#
# Get the data from a file
#
> source("Father_Son.txt")
# 
# Summarize the data set
#
> summary(father.son)
    fheight         sheight     
 Min.   :59.01   Min.   :58.51  
 1st Qu.:65.79   1st Qu.:66.93  
 Median :67.77   Median :68.62  
 Mean   :67.69   Mean   :68.68  
 3rd Qu.:69.60   3rd Qu.:70.47  
 Max.   :75.43   Max.   :78.36  
#
#
> dim(father.son)
[1] 1078    2
#
#
> attach(father.son)
#
# Fit a simple linear regression model
#
> fit.1 <- lm(sheight~fheight)
> fit.1

Call:
lm(formula = sheight ~ fheight)

Coefficients:
(Intercept)      fheight  
    33.8866       0.5141  

> summary(fit.1)

Call:
lm(formula = sheight ~ fheight)

Residuals:
      Min        1Q    Median        3Q       Max 
-8.877151 -1.514415 -0.007896  1.628512  8.968479 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 33.88660    1.83235   18.49   <2e-16 ***
fheight      0.51409    0.02705   19.01   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Residual standard error: 2.437 on 1076 degrees of freedom
Multiple R-Squared: 0.2513,     Adjusted R-squared: 0.2506 
F-statistic: 361.2 on 1 and 1076 DF,  p-value: < 2.2e-16 
# 
# compute correlations
#
> cor(father.son)
          fheight   sheight
fheight 1.0000000 0.5013383
sheight 0.5013383 1.0000000
#
# add a quadratic term
#
> f2 = fheight^2
> fit.2 <-lm(sheight~fheight+f2)
> summary(fit.2)

Call:
lm(formula = sheight ~ fheight + f2)

Residuals:
     Min       1Q   Median       3Q      Max 
-8.86743 -1.50728 -0.01277  1.62595  8.97530 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)
(Intercept) 39.779150  33.264006   1.196    0.232
fheight      0.339381   0.985137   0.345    0.731
f2           0.001293   0.007287   0.177    0.859

Residual standard error: 2.438 on 1075 degrees of freedom
Multiple R-Squared: 0.2514,     Adjusted R-squared:  0.25 
F-statistic: 180.5 on 2 and 1075 DF,  p-value: < 2.2e-16 
#
#  Notice that neither coefficient is significant!
#  BUT the overall F test is overwhelmingly significant
#  This happens when the two estimated coefficients are
#   highly correlated;  
#  In fact the variables fheight and f2=fheight^2 are
#  highly correlated.
> anova(fit.2)
Analysis of Variance Table

Response: sheight
            Df Sum Sq Mean Sq  F value Pr(>F)    
fheight      1 2144.6  2144.6 360.9096 <2e-16 ***
f2           1    0.2     0.2   0.0315 0.8592    
Residuals 1075 6387.8     5.9                    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
#
#  The ANOVA table says you probably don't need f2
#


> cor(fheight,f2)
[1] 0.9996227