Section 25 MLR: Prediction

Multiple Linear Regression: Prediction


25.1 Prediction from the model


\[ \large fm \leftarrow lm(SBP \sim BMI + Age, \space data=BP) \]

Confidence Interval

\[ \large predict(fm, \space newdata=X, \space se.fit=TRUE, \space interval=`confidence`) \]


Prediction Interval

\[ \large predict(fm, \space newdata=X, \space se.fit=TRUE, \space interval=`prediction`) \]



Call:
lm(formula = SBP ~ BMI + Age, data = BP)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.6030 -2.0345  0.1196  1.9800  6.8630 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -10.83438    6.91013  -1.568    0.118    
BMI           2.33147    0.07108  32.799  < 2e-16 ***
Age           1.09032    0.15678   6.954 1.12e-11 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.633 on 497 degrees of freedom
Multiple R-squared:  0.8175,    Adjusted R-squared:  0.8168 
F-statistic:  1113 on 2 and 497 DF,  p-value: < 2.2e-16


25.2 Explanation

Statistical Model

\[ \large y_{i} = a + \beta_1 x_{1i} + \beta_2 x_{2i} + \epsilon_{i} \]


Prediction

\[ \large E(y|X=x^*) = \hat y^* = \hat\beta_0 + \hat\beta_1x_{1i}^* + \hat\beta_2x_{2i}^* \]


25.3 Predicted Mean and SE for calculating Confidence and Prediction Intervals