Comparison of various formulae for estimating lowdensity lipoprotein cholesterol by a combination of ages and genders in Taiwanese adults
 ChungHuang Tsai^{1},
 HsinHung Wu^{2} and
 ShaoJen Weng^{3}Email author
DOI: 10.1186/1471226114113
© Tsai et al.; licensee BioMed Central Ltd. 2014
Received: 8 May 2014
Accepted: 22 August 2014
Published: 2 September 2014
Abstract
Background
The accuracy and precision of the Friedewald formula for estimating lowdensity lipoprotein cholesterol (LDLC) is questionable. Although other formulae have been developed, only a few studies compare them. Thus, we compared the efficiencies of various formulae, based on the age and gender of adults, to determine which ones yield more accurate estimations in terms of mean squared error, and which formulae underestimated and overestimated LDLC performance.
Methods
This study compares various formulae in terms of mean squared error (MSE), as well as underestimation and overestimation of LDLC concentrations, using subjects of various ages and both genders. 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.
Results
The results show that the Friedewald formula has relatively low accuracy, and while its performance among older (aged 45 and above) women with triglyceride concentrations ≤ 400 mg/dL is better than that with other groups, it is still more inaccurate than the other formulae. In terms of prediction errors and mean squared errors, Tsai’s formula (TF) and a calibrated TF provide the most accurate results with regard to the LDLC concentration. Moreover, based on a crossvalidation of age and gender, these two formulae provide highly accurate results for the LDLC concentrations of all the studied groups, except for women aged 20–44 years.
Conclusions
Based on the experimental results, this study provides a set of benchmarks for the formulae used in LDLC tests when considering the factors of age and gender. Therefore, it is a valuable method for providing formula benchmarking.
Keywords
Lowdensity lipoprotein cholesterol Residual cholesterol Friedewald formula TriglycerideBackground
Medical research and clinical trials have shown that the lowdensity lipoprotein cholesterol (LDLC) concentration is causally related to an increased risk of coronary artery disease [1, 2]. In addition, a report by the National Cholesterol Education Program Adult Treatment Panel III notes that the level of LDLC is the primary variable that is used to predict cardiovascular disease [1]. One wellknown formula for calculating this, the Friedewald formula (FF), is of doubtful accuracy and precision, and thus other approaches have been developed, such as DeLong’s formula (DF) [3, 4], Teerakanchana’s multiple regression (MR) [5], Balal’s formula (BF), which is derived from the FF [6], Tsai’s formula (TF) [7], calibrated from TF (CTF) [8], and Tsai’s multiple regression (TMR) [8]. All of these formulae measure the LDLC concentrations based on total cholesterol (TC), highdensity lipoprotein cholesterol (HDLC), and triglyceride (TG) concentrations [9–12]. Several studies compare the various methods used to assess the LDLC concentration, and this is likely due to rising healthcare expenditures as well as an increasing demand for quality healthcare. It is thus highly desirable to identify an accurate, a costeffective method to determine the LDLC concentration.
Most clinical trials employ the FF [3], which uses TC, HDLC, and TG to measure the levels of LDLC [5]; thus, it can be applied to the clinical treatment and prevention of atherosclerotic disease [6, 8]. However, the FF has produced inaccurate results in some cases, and it is not recommended for use in the presence of hypertriglyceridemia (>400 mg/dL) or type III hyperlipoproteinemia [13]. This method also tends to underestimate LDLC concentrations [6, 14–18] when the triglyceride concentration is normal [19, 20] or less than 400 mg/dL [4, 6, 21, 22]. Balal et al. [6] thus revised the FF for use with renal transplant recipients by considering those with TG concentrations lower than 400 mg/dL to calculate LDLC levels. Teerakanchana et al. [5] developed a multiple regression formula by using a multiple linear regression model to test different data sets. Tsai et al. [8] further took into account residual cholesterol (RC), which consists of highdensity lipoprotein cholesterol (HDLC), and revised the FF by using TG = 1/8 instead of TG = 1/5, which represents verylowdensity lipoprotein cholesterol (VLDLC).
Comparison of seven LDLC formulae
Author  Formula 

