Utility of a panel of sera for the alignment of test results in the worldwide multicenter study on reference values

Introduction

Plans for a worldwide multicenter study on reference values were developed progressively since 2010 by the Committee on Reference Values and Decision Limits (C-RIDL) of the International Federation of Clinical Chemistry and Laboratory Medicine (IFCC), with the participation of the USA, the UK, Turkey, Japan, China, Saudi Arabia, India, South Africa, and the Philippines. Its objectives are (1) to establish country-specific RIs from 500 or more healthy individuals in a harmonized manner using a common protocol (2) to make test results comparable across participating countries through the common measurement of a specified panel of sera on the basis of linear regression analysis, and (3) to explore sources of variation of the aligned test results using information obtained from a questionnaire given to each individual in the pool.

The detailed protocol, standard operating procedures (SOPs) and the questionnaire to be used for subject recruitment, sample collection, specimen processing, measurements, and data analysis have been discussed in the companion paper [1]. The schemes for implementing the adopted study were as follows:

1. Each country will conduct its own multicenter study to derive country-specific RIs and explore sources of variation of the test results relevant to that country. The target analytes and questionnaire items, which are not included in the general protocol, can be varied according to local needs for the survey.

2. Collaborating local laboratories in each country will recruit appropriate healthy volunteers and collect and process specimens as per the SOPs.

3. A centralized measurement scheme will be used to eliminate variation attributable to differences in analytical methods. One laboratory (or two) will act as a central laboratory in each country, receive specimens from local laboratories, and carry out collective measurement. The central laboratory can use any assay platform.

4. For standardized analytes, it is requested that the laboratory ensure that RIs are traceable to the RMPs for standardized analytes through the measurement of certified reference materials (CRMs) or value-assigned sera provided by reference laboratories listed by the Joint Committee for Traceability in Laboratory Medicine.

5. For non-standardized analytes, RIs determined centrally will be converted to those of each participating laboratory based on the results of local comparative measurements (cross-check testing) using aliquots of the specimens taken for the study.

6. For worldwide comparison, test results will be made comparable across the collaborating countries by the measurement of the same panel of sera by all laboratories.

Among these various schemes, the successful alignment of test results among countries is of utmost importance to allow the comparison of country- or region-specific reference intervals (RIs). This will depend largely on the reliability of the comparison of test results on the panel of sera.

In this study, a panel of sera from 80 healthy volunteers was freshly prepared and then distributed to four reference laboratories in Turkey, Japan, and the US (two sites). All samples were assayed twice in each laboratory, 1 week apart, for 45 commonly tested analytes (general chemistry and immunoassays). The practicability of converting test results or RIs from one laboratory to another was evaluated based on variability of the slope and converted reference limits as a function of the required sample size in the panel.

Materials and methods

Data analysis

Cross-comparison of test results among laboratories

There are a variety of ways for expressing the linear functional relationship of test results between any two laboratories [2–7]. Principal component regression (PCR) requires the assumption that the slope of the regression line equals 1.0 and the use of equivalent scales and units of measurement in both laboratories for use in method comparison, whereas reduced major axis regression (MAR) does not require these assumptions [2, 3]. Deming regression (DR) [7] overcomes the problem of PCR by introducing an error ratio (λ) of test results of one laboratory to the other. However, determining λ requires replicate measurements of each specimen in both laboratories. Therefore, the most appropriate and generally applicable method of regression for the purpose of this study was MAR [2]. It is also known as geometric mean regression [6] or least-square product regression [2] and is expressed as follows, with , , and b denoting the means of x (test results from laboratory X), y (corresponding results from laboratory Y), and the slope, respectively:

Slope b is determined by the following formula:

where n, S_xx, and S_yy denote the sample size and sums of squared deviations of values from laboratories X and Y, respectively.

Using the regression equation (1), the lower limit (LL₀) and upper limit (UL₀) of an RI derived using test results (x) of laboratory X can be converted to those of laboratory Y (LL and UL) using the following formulae:

The precision of the conversion can be evaluated from the magnitude of standard error (SE) of the slope b [SE(b)], LL [SE(LL)], or UL [SE(UL)], where the SE in this context implies standard deviation (SD) of any summary value X (statistics, such as b, r, LL, UL).

