logLik_o: Log-Likelihood for the orthogonal residuals

In onls: Orthogonal Nonlinear Least-Squares Regression

Log-Likelihood for the orthogonal residuals

Description

Returns the log-likelihood as calculated from the orthogonal residuals obtained by residuals_o, including a Jacobian/normalizing-constant correction for the response and predictor precisions (weights, sigma_x, sigma_y) used by onls, so that the resulting value (and hence AIC/BIC) is comparable across onls fits made with different weighting schemes.

Usage

logLik_o(object) 

Arguments

object

an object returned from onls.

Details

Let d_i be the weighted orthogonal distance of observation i at the fitted parameters (residuals_o, i.e. object$resid_o), and N the number of observations. A naive treatment of the d_i as i.i.d. N(0,\sigma^2) draws gives the usual concentrated Gaussian log-likelihood

\ell_0 = -\frac{N}{2}\left(\log(2\pi) + 1 - \log(N) + \log\left(\sum_i d_i^2\right)\right),

which is what earlier versions of logLik_o returned. However, since onls already folds the response precision Qyy_i and predictor precision Qx_i into d_i itself (see 'Details' in onls), \ell_0 alone omits the normalizing-constant (Jacobian) term that these precisions contribute to the underlying Gaussian density – exactly as plain logLik/AIC on unweighted residuals from a weighted lm/nls fit would, if computed without R's own +\frac{1}{2}\sum_i \log(w_i) correction. logLik_o therefore adds the corresponding correction,

\ell = \ell_0 \;+\; \frac{1}{2}\sum_{i=1}^{N} \log(Qyy_i) \;+\; \frac{1}{2}\sum_{i=1}^{N} \log\left|Qx_i\right|,

where |Qx_i| is the determinant of the (possibly per-observation) predictor precision matrix used by onls – a scalar for single-predictor models, and the full p \times p determinant (correctly accounting for any predictor correlation) for multivariate models. For the default, fully unweighted onls fit, Qyy_i = 1 and Qx_i = I_p for every i, so both correction terms are exactly 0 and \ell = \ell_0; the correction only changes the value for fits using weights, sigma_x, and/or sigma_y.

Note that d_i itself remains a simplification: it is the square root of a sum of p+1 squared, precision-weighted Gaussian terms (one from the response, p from the predictors), not a single univariate normal draw, so treating \sum_i d_i^2 / N as the MLE of a common residual variance (as \ell_0 does) is itself an approximation, inherited unchanged from the original (unweighted) formula. The correction above addresses only the missing weighting/precision Jacobian term, not this deeper simplification.

Value

The log-likelihood as calculated from the orthogonal residuals, with attributes "df" (1 + q_{free}, the number of free model parameters plus one for the residual scale) and "nobs"/"nall" (the number of observations).

Note

logLik_o has no other generic functions on top, so for calculating AIC, one has to apply AIC(logLik_o(model)), see 'Examples'. The usual logLik applies to the vertical residuals of the orthogonal model. Because logLik_o includes the response/predictor precision correction described in 'Details', its value (and AIC/BIC derived from it) can be validly compared between two onls fits on the same data even if they used different weights, sigma_x, or sigma_y, provided the response variable and the number of observations are the same.

Author(s)

Andrej-Nikolai Spiess

Examples

DNase1 <- subset(DNase, Run == 1)
DNase1$density <- sapply(DNase1$density, function(x) rnorm(1, x, 0.1 * x))
mod1 <- onls(density ