Secondary Logo

Journal Logo


Total Cholesterol and Vascular Mortality

A Meta-regression

Takagi, Hisato; Umemoto, Takuya

Author Information
doi: 10.1097/EDE.0b013e31829f59ec
  • Free

To the Editor:

In a landmark meta-analysis,1 individual participant data were obtained from 61 prospective observational studies consisting of almost 900,000 adults without previous disease and with baseline measurements of total cholesterol and blood pressure. During nearly 12 million person-years at risk between the ages of 40 and 89 years, there were >55,000 vascular deaths. Total cholesterol was positively associated with ischemic heart disease mortality in both middle age and old age and at all blood pressure levels. Conventional “linear” regression, however, assumes a linear relationship. We explored the possibility of a nonlinear relationship by carrying out a “flexible” meta-regression, based on fractional polynomials2 and using summary statistics from the individual studies.

The baseline total cholesterol level (mmol/l) (= x) and the vascular mortality rate were abstracted from each of the 61 studies included in the earlier meta-analysis.1 The family of first- and second-order fractional polynomials models is defined as the forms, mortality (%) = β1 + β2xp and β1 + β2xp + β3xq, respectively. In particular, by choosing p and q from a predefined set {–2, –1, –0.5, 0, 0.5, 1, 2, 3}, the model can accommodate a rich set of possible functions, including some U-shaped and J-shaped relationships.2 The powers are expressed according to the Box-Tidwell transformation,3 in which xp denotes xp if p ≠ 0 and log x if p = 0. When p = q, the model becomes β1 + β2xp + β3(xp log x). 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.2 It is convenient to use the deviance associated with a reference model (eg, with the conventional linear model [p = 1 in the first-order models]) as the baseline for reporting the deviances of other models. Thus, the gain for a given model is defined as the deviance associated with the reference model minus that for the model in question.2 A larger gain indicates a better fit.4

The conventional quadratic model (p = 1, q = 2; solid curve in the Figure) fit the data better than the linear model (straight line in the Figure), with a gain in deviance of 599. The best-fitting model (p = –2, q = –1; dotted curve in the Figure) offered a gain in deviance of 652 with respect to the reference linear model, with a J-shaped dose-response relationship between the baseline total cholesterol level and the vascular mortality rate (nadir at 5.0-mmol/l baseline total cholesterol levels).

Relation between baseline total cholesterol levels (mmol/l) and vascular mortality rates (%). The area of each circle is proportional to the precision (reciprocal standard error) of the vascular mortality rate. Straight line, linear model (p = 1 in the first-order models); solid curve, conventional quadratic model (p = 1, q = 2); dotted curve, best-fitting fractional polynomials model (p = –2, q = –1).

Attempts to represent nonlinearity are often made by means of polynomial models,2 typically quadratic models.5,6 Fractional polynomials7 are a family of models that consider, as covariates, power transformations of a continuous exposure variable restricted to a small, predefined set of integer and noninteger exponents.2,3 Such models provide great flexibility for meta-regression (meta-analysis of dose-response aggregate data) and are especially valuable when important nonlinearity is anticipated.2 Such an approach, however, is underused in epidemiologic research2 and has seldom8 been applied to meta-regression; accordingly, fractional polynomials would be expected to be extensively used in meta-analysis.

Our results argue strongly against a linear dose-response, but are perhaps less convincing in concluding that vascular mortality actually increases at low levels of cholesterol. We doubt, on the basis of the present analysis, that a quadratic dose-response can be distinguished from a threshold in which mortality is constant at lower levels of cholesterol and then rises at higher levels. Although a quadratic regression may fit better than a linear regression, there probably is little evidence here for an actual increased risk at the lowest cholesterol levels. Further analyses would be required to explore that possibility.

Hisato Takagi

Departments of Clinical Research and

Cardiovascular Surgery

Shizuoka Medical Center

Shizuoka, Japan

[email protected]

Takuya Umemoto

Departments of Clinical Research and

Cardiovascular Surgery

Shizuoka Medical Center

Shizuoka, Japan


1. Lewington S, Whitlock G, et al. Blood cholesterol and vascular mortality by age, sex, and blood pressure: a meta-analysis of individual data from 61 prospective studies with 55,000 vascular deaths. Lancet. 2007;370:1829–1839
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;159:1077–1086
3. Royston P, Altman DG.. Regression using fractional polynomials of continuous covariates: parsimonious parametric modelling. Appl Stat. 1994;43:429–467
4. Royston P. Flexible parametric alternative to the Cox model, and more. Stata J. 2001;1:1–28
5. Berlin JA, Longnecker MP, Greenland S. Meta-analysis of epidemiologic dose-response data. Epidemiology. 1993;4:218–228
6. Friedenreich CM. Methods for pooled analyses of epidemiologic studies. Epidemiology. 1993;4:295–302
7. Royston P. A strategy for modelling the effect of a continuous covariate in medicine and epidemiology. Stat Med. 2000;19:1831–1847
8. Bellocco R, Pasquali E, Rota M, et al. Alcohol drinking and risk of renal cell carcinoma: results of a meta-analysis. Ann Oncol. 2012;23:2235–2244
© 2013 by Lippincott Williams & Wilkins, Inc