The computer simulations aimed to demonstrate that the sources of bias variation described in the literature are the true causes of the increased SD in more extended time frames and to confirm the validity of Equation 2. The real-life QC examples demonstrate that computer simulations are grounded in reality.

In the computer simulation of a single calibration (a single shift in the mean), the graph of the run mean is a horizontal line with bias=0 before the mean shift (calibration) and a horizontal line with mean=bias after the mean shift (calibration). The results are randomly dispersed around the run mean with SD of 1 (=s_r) (Figure 1). The SDs calculated from 500 data before and from 500 data after the shift are 1 (=s_r), while the SD calculated from all data (s_RW=1.43) is significantly bigger according (F_{0.95, 500,500}=1.43). The SD calculated from runs 480‐520 (including the shift) is 1.55, suggesting that the bias variation causes the increase of the SD (s_RW). A sudden change of 1 SD (1s_r) in the mean causes an increase of only 12% in the overall SD (s_RW), and it is difficult to observe visually such minimal increases.

Representing the sRW2 values as a function of sVCSE2 , a linear graph with slope ≈1, consistent with Equation 2, was obtained, confirming its validity (Figure 2).

An example of magnesium obtained in March 2021 on a Cobas Pro analyzer (2 × 7 runs, 2 levels, and one calibration after 7 runs) is presented in Figure 3 to exemplify real-life data. The results were represented in %, not as absolute values, to reduce the influence of the s_RW variability.

The graph has an insignificant drift on both levels. Calculations presented in Table 3 show that before and after calibration, the coefficient of variation (CV) is consistent with the CV_r (an F test did not reveal significant differences), and the increase in the CV_RW (CV measured in variable, reproducibility within laboratory conditions) is due to the shift in the mean. Equation 2 is valid. From the s_VCSE calculated from the mean variation and the s_r values, it was possible to predict the value of the CV_RW. The F test did not find significant differences between the CV of all data, the predicted CV (Equation 2), and the actual CV_RW. The actual CV_RW (determined from one month’s data) is slightly bigger because it includes more calibrations and reagent changes.

Figure 4 shows the simulation graph of more random changes in the mean. Without computer assistance, we can visually detect only the significant mean variation (run 11). As shown in Table 4, the s_r values are quasi-constant. Simultaneously, the s_RW values depend on the time frame (variations from 1.10 to 1.94). The bigger the mean change, the bigger the s_RW. The validity of Equation 2 is maintained (compare line 4 with line 10).

In the computer simulation, the graph of the daily mean was an oblique line with decreasing tendency, with slope=−0.001 × _maxBias. _maxBias is the maximum bias in absolute values in each simulation. The SD calculated from the daily means was s_VCSE=Biasmax2*3, corresponding to a uniform distribution. The deviation of the results from the daily means had an SD ≈1 (=1s_r) in all simulations. The SD calculated from all 1001 data (s_RW) was bigger than 1s_r. A bias variation of 1.5s_r caused an increase in s_RW of only 10%, which was difficult to observe visually.

If we represent the sRW2 values as a function of sVCSE2, we obtain an identical graph, as shown in Figure 2, consistent with Equation 2 (a linear graph with slope ≈1 and intercept ≈1).

Figure 5 shows a 35-run real-life chart (glucose, Cobas 6000 analyzer, July 2023). The period includes 2 reagent changes (corresponding to the shifts in the mean between runs 14‐15 and 25‐26). No calibrations were made. In the periods between reagent changes, the means are similar in all time frames (0.22%/run, 0.23%/run, and 0.20%/run), consistent with the degradation tendency of the reagent. Most data are within the estimated mean (SD 2CV_r) limits, suggesting that CV_r (s_r) is the true measure of the RE.

The CV_r values calculated from the deviations of the percent expressed results from the estimated mean are similar to the CV_r value calculated from the half differences between the percent expressed results obtained on the 2 control levels (Table 5; a Cochran F test for equality of 2 variances did not find significant differences between the CV_r values).

A control material handling error (reused control material) in run 26 (false simultaneous increase) caused the slightly bigger s_r in runs 26‐35.

Another example with total bilirubin was published by Vandra [27] in a preprint paper.

“The bias” is a definitional uncertainty. The same distinction is necessary between the biases obtained in repeatability and respective reproducibility within laboratory conditions, as in the case of SDs. The need for standardization imposes similar notations. We must highlight the time-variable function character of the bias as well. The author proposes the following notations:

We can obtain only a mean bias value in more extended time frames.

The difference between B_r(t) and B-RW is VCSE(t), a time-variable function.

Variations in the mean caused by reagent property changes cause drifts. The VCSE(t) cannot increase indefinitely (in absolute values) due to human interventions. It may have only cyclical variations. The cycles depend on external factors (eg, the rhythm of reagent use, frequency of human interventions, and the size of random calibration errors). They have different amplitudes, means, and lengths.

In some cases, a cycle may last even a month. The graphs of the daily means (not of the results) have sawtooth shapes masked by the noise of the RE (easily observed in the case of the unstable reagents, eg, Figure 5). In short or medium time frames, the B-RW values may have variations. The longer the time frame, the less uncertainty there is for the B-RW values. Only yearly B-RW values can be considered quasi-constant [21] and used for accurate corrections. In a chosen time frame, we can identify B-RW with the CCSE. Consequently, the mean of the VCSE(t) is 0. If we calculate the long-term mean of the B_r(t) values:

We obtain the same value for the long-term mean of TE (TE-RW) because ∑t=1nRE(t)=0. Similarly, the SD can be calculated from long-term data (s_RW):

Which confirms the validity of Equation 2 (because the long-term mean of RE and VCSE(t) is 0, ∑t=1n(RE(t)∗VCSE(t))≈0). Regrouping the terms in Equation 5 can be calculated using s_VCSE.

Regrouping Equation 3 and adding RE to both parts of the equation yields:

Equations 3, 5, and 6 define a new error model, which is presented in Figure 6.

Figure 6 shows that both s_RW and B_r(t) include VCSE(t) in a hidden form.

We obtain 2 sets of error parameters by separating the bias into a constant and a variable component and distinguishing bias measured in repeatability and reproducibility within laboratory conditions. According to quintessential principle 1, UM calculations must be based on parameters determined in reproducibility within laboratory conditions (s_RW, B-RW). In the meantime, the internal QC decisions must be based on parameters determined in repeatability conditions (s_r, B_r(t)). The second conclusion contradicts the recommendations of Westgard et al [1] to design Levey-Jennings charts with an SD calculated from long-term control data (s_RW).

Consistent with the VIM 2.17 definitions [14], we can define the bias components as:

Note 1: The CCSE is the long-term mean bias B-RW, depending on the time frame.

Note 2: VCSE(t) is a time-variable function.

Note 3: VCSE(t) is hidden in B_r(t) and s_RW.

The simultaneous use of B_r(t) and s_RW causes a redundant use of VCSE(t) in equations—for example, _maxTE=B_EQA + z × s_RW. B_EQA is the bias measured in the last EQA round in repeatability conditions, and _maxTE is the TE limit, which includes all TE values with confidence corresponding to z, the confidence factor.

If bias is variable, TE is also variable (contradicting the graphical model of Theodorsson et al [21]). A distinction is necessary between:

Internal QC decisions must be based on _maxTE(t), the maximum value of the TE at the moment t of decisions with a chosen confidence level, where z is the confidence factor.

Where _maxTE_RW is the maximum TE value in long time frames, with a chosen confidence. It must be used when setting limits and is a starting point for uncertainty of measurement (UM) calculations.

TE is also an ambiguous term. It is necessary to specify which TE is mentioned.

TE was the dominant paradigm until the emergence of UM after the publication of GUM in 1993 [11,12,28].

UM mathematically expresses our lack of knowledge about the accuracy of the result. According to VIM 2.26 [14]:

The definition is also mentioned in ISO 15189. According to ISO 15189, 5.6.2 [29]:

Surprisingly, neither the calibration error nor the reagent instability is mentioned among the uncertainty sources. According to the Hong Kong Association of Medical Laboratories, “the IQC procedure is designed to detect variations in reagents or calibrators” [30].

According to Magnusson and Ellison [28]:

A prerequisite for the application of the GUM [11] is that:

Using either a correction (GUM B.2.23) or a correction factor (GUM B.2.24) [11]. Then, the uncertainty of the correction is included in the uncertainty budget. Unfortunately, according to Magnusson and Ellison [28]:

The uncorrected bias must be included in the uncertainty budget, and due to the VCSE, it is not negligible. There is a debate in the literature about incorporating the uncorrected bias in the expression of total uncertainty (eg, Magnusson and Ellison [28], Westgard [31]). A review of this debate is not the task of this study.

UM equations start from the same error model, and TE equations (TE=SE+RE). They substitute the error parameters with the uncertainties caused in patient results.

Where Utot_SE is the total uncertainty of the patient’s result caused by the SE, and Utot_RE is the total uncertainty caused by the RE.

There are 2 types of uncertainty in the case of both parameters: the uncertainty of the result because the error parameters exist, and our uncertainty about the value of the parameters. For example, the uncertainty of a patient’s result, caused by the RE, is as follows:

Unfortunately, the SD value is not accurate. Therefore:

where U_SD is the uncertainty of the SD value, and SD_max is the maximum value of the SD. However, the adepts of the UM critique TE theory because TE equations do not include the uncertainty of the error parameters, nor do UM equations include the uncertainty of the SD. However, s_RW has big monthly variations [5].

The uncertainty caused by the SE (bias) equals the bias value. U_SE=B. Because the bias value is uncertain, it must be added to the U_B term.

According to the GUM recommendations, all discovered bias sources must be corrected. The bias becomes insignificant, and the B term can be neglected.

Applying the UM, the first step is to correct for bias (if possible and recommended). Having 2 sets of error parameters, according to the presented error model and 2 TE equations, 2 different UM equations can be obtained.

The first, calculated in repeatability conditions, starts from Equation 7. The bias, which must be corrected in the first step, is the average of the bias measurements in the same EQA round (B-EQA). The bias value after correction can be considered negligible. The uncertainty of the correction can be determined in 2 ways (bottom-up and top-down methods). In the bottom-up approach, the bias uncertainty is calculated as the sum of uncertainties of the reference value and the uncertainty of the measurement in repeatability conditions (s_r); in the top-down method, as the sum of the uncertainty of the reference value and the root mean square of the corrected bias values ((BEQAi -B-EQA) (Where BEQAi are the individual bias results.) The 2 methods give similar results (within the statistical methods’ limits) because RMS(BEQAi -B-EQA)≈s_r.

Where n is the number of measurements, srmax is the estimated maximum value of the s_r, uCref is the uncertainty of the nominal value of the reference material, and urec is the uncertainty of its reconstitution, equal to the uncertainty of 2 volume measurements (≈2 × 0.5% — the accuracy of the actual pipettes is 0.5%‐0.6%). However, u_rec is not a negligible value; the recommended uncertainty equations do not include it. The division of the uncertainty of the bias with n was necessary because the bias value is a mean. As the number of measurements increases, the uncertainty of a mean value decreases n times). An equivalent equation was published by White [32] and in Nordtest TR 537 [22], except for the neglected u_rec value.

In repeatability conditions, the bottom-up and top-down methods, within the limits of the statistical measurements, give similar results for the uncertainty because RMS(BEQAi-B-EQA)2n≈ sr2n. The SD is an RMS of the deviations from the mean, with a correction: n is substituted with n-1. If a calibration is made between measurements, the top-down uncertainty will be bigger due to the bias variability. This is similar to the case of Mg: Table 3 and Figure 3.

Unfortunately, Equation 13 has no practical value in the clinical laboratory. There is a significant delay between the measurement and the moment when the results are obtained. In the meantime, reagent changes and calibrations are done, and the bias is changed. A constant cannot correct a variable. In addition, there is insufficient information to determine whether the bias is constant or proportional. Due to bias variability, the calculated uncertainty value cannot be used for extended time frames. UM is a long-term parameter.

The situation changes over time. The B-RW=CCSE is a constant, which can be corrected without contradicting quintessential principle 3.

Each EQA round measures a different bias using different reference materials with different u_Cref and with varying errors of reconstitution. The average of the measured bias values in different rounds is B-RW (absolute mean bias). Starting from Equation 8, with bottom-up and top-down approaches, we obtain:

Actual recommendations suggest calculating the uncertainty of the bias correction as the root mean square (RMS) of the bias values [22], but this equation assumes “…a variance of bias based on assumed mean of zero” [28].

The assumption is only valid, and the equation is correct if the bias is corrected efficiently. If not, RMS_bias is not only u_B but includes the mean bias in its expression.

Which is only correct if we accept the quadratic addition law between bias and its uncertainty (questioned by the debates in the literature).

If n=1 and the B-RW term is added quadratically to the other terms under the square root, the top-down term of Equation 14 is equivalent to the equation proposed by Nordtest TR 537, except for the missing RMSuCref2term. The equation in Nordtest TR 537 expresses the uncertainty of a single value, not the uncertainty of a mean (n=1) [22].

In repeatability conditions, the u_Cref and u_rec caused an unknown bias in the bias value, and these terms expressed our uncertainty about this value. Making more measurements decreases the influence of random errors; however, our uncertainty about the reference value remains unchanged. In the case of different EQA rounds, these biases of the bias values are variable and contribute to the bias variability. Therefore, to avoid redundancy, the RMSuCref2 term (included in RMS_B) must be eliminated from the top-down equation. While uCref orRMSuCref,) are bottom-up parameters, whereas the RMS_B is a top-down parameter, considering the consequences of the individual sources. Their mix causes redundancy in equations.

While the uncertainty caused by the bias variability (the s_VCSE) term) is included in both expressions in the sRWmax, the top-down values are significantly bigger than the bottom-up ones. In the meantime, in the case of calculations based on internal QC data, there are no significant differences (as in the case of EQA in a single round).

Table 6 presents the differences between the bias uncertainty results obtained with top-down and bottom-up methods. Similar calculations based on internal QC data and those from a single EQA round are provided for comparison. The number of measurements is considered n=1 in all calculations for the sake of better comparison.

In long time frames (more EQA rounds), the uncertainty is more significant than in a single round because variable bias values are measured. The differences between internal QC and the bottom-up method are not significant and are caused by the u_Cref and u_rec included in the bottom-up uncertainty. Except for potassium, in almost all cases, the top-down method gives a bigger value due to the difference between the declared and true u_Cref values.

In the bottom-up equation, the declared uCrefvalue is substituted. In the meantime, the top-down equation includes the real one in the RMS_B term, causing the differences. There are 2 conditions for a correct EQA. The sample must be commutable and must have predetermined values [33]. Neither of these conditions is fulfilled in EQA with surrogate reference values (the mean of participants). The equation used to evaluate the uncertainty of the reference value may only be correct if the peer groups are homogeneous, and they are not [34]. This error causes an additional and significant uncertainty.

The uncertainty equations can be corrected by eliminating the confusion hidden in the bias definitional uncertainty. A key conclusion: only the long-term mean biases can be corrected efficiently. Correcting individual values is risky due to the variability of bias and delay. The actual EQA bias determinations conceal a significant source of uncertainty: the uncertainty of the surrogate reference values. Not even the bias variability can explain the differences between the uncertainties calculated in single and multiple EQA rounds, as well as between bottom-up and top-down methods. Following studies are necessary to sustain the former theoretical conclusions; the proofs and discussion do not fit within the limits of this study.

The existence of the VCSE suggests a change in the point of view. Even after correction, the bias reappears due to its variable properties. The confusion between the bias and the mean of the variable bias is a source of error.

The (immediately) incorrigible biases bring to attention the debates about including uncorrected biases in uncertainty equations. If they are not corrected immediately, the mean bias must be included in the uncertainty budget.

We cannot quantify the preanalytical and postanalytical errors in the QC, nor can we measure the method and matrix errors only in EQA. The analytical errors detectable in IQC are:

In the case of a laboratory with air conditioning, using liquid phase reactions in thermostated conditions, the influence of the environment is quasi-negligible. The laboratory and human errors are redundant in the list. Neither specific laboratory nor specific human errors exist. Laboratory and human errors are a sum of preanalytical, noninstrumental, and instrumental errors.

We can include rounding errors in the instrumental error category. They have similar properties (both are nonspecific and time-invariable).

The instrumental errors are linked to the construction and functionality of the analyzer. They are always constant and nonspecific (assumptions 4 and 5). An instrumental failure will influence all measurements in an aberrant manner. Instrumental errors may be the sources of the constant error components, but never of the variable ones.

There are only 2 noninstrumental error sources: the reagent stability and the calibration graph (see quote from HKALM recommendations [30]). Both are specific and variable. Each measurement has its specific reagents with variable properties. Producers only guarantee that we can successfully recalibrate the reagents in the validity term, not that the properties remain constant. Random changes in the reagent properties contradict the laws of chemistry. The changes are always unidirectional and gradual. The variation is not perfectly linear; however, linearity is an acceptable approximation in short intervals. The phenomenon is consistent with the linear bias variation model of J. Krouwer (B=B₀+b₁t) [19]. It applies only to time frames that do not include human interventions (such as calibrations, reagent changes, or control bottle changes).

The noise of the RE usually covers the drift. We can observe only significant drifts (if the mean change is >1.5s_r); however, all contribute to the increase of the s_RW. The significant drifts cause R_7T, R_2-2S, and R_1-3S violations.

Many authors consider the calibration a quasi-perfect process [35]. Raúl Girardi, on an IFCC webinar (Metrology and uncertainty, August 21, 2021), even presented an alternative equation that reduced the bias uncertainty to the nominal value uncertainty of the reference material. Other authors share similar opinions [29]. Such an attitude neglects the most significant causes of the calibration graph error. On one hand, the measured reference material does not have the same composition as the material analyzed by the producer. It undergoes a lengthy process before being measured. Even if we neglect human errors (stability, homogenization, temperature errors), the reconstitution includes 2 volume measurements: one at the producer and another at the user. Badrick [36], referring to the Tietz Textbook of Clinical Chemistry [37], underlines:

Each reconstituted reference material bottle has a different concentration. We generate similar systematic errors until we use the same reconstituted calibrator bottle.

On the other hand, calibration is a measurement subject to systematic and random errors. In a linear calibration, we make 2 × 2 measurements and calculate the slope factor as a difference. We make calibrations in repeatability conditions. The average calibration random error is ≈2∗12CVr=1CVr (the error of the null-point absorption A₀ was neglected in the former estimation).

The calibration error introduces a systematic error in measurements, which remains constant within the time frame between calibrations, but each calibration induces an unpredictable variation in the systematic error. The result is a randomly variable systematic error. The phenomenon is consistent with the models presented by Marquise [16] and Magnusson et al [22]. We can observe only the significant shifts in the mean (>1 sr).

Because we can observe only the significant drifts and shifts, we tend to consider the bias variations unpredictable (case b of unpredictable), contradicting the bias definition (predictable).

The mostly predictable character of the bias suggests that a focus change in the internal QC is necessary. The QC system must also have a strategy to predict bias variations and detect unpredictable changes.

The VCSE(t) is a time-variable function that describes the bias variations around the CCSE. It is a variable error component but different from RE. The RE changes unpredictably from measurement to measurement; meanwhile, VCSE(t) remains quasi-constant on a given day. The bias variations have unequal cycles, while the long-term mean of VCSE(t) is 0. Its values are not normally distributed.