Friedewald et al. [3]  FF: 
LDLC = TC (HDLC)  (TG/5)  
Balal et al. [6]  BF: 
LDLC =8.018 + 0.99(LDLC predicted by FF)  
Delong et al. [4]  DF: 
LDLC = TC (HDLC) 0.16TG  
Teerakanchanna et al. [5]  MR: 
LDLC = 0.910TC  0.634(HDLC)  0.111TG  6.755  
Tsai et al. [7]  TF: 
LDLC = TC (HDLC)  (TG/8)  
Tsai et al. [8]  CTF: 
LDLC =0.276 + 0.997(LDLC predicted by TF)  
Tsai et al. [8]  TMR: 
LDLC =0.988TC  0.853(HDLC)  0.107TG  8.703 
Methods
Study population
Baseline characteristics of lipid profile
TC  HDLC  LDLC  TG  

Whole set of the data (n = 3532)  
Mean  183.8  49.9  112.1  159.3 
SD  40.1  15.3  34.3  110.3 
Min.  57  3  20  22 
Max.  569  126  444  1252 
Q_{1}  157  39  89  90 
Median  181  48  110  130 
Q_{3}  208  58  132  192 
Cases with TG ≤ 400 (n = 3395)  
Mean  182  50.4  111.8  143.8 
SD  38.4  15.3  33.6  73.6 
Min.  57  3  20  22 
Max.  395  126  312  399 
Q_{1}  156  40  89  89 
Median  179  48  110  125 
Q_{3}  205  59  132  182 
Cases with TG > 400 (n = 137)  
Mean  227.4  38.6  119.2  544.8 
SD  54.2  10.1  49.7  158.1 
Min  120  8  36  401 
Max  569  73  444  1252 
Q_{1}  191  33  88  438 
Median  223  38  115  486 
Q_{3}  252  44  140  594 
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).
Measurement
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 LDLC 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.
Results
This study carried out six experiments based on various combinations of age and gender. The results are presented in two parts, as follows:
Study 1: Comparison of LDLC MSE
Study 2: Comparison of LDLC underestimation/overestimation
Formulae benchmarking based on the crossvalidation of age and gender
Gender  