Assuming equal variance of the data points (homoscedasticity) around the regression line for the range of regression and an equal ratio of inherent errors on both x and y, SE(b) can be computed by the following formula using Pearson correlation coefficient r and the data size n. This formula has not been described in the past [2–6]; it was derived in this study, as shown in Appendix A. SE(b) can be standardized by taking its ratio to b and expressed as CV(b) (Figure 1)

Figure 1

The concept for CV(b) as a measure of precision of regression line.

In estimating the regression line using reduced MAR, the SE of b is given as a function of the correlation coefficient r and data size n. CV(b) can be a marker in predicting the precision of converting the RI. In the graph, the range of fluctuation of the regression line according to b±SE(b) is shown by the red line.

SE(LL) and SE(UL) can also be calculated theoretically using the formulae shown in Appendix B by specifying x=LL₀ and x=UL₀, respectively, together with well-known parameters of linear regression analysis: r, n, SD of the observed x (s_x), and SD around regression line computed by the ordinary least-squares method (s_LS, often denoted as s_y,x). The computed values of the precision parameters CV(b), SE(LL), and SE(UL) were compared with those obtained empirically by the conventional method using the bootstrap method (8). In the computation, LL₀ and UL₀ were derived from the test results (n=80) of laboratory X as x̅ ±1.96s_x.

To standardize the variability of SE(LL) or SE(UL), we considered two methods. One is to take its ratio to the SD comprising RI (SD_RI), and we designated it as the SD ratio (SDR) of LL or UL (SDR_LL and SDR_UL, respectively) (Figure 2).

Figure 2

The concept of SDR as a measure of precision for converting the RI.

The RI determined by laborator X (LL₀–UL₀) is converted to that of laboratory Y as LL–UL using the reduced major-axis regression line. The SEs of LL and UL [SE(LL) and SE(UL), respectively] can be estimated by the theoretical formulae. SE(LL) and SE(UL) are standardized as a ratio to the SD comprising the RI (SD_RI). We call the ratios SDR_LL and SDR_UL.

where SD_RI can be obtained as the width of the RI after conversion, UL−LL, divided by 3.92. SD_RI corresponds to , to specify the level of analytical bias (B_A) in clinical chemistry. SD_I and SD_G denote within- and between-individual SDs of test results. The following thresholds of B_A specified by Fraser [9] are commonly used in clinical chemistry: B_A≤0.25 is regarded as desirable performance, and B_A≤0.125 as optimal performance. In analogy, we regarded SDR≤0.25 as the level of precision allowable for converting the RI, and SDR≤0.125 as optimal precision.

To avoid the influence of outlying points in the regression, the SD of data points along the minor axis of the ellipse (SD_Min) perpendicular to the regression line was computed, and any point four times SD_Min or more away from the center along the minor axis was excluded from the computation.

When the distribution of observed test results deviates from a Gaussian distribution, the theoretically derived CV(b), SDR_LL, and SDR_UL can be biased partly due to heteroscedasticity of data points around the regression line (more scattering of data points toward the tail of the distribution). To examine the effect of Gaussian transformation on the regression statistics, we chose the following analytes whose values usually show a distribution with a long right tail: TBil, Glu, TG, AST, ALT, LD, GGT, CK, Tf, VB12, AFP, CEA, CA19-9, CA125, PSA, estradiol, PRL, LH, FSH, TSH, FT4, and insulin.

SDR_LL or SDR_UL based on the regression line derived after logarithmic (log) transformation can be computed from LL, UL, SE(LL), and SE(UL) using the following formulae involving exponential function for back-transformation, where LL^T and UL^T represent LL₀ and UL₀ converted using the regression line and y̅ and x̅ represent the averages of log(x) and log(y) for all data points.

LL=exp(LL^T)UL=exp(UL^T)

When a general power transformation (x^p) is used to convert the distribution of test values into the Gaussian form, the above formulae should be modified at the back-transformation step using the value to the power of 1/p.

