Section 40 MLR - Polynomial terms: lm outputs
40.1 Statistical Model
\[ \large y_{i} = \beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i} + \beta_3 x_{2i}^2 + \epsilon_{i} \]
\[ \large fm \leftarrow lm(SBP \sim BMI + Age + I(Age \hat \space 2), \space data=BP) \]
\[ \large anova(fm) \]
| Df | Sum Sq | Mean Sq | F value | Pr(>F) | |
|---|---|---|---|---|---|
| BMI | 1 | 15103.0386 | 15103.0386 | 2188.3255 | 0.0000 |
| Age | 1 | 335.4035 | 335.4035 | 48.5976 | 0.0000 |
| I(Age^2) | 1 | 23.5427 | 23.5427 | 3.4112 | 0.0654 |
| Residuals | 496 | 3423.2143 | 6.9016 | NA | NA |
| Total | 499 | 18885.1991 | NA | NA | NA |
40.2 Estimates: Effects
\[ \large summary(fm) \]
| Estimate | Std. Error | t value | Pr(>|t|) | |
|---|---|---|---|---|
| (Intercept) | -441.6343 | 233.3528 | -1.8926 | 0.0590 |
| BMI | 2.3322 | 0.0709 | 32.8882 | 0.0000 |
| Age | 18.3479 | 9.3452 | 1.9634 | 0.0502 |
| I(Age^2) | -0.1728 | 0.0935 | -1.8469 | 0.0654 |
40.3 Estimates: Effects with centered BMI and Age
BP <- read.csv('data/BP.csv')
BP$cBMI <- scale(BP$BMI, center=mean(BP$BMI))
BP$cAge <- scale(BP$Age, center=mean(BP$Age))
fm <- lm(SBP ~ cBMI + cAge + I(cAge^2), data=BP)\[ \large summary(fm) \]
| Estimate | Std. Error | t value | Pr(>|t|) | |
|---|---|---|---|---|
| (Intercept) | 102.1335 | 0.1443 | 707.6460 | 0.0000 |
| cBMI | 4.8743 | 0.1482 | 32.8882 | 0.0000 |
| cAge | 1.0025 | 0.1491 | 6.7221 | 0.0000 |
| I(cAge^2) | -0.1551 | 0.0840 | -1.8469 | 0.0654 |
40.4 Plot
Note the curvilinear pattern due to Age for different values of BMI. As shown in the plot as well as from the model outcomes, the evidence is not strong (p>0.05). We will possibly remove the polynomial term from the final model. At this stage, however, we need to investigate the model further along with other predictors. The green dashed vertical lines shows the mean Age of the population. The mean BMI of the population is approximately 25.