VCSE(t) has 2 primary sources. Both are noninstrumental and specific. The variation in reagent quality follows a predictable pattern, and this variation is also predictable. After the calibration, we can predict the mean and bias variation from the old and new calibration parameters.

cbefore calibrationcafter calibrationFcalbefore calibrationFcalafter calibration

VCSE(t) is a mostly predictable phenomenon. We can correct it for a moment, but not definitively eliminate it. In repeatability conditions, the VCSE(t) is nonsense. The differences between CCSE (B_RW) (B_RW: long-term mean bias, measured in RW conditions, a constant), VCSE(t), and RE are presented in Table 7.

We cannot ignore the differences between VCSE(t), RE, and CCSE. If, and only if, we are conscious that both B_r(t) and s_RW contain VCSE(t), it is not an erroneous practice to measure RE and VCSE(t) together and to include VCSE(t) in s_RW. The origins of the equations must be known, as well as the risk of redundant use.

The determination of CCSE≡B-RW is possible using the control results and Equation 4. Such CCSE values only show the difference between the mean of control measurements and the target specified by the producer. We can obtain an absolute value of CCSE from the percent expressed EQA results. Due to the low number of measurements, the value has significant uncertainties.

The comparison between the 2 types of CCSE values is not the task of this study. A single mention: the difference between the 2 CCSE-s is predictably constant until we use the same control material. Another study is necessary to verify this prediction.

As a consequence of the constant difference, the VCSE(t) measured in internal QC and EQA predictably is the same; however, the statement needs confirmation. The accurate determination of the VCSE(t) function has a high cost-effectiveness ratio and negligible practical importance, mainly due to its short validity term. The computer-assisted estimation of the run means (Figure 5) is a promising solution, but needs a separate study to confirm its efficiency.

The same statement applies to the s_VCSE. To estimate s_VCSE using Equation 2 is more practical than calculating it from daily VCSE(t) values [27].

The increased s_VCSE/s_r ratio indicates wrong internal QC decisions (delayed calibrations); however, we can also use the s_RW/s_{r ratio} without calculating s_VCSE [23].

The paramount importance of VCSE(t) and s_VCSE lies in the distinction between SD and bias types, not their absolute value. We do not need their accurate values; we do not make decisions based on them. These 2 parameters are always included in B_r(t) or s_RW. However, we must be aware of where they are hidden. Highlighting VCSE(t) and s_VCSE in equations helps us avoid redundant use.

The original aim of this study was to draw attention to the neglected VCSE(t) and s_VCSE. The proposed new error model (Figure 6, Equations 2, 3, 5) also uncovers the weaknesses of the actual Westgard rules–based internal QC system. By distinguishing the biases measured in repeatability and reproducibility within conditions (B_r[t] and B_RW), 2 sets of error parameters are obtained (B_r[t] and s_r, respectively, for B_RW and s_RW). The link between them is VCSE(t) and s_VCSE (which are usually hidden in the B_r(t) and s_RW). Avoiding redundant use by highlighting VCSE(t) and s_VCSE in equations is not the only advantage of the proposed error model. The non-Gaussian distribution of the VCSE(t) values explains the non-Gaussian distribution of the long-term QC data [3,4] and the significant monthly variability of s_RW [5], which contradicts the laws of the normal distribution. The Gauss-Laplace equation is valid only under constant repeatability conditions (if the mean remains constant). Therefore, s_r is the correct estimator of the Gaussian σ parameter and the mean RE. While the sources of specific bias variability (reagent property and calibration parameter changes) are known [16,18,19,22,23], the sources of specific RE variability cannot be identified. All identifiable RE sources are linked to the inconsistent functionality of the instrument and, therefore, are constant (nonvariable) and nonspecific [38]. In contrast to s_RW, s_r is invariant within the limits of accuracy of the statistical methods (Vandra’s unpublished data [38]).

