Six Sigma methodology has been used successfully in industry since the mid-1980s. Unfortunately, the same success has not been achieved in laboratory medicine. In this case, although the multidisciplinary structure of laboratory medicine is an important factor, the concept and statistical principles of Six Sigma have not been transferred correctly from industry to laboratory medicine. Furthermore, the performance of instruments and methods used in laboratory medicine is calculated by a modified equation that produces a value lower than the actual level. This causes unnecessary, increasing pressure on manufacturers in the market. We concluded that accurate implementation of the sigma metric in laboratory medicine is essential to protect both manufacturers by calculating the actual performance level of instruments, and patients by calculating the actual error rates.
Six Sigma methodology is the latest version of total quality management and has been widely used in industry since the mid-1980s. One of the most powerful aspects of Six Sigma is its universal applicability to various fields including business, health care, and laboratory medicine. Unfortunately, despite great success of the Six Sigma methodology in industry, the same success has not been achieved in laboratory medicine.
Quality principles have not been applied to laboratory medicine as rigorously as industry. Interestingly (probably psychologically), the expected success or error rates in industry and laboratory medicine are not the same. For example, in laboratory medicine, 4 Sigma quality is accepted as a success, but in the aviation sector the target is 7 Sigma.
Although the principles of the Six Sigma methodology are universal, calculations of the sigma metric (SM) in industry and laboratory medicine differ. The reasons for the different calculation of SM in laboratory medicine need to be investigated. In our previous studies, we have shown that the conventional equation used to calculate SM in laboratory medicine is different from the equation used in industry (
In this paper, we aimed 1) to explain the statistical techniques of how engineers calculate short and long-term SM and error rates in industry and business, and 2) to show the defects of the SM equation used in laboratory medicine.
In industry, SM is calculated as given below (Equation (Eq.) 1):
The SM value indicates the performance of the process; the lower the SM value the process has, the higher error rate the process produces. To illustrate further, we explain the meaning of SM in terms of errors. For this purpose, we use defects
The SMs and corresponding DPMOs are given in
(0.493790335) | (- 0.191462461)* | (0.3023278735) | 697,670 | (0.3413447458) | (0.3413447458) | (0.6826894917) | 317,310 | |
0.498650102 | 0.0000000000 | 0.498650102 | 501,350 | 0.4331927985 | 0.4331927985 | 0.8663855971 | 133,610 | |
0.499767371 | 0.191462461 | 0.6912298320 | 308,770 | 0.4772498679 | 0.4772498679 | 0.9544997359 | 45,500 | |
0.499968329 | 0.341344746 | 0.8413130746 | 158,690 | 0.4937903346 | 0.4937903346 | 0.9875806693 | 12,420 | |
0.499996602 | 0.433192799 | 0.9331894009 | 66,810 | 0.498650102 | 0.498650102 | 0.9973002039 | 2700 | |
0.499999713 | 0.477249868 | 0.9772495813 | 22,750 | 0.499767371 | 0.499767371 | 0.499767371 | 465 | |
0.499999981 | 0.493790335 | 0.9937903156 | 6210 | 0.499968329 | 0.499968329 | 0.9999366575 | 63 | |
0.499999999 | 0.498650102 | 0.9986501010 | 1350 | 0.499996602 | 0.499996602 | 0.9999932047 | 6.8 | |
0.4999999999 | 0.499767371 | 0.9997673709 | 233 | 0.499999713 | 0.499999713 | 0.9999994267 | 0.57 | |
0.4999999999 | 0.499968329 | 0.9999683278 | 32 | 0.499999981 | 0.499999981 | 0.9999999620 | 0.04 | |
0.4999999999 | 0.499996602 | 0.9999966013 | 3.4 | 0.499999999 | 0.499999999 | 0.9999999980 | 0.002 | |
We use z score and z table to calculate the DPMOs corresponding to SMs. Area under the curve (AUC) obtained from z table. *Due to 1.5 SD shift in long term SM, the Z_{UTL} is higher than UTL and therefore the AUC of Z_{UTL} was subtracted from the AUC of Z_{LTL}. |
Next, we examine how engineers calculate the DPMO corresponding to short and long-term SM. The calculation of DPMO (both short and long term) is based on the normal distribution curve. The equation of the normal distribution function is given below (Eq. 2).
