Does cigarette smoking modify the association between change in body mass index during young adulthood and risk of coronary mortality during middle-age in men? Twenty-five year follow-up results from the Chicago Western Electric Study

Many persons in the U.S. gain weight during young adulthood, and the prevalence of obesity has been increasing among young adults. Although obesity and physical inactivity are generally recognized as risk factors for coronary heart disease (CHD), the magnitude of their effect on risk may have been seriously underestimated due to failure to adequately handle the problem of cigarette smoking. Since cigarette smoking causes weight loss, physically inactive cigarette smokers may remain relatively lean because they smoke cigarettes. We hypothesize cigarette smoking modifies the association between weight gain during young adulthood and risk of coronary heart disease during middle age, and that the true effect of weight gain during young adulthood on risk of CHD can be assessed only in persons who have not smoked cigarettes. Specifically, we hypothesize that weight gain during young adulthood is positively associated with risk of CHD during middle-age in nonsmokers but that the association is much smaller or absent entirely among cigarette smokers. The purpose of this study was to test this hypothesis. The population for analysis was comprised of 1,934 middle-aged, employed men whose average age at the baseline examination was 48.7 years. Information collected at the baseline examinations in 1958 and 1959 included recalled weight at age 20, present weight, height, smoking status, and other CHD risk factors. To decrease the effect of intraindividual variation, the mean values of the 1958 and 1959 baseline examinations were used in analyses. Change in body mass index ($\Delta$BMI) during young adulthood was the primary exposure variable and was measured as BMI at baseline (kg/m$\sp2)$ minus BMI at age 20 (kg/m$\sp2).$ Proportional hazards regression analysis was used to generate relative risks of CHD mortality by category of $\Delta$BMI and cigarette smoking status after adjustment for age, family history of CVD, major organ system disease, BMI at age 20, and number of cigarettes smoked per day. Adjustment was not performed for systolic blood pressure or total serum cholesterol as these were regarded as intervening variables. Vital status was known for all men on the 25th anniversary of their baseline examinations. 705 deaths (including 319 CHD deaths) occurred over 40,136 person-years of experience. $\Delta$BMI was positively associated with risk of CHD mortality in never-smokers, but not in ever-smokers (p for interaction = 0.067). For never-smokers with $\Delta$BMI of stable, low gain, moderate gain, and high gain, adjusted relative risks were 1.00, 1.62, 1.61, and 2.78, respectively (p for trend = 0.010). For ever-smokers, with $\Delta$BMI of stable, low gain, moderate gain, and high gain, adjusted relative risks were 1.00, 0.74, 1.07, and 1.06, respectively (p for trend = 0.422). These results support the research hypothesis that cigarette smoking modifies the association between weight gain and CHD mortality. Current estimates of the magnitude of effect of obesity and physical inactivity on risk of coronary mortality may have been seriously underestimated due to inadequate handling of cigarette smoking. ^

DISTRIBUTION FUNCTIONS, MEAN SQUARED ERRORS, AND CONFIDENCE LIMITS FOR RIDGE REGRESSION ESTIMATORS

A large number of ridge regression estimators have been proposed and used with little knowledge of their true distributions. Because of this lack of knowledge, these estimators cannot be used to test hypotheses or to form confidence intervals.^ This paper presents a basic technique for deriving the exact distribution functions for a class of generalized ridge estimators. The technique is applied to five prominent generalized ridge estimators. Graphs of the resulting distribution functions are presented. The actual behavior of these estimators is found to be considerably different than the behavior which is generally assumed for ridge estimators.^ This paper also uses the derived distributions to examine the mean squared error properties of the estimators. A technique for developing confidence intervals based on the generalized ridge estimators is also presented. ^

THE IMPROVEMENT REGION IN RIDGE REGRESSION

One of the difficulties in the practical application of ridge regression is that, for a given data set, it is unknown whether a selected ridge estimator has smaller squared error than the least squares estimator. The concept of the improvement region is defined, and a technique is developed which obtains approximate confidence intervals for the value of ridge k which produces the maximum reduction in mean squared error. Two simulation experiments were conducted to investigate how accurate these approximate confidence intervals might be. ^