The constant RE (s_r) questions the efforts of Westgard et al [1,8,39] to detect variations in RE. The primary objective of internal QC is to detect risky variations in bias, and, by definition, the bias between human interventions is predictable [14]. Anyway, according to Westgard JO [40], the QC rules cannot be applied across corrective actions. The objective change changes the way of thinking in QC. The focus is not on the immediate detection of unpredictable changes, but rather on following tendencies in bias to predict the moment when the run bias will reach a critical value.

There are 4 different mechanisms to reach a critical bias, imposing different decision strategies, because the QC rules (especially the cross-run rules: R_4-1S and R_10X) have different efficiencies in each case.

The immediate error detection is compulsory only in cases 1 and 4. In cases 2 and 3, bias is predictable. However, the QC system must be able to detect changes in the tendencies.

GRD Jones was the first to notice the difference between cases 1 and 4 [41], highlighting that in case 1, the cross-run rules (R_4-1S and R_10X) cannot be applied due to a lack of data. However, he did not observe the hidden assumption in Westgard’s calculations, which falsely assumes a constant bias in all runs. While focusing on immediate error detection in case 4, the calculations are based on case 2 (constant bias). If the cross-run rules detect a constantly critical bias, it indicates delayed, rather than immediate, error detection. In cases 3 and 4, the previous bias value is lower than in the last run, and the efficiency of the cross-run rules was overestimated.

In cases 2 and 3, the QC rules are applied repeatedly, increasing the efficiency of error detection. Instead of applying the R_1-3S rule, the R_{1 of n-3S} rule is used de facto. All runs are only accepted if neither of them violates the 3 SD decision limit.

The former observations impose the reevaluation of the efficiency of the Westgard rules in a subsequent study.

The Westgard rules are only correctly applied if the QC graphs are designed with σ or the correct estimator. As previously concluded, the correct estimator of the σ parameter and the mean RE is s_r, and Westgard’s assumption that s_RW≈σ is false. Not else, but Westgard and Groth [39] acknowledged that:

Considering the s_RW//s_r ratio, this results in an overestimation of the decision limits 1.5‐2 times. Respecting Westgard’s recommendations, intending to apply the R_1-3S rule, de facto, we use the R_1-4.5S or the R_1-6S rule (3s_RW ≈ 4.5‐6s_r.) This contradiction and overestimation explain the existence of the statistically impossible graphs observed in practice (mentioned in the Introduction).

Correcting the estimator of σ (from s_RW to s_r) requires recalculating all parameters that include SD in their equations: TE, MU, sigma metrics, the critical SE, not just a change in the design of the QC graphs. This means an entirely new QC system, using different rules and strategies.

Sounds bizarre, but according to calculations based on normal distribution tables, a correctly applied Westgard rules–based QC system (designing the graphs with σ) would be dysfunctional due to several false alarms. Despite the efforts to correct them, half of the monthly biases measured in the internal QC are around 1 s_RW or bigger, and two-thirds of them are bigger than 1s_r. According to quintessential principle 2, it cannot be corrected by calibration for smaller biases than the average calibration error, questioning another assumption of Westgard et al [39]: the assumption of error-free calibrations. According to Vandra [38], the average calibration error is ≈1‐2 s_r (consistent with the observed monthly biases). If such biases are incorrigible, the QC rules must avoid alarms in these cases. The correctly applied Westgard rules alarm in the first run only by exception if the bias is 0. The statement is not valid if B>1s_r and the rules are applied in several runs.

This study is a theoretical one. It aims to draw the attention of the scientific community to the fact that the VCSE is a neglected phenomenon and a source of several errors. Because it is hidden in the inaccurately defined bias and the s_RW, there is the risk of its redundant use in equations. This study also aimed to uncover the primary sources of bias variations (both present in the literature in mosaic pieces), propose corrected equations, and describe the properties of the VCSE. Because several problems were uncovered, the proofs, based on computer simulations and real-life data for each issue, neither fit within the limits of a single study nor are consistent with the declared aims. To analyze them, subsequent studies will be necessary in the future. This study intends to be a starting point for building a new QC system based on a different error model, a different strategy, and a rule system. The theoretical foundations, description, proofs with computer simulation, and real-life data do not fit within the limits of this study.

