Question 11.1: The body mass index (BMI) and the systolic blood pressure of......
The body mass index (BMI) and the systolic blood pressure of 6 people were measured to study a cardiovascular disease. The data are as follows:
(a) The research hypothesis is that a high BMI relates to a high blood pressure. Estimate the linear model where blood pressure is the outcome and BMI is the covariate. Interpret the coefficients.
(b) Calculate R² to judge the goodness of fit of the model.
| Body mass index | 26 | 23 | 27 | 28 | 24 | 25 |
| Systolic blood pressure | 170 | 150 | 160 | 175 | 155 | 150 |
(a) Calculating \bar{x}=\frac{1}{6}\left(26 + 23 + 27 + 28 + 24 + 25\right) = 25.5 and \bar{y}=\frac{1}{6}\left(170 + 150 + 160 + 175 + 155 + 150\right) = 160, we obtain the following table needed for the estimation of \hat{\alpha } and \hat{\beta }
With \Sigma _{i}v_{i}=\Sigma _{i}\left(x_{i}-\bar{x} \right) \cdot \left(y_{i}-\bar{y} \right) = 80, it follows that S_{xy} = 80. Moreover, we get S_{xx} =\Sigma _{i}\left(x_{i}-\bar{x} \right)^{2} = 17.5 and S_{yy} = \Sigma _{i} \left(y_{i}-\bar{y} \right)^{2} = 550. The parameter estimates are therefore
\hat{\beta } =\frac{S_{xy}}{S_{xx}} =\frac{80}{17.5} \approx 4.57,
\hat{\alpha }=\bar{y} -\hat{\beta }\bar{x} =160 − 4.57 · 25.5 = 43.465.
A one-unit increase in the BMI therefore relates to a 4.57 unit increase in the blood pressure. The model suggests a positive association between BMI and systolic blood pressure. It is impossible to have a BMI of 0; therefore, \hat{\alpha } cannot be interpreted meaningfully here.
(b) Using (11.14), we obtain R² as
R^{2}=r^{2}=\left(\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}} } \right)^{2} = \left(\frac{80}{\sqrt{17.5 · 550} } \right)^{2}\approx 0.66.
Thus 66%of the data’s variability can be explained by the model. The goodness of fit is good, but not perfect.
| Body mass index | Systolic blood pressure | v_{i} | |||||
| x_{i} | x_{i}-\bar{x} | \left(x_{i}-\bar{x}\right)^{2} | y_{i} | y_{i}-\bar{y} | \left(y_{i}-\bar{y}\right)^{2} | ||
| 26 | 0.5 | 0.25 | 170 | 10 | 100 | 5 | |
| 23 | -2.5 | 6.25 | 150 | -10 | 100 | 25 | |
| 27 | 1.5 | 2.25 | 160 | 0 | 0 | 0 | |
| 28 | 2.5 | 6.25 | 175 | 15 | 225 | 37.5 | |
| 24 | -1.5 | 2.25 | 155 | -5 | 25 | 7.5 | |
| 25 | -0.5 | 0.25 | 150 | -10 | 100 | 5 | |
| Total | 153 | 17.5 | 960 | 550 | 80 | ||