## Demonstrate the relationship between the partial R2 for a covariate
## and the significance level of the covariate.

M = NULL

## Use bivariate regression with cor(X1,X2) = r.
r = 0.4

for (n in c(20,50,100,200,500)) {

  sig_level = NULL
  p_r2 = NULL
  
  ## Do replications to get a more stable result.
  for (k in 1:100) {
    
    ## Generate the design matrix.
    X1 = rnorm(n)
    U = rnorm(n)
    X2 = r*X1 + sqrt(1-r^2)*U
    X = cbind(X1, X2)
    
    ## Simulate the response.
    Y = X1 + X2 + 4*rnorm(n)
    
    ## Calculate the full model R^2.
    m_full = lm(Y ~ X)
    R2_full = cor(Y, m_full$fitted.values)^2

    ## Get the significance level.
    XX = cbind(array(1,n), X)
    qrd = qr(XX)
    R = qr.R(qrd)
    H = solve(t(R) %*% R)
    sigma2 = sum( (Y - m_full$fitted.values)^2 ) / (n-3)
    se = sqrt(sigma2 * H[2,2])
    sig_level[k] = m_full$coeff[2] / se

    ## Calculate the submodel R^2 for X2 only.
    m_2 = lm(Y ~ X2)
    R2_2 = cor(Y, m_2$fitted.values)^2
    
    ## The significance level and partial R^2.
    p_r2[k] = (R2_full-R2_2)/(1-R2_2)
  }
    
  M = rbind(M, c(n, mean(sig_level), mean(p_r2)))
}