This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Based on the 40-year follow-up of the Framingham Heart Study (FHS), we used logistic regression models to demonstrate that different designs of an observational study may lead to different results about the association between BMI and all-cause mortality. We also used dynamic survival models to capture the time-varying relationships between BMI and mortality in FHS. The results consistently show that the association between BMI and mortality is dynamic, especially for men. Our analysis suggests that the dynamic property may explain part of the heterogeneity observed in the literature about the association of BMI and mortality.

Obesity is a major health concern in the United States and in the world. More and more evidence shows that it is related to a high incidence of diseases such as coronary heart disease (CHD), diabetes, and cancer, and mortality from those diseases. However, there appears to be no consensus on the relationship between body weight and mortality in epidemiological studies. For example, conclusions about the relationship between BMI and all-cause mortality include no association [

Several factors make it difficult to determine the true relationship between BMI and mortality using traditional observational studies and analysis methods. These factors include: (1) BMI often changes during a person’s lifetime but in most studies only BMI at baseline (the beginning of the follow-up period) is used for analysis; (2) a longer follow-up period covers more long-term effects and the length of follow-up may vary from several years to several decades; (3) analytic methods vary but have included cross-tabulations, logistic regression models, and the Cox survival models. Each method gives one or several numerical values (odds ratio, relative risk, or hazard ratio) summarizing the relationship between BMI (measured at a particular time point: baseline) and mortality (within a particular follow-up period).

The design of a study about BMI and mortality can be illustrated with

The objective of this study is to examine and identify the relationship between BMI and mortality from a dynamic prospective. Our analysis is based on the first cohort of the Framingham Heart Study (FHS), which has a long follow-up period and a large number of repeated measures to support our analysis. We first use logistic regression models to demonstrate that different designs on the same population may lead to different results; then we use dynamic survival models to capture the time-varying relationship between BMI and mortality in FHS.

The Framingham Heart Study (FHS) is an epidemiologic study funded by the National Heart, Lung, and Blood Institute (NHLBI). Multiple cohorts have been recruited into the large study. The original cohort of FHS consists of 5209 men and women between the ages of 30 and 62 from Framingham, Massachusetts. Extensive examinations were carried out every other year on this cohort since 1948 and exam 29 began in April of 2006. In this study, we use information of the first 20 exams (about 40 years of follow-up) of the original cohort for analysis. The exact survival time of each individual is available.

We first demonstrate that different study designs shown in

The purpose of this analysis is to show whether the result may be different for a study on the same group of subjects when BMI at a different time point is used. We choose the time period from year 30 (exam 16) to year 40 of FHS as the follow-up period. Each measurement of BMI from exam 1 to exam 16 is used to model the mortality within the 10-year period with a logistic regression model. The ORs of BMI of all models within each gender are plotted together against the time difference between the BMI measurement and the baseline of the follow-up period (year 30 of FHS). Only 1,582 (639 men and 943 women) subjects who were alive at exam 16 and had all 16 BMI measurements (exam 1 to exam 16) available are included in the analysis. Subjects excluded are 1762 participants who died before exam 16 and 1865 participants who have at least one missing BMI measurement from exam 1 to exam 16. This strategy is adopted to make sure that all models within each gender are built upon the same group of individuals. The comparison of BMI ORs across all models within each gender demonstrates the effect of using BMI at different time points on the result about the association between BMI and mortality.

The purpose of this analysis is to show whether the result may be different for a study on the same group of subjects when the follow-up length is different. We set the baseline of follow-up at year 0 (exam 1) and increase the length of follow-up from 6 to 40 years. BMI measured at exam 1 is used for all models. Except 8 subjects with missing BMI measurements at exam 1, 5201 subjects are included in the analysis. There are 18 logistic regression models estimated for each gender. The comparison of BMI ORs across all models within each gender demonstrates the effect of the length of follow-up on the result about the association between BMI and mortality.

The purpose of this analysis is to determine whether short studies using the traditional design in

The Cox model is a commonly used method in epidemiological studies when survival times are observed. A Cox model has a hazard function in the following form:

When the PH assumption is violated, that means the association between BMI and the risk of death is not constant but changing within the box in

In this part, one model is used to capture the association between BMI and mortality for each gender. Besides gender and age, smoking status (current smoker, yes or no) and systolic blood pressure (SBP, mmhg) are also considered in the analysis. The data set with all this information available is the the original cohort of FHS starting from exam 4 to the end of the 40-year follow-up. The values of BMI, age, smoking status, and SBP are measured at exam 4. The total sample size is 4526 (2005 men and 2521 women).

The Cox model is used at first to examine the relationship between BMI and the risk of death for males and females, separately. The effect of age, blood pressure, and smoking status are controlled. When the test of PH assumption shows significant results, the two different approaches for dynamic survival models are applied to model the associations between BMI and mortality as time-varying functions. All the other variables are allowed to have time-varying coefficients if the PH assumptions for those variables are violated.

