3.14.1. The Framingham Example
When we did simple linear regressions of log[SBP] against log[BMI] for men and women we obtained slope estimates of 0.273 and 0.399 for men and women, respectively. The multiple regression model (3.19) gives a single slope estimate of 0.2626 for both sexes, but finds that the effect of increas- ing age on log[SBP] is twice as large in women than men. That is, for women this slope isβ2+β5 =0.0035+0.0049=0.0084 while for men it is β2=0.0035. How reasonable is our model? In Section 3.2 we said that the parameter for a covariate in a multiple regression model measures the slope of the relationship between the response variable and this covariate when all other covariates are held constant. One way to increase our intuitive un- derstanding of the model is to plot separate simple linear regressions of SBP against BMI in groups of patients who are homogeneous with respect to the other variables in the model. Figure 3.5 shows linear regressions of log[SBP]
against log[BMI] in subgroups defined by sex and 10-year age groups. These regressions are restricted to subjects whose log[SCL] lies in the inter-quartile range for this variable, which is from 5.28 to 5.42. The vertical and horizontal lines show the mean log[BMI] and log[SBP] in each panel. The black re- gression lines plot the simple linear regression of log[SBP] against log[BMI]
for the patients in each panel. The thick gray lines are drawn through each panel’s joint mean value for log[SBP] and log[BMI] and have slope 0.263 (the estimated parameter for log[BMI] from model (3.19)). A dashed line is also drawn through the joint mean values in the panels for women and has slope 0.399. This is the slope of the simple linear regression of log[SBP]
against log[BMI] restricted to women (see Section 2.19.1). Note that the slopes of the black and gray lines are almost identical in all of the pan- els except for women aged 30–40 and 40–50. For women aged 30–40 the black simple regression slope for this panel is less than both the gray multi- ple regression slope and the dashed simple regression slope for all women.
The gray multiple regression slope comes much closer to the simple regres- sion slope for this panel than does the dashed simple regression line for all women. For women aged 40–50 the simple regression slope exceeds the multiple regression slope and comes close to the dashed line for all women.
However, by and large, this figure supports the finding that there is little variation in the rate at which SBP increases with BMI among people of the same sex and similar age and SCL.
Women
Body Mass Index Body Mass Index
Systolic Blood PressureSystolic Blood PressureSystolic Blood PressureSystolic Blood Pressure
Men
20 30 40
100 150 200 250
20 30 40
60 ≤ Age < 70
50 ≤ Age < 60
30 ≤ Age < 40 30 ≤ Age < 40
40 ≤ Age < 50 50 ≤ Age < 60 60 ≤ Age < 70
40 ≤ Age < 50
100 150 200 250 100 150 200 250 100 150 200 250
Figure 3.5 The black sloping lines in these panels are simple linear regressions of log systolic blood pressure (SBP) against log body mass index (BMI) in men and women of similar age and serum cholesterol (SCL) levels from the Framingham Heart Study. The thick gray lines have the slope of the log[BMI] parameter in the multiple linear regression model (3.19). The dashed lines have the slope of the simple linear regression of log[SBP] against log[BMI] among women in this study. This graph confirms the finding of model (3.19) that the relationship between log[SBP] and log[BMI] is similar among men and women of similar age and SCL levels (see text).
87 3.14. Intuitive understanding of a multiple regression model
Body Mass Index
24 25 26 27
120 130 140
150 60 ≤ Age < 70
24 25 26 27
120 130 140 150
24 25 26 27
120 130 140 150
24 25 26 27
120 130 140 150
50 ≤ Age < 60
40 ≤ Age < 50
30 ≤ Age < 40 Women
Systolic Blood PressureSystolic Blood PressureSystolic Blood Pressure
Men
Body Mass Index
Systolic Blood Pressure
24 25 26 27
120 130 140 150
24 25 26 27
120 130 140 150
24 25 26 27
120 130 140 150
24 25 26 27
120 130 140 150
60 ≤ Age < 70
50 ≤ Age < 60
40 ≤ Age < 50
30 ≤ Age < 40
Figure 3.6 The mean systolic blood pressure and body mass index of patients from the Framingham Heart Study are indicated by horizontal and vertical lines in panels defined by age and sex. This figure illustrates the marked interaction between gender, body mass index, and age on systolic blood pressure.
The interrelationship between SBP, sex, BMI and age is better illustrated in Figure 3.6. In this figure SBP and BMI are drawn on a linear scale. In each panel the vertical and horizontal lines mark the mean SBP and BMI for all subjects with the gender and age range specified for the panel. In their thirties men, on average, are fatter than women and have higher systolic
88 3. Multiple linear regression
blood pressures. The average increase in BMI with increasing age among men, however, is modest. In contrast, the mean BMI increases in women from 23.8 in their thirties to 27.5 in their sixties. This corresponds to an average increase in weight of 9.5 kg (21 lb) for a woman 160 cm (5 ft 3 in) tall.
Moreover, SBP increases much faster with age for women than men, and by their sixties, women have a higher mean SBP than their male counterparts.
Thus, Figure 3.6 is consistent with our analysis model (3.19), which found that there is a pronounced interaction of sex and age on log[SBP] but no evidence of interaction between sex and log[BMI] on log[SBP].
A factor that should be considered in interpreting Figures 3.5 and 3.6. is that these figures do not take differential mortality rates between men and women into account. Hence, the comparatively modest BMI of men in their sixties is, in part, influenced by the fact that some of the fatter members of their birth cohort died before age 60. We will discuss how to analyze mortality data in the Chapters 6 and 7.