The DPMO corresponding to SM is derived from the area under the curve (AUC) restricted by LTL and UTL as calculated below (Eq. 3):
This equation gives AUC but is rather complex to be used in daily practice. For simplicity, we use the standard normal distribution curve. In this curve, the target is 0 and SD is 1.
To calculate the DPMO corresponding to SM, in the first step, we calculate the AUC restricted by LTL and UTL. The total AUC (-∞ to +∞) is 1. The error rate is 1-AUC and therefore the DPMO can be calculated as given below (Eq. 4):
Although this equation is very simple, we still must calculate the restricted AUC, which is not easy. To overcome this problem, we use tables instead of complex equations. For the standard normal distribution curve, we use the z table.
To calculate the DPMO corresponding to SM, we find the AUC from the LTL to UTL from the z table and use Eq. 4. For example, to calculate the DPMO corresponding to 5 SM, we find the AUC restricted by - 5 to 5 from z table and then use Eq. 4 as given below:
AUC from − 5 to 0 = 0.4999997133
AUC from 0 to 5 = 0.4999997133
AUC from − 5 to 5 = 0.9999994266
DPMO = 10^{6} (1 − 0.9999994266) = 0.57.
This is the short-term DPMO, but in daily practice we use the long-term SM and corresponding DPMO. To calculate the long term DPMO we include the shift in the calculations. In this case we find the AUC from (- shift - LTL) to (UTL + Shift) from the z table. For example, if we want to calculate the long-term DPMO of 5 SM we find the AUC from − 6.5 to 3.5. Due to the 1.5 SD shift, the limit of the left tail of the curve is − 6.5 (−1.5 - 5) and the limit of the right tail of the curve is 3.5 (5 -1.5) (
To calculate the long-term DPMO of 5 SM we find the area under the curve from − 6.5 to 3.5. Due to the 1.5 SD shift, the limit of the left tail of the curve is − 6.5 (− 1.5 - 5) and the limit of the right tail of the curve is 3.5 (5 - 1.5). The DPMO corresponding to SMs are calculated using Eq. 4.
The DPMO corresponding to SMs are calculated using Eq. 4 as given below:
AUC from − 6.5 to 0 = 0.4999999999
AUC from 0 to 3.5 = 0.4997673709
AUC from − 6.5 to 3.5 = 0.9997673708
DPMO = 10^{6} (1 - 0.9997673708) = 233.
From these simple examples we can see that both short and long term DPMOs are calculated using the standard normal distribution curve (
In laboratory medicine literature, the scientific background of Six Sigma methodology is not well understood. In our previous studies we aimed to implement the corrected SM in laboratory medicine and calculate the performance of the instruments accurately, but it seems that there are still misunderstandings in the field (
In laboratory medicine, a different equation is used to calculate SM as given below (Eq. 5):
TR* is the range from the center (target) to UTL or LTL. This equation is different from the main SM equation (Eq. I), it includes bias and structurally similar to process capability index (Cpk) as given below (Eq. 6):
It should be noted that SM and Cpk are not the same. Sigma metric is linked to DPMO and must be interpreted correctly. The difference between SM and Cpk is the subject of another paper and cannot be summarized in this short paper.
There are two major drawbacks of calculating SM from Eq. 5.
1. Mathematical defects and illogical results. We derive SM and DPMO from the normal distribution curve but the relation between these parameters is not linear. A linear relation is present in uniform distributions, not in normal distributions (
A linear relation is present in uniform distributions (A), but not in normal distributions (B). Moving the mean to the right or left increases or decreases the AUC linearly in the uniform distribution, but not in the normal distribution. Therefore, inclusion of bias as a linear component in Eq. 5 is mathematically not valid. The SD of uniform distribution is [(b-a)/12]^{1/2}.