Resampling studies for evaluating the minimum sample size required for the panel

For each of the pairwise comparisons among the four laboratories, consisting of 80 pairs of test results, a subset of the results with data size of 20, 30, 40, 50, 60, or 70 was randomly sampled with replacement, and CV(b), SDR_LL, and SDR_UL were computed by reduced MAR. In any case, the LL₀ and UL₀ required for computing LL and UL were fixed to the mean±1.96 SD derived from all test results (n=80) of laboratory X.

Results

Comparison of test results across the four laboratories

Typical examples of the comparative measurements of the panel (n=80) among the four laboratories are shown in Figure 3 for two analytes, HDL-C and TSH. The full results are available in Supplementary Figure 1 composed of 45 correlation matrix (8×8) graphs. Because each laboratory (A, B, C, and D) measured the panel twice 1 week apart, the number 1 or 2 attached to the laboratory name indicates the first or second measurement. The regression lines (depicted in solid lines) were computed by reduced MAR. The broken lines represent the line of concordance (y=x). The colored blocks of the matrix indicated reproducibility of test results within the same laboratory.

Figure 3

Examples of correlation matrix graphs of method comparison among the four laboratories.

Laboratories A, B, C, and D measured the panel of 80 sera twice, 1 week apart. The colored blocks indicate the within-laboratory variability of the test results. Within each graph, the line of concordance (y=x) is depicted by the broken line.

The following findings can be noted from Supplementary Figure 1:

• The reproducibility of two measurements obtained over a 1-week period within the same laboratory is excellent by visual inspection except for those of Na, Cl, CA19-9, FT4, and FT3 for which a large scatter around the regression lines was observed.

• The regression lines drawn between the results from any two laboratories were very close to y=x for almost all of the standardized analytes except for those showing variable degrees of bias, judging from a wide range of slopes (cf. Table 2), such as Na, Cl, Ca, HDL-C, LDL-C, ALP, and C4.

• For the non-standardized analytes, regression were approximately y=x for IP, Fe, C3, AFP, CEA, CA125, and PSA, whereas variable degrees of bias were noted for ferritin, folate, VB12, CA19-9, estradiol, PRL, LH, FSH, FT4, PTH, and insulin.

• Although most points are close to the regression line, a few to several outlying points were observed in the plots for CEA, CA125, folate, PRL, and insulin.

Comparison of regression parameters representing the precision of converting RI

Table 2 shows the summary statistics obtained from the dataset shown in Figure 3 (and Supplementary Figure 1), which represent the closeness of fit of data to the regression lines and precision of conversion of the RIs when all test results (n=80) were used for the comparison. For each comparison between laboratory X and laboratory Y (X, Y=A, B, C, or D), statistics r, b, CV(b), SDR_LL, and SDR_UL were computed.

Table 2

Summary of regression line statistics for all pairwise comparisons of test results.

All sera in the panel (n=80) were measured twice by the four laboratories (or by three laboratories for test items marked by superscript a), and all pairwise comparisons were made excluding within-laboratory comparison. Therefore, the total number of comparisons for each analyte was either 24 or 12. Columns 2, 4–6, and 10–12: The statistics with gray headers are medians of those obtained by the all pairwise comparisons. Column 4: The CV of the regression line slope, CV(b). Values of 5.5%–11% and >11% are marked by light and dark gray, respectively. Columns 5–7: The SDR_LL and SDR_UL represent the ratios of SD of the lower and upper limits (LL and UL) to the SD comprising RI, or (UL−LL)/3.92. The aveSDR represents the average of SDR_LL and SDR_UL. Values of 0.125–0.25 and >0.25 are marked by light and dark gray, respectively. Column 8: The ratio of SDR_UL to SDR_LL above 1.2 is marked. Column 9: Skewness (Sk) was determined from the distribution of test values measured by each laboratory. The average of Sk of the four laboratories is shown. Graded gray colors indicate 0.4<Sk≤0.8 and 0.8<Sk. Columns 10–12: CV(b), SDR_LL, and SDR_UL were determined by the bootstrap method by resampling 80 points for 500 times. Columns 13–15: The ratios of statistics obtained by the bootstrap method to those computed by the theoretical formulae. Values ≤0.8 or ≥1.2 are marked by gray background.

