Tests: The Linear Regression Model (Projection Matrix)

Ivan Jacob Agaloos Pesigan

2020-12-31

# The Linear Regression Model: Projection Matrix {#linreg-estimation-projection-example}

Data

See jeksterslabRdatarepo::wages.matrix() for the data set used in this example.

X <- jeksterslabRdatarepo::wages.matrix[["X"]]
# age is removed
X <- X[, -ncol(X)]
y <- jeksterslabRdatarepo::wages.matrix[["y"]]
head(X)
#>      constant gender race union education experience
#> [1,]        1      1    0     0        12         20
#> [2,]        1      0    0     0         9          9
#> [3,]        1      0    0     0        16         15
#> [4,]        1      0    1     1        14         38
#> [5,]        1      1    1     0        16         19
#> [6,]        1      1    0     0        12          4
head(y)
#>      wages
#> [1,] 11.55
#> [2,]  5.00
#> [3,] 12.00
#> [4,]  7.00
#> [5,] 21.15
#> [6,]  6.92

Projection Matrix / Hat Matrix

Pmatrix <- P(X = X)
result_yhat <- as.vector(Pmatrix %*% y)

Residual Maker Matrix

Mmatrix1 <- M(X = X)
Mmatrix2 <- .M(
  X = X,
  P = Pmatrix
)
result_epsilonhat1 <- as.vector(Mmatrix1 %*% y)
result_epsilonhat2 <- as.vector(Mmatrix2 %*% y)

lm() function

lmobj <- lm(
  wages ~ gender + race + union + education + experience,
  data = jeksterslabRdatarepo::wages
)
lm_yhat <- as.vector(predict(lmobj))
lm_epsilonhat <- as.vector(residuals(lmobj))

context("Test linreg-estimation-projection")
test_that("Py = yhat.", {
  expect_equivalent(
    length(result_yhat),
    length(lm_yhat)
  )
  for (i in seq_along(result_yhat)) {
    expect_equivalent(
      result_yhat[i],
      lm_yhat[i]
    )
  }
})
#> Test passed 🎊
test_that("My = epsilonhat.", {
  expect_equivalent(
    length(result_epsilonhat1),
    length(result_epsilonhat2),
    length(lm_epsilonhat)
  )
  for (i in seq_along(result_epsilonhat1)) {
    expect_equivalent(
      result_epsilonhat1[i],
      lm_epsilonhat[i]
    )
  }
  for (i in seq_along(result_epsilonhat2)) {
    expect_equivalent(
      result_epsilonhat2[i],
      lm_epsilonhat[i]
    )
  }
})
#> Test passed 🥳