Equation 5 is a one-sided and linear treatment of bias open to illogical results such as negative Sigma. If the bias is larger than UTL/LTL, according to Eq. 5, SM will be a negative value (
A normal distribution curve is a two-sided curve (from −∞ to +∞) and the tails of the curve do not intersect the x-axis. Bias might be on the right or left side of the mean. Even if the bias is larger than UTL or LTL, the performance of a working process is always higher than zero. Normal distribution curve is the mathematical reference of both SM and DPMO. Negative SM cannot be obtained from normal distribution curve.
Mathematically we can show that the performance of a working process is always > 0 (
In industry for a given SM value, we have one short (no shift) and one long term (shift is 1.5 SD) DPMO. But according to Eq. 5, for a given SM value we have one short (no shift) but can obtain a lot of long term DPMOs. For example, if SM = 1, the short-term DPMO will be 317,310 (
24 | 20 | 3.0 | 1.0 | - 7.0 (0.4999999999) | 1 (0.3413447458) | - 7 to + 1 |
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24 | 20 | 2.0 | 2.0 | - 3.0 (0.4986501019) | 1 (0.3413447458) | - 3 to + 1 |
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24 | 20 | 0.0 | 4.0 | - 1.0 (0.3413447458) | 1 (0.3413447458) | - 1 to + 1 |
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112 | 100 | 3.0 | 3.0 | - 5.0 (0.4999997133) | 3 (0.4986501020) | - 5 to + 3 |
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112 | 100 | 1.5 | 3.5 | - 3.9 (0.4999433065) | 3 (0.4986501020) | - 3.9 to + 3 (0.9985934084) | |||
112 | 100 | 0.0 | 4.0 | - 3.0 (0.4986501020) | 3 (0.4986501020) | - 3 to + 3 (0.9973002039) | |||
For the same SM, we can obtain various DPMOs. In this table for simplicity we show only three different DPMOs corresponding to the same SM. For the same test, the DPMO of (**) is approximately 2 times higher than the DPMO of (*). LTL - lower tolerance limit. UTL – upper tolerance limit. CV – coefficient of variation. SM – Sigma metric. DPMO – defects |
2. Application defect. If we include bias directly, we must use the short-term DPMO, not the long-term DPMO. For example, if the TR, Bias, and SD of a process are 12, 4, and 2 respectively, then according to Eq. 5 (although mathematically it is not correct) the SM will be 4. Now the question is what is the DPMO corresponding to 4 Sigma. The answer is 63, not 6200 (
As mentioned previously, engineers in industry use Eq. 1, in which bias is not directly included, because mathematically it is not valid. Instead of directly including bias, they prefer to use the long-term DPMO corresponding to SM. This approach has two important advantages: first, the equation is very simple and it is easy to calculate the performance of a process using the TR and SD. Second, in daily practice we do not know whether bias exists or not, and therefore we accept the presence of a 1.5 SD shift in any case. This assumption protects the process performance evaluation from the possible presence of biases.
The approach used in laboratory medicine has serious defects. First, it includes bias measured in laboratory and uses DPMO corresponding to long term SM. It should be noted that, in laboratory medicine, the bias measurement methods differ among laboratories and some of them are defective, thus the results may not be reliable. The SM calculated using Eq. 5 will be very low; consequently the DPMO will be very high. This results from two biases: the measured bias, and the 1.5 SD shift. In this case, the calculated performance of the process will be incorrect, significantly lower than the actual performance level, causing serious trouble for manufacturers in the market. To overcome this problem we have two choices. First, we can perform the calculations as engineers do in industry, by only using TR and SD to calculate SM, and then use the long-term DPMO. Second, if we include bias we should use z score and z table to calculate short term DPMO.
None declared.