The total number of pairwise comparisons made (excluding within-laboratory comparison) was 24 [=[8×(8- 1)−4×2]/2] for the analytes measured by all four laboratories and 12[=[6×(6−1)−3×2]/2] for those measured only by three laboratories. The statistics (listed in columns with gray headers numbered 2, 4–6, and 10–12) are the medians of those obtained by all comparisons.

The correlation coefficients r were generally very high (r>0.95). Analytes with r<0.9 include Alb, Na, Cl, Ca, CA19-9, FT4, and FT3, indicated by a gray background. The wider 90% range of variation of the slope b shown in column 3 implies either a lack of close correlation or the presence of between-method bias in the test results or both. The presence of both conditions implies that the conversion of test results between laboratories is not feasible.

The skewness (Sk) of distribution was computed from the test values of each laboratory, and the average Sk of the 4 (or 3) laboratories is shown in column 3. The distribution was regarded as nearly symmetrical when |Sk|≤0.4, and values of Sk>0.4 are shown with a gray background.

SDR_LL and SDR_UL, which were above the optimal (0.125) or allowable (0.25) limits, are also shown with a gray background. Analytes with either of the SDRs above the allowable limits using all test results (n=80) include Na and CA19-9; additional analytes with either of SDRs above the optimal limits are Cl, CRP, FT4, and FT3.

The ratio of SDR_UL to SDR_LL shown in column 8 was noticeably higher when Sk is >0.4. Close correlation (r=0.874) was observed between Sk and SDR_UL/SDR_LL, implying that SDRs depend on the distribution pattern.

Meanwhile, it was noted that CV(b) shows a high correlation with both SDR_LL and SDR_UL. Furthermore, the correlation is dependent on Sk of the distribution of test results, as shown by analyzing the correlation after stratification of 45 analytes by Sk (Figure 4). It was also found that there is a closer correlation of CV(b) with the average of SDR_LL and SDR_UL (aveSDR, shown in column 7), which was less affected by Sk than was the individual SDR. The relationship between the two statistics was computed as follows using reduced MAR:

aveSDR=0.001+0.0222 CV(b)

Figure 4

Relationships among SDRs and CV(b).

From the data listed in Table 2, the relationships between CV(b) and SDRs (SDR_LL, SDR_UL, or the average of the two) were examined after partitioning the data for 45 analytes by average skewness (Sk) of their distributions into three groups (|Sk|≤0.4, 0.4<Sk≤0.8, and 0.8<Sk). The regression lines drawn were computed by reduced MAR.

From this equation, SDRs of 0.125 (optimal limit) and 0.25 (allowable limit) was calculated as equivalent to CV(b) of approximately 5.5% and 11%, respectively.

For validation of the theoretically computed values, CV(b), SDR_LL, and SDR_UL were also estimated by the bootstrap method by resampling 80 data points 500 times. The results were designated as bCV(b), bSDR_LL, and bSDR_UL and are shown in columns 10–12.

Their ratios to respective values calculated by the formulae shown in Appendix A are listed in columns 13–15. It was found that bCV(b)/CV(b) and bSDR_UL/SDR_UL depend on Sk; when the distribution is not skewed (|Sk|<0.4), both ratios are nearly 1.0, indicating that the value derived by the bootstrap method matches well with the calculated value from the formula. When the distribution is skewed to the higher side (Sk>0.4), both ratios increase in proportion to Sk.

Meanwhile, bSDR_LL/SDR_LL is not much associated with Sk. This implies that skewness to the lower side of values does not affect the regression in estimating LL because of clustering of data to the lower side.

Resampling study to examine the effect of data size on precision of the conversion

From the original sets of 80 test results from laboratories X and Y, a sample subset (n=20, 30, 40, 50, 60, 70, or 80) was randomly drawn (resampled) with replacement, and the slope b by reduced MAR and converted LL₀ and UL₀ (LL and UL) were computed. This process was repeated 500 times, and CV(b), SDR_LL, and SDR_UL were computed using the theoretical formula.

