be recorded to the nearest days, but if two or more events (for example, inpatient falls) can occur on the same day, the hours should also be recorded. Be careful not to have any zeros in the data (for example, two rare events occurring at exactly the same time). Increase precision (hours vs days, minutes vs hours and so forth) when any time intervals will be recorded as zero (or add 0.5 units to the calculated time for the second data point recorded at the same time).

Because the calculations are based on the I-chart for individual measurements, the chart limits are affected by too few data points. Try to get at least 20 incidents before calculating control limits.

Method

Figure 5 shows data collected for sentinel events (death or serious injury). The number of days between each event is the focus of the chart.

1. the range B2:B21 lists the time-intervals between 20 incidents.
   t = time between incidents
2. the range C2:C21 transforms the time-intervals using the Weibull distribution
   y = transformed time
   → C2 = POWER(B2, 1/3.6) = POWER(60, 1/3.6) = 3.118
3. the range D3:D21 calculates the moving range (MR) between pairs of events as the absolute (negative values ignored) difference between the two events
   → D4 = ABS(C4 - C3) = ABS(-0.933) = 0.933
4. D22 calculates the provisional average moving range (MR_bar)
   → MR_bar = D22 = AVERAGE(D3:D21) = 0.513
5. E1 holds the provisional upper limit of the moving range (UL_MR) difference between two events
   → UL_MR = E1 = 3.27 * D22 = 3.27 * 0.513 = 1.678
6. the range E3:E21 calculates the adjusted moving range (MR') as for individual measurement I-charts by removing any values from the MR column that exceed 3.27 * UL_MR. Each line in E is calculated as:
   → D4 = IF(D4 > $E$1, "", D4)
7. E22 calculates the adjusted average moving range (MR_bar^new)
   → MR_bar^new = E22 = AVERAGE(E3:E21) = 0.513
8. Columns G, H, I calculate CL, UCL, LCL for transformed data:
   → CL(y) = POWER($B$22, 1/3.6) => $G$1 = 3.266
   → LCL(y) = G3 - 2.66 * $E$22
   → UCL(y) = G3 + 2.66 * $E$22
9. Columns K, L, M calculate CL, UCL, LCL for limits transformed back to the original units:
   → CL(t) = $B$22
   → LCL(t) = POWER(H3, 3.6)
   → UCL(t) = POWER(I3, 3.6)

Interpretation

Figure 3 (lower) shows the T-chart (limits transformed back to time scale) with the original days between plotted. Figure 3 (upper) is the same information, but the y-axis uses a logarithm (log10) scale. The run of 9 consecutive points above the center line is a signal of improvement. Note that both charts tell exactly the same story, with different visual emphasis, so that the users can decide which chart they prefer for presentation or decision making. The default should be to present the chart on the original time scale.

Figure 4 shows the charts from Figure 3 with the addition of trendlines. Since the time-between data is exponential, the lower graph (original units) uses exponential regression. The trend is more obvious on the logarithmic scale, where the trendline becomes linear. If the quality improvement is stable, the trendlines can be taken into consideration when setting goals for the near future.

Note how the logarithmic scale makes it much easier to identify problems around the LCL, such as #8 data point. One of the goals of planning data collection is to ensure that LCL > zero so that warning signals can be recognized, and the logarithmic scale is of assistance with this. Otherwise, on the units before transformation, the LCL may be zero, or so close to zero, that it is difficult to see whether data points are above, below, or on the LCL.