The weighting factor of exponentially weighted moving average chart

Hikmet Can Çubukçu

doi:10.1515/tjb-2019-0368

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The weighting factor of exponentially weighted moving average chart

Hikmet Can Çubukçu

Published/Copyright: September 18, 2020

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From the journal Turkish Journal of Biochemistry Volume 45 Issue 5

Keywords: exponentially weighted moving average; quality control; weighting factor

To the Editor,

The exponentially weighted moving average (EWMA), firstly described by Roberts in 1959, is known to be beneficial for determining small shifts in the quality control process [1]. Therefore, it can help laboratorians to take preventive action before the process gets out of control.

Exponentially weighted moving average (EWMA) chart can be drawn by the following formula [2]:

zi=λ×xi+(1− λ)×zi−1

x_i: Current result
z_i: ith EWMA result
z_i−1: (i − 1)th EWMA result
λ: The weighting factor which determines the weight to be given to the current and previous quality control results.
Upper and lower control limits are calculated by following formulas [2]:
Upper control limit(UCL) = μ+Lσ((λ)(1−(1− λ)(2i))(2− λ)
Lower control limit(LCL)= μ−Lσ((λ)(1−(1− λ)(2i))(2− λ)
μ=z₀: Target value
σ: Standard deviation
L: The factor determines the width of upper and lower control limits.

When the value of z_i exceeds control limits, the process is considered out of control.

The weighting factor (λ) represents the most crucial part of the EWMA chart. EWMA chart was found to be superior to the Shewart chart for detecting small shifts in mean and variance when a small weighting factor is used (especially lower than 0.20) [3].

Weighting factor takes a value between 0 and 1. When the weighting factor is chosen as 1, the EWMA chart turns into the Shewart chart [4]. Different approaches have been described in the literature about the utility of the weighting factor. Small weighting factors (0,05–0,20) are preferred for determining small shifts that comply with fitness for the purpose [3]. However, this approach can’t quickly respond to a sudden large reverse shift due to the inertia effect. Thus, the Shewart chart should be used along with the EWMA chart when a small weighting factor is selected [5]. Linnet stated that weighting factor 0.5 could be more suitable for detecting larger shifts as 2 s encountered in daily laboratory practice. This approach performs better than traditional quality control rules, as indicated previously [6].

Another approach described by Neubauer is based on determining the weighting factor by adjusting shift sensitivity. Firstly, the most sensitive shift is determined using designated critical error (shift) which can be calculated by the following formulas [4], [7]:

critical error(shift) = total error−biasstandard deviation−1.65

or

critical error(shift)=total error% − bias%coefficient of variation%−1.65

or

critical error(shift) = sigma metric value – 1.65

The critical shift corresponds to the most sensitive shift.

Secondly, the corresponding weighting factor is chosen from the chart offered by Neubauer, as shown in Figure 1 [4]. For example, the allowable total error of albumin is 8%, according to Clinical Laboratory Improvement Amendments 2019 criteria. If the bias is 1% and analytical coefficient of variation is 1.24%, the critical error becomes 4% ((8 − 1)/1.24) − 1.65 = 4. In this case, the corresponding weighting factor is 0.9. When the EWMA chart with the weighting factor “0.1” applied to 1% positively biased simulation results, the first out of control result detected in the 30th day, as shown in Figure 2. However, when the weighting factor “0.9” preferred, there was no out of control result detected, as shown in Figure 3.

Figure 1:

Optimal weighting factor corresponding to the shift according to Neubauer [4].

Figure 2:

EWMA chart with weighting factor = 0.1 applied to 1% positively biased simulation results.

Figure 3:

EWMA chart with weighting factor = 0.9 applied to 1% positively biased simulation results.

This high weighting factor hinders EWMA’s efficiency in detecting small systematic shifts. Therefore in this problematic approach, high sigma metric values will lead to the selection of high lambda values. The power of the EWMA chart to detect small shifts will decrease, which is not fit for the purpose.

Hence, the utility of critical error to determine the weighting factor seems to be unreasonable. The most sensible approach might be choosing a low weighting factor for the EWMA chart to work in fitness for the purpose and the usage of it in combination with the Shewart chart. Simulation studies about EWMA should be carried out to determine optimum weighting factors.

Corresponding author: Hikmet Can Çubukçu M.D, Department of Medical Biochemistry, Erzurum Maresal Cakmak State Hospital, Erzurum, Turkey. Phone: +905376728807, E-mail: hikmetcancubukcu@gmail.com

References

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Received: 2019-09-04

Accepted: 2020-02-14

Published Online: 2020-09-18

Articles in the same Issue

https://doi.org/10.1515/tjb-2019-0368

Keywords for this article

exponentially weighted moving average; quality control; weighting factor