Table 3

The effect of logarithmic transformation on the regression statistics related to the precision of the method comparison.

Skewness (Sk) was determined from distribution of test values measured by each lab. The average of Sk of the 4 lab is shown. Graded gray colors indicate 0.4<Sk≤0.8 and 0.8<Sk. CV(b) represents CV of the regression line slope computed by the theoretical formula. The median CV(b) of all pairwise comparison among 4 labs is shown. Values of 5.5–11%, and >11% are marked by light and dark gray. bCV(b) represents CV of the regression line slope empirically derived by the bootstrap method. The same scheme as CV(b) is used for listing of the values. SDR_LL and SDR_UL represent the ratio of SE(LL) and SE(UL) to the SD comprising RI, respectively, which were computed by the theoretical formula. The median SDRs of all pairwise comparison among 4 labs are shown. Values of 0.125–0.25 and >0.25 are marked by light and dark gray. bSDR_LL and bSDR_UL represent the ratio of SE(LL) and SE(UL) to the SD comprising RI, respectively, which were computed by the bootstrap method. The same scheme as SDR_LL and SDR_UL is used for listing of the values.

Table 4 summarizes the results showing the effect of sample size on three statistics of conversion. It tabulates the medians SDR_LL, SDR_UL, and CV(b) of all pairwise comparisons for each data size (12 combination for analytes measured twice by three laboratories, and 24 combinations for those measured twice by four laboratories). From our criterion for an allowable level (SDR≤0.25), the percentage of analytes with that level of precision does not change much for 20≤n≤60. Meanwhile, the percentage of analytes with the optimum level of precision (SDR≤0.125) declines appreciably for n=20. For CV(b), applying the above-mentioned criteria [allowable CV(b) as ≤11%; optimum CV(b) as ≤5.5%], a similar tendency was observed to that for SDRs.

Table 4

The sample size of the panel and precision of converting RIs [SDR_LL, SDR_UL, and CV(b)].

*1 SDR_LL or SDR_UL values between 0.125 and 0.25, and above 0.25 are marked by graded gray background. *2 CV(b) values between 5.5% and 11%, and above 11% are marked by graded gray background colors.

Assuming 40 sera as a practical size of the panel for use in the worldwide study, based on the allowable limits of CV(b), the reference values of Na, Cl, CA19-9, FT4, and FT3 (or 11% of the analytes examined) were not acceptable for use in comparison across the countries. At the optimal level of CV(b), 20% of the analytes, with the addition of TP, Alb, Ca, and CA125 to the above analytes, were regarded as not acceptable for comparison.

Discussion

The key strategy of the worldwide study on reference values is to measure a panel of sera in common and to align test results across countries on the basis of linear regression analysis. Thus, the objective of this preliminary investigation was to determine the most appropriate parameter along with its threshold value to judge its utility in converting RI among the collaborating laboratories. This study also intends to determine the most practical number of specimens (sample size, n) in the panel for use in the worldwide study.

There are several options for the statistical approach for the method comparison [2–7]. In the field of clinical chemistry, DR [7] has generally been considered as the most rational one because it can deal with the imbalance in the inherent errors of variables x and y. However, it requires an appropriate estimate of the error ratio (λ), especially when x and y variables are measured in different units or scales. Therefore, a replicate measurement is expected at the time of method comparison, which is not a common laboratory practice. Nevertheless, it is not reliable to estimate λ from the CV_A of each variable if not determined concurrently. Furthermore, λ is usually not constant over the range of regression. This situation necessitates the use of weighted DR [7] and proper derivation of the weighting factor, which further complicates the problem by requiring a larger number of measurements in replicate. Another downside of DR is that there is no mathematical formula to derive the SEs of the slope and other parameters. Their derivation depends on the use of the bootstrap method [8], which may not be reliable with a small sample size.

