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Figure. Relation between logarithmic relative risk for major vascular events and mean absolute reduction in low-density lipoprotein cholesterol (LDL-C) level. The area of each circle is proportional to the precision (inverse variance) of the relative risk. Black line, linear model; red curve, best-fitting fractional polynomials model. To convert LDL-C from milligrams per deciliter to millimoles per liter, multiply by 0.0259.

Figure. Relation between logarithmic relative risk for major vascular events and mean absolute reduction in low-density lipoprotein cholesterol (LDL-C) level. The area of each circle is proportional to the precision (inverse variance) of the relative risk. Black line, linear model; red curve, best-fitting fractional polynomials model. To convert LDL-C from milligrams per deciliter to millimoles per liter, multiply by 0.0259.

1.
Delahoy PJ, Magliano DJ, Webb K, Grobler M, Liew D. The relationship between reduction in low-density lipoprotein cholesterol by statins and reduction in risk of cardiovascular outcomes: an updated meta-analysis.  Clin Ther. 2009;31(2):236-244PubMedArticle
2.
Bagnardi V, Zambon A, Quatto P, Corrao G. Flexible meta-regression functions for modeling aggregate dose-response data, with an application to alcohol and mortality.  Am J Epidemiol. 2004;159(11):1077-1086PubMedArticle
3.
Maclure M, Greenland S. Tests for trend and dose response: misinterpretations and alternatives.  Am J Epidemiol. 1992;135(1):96-104PubMed
4.
Berlin JA, Longnecker MP, Greenland S. Meta-analysis of epidemiologic dose-response data.  Epidemiology. 1993;4(3):218-228PubMedArticle
5.
Royston P. A strategy for modelling the effect of a continuous covariate in medicine and epidemiology.  Stat Med. 2000;19(14):1831-1847PubMedArticle
6.
Bellocco R, Pasquali E, Rota M,  et al.  Alcohol drinking and risk of renal cell carcinoma: results of a meta-analysis.  Ann Oncol. 2012;23(9):2235-2244PubMedArticle
7.
Cholesterol Treatment Trialists' (CTT) Collaboration. Baigent C, Blackwell L, Emberson J,  et al.  Efficacy and safety of more intensive lowering of LDL cholesterol: a meta-analysis of data from 170,000 participants in 26 randomised trials.  Lancet. 2010;376(9753):1670-1681PubMedArticle
8.
Sniderman A, Thanassoulis G, Couture P, Williams K, Alam A, Furberg CD. Is lower and lower better and better? a re-evaluation of the evidence from the Cholesterol Treatment Trialists' Collaboration meta-analysis for low-density lipoprotein lowering.  J Clin Lipidol. 2012;6(4):303-309PubMedArticle
Research Letter
June 10, 2013

Limit to Benefits of Large Reductions in Low-Density Lipoprotein Cholesterol LevelsUse of Fractional Polynomials to Assess the Effect of Low-Density Lipoprotein Cholesterol Level Reduction in Metaregression of Large Statin Randomized Trials

Hisato Takagi, MD, PhD; Takuya Umemoto, MD, PhD; for the ALICE (All-Literature Investigation of Cardiovascular Evidence) Group
Author Affiliations

Author Affiliations: Departments of Cardiovascular Surgery, Shizuoka Medical Center, Shizuoka, Japan.

JAMA Intern Med. 2013;173(11):1028-1029. doi:10.1001/jamainternmed.2013.659

A recent metaregression1 of 25 large statin randomized trials involving 155 613 participants and 23 791 major vascular events reported a significant reduction in the risk of major vascular events associated with a reduction in low-density lipoprotein cholesterol (LDL-C) level. The question that naturally follows is whether there is a threshold for the benefit of LDL level reduction that can be achieved with statins or whether greater reductions in LDL level would bring greater reductions in vascular events.

Conventional metaregressions such as the one by Delahoy et al,1 however, rely on “linear” modeling, which assumes that the association fits a line (a constantly increasing or decreasing risk as the exposure increases or decreases) and does not allow for alternative associations such as threshold effects. We performed a “flexible” (not “linear”) unrestricted maximum-likelihood metaregression (inverse variance-weighted regression) based on fractional polynomials2 of the reduction in LDL-C level on the logarithmic relative risk (RR) for major vascular events.

Methods

The mean absolute reduction in LDL-C level at 1 year and the RR for major vascular events were abstracted from each individual randomized trial included in the recent metaregression.1 First-order and second-order fractional polynomial models take the forms log RR = β1 + β2xp and β1 + β2xp + β3xq, respectively. By choosing p and q from the predefined set {–2, –1, –0.5, 0, 0.5, 1, 2, 3}, a rich set of possible functions, including some so-called U-shaped and J-shaped relations, may be accommodated. The powers are expressed according to the Box-Tidwell transformation, in which xp denotes xp if p ≠ 0 and log x if p = 0.2 When p = q, the model becomes β1 + β2xp + β3(xp log x).

For each set of powers (p, q), we calculated β1, β2, and β3 that minimized the deviance (sum of inverse variance-weighted squared residuals) using Microsoft Excel Solver (Microsoft Corp). The best fit among the family of models thus generated is defined as that with the highest likelihood or, equivalently, that with the lowest deviance. The gain for a given model is defined as the deviance associated with the reference linear model (β1 = 0, p = 1; applied in the recent metaregression1) minus that for the model in question; accordingly, a larger gain indicates a better fit.2

Results

The conventional quadratic model (p = 1, q = 2) better fitted the data than the linear model (black line, Figure), with a gain in deviance of 12.87. The best-fitting model (p = –2, q = –2; red curve,Figure) offered a gain in deviance of 13.30 with respect to the reference linear model, representing an almost horizontal line when the reduction in LDL-C level is more than approximately 40 mg/dL (to convert to millimoles per liter, multiply by 0.0259) (RR [log RR] of 0.80 [–0.23], 0.79 [–0.24], 0.78 [–0.25], and 0.77 [–0.26] at the LDL-C level reductions of 40, 50, 60, and 70 mg/dL, respectively).