With limited data, this paper focuses on showing some evidence that the association between BMI and mortality may change with time rather than modeling the exact changes. For all analyses, BMI and SBP are used as continuous variables and only linear terms are included. The time-varying coefficients survival model shows the change of the linear association between BMI and mortality with time. From the the literature, the association between BMI and mortality is likely to be nonlinear rather than linear. However, a large number of parameters are needed to capture a nonlinear relationship, which leaves limited flexibility for modeling the change of the relationship with time using this small data set. The changes of a linear association show some evidence of the changes of the underlying nonlinear association. This strategy simplifies the analysis and serves the purpose of this preliminary research. Larger data sets and more sophisticated methods will be used for future analysis.

The analysis using logistic regression models shows that the result of the association between BMI and mortality may differ when the design of a study changes, such as using BMI measured at a different time or changing the length of the follow-up.

Each measurement of BMI at exam 1 to exam 16 is used to model the mortality in year 30 to year 40 of FHS.

Both plots in

The analysis in part (a) suggests that the more recent BMI is used, the more likely it is to obtain no linear association for women or an inverse association for men. The analysis in part (b) suggests that the shorter the length of follow-up, the more likely it is to obtain the same result. We design the third scenario to examine whether short studies using baseline BMI tend to obtain similar results.

With no obvious changing patterns in either plot of

First a Cox model is used to model the association between BMI and mortality with age, smoking status, and blood pressure controlled. We tested the proportional hazards (PH) assumptions for each covariate in the models. The test results using STATA 10 are in

In

The results using dynamic survival models clearly show the changing pattern of the association between BMI and mortality with the follow-up time, especially for men.

The purpose of our analyses is to examine and identify some evidence that the association between BMI and mortality may not be static so that studies with different designs will capture different pictures of the association of BMI and mortality. We demonstrated, using a single data set, that the relationship between BMI and mortality is, for this data, a complex dynamic relationship where the results obtained depend heavily on the designs of the study and that the relationship does not appear to satisfy the proportional hazards assumption. We also observed from this study that the dynamic feature is stronger for men than for women. However, the same analysis may not be repeatable in other studies if they do have have a large number of repeated measures of BMI within a long follow-up period. Our conclusion is limited to this particular study and population.

There are some findings in the literature that are consistent with our results from different aspects. Some published studies include all individuals alive at baseline for analysis, while others excluded deaths within the first several years of follow-up from analysis [

Our analyses are limited in another way in that only the linear association between BMI and mortality is under consideration. Our analysis does not show directly, but suggests the change of the shape of the association between BMI and mortality. In scenario b) with fixed BMI and changing length of follow-up (

To our knowledge, despite its limitations, our study is the first study to examine the association of BMI and mortality from a dynamic prospective that it provides a unified view of the impact of the study design on the result. If different designs lead to different results, then which design better capture the causal effect of body weight on mortality? For an observational study about body weight and mortality, researchers may consider obtaining BMI at multiple time points to compare the differences, having a relatively longer follow-up period, and applying appropriate alternative models when the proportional hazards assumption is violated.

The authors thank Jonathan Mahnken of the University of Kansas Medical Center for his helpful comments and suggestions. Data from the Framingham Heart Study was obtained from the National Heart, Lung, and Blood Institute. The views expressed in this paper are those of the authors and do not necessarily reflect the views of this Institute.

The design of an observational study about BMI and mortality.

The estimated ORs of BMI for all-cause mortality and their 95% pointwise confidence intervals when measurements made at different time points are used to model the mortality within year 30 to year 40 of the Framingham Heart Study. Death rates: male = 265/639, female = 279/943. Time 0 denotes year 30 of FHS, the beginning of the follow-up period.

The estimated odds ratios of BMI measured at exam 1 and their 95% pointwise confidence intervals for all-cause mortality in different follow-up periods. Sample size: 2333 men and 2868 women.

The estimated odds ratios of BMI for all-cause mortality and their 95% pointwise confidence intervals when short follow-up periods and baseline (the beginning of each follow-up period) measurements of BMI are used.

The estimated time-varying log hazard ratio functions of BMI. 1) thin lines: Cox model with interactions with time. 2) thick curves: Time-varying coefficient survival models. The effects of age, SBP, and smoking status are controlled.

Cox models and the tests of the proportional hazards assumptions for the Framingham Heart Study.

Analysis Result for Men | |||

Variables | Hazards Ratio | p-value | Test of PH assumption (p-value) |

Age (5 years) | 1.57 | < 0.001 | 0.034 |

sbp (10 mmHg) | 1.15 | < 0.001 | 0.026 |

Smoking | 1.42 | < 0.001 | < 0.0001 |

BMI | 1.00 | 0.876 | 0.0001 |

Analysis Result for Women | |||

Variables | Hazards Ratio | p-value | Test of PH assumption (p-value) |

Age (5 years) | 1.58 | < 0.001 | 0.001 |

sbp (10 mmHg) | 1.12 | < 0.001 | 0.017 |

Smoking | 1.42 | < 0.001 | 0.570 |

BMI | 1.02 | 0.011 | 0.060 |