Meanwhile, the reduced MAR [2–7] gives the line of structural relationship equal to the reduced major axis of the ellipse delineating the scatter of data points. Its preferable property is that the regression is not influenced by the scale of measurement [2, 3] and thus can be used for comparison of test results for non-standardized analytes measured using two different units or scales. Furthermore, the SEs of the slope and point estimation based on the regression can be mathematically derived as we demonstrated in the appendices. Therefore, we regarded MAR as a practical method of choice for the purpose of the worldwide study.

The precision of the conversion using MAR depends on the sample size n and the degree of correlation of test results from the laboratories under comparison. Both parameters are contained in the mathematical formula to derive the CV of the slope:

CV(b), derived by the bootstrap method, was used in the 2009 Asian study to judge whether it was permissible to convert the centrally derived RI to that of each local laboratory [10, 11]. This conversion of RIs was provided as an option to the collaborating laboratories on an experimental basis. The number of sera actually measured for comparison in each local laboratory varied widely from nine to 73, with an average of 22.2. Therefore, we could not rely solely on the degree of correlation r to judge the appropriateness of converting the RI and found CV(b) to be useful in the judgment based on the knowledge that the CV of the slope by the least-squares method is well known to reflect both r and n. We empirically set the allowable level of conversion as CV(b)<10%, which roughly corresponds to r>0.96 (with n=10), r >0.90 (with n=25), or r>0.85 (with n=40), after inspection of all results of cross-check testing between the central laboratory and 44 local laboratories with regard to the stability of the regression lines. Incidentally, the cutoff value of 10% is very close to the allowable limit set in this study for CV(b) computed by the mathematical formula.

However, there was some criticism about judging the allowable level of conversion simply on the basis of r and n. Therefore, in this study, we also evaluated the SE of the converted reference limits LL and UL, SE(LL) and SE(UL), respectively, which were computed based on the mathematical formula we have derived in this study (see Appendix B for its proof). To standardize them, we opted to use their ratio to SD_RI, which is approximately 1/4 of the width of the converted RI and corresponds to gross between-individual SD, . The cutoff values for the ratios SDR_LL and SDR_UL were set in analogy to setting limits for analytical bias, with SDRs=SE/SD_RI≤0.25 as an allowable level and SDRs=SE/SD_RI ≤0.125 as an optimal level [9].

At the threshold of allowable precision (SDR_UL=0.25), assuming a standardized Gaussian distribution of test values (z) and the converted RIs of LL=−1.96 and of UL=1.96, the 90% CI of UL is computed as z=1.55–2.37 (1.96±1.645×0.25). This implies that when UL is computed as z=1.55 as an extreme case, we would expect a 3.56% absolute increase (or 2.42-fold relative increase) in false-positive cases using the RI: P(z≥1.55)−P(z≥1.96)=0.0606−0.0250=0.0356 (or 0.0606/0.025=2.42). Similarly, the UL of z=2.37 as an opposite case gives a 1.61% (0.0250−0.0089) absolute increase (2.81-fold relative increase) of false-negative cases. At the threshold of optimal precision (SDR_UL=0.125), the 90% CI of the UL is z=1.75–2.16, and thus, we would expect a 1.51% (0.0401−0.0250) absolute increase (1.60-fold relative increase) in false-positive cases at z=1.75, and 0.96% (0.025−0.0154) absolute increase (1.62-fold relative increase) in false-negative cases at z=2.16.

From this outcome analysis of conversion error, the use of conversions at the optimal level of precision seems better in a practical sense, giving smaller rates of misclassification in using the RI than the larger error rates associated with use of conversions at the allowable level of precision.

An important issue we found in applying SDRs in the judgment of convertibility was that SDRs depended on the distribution pattern of test results for the panel. When the test results showed a symmetrical or Gaussian distribution (|Sk|<0.4), SDR_LL and SDR_UL were equivalent, but when the distribution was skewed toward the higher values, as is commonly seen in test results for TG, ALT, or GGT, SDR_UL tended to become larger relative to SDR_LL. Therefore, judgment on convertibility depended solely on the value of SDR_UL.

