Calculation of the likelihood in the Boxcox transformation

Question

I have an outcome which may be transformed in cases of egregious departures from normality to its Box-Cox optimal normal transformation in an unconditional model.

How is the likelihood calculated for the $\lambda$ parameter?

For instance, if I generate data according to a skewnormal model:

set.seed(123)
y <- rlnorm(100, 1, 1)
x <- rep(1, 100)
o <- boxcox(y ~ x)

I get as the optimal $\lambda$ a value close to 0 as expected which achieves a maximum of $\log(L(x, y,\lambda)) \approx -200$ .

However, when I evaluate the log-likelihood of the log transformed response using logLik(glm(y ~ 1, family=gaussian(link=log))) the value is:

'log Lik.' -295.7451 (df=2)

I would expect those values to be the same.

Answer 1

In the source code for getS3method('boxcox', 'default'). It seems what they call the log-likelihood is calculated like this

loglik[i] <- -n/2 * log(sum(qr.resid(xqr, yt)^2))

whereas logLik.lm uses

val <- 0.5 * (sum(log(w)) - N * (log(2 * pi) + 1 - log(N) + log(sum(w * res^2))))

which if we have unity weights is equivalent to

val <- 0.5 * ( - N * (log(2 * pi) + 1 - log(N) + log(sum(res^2))))

so basically the boxcox procedure drops the additive constant of 0.5 * ( - N * (log(2 * pi) + 1 - log(N))), which gives the same optimal solution, but the graphic is incorrect in that the raw values are not the actual log-likelihood.