Male  Female  
Age  MSE  Under/Over  Cross  MSE  Under/Over  Cross 
Validation*  Validation*  
2044  MR  MR  TF  TF  BF  MR 
TF  TF  CTF  CTF  MR  TF  
CTF  CTF  TMR  TMR  TF  CTF  
TMR  TMR  CTF  TMR  
TMR  
4564  MR  MR  TF  TF  TF  TF 
TF  TF  CTF  CTF  CTF  CTF  
CTF  CTF  
TMR  TMR  
65+  MR  MR  TF  TF  TF  TF 
TF  TF  CTF  CTF  CTF  CTF  
CTF  CTF  TMR  TMR  TMR  TMR  
TMR  TMR 
Discussion
The results shown in Figures 1 and 2 indicate that the FF has relatively low accuracy. Although it exhibits relatively good performance among older women (aged 45 and above) with TG ≤ 400 mg/dL, its overall performance is worse than that of the other formulae. The formula with the best performance is TMR, followed by TF, CTF, and MR, with no significant differences among them, and the TF and CTF values in particular being virtually identical. Due to the properties of the multiple regression equation, the coefficients are more complex for MR and TMR. In terms of ease of use, TF is the preferred formula.
According to Tsai’s analyses, the FF tends to underestimate LDLC concentration by 10.1 mg/dL on average [7], while Balal et al. [6] report that the FF underestimates it by 8 mg/dL, and other studies have shown similar results [14–18]. Tsai’s results also showed that the difference in the maximum and minimum for the FF is larger than that of the other formulae, and concluded that it is unsuitable for research on epidemiological or causal relationships [7].
For all cases examined with/without TG ≥ 400 mg/dL in this study, BF, the formula proposed by Balal et al. [6], provided better results than the FF, although it was still not as good as the other formulae. Tsai et al. [7] report that BF has exactly the same R^{2} as the FF, suggesting that BF only calibrated the underestimation of the FF. These results demonstrate that while the calibrated formula, acquired from the regression of the estimated value and the measured value, could produce an average estimated error that approaches zero and hence reduce the estimated bias, this still would not make the estimation more precise [7]. In addition, an LDLC formula is primarily used to precisely estimate the LDLC concentration for individuals, and while reducing the group estimated bias is important, this only reduces part of the individual estimated bias by expanding another part of it, and the standard deviation of estimated error is not improved. As shown in this study, BF is not able to replace the FF or improve its shortcomings.
As noted above, the best performance for the FF was in subjects with TG ≤ 400 mg/dL, although even among these it was outperformed by the other formulae, which provided stable results when age and gender were taken into account.
Based on a multiple linear regression analysis of 1,016 cases, Teerakanchana et al. [5] obtained the formula LDLC = 0.910TC  0.634(HDLC)  0.111TG  6.755. Tsai et al. [8] also analyzed training data with multiple linear regression, and found that LDLC = 0.9882TC  0.8526(HDLC)  0.1065TG  8.7029, with an R^{2} value similar to that of MR (R^{2} = 0.9649) and TF (R^{2} = 0.9608). In the present study, the R^{2} values for MR and TMR were determined to be 0.9648 and 0.9597, respectively; thus, there was no substantial difference between them in this respect. Since multiple linear regression analysis, TMR, is far more complex than TF, it is suggested that TF be used in most cases.
Because LDLC tests tend to be timeconsuming and inconvenient, the FF of LDLC = TC  (HDLC)  (VLDLC) is often clinically applied to produce estimates of this value [3]. This formula assumes that the VLDLC of healthy adults, except those with type III hyperlipidemia, is TG/5 [3, 23, 24] without chylomicrons. However, when using FF, VLDLC would be overestimated, causing the underestimation of LDLC, when TG chylomicrons and related remnants appear in plasma [25]. FF also assumes that TC only contains LDLC, HDLC, and VLDLC, although it likely contains other constituents as well. For example, it has been shown that TC also contains intermediatedensity lipoprotein cholesterol (IDLC), chylomicrons, VLDLC remnants, lipoprotein(a) [Lp(a)], LpX, and some fats that cannot be quantified with current methods [26]. In this case, when the contents other than HDLC and LDLC in TC are defined as RC, then RC = TC  (HDLC)  (LDLC) would be more accurate than using VLDLC to estimate the RC. When TG has a specific relationship with RC, it would be more reasonable to estimate RC using TG [8]. When the regression analysis takes into account that TC contains LDLC, HDLC, VLDLC, IDLC, chylomicrons, Lp(a), LpX, and other nonquantifiable fats, Tsai et al. suggest revising the FF using TG = 1/8 instead of TG = 1/5 [8].
Research limitations
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 lipidmodifying 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.
Conclusions
Advances in current testing technology have resulted in efficient quantification of LDLC concentration, although the costs of these technologies are relatively high. In contrast, estimating LDLC concentration using formulae can produce reliable results at a relatively low cost, particularly when carrying out a large number of tests. We compared the results of direct homogeneous LDLC assay with the FF, DF, MR, BF, TF, CTF, and TMR for determination of LDLC based on underestimates/overestimates and MSE, using various combinations of age and gender. In terms of prediction errors and MSE, TF and CTF were the most accurate with regard to LDLC concentration, except for women aged 20–44. Table 3 provides details for benchmarking the formulae when considering age and gender, and this could be a valuable reference for clinical practitioners deciding on the best estimation method for their particular situation.
Abbreviations
 FF:

Friedewald formula
 LDLC:

Lowdensity lipoprotein cholesterol
 MSE:

Mean squared error
 TG:

Triglyceride
 CTF:

Calibrated from TF
 DF:

DeLong’s formula
 MR:

Teerakanchana’s multiple regression
 BF:

Balal’s formula, which is calibrated from FF
 TF:

Tsai’s formula
 TMR:

Tsai’s multiple regression
 TC:

Total cholesterol
 HDLC:

Highdensity lipoprotein cholesterol
 VLDLC:

Verylowdensity lipoprotein cholesterol.
Declarations
Acknowledgements
The data used in this study were provided by the Cheng Ching General HospitalChung Kang Branch, Taichung City, Taiwan.
Authors’ Affiliations
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