A.12 The Box–Cox Method for Transformations
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A.12
313
the box–cox method for transformations
Many statistical packages will have routines that will provide a graph of
RSS(λ) versus λ, or of (n/2) log(RSS(λ)) versus λ as shown in Figure 8.7, for
the highway accident data. Equation (A.40) shows that the confidence interval
for λ includes all values of λ for which the log-likelihood is within 1.92 units
of the maximum value of the log-likelihood, or between the two vertical lines
in the figure.
A.12.2 Multivariate Case
Although the material in this section uses more mathematical statistics than
most of this book, it is included because the details of computing the multivariate extension of Box–Cox transformations are not published elsewhere. The
basic idea was proposed by Velilla (1993).
Suppose X is a set of p variables we wish to transform and define
ψ M ( X , l ) = (ψ M ( X 1, λ1 ), … , ψ M ( X k , λ k ))
We have used the modified power transformations (8.5) for each element of
X, but the same general idea can be applied using other transformations such
as the Yeo–Johnson family introduced in Section 8.4. In analogy to the univariate case, we assume that for some λ, we will have
ψ M ( X , l ) ~ N( m , V)
where V is an unknown positive definite symmetric matrix that needs to be
estimated. If xi is the observed value of X for the ith observation, then the
likelihood function is given by
n
L( m , V, l|X ) =
1
∏ (2π V )
1/2
i =1
1
× exp − (ψ M (x i, l ) − m )′ V −1 (ψ M (x i, l ) − m )
2
(A.41)
where |V| is the determinant.3 After rearranging terms, the log-likelihood is
given by
n
n
log(L( m , V, l|X )) = − log(2π ) − log ( V )
2
2
−
3
1
2
n
∑
V −1 (ψ M (x i, l ) − m )(ψ M (x i, l ) − m )′
i =1
The determinant is defined in any linear algebra textbook.
(A.42)
314
appendix
If we fix λ, then (A.42) is the standard log-likelihood for the multivariate
normal distribution. The values of V and μ that maximize (A.42) are the
sample mean and sample covariance matrix, the latter with divisor n rather
than n − 1,
1
m(l ) =
n
V( l ) =
1
n
n
∑ψ
M
(x i , l )
i =1
n
∑ (ψ
M
(x i, l ) − m ( l ))(ψ M (x i, l ) − m ( l ))′
i =1
Substituting these estimates into (A.42) gives the profile log-likelihood
for λ,
n
n
n
log(L( m ( l ), V( l ), l|X )) = − log(2π ) − log ( V( l ) ) −
2
2
2
(A.43)
This equation will be maximized by minimizing the determinant of V(λ) over
values of λ. This is a numerical problem for which there is no closed-form
solution, but it can be solved using a general-purpose function minimizer.
Standard theory for maximum likelihood estimates can provide tests concerning λ and standard errors for the elements of λ. To test the hypothesis that
λ = λ0 against a general alternative, compute
G 2 = 2 log(L(m (lˆ ), V(lˆ ), lˆ )) − log(L(m (l0 ), V(l0 ), l0 ))
and compare G2 with a chi-squared distribution with p df. The standard error
of lˆ is obtained from the inverse of the expected information matrix evaluated
at lˆ . The expected information for lˆ is just the matrix of second derivatives
of (A.43) with respect to λ evaluated at lˆ . Many optimization routines, such
as optim in R, will return the matrix of estimated second derivatives if
requested; all that is required is inverting this matrix, and then the square roots
of the diagonal elements are the estimated standard errors.
A.13 CASE DELETION IN LINEAR REGRESSION
Suppose X is the n × p′ matrix of regressors with linearly independent columns.
We use the subscript “(i)” to mean “without case i,” so that X(i) is an (n − 1) × p′
matrix. We can compute (X (′i ) X (i ) )−1 from the remarkable formula
(X (′i ) X ( i ) )−1 = (X ′X)−1 +
(X ′X)−1 x i x ′i (X ′X)−1
1 − hii
(A.44)
A.13
315
case deletion in linear regression
where hii = x i′(X ′X)−1 x i is the ith leverage value, a diagonal value from the hat
matrix. This formula was used by Gauss (1821); a history of it and many variations is given by Henderson and Searle (1981). It can be applied to give all the
results that one would want relating multiple linear regression with and
without the ith case. For example,
(X ′X) x i eˆi
bˆ (i ) = bˆ −
1 − hii
−1
(A.45)
Writing ri = eˆi /σˆ 1 − hii , the estimate of variance is
−1
n − p′ − 1
σˆ (2i ) = σˆ 2
n − p′ − ri2
(A.46)
and the studentized residual ti is
n − p′ − 1
ti = ri
n − p′ − ri2
1/ 2
(A.47)
The diagnostic statistics examined in this book were first thought to be
practical because of simple formulas used to obtain various statistics when
cases are deleted that avoided recomputing estimates. Advances in computing
in the last 30 years have made the computational burden of recomputing
without a case much less onerous, and so diagnostic methods equivalent to
those discussed here can be applied to problems other than linear regression
where the updating formulas are not available.
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