The time variability of bias is a well-known but neglected phenomenon. A variable bias does not fit into the classical error model. If bias has variations, a question arises: Which bias is being referred to? A new error model was obtained by (1) separating the bias into a constant and a variable subcomponent and (2) distinguishing between bias measured in repeatability and reproducibility within laboratory conditions. The error model is consistent with similar attempts found in the literature; however, it questions the theory of transformation of variable biases into random errors (based on an inaccurate definition of ‘random’ in VIM), which forces the VCSE into the Procrustes’ bed of the old error model. The author proposed definitions consistent with the VIM 2.17 definition of the SE and abbreviations consistent with those used for SD (B-RW, B_r(t)).

The bias variability has 2 sources. Both are noninstrumental and specific to each measurement, and neither causes normally distributed biases. One is reagent instability, and the other is human intervention, including reagent changes and calibrations. Reagent instability causes gradually increasing, quasilinear biases, whereas calibrations result in alternation between constant periods with random shifts in the calibration parameters. Computer simulations and real-life QC data presented in this study support that these are real sources of bias variability.

The 2 phenomena occur simultaneously, resulting in sawtooth-like variations in bias. In the time frames between human interventions, the biases are predictable. However, they are hidden behind the noise of the RE. Without computer assistance, we can observe only significant shifts and drifts. For this reason, the increase of the SD in longer time frames was erroneously considered unpredictable, with an unknown cause (type b of unpredictable). An unpredictable bias contradicts its definition in VIM.

We must change our way of thinking in the QC by focusing on predictive actions instead of corrective ones.

The properties of the CCSE, the VCSE(t) function, and the RE differ, justifying the distinction between them. Accurately determining the SE subcomponents theoretically is possible; however, it has a high cost/effectiveness ratio. The significance of their separation is that they help us avoid the redundant use of the VCSE(t) classically hidden in B_r(t) and s_RW.

Two sets of error parameters are obtained by separating biases measured in repeatability and reproducibility within laboratory conditions. We must determine the parameters under the same conditions we use them. UM calculations must be based on parameters determined under reproducibility within laboratory conditions, whereas internal QC decisions must be based on parameters determined under repeatability conditions. This conclusion is thought-provoking because it contradicts the recommendations for designing the Levey-Jennings graphs based on the SD calculated from long-term control data. In the meantime, the calculations behind the Westgard rules assume pure RE.

The actual Westgard rules–based internal QC system is not consistent with two quintessential principles valid in all sciences:

The proposed error model uncovered several false assumptions behind the actual Westgard rules–based QC system.

The false assumptions 6 and 7 cause 2 compensating errors. The compensation explains the long-term success of the Westgard rules. If we use s_RW in the design of the Levey-Jennings graphs, we use larger, increased decision limits, de facto applying different rules (eg, the R_1-5S rule instead of the intended R_1-3S). As a consequence, the alarms for incorrigible biases become less frequent. However, this compensation is not accurate. The observed statistically impossible QC graphs sustain the overestimation of the RE by the s_RW.

Based on the proposed error model, correcting the former false assumptions, and considering the 4 different decision situations, the Westgard rules–based QC system must be mathematically reevaluated. It can be predicted that patching it is not a solution, and a new QC system is necessary, based on the s_r, and the avoidance of alarms in the case of incorrigible biases.

The proposed error model also suggests corrections to the MU equations. MU is a long-term parameter, and therefore, its equation must be based on long-term parameters. The uncertainty of the inaccurately defined bias (Which one?) must be substituted with the uncertainty of the long-term mean bias, measured in reproducibility within laboratory conditions (U(B-RW )), and must be considered the uncertainty caused by the variability of s_RW, substituting it with its maximal value in the MU equation.

Furthermore, the proposed error model, together with quintessential principle 1 (that all parameters must be determined under the same conditions under which they are used), explains why the more correct MU theory cannot substitute for TE in internal QC decisions. MU is a long-term parameter, while internal QC decisions are made under repeatability conditions.