The data used in this study was collected from Cheng Ching General Hospital in Taiwan in 2011, with 3,532 valid samples obtained for measurement of LDL-C concentration. All subjects were 20 to 95 years old, with TG concentrations ≤ 400 mg/dL (n = 3,395; 96.1%) and > 400 mg/dL (n = 137; 3.9%). The subjects were classified into three groups according to age, i.e., younger (20–44 years old), middle-aged (45–64 years old), and elderly (65 years old and above). The subjects’ basic information with regard to TC, HDL-C, LDL-C, and TG is summarized in Table 2. The maximum and minimum of TG are 1252 and 22 mg/dL. Moreover, the maximum values of TC, HDL-C, and LDL-C are 569, 126, and 444 mg/dL, respectively, whereas the respective minimum values are 57, 3, and 20 mg/dL. Blood samples were taken from all the subjects, and after clotting at room temperature these were then centrifuged at 3000 rpm for 10 minutes, and the supernatants were analyzed colorimetrically using a Hitachi 7600 analyzer. Ethical approval for this study was obtained from the Institutional Review Board of Cheng Ching General Hospital in Taiwan (IRB No: HP140014).

In summary, six groups were examined in this study based on age and gender: males 20–44 years old, 45–64, and 65 and above, and females in the same three age ranges. A total of 3,532 participants enrolled in the present study (2,152 men and 1,380 women).

Two approaches are typically employed to evaluate model adequacy. The first approach is to compare MSE, which measures the dispersion around the true value of the parameter. The lower the MSE value, the more accurate the formula. The second approach is to compare the underestimated and overestimated LDL-C values with the real values based on the existing formulae. An overestimate is defined as when the predicted value is greater than the true value whereas an underestimate is when the true value is greater than the predicted value.

Figure 1 displays the MSE performance of all formulae with/without TG ≥ 400 mg/dL observations. These data show that the FF and BF that exclude TG ≥ 400 mg/dL have cutoffs of approximately 34% and 43%, respectively, while the other formulae are less affected by TG concentration. The FF is more accurate and precise when only observations with TG ≤ 400 mg/dL are considered [3]. In order to provide generalized benchmarking of the formulae, our experimental data include all levels of TG, i.e., all subjects were considered in the experimental analysis. The experimental results (Figure 2a-2f) demonstrate that the FF has the largest MSE value, which indicates that it has the greatest differences between predictions and real observations. In fact, several studies have noted that the FF is known to underestimate the LDL-C concentration [4, 20]. In addition, the values predicted by MR, TF, CTF, and TMR, which have lower MSE values than the other formulae, are approximately half that of the vales predicted by the FF formula for both age and gender categories in our study. Furthermore, these four formulae also produce less variability in the error bars, and therefore less uncertainty in their predicted values (Figure 2a-2f).

Figure 3 shows the underestimated/overestimated performance of all formulae with/without TG ≥ 400 mg/dL. It is notable that the FF and BF have cutoffs of approximately 4% and 7% for the underestimated index without TG ≥ 400 mg/dL, respectively. As has been previously reported [3], FF is more accurate and precise when the observations only consider a TG ≤ 400 mg/dL. To provide generalized benchmarking for the formulae, our experimental data include all levels of TG. The dotted lines in Figure 4a-4f represent an underestimated LDL-C prediction, i.e., when the predicted value is lower than the result of a medical test. The solid lines represent an overestimated LDL-C prediction, in which the predicted value is higher than the result from the test. These results show that the FF and DF tend to underestimate the LDL-C concentrations. These two formulae were the most consistent in terms of underestimating the LDL-C concentration in all six groups, and their predictions were affected by age and gender. BF and MR produced similar results to the FF and DF, in that they underestimated the LDL-C concentration in most cases. However, BF and MR provided fewer overall underestimated values compared to the FF and DF.One finding of particular interest is that TF and CTF both produce not only similar numbers of observations but also produce more overestimates than underestimates (Figure 4a-4f). Therefore, if there is a preference for patient safety by virtue of overestimated predictions of LDL-C, then TF and CTF can provide safer and more accurate results. That is, underestimates of LDL-C suggest that patients are in better health than they really are, while overestimates can provide an early warning for patients, so that they may choose to have more advanced medical tests performed.

Based on the MSE performance findings of this study, MR, TF, CTF, and TMR are preferable for estimating LDL-C concentration, as there is less variability in their results. We then assessed these formulae based on the degree to which they overestimate or underestimate the actual results. Tsai’s formulae, TF and CTF, performed the best in the LDL-C concentration estimations for all groups except females aged 20–44, with Teerakanchana’s multiple regression (MR) providing better results for this group. Moreover, TMR is a formula that can be easily applied to all groups, even though its performance among men and women aged 45–64 was slightly inaccurate than that of both TF and CTF, because these latter two approaches tend to underestimate the LDL-C concentration. Note that CTF is a revised version of TF, and if both formulae are recommended then TF should be used, because it is simpler than CTF. It is anticipated that clinical practitioners will be able to utilize the formulae benchmarking table produced in this work (Table 3), in order to choose the appropriate method for estimating LDL-C concentration when age and gender are taken into consideration.

In this study, participants with diabetes, secondary dyslipidemias (e.g., dyslipidemia due to renal, liver, or thyroid disease), and those who were taking statins or other lipid-modifying agents at the time of the enrollment were not excluded. In addition, the extrapolation of findings to other populations could introduce errors. The experimental benchmarking is therefore deemed specific for the Taiwanese cohort in this study.

Some subjects with heritable hyperlipidemia have extremely high TG. However, the current study had few cases with TG > 1500 mg/dL; these were not included in the analyses. In addition, some related studies were carried out after the subjects had fasted for 12 hours [27, 28], while in this study the subjects fasted for 8 hours, and this may have produced some discrepancies with previous results, which is an issue that requires further examination.