7.3 Multiple linear regression

We can, of course, extend this to include multiple continuous explanatory variables of interest just as we did with ANOVA for multiple categorical explanatory variables!

Here is an example to whet your appetite. Let’s say we want a multiple regression model that includes both Education and Catholic?

multiple_mod <- lm(Fertility ~ Education + Catholic, data = swiss)

summary(multiple_mod)

## 
## Call:
## lm(formula = Fertility ~ Education + Catholic, data = swiss)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -15.042  -6.578  -1.431   6.122  14.322 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 74.23369    2.35197  31.562  < 2e-16 ***
## Education   -0.78833    0.12929  -6.097 2.43e-07 ***
## Catholic     0.11092    0.02981   3.721  0.00056 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.331 on 44 degrees of freedom
## Multiple R-squared:  0.5745, Adjusted R-squared:  0.5552 
## F-statistic:  29.7 on 2 and 44 DF,  p-value: 6.849e-09

Or if we really want to get crazy with the hot sauce:

full_mod <- lm(
  Fertility ~ Agriculture + Examination + Education + Catholic,
  data = swiss
  )

summary(full_mod)

## 
## Call:
## lm(formula = Fertility ~ Agriculture + Examination + Education + 
##     Catholic, data = swiss)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -15.7813  -6.3308   0.8113   5.7205  15.5569 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 91.05542    6.94881  13.104  < 2e-16 ***
## Agriculture -0.22065    0.07360  -2.998  0.00455 ** 
## Examination -0.26058    0.27411  -0.951  0.34722    
## Education   -0.96161    0.19455  -4.943 1.28e-05 ***
## Catholic     0.12442    0.03727   3.339  0.00177 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7.736 on 42 degrees of freedom
## Multiple R-squared:  0.6498, Adjusted R-squared:  0.6164 
## F-statistic: 19.48 on 4 and 42 DF,  p-value: 3.95e-09