Discussion

Our fractional polynomials metaregression suggests almost no additional benefit in the use of statins beyond a 40 mg/dL decrease in LDL-C level in preventing major vascular events.

A traditional method of summarizing dose-response relations across studies is to estimate the change in the logarithm of the RR per unit of exposure within each study and to combine these estimates across studies. Such an approach could be misleading, however, because it assumes that the dose-response relation follows a specific model form, usually linear.3 Polynomial models, typically quadratic models, are used to represent nonlinearity.4 An alternative curve-fitting method, fractional polynomial regression,5 has been described. Fractional polynomials are a family of models to consider, as covariates, power transformations of a continuous exposure variable restricted to a small, predefined set of integer and noninteger exponents.2 Such approaches are underused in epidemiologic research and have seldom6 been compared in a meta-analysis of dose-response aggregate data. Fractional polynomials provide great flexibility for a meta-analysis of dose-response aggregate data and are especially valuable when important nonlinearity is anticipated.2 Furthermore, they are easier to communicate mathematically, require the estimation of fewer parameters, and are less influenced by arbitrariness in the choice of the model than traditional approaches.

On the basis of the Cholesterol Treatment Trialists' meta-analysis,7 Sniderman et al8 calculated that any potential gain from increasing the dose of atorvastatin calcium from 40 to 80 mg would be small, at best an additional 2% reduction in clinical events. The increase in dose, unfortunately, would likely be associated with increased adverse effects and decreased adherence. Accordingly, whether net benefit would be demonstrable cannot be assumed. It follows that definitive evidence supporting maximal lowering of LDL-C level or maximal dose of statins is still lacking and that guidelines, if they are to be evidence based, should acknowledge this uncertainty.8 Although we found, on the basis of flexible (not linear) metaregression, that using statins to reduce LDL-C level by more than approximately 40 mg/dL could produce almost no additional reduction in the risk of major vascular events, further analysis would be required to confirm our findings.

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Article Information

Correspondence: Dr Takagi, Departments of Cardiovascular Surgery, Shizuoka Medical Center, 762-1 Nagasawa, Shimizu-cho, Sunto-gun, Shizuoka 411-8611, Japan (kfgth973@ybb.ne.jp).

Published Online: April 29, 2013. doi:10.1001/jamainternmed.2013.659

Author Contributions: Dr Takagi had full access to all the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis. Study concept and design: Umemoto. Acquisition of data: Takagi. Analysis and interpretation of data: Both authors. Drafting of the manuscript: Takagi. Critical revision of the manuscript for important intellectual content: Umemoto. Statistical analysis: Takagi. Administrative, technical, and material support: Umemoto. Study supervision: Umemoto.

Group Information: The members of the ALICE Group are Hisato Takagi, MD, PhD; Yusuke Mizuno, MD; Hirotaka Yamamoto, MD; Masao Niwa, MD; Shin-nosuke Goto, MD; Masafumi Matsui, MD; and Takuya Umemoto, MD, PhD.

Conflict of Interest Disclosures: None reported.

References
1.
Delahoy PJ, Magliano DJ, Webb K, Grobler M, Liew D. The relationship between reduction in low-density lipoprotein cholesterol by statins and reduction in risk of cardiovascular outcomes: an updated meta-analysis.  Clin Ther. 2009;31(2):236-244PubMedArticle
2.
Bagnardi V, Zambon A, Quatto P, Corrao G. Flexible meta-regression functions for modeling aggregate dose-response data, with an application to alcohol and mortality.  Am J Epidemiol. 2004;159(11):1077-1086PubMedArticle
3.
Maclure M, Greenland S. Tests for trend and dose response: misinterpretations and alternatives.  Am J Epidemiol. 1992;135(1):96-104PubMed
4.
Berlin JA, Longnecker MP, Greenland S. Meta-analysis of epidemiologic dose-response data.  Epidemiology. 1993;4(3):218-228PubMedArticle
5.
Royston P. A strategy for modelling the effect of a continuous covariate in medicine and epidemiology.  Stat Med. 2000;19(14):1831-1847PubMedArticle
6.
Bellocco R, Pasquali E, Rota M,  et al.  Alcohol drinking and risk of renal cell carcinoma: results of a meta-analysis.  Ann Oncol. 2012;23(9):2235-2244PubMedArticle
7.
Cholesterol Treatment Trialists' (CTT) Collaboration. Baigent C, Blackwell L, Emberson J,  et al.  Efficacy and safety of more intensive lowering of LDL cholesterol: a meta-analysis of data from 170,000 participants in 26 randomised trials.  Lancet. 2010;376(9753):1670-1681PubMedArticle
8.
Sniderman A, Thanassoulis G, Couture P, Williams K, Alam A, Furberg CD. Is lower and lower better and better? a re-evaluation of the evidence from the Cholesterol Treatment Trialists' Collaboration meta-analysis for low-density lipoprotein lowering.  J Clin Lipidol. 2012;6(4):303-309PubMedArticle
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