To confirm the appropriateness of the mathematical formulae to derive CV(b) and SDRs, the theoretical values were compared with values derived empirically using the bootstrap method. As expected, both values matched very well when test values used for the method comparison distribute in a symmetrical or Gaussian manner. However, when the distribution was skewed with a long tail toward higher values, the calculated CV(b) and SDRs tended to dissociate from those empirically derived by the bootstrap method.

From the study of log transformation of test results with skewed distributions, it was found that SDR_UL computed by the theoretical formula tended to underestimate the value compared with that derived by the bootstrap method. Therefore, for a skewed distribution, it is recommended to calculate the SDR_UL after the Gaussian transformation of test results or to estimate it based on the bootstrap method. Meanwhile, the SDR_LL for a skewed distribution was found to be overestimated by either of the methods, and thus, it is necessary to convert it into a Gaussian distribution for more reliable derivation of SDR_LL.

In contrast, the calculated CV(b) appears to be more robust and less affected by skewness than the SDRs. Therefore, CV(b) appears to be an index of choice in judging the practicability of converting the RI based on the regression line.

No rational theory for setting the threshold level of CV(b) is available at present. However, a very close relationship was found in this study between CV(b) and SDRs. We empirically derived a linear equation between CV(b) and aveSDR (average of SDR_LL and SDR_UL) from the entire set of data on method comparison for all 45 analytes. The equation can be approximated as SDR=0.22 CV(b), and SDR=0.25 and SDR=0.125 are equivalent to CV(b)≈11% and CV(b)≈5.5% in judging the allowable and optimal precision, respectively. It is important to note that this relationship holds true only when the values of the mean±1.96SD, as LL₀ and UL₀, are converted to obtain LL and UL (see Appendix B).

It is obvious that the more specimens there are in the panel, the higher the precision of the conversion. From the results of our simulation study shown in Table 4, we found that a majority of the analytes had optimum precision for conversion either judged by CV(b) or by SDRs with n≥60. The precision declines gradually over the range of 30≤n≤50. However, the average CV(b) and SDRs increase more sharply with n≤30.

The analytes we plan to measure in the worldwide study are numerous, and they are heterogeneous in their behavior in the method comparison. Therefore, there is no particular best sample size to cope with all situations. Linnet evaluated the sample size necessary for method comparison studies by simulation [12] and reported that the range (LL–UL) of values under comparison is very critical in determining the sample size. He concluded that a range ratio (maximum value divided by minimum value) of <2.0 requires a larger sample size for proper comparison. In our study, among the analytes whose CV(b) or SDR was relatively high, TP, Alb, Na, Cl, and Ca showed r<0.9, and the ratio of UL to LL, which corresponds to the range ratio, was <1.3. Therefore, the test results for those analytes must be compared using a large number of specimens, which may be an impractical number (n) as predicted from the theoretical formula for CV(b); i.e., assuming r=0.5, 0.6, 0.7, or 0.8, to achieve the allowable limit of CV(b)=11%, the n required is 250, 149, 99, or 49; for the optimal limit of CV(b)=5.5%, the n required is 994, 590, 346, or 188. As a result, these analytes are likely to be excluded from the comparison when the observed r is <0.8. We should rather standardize the test results of each testing center and compare the results without making an attempt to adjust values by method comparison.

The sample size of 40 that we adopted is equivalent to that stipulated in the EP9-A2 for Method Comparison and Bias Estimation Using Patient Samples [13]. The measurement of each specimen in duplicate is also suggested, in consideration of within-day variations of the assay. However, we consider it more important to measure the panel on separate days after dividing them into multiple parts to include both within- and between-day variations in the test results. In fact, we observed occasional bias in the repeated measurement of the panel 1 week apart, and the between-day components of error outweighed the within-day components. Therefore, we strongly recommend that the 40 sera be measured on 4–8 separate days in the worldwide study.

At the end of this article, we have to admit that our strategy to use the panel of sera for the alignment of reference values across the countries remains largely exploratory, although we provided theoretical formulation of the statistical approach for converting the values. Therefore, the approach may require a reevaluation from the practical point of view after their implementation in the ongoing worldwide study.