Multiple regression refers to the method of predicting one variable as a linear function of more than one predictor.
LA homes example:
\[ \widehat{price} = \beta_0 + \beta_1 sqft + \beta_2 bath \]
But we could also introduc as the additional predictor a polynomial term of the existing predictor.
\[ \widehat{price} = \beta_0 + \beta_1 sqft + \beta_2 sqft^2 \]
Recall what happened when we fit this model:
\[ \widehat{logprice} = \beta_0 + \beta_1 logsqft \]
We could consider adding a quadratic term to our model:
\[ \widehat{price} = \beta_0 + \beta_1 sqft + \beta_2 sqft^2 \]
m2 <- lm(log_price ~ log_sqft + I(log_sqft^2), data = LA) summary(m2)$coef
## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 10.2474 1.17846 8.696 8.438e-18 ## log_sqft -0.5482 0.30914 -1.773 7.638e-02 ## I(log_sqft^2) 0.1301 0.02017 6.449 1.490e-10
summary(m1)$coef
## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 2.703 0.14369 18.81 1.972e-71 ## log_sqft 1.442 0.01954 73.79 0.000e+00
summary(m2)$coef
## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 10.2474 1.17846 8.696 8.438e-18 ## log_sqft -0.5482 0.30914 -1.773 7.638e-02 ## I(log_sqft^2) 0.1301 0.02017 6.449 1.490e-10
The residual plots for the second (more complex) model seem slightly better, so we're inclined to use that model. We can also compare the explanatory power of the models by looking at \(R^2\).
summary(m1)$adj
## [1] 0.7736
summary(m2)$adj
## [1] 0.7793
These two models are very similar - both are quite good in terms of validity and explanatory power - but the quadratic one edges out the simple linear one.
m3 <- lm(log_price ~ log_sqft + rnorm(length(LA$log_price)), data = LA) summary(m1)$r.squared
## [1] 0.7738
summary(m3)$r.squared
## [1] 0.7738
summary(m1)$adj
## [1] 0.7736
summary(m3)$adj
## [1] 0.7735