Section 23 MLR: Coefficients
Multiple Linear Regression: Coefficients
23.1 Estimates: Effects
\[ \large fm \leftarrow lm(SBP \sim BMI + Age, \space data=BP) \]
\[ \large summary(fm) \]
| Estimate | Std. Error | t value | Pr(>|t|) | |
|---|---|---|---|---|
| (Intercept) | -10.8344 | 6.9101 | -1.5679 | 0.1175 |
| BMI | 2.3315 | 0.0711 | 32.7990 | 0.0000 |
| Age | 1.0903 | 0.1568 | 6.9544 | 0.0000 |
23.2 Estimates: Effects with centered BMI & Age
| Estimate | Std. Error | t value | Pr(>|t|) | |
|---|---|---|---|---|
| (Intercept) | 101.9787 | 0.1178 | 865.8992 | 0 |
| cBMI | 2.3315 | 0.0711 | 32.7990 | 0 |
| cAge | 1.0903 | 0.1568 | 6.9544 | 0 |
23.3 Interpretation
Statistical Model
\[ \large y_{i} = a + \beta_1 x_{1i} + \beta_2 x_{2i} + \epsilon_{i} \]
Check if the estimate of \(\beta_0\) is meaningful. An interpretable estimate of \(\beta_0\) can be obtained by centering the \(x\) variables.
The ideal scenario is when the predictors are uncorrelated.
A regression coefficient \(\beta_p\) estimates the expected change in \(y\) per unit change in \(x_k\), in otherwise identical conditions for other predictors.
In other words, a regression coefficient \(\beta_k\) suggests that a unit change in \(x_k\) is associated with \(\beta_k\) change in \(y\) while all other predictors held fixed.
Correlations amongst predictors cause problems; changes in one variable may change in others.
Claims of causality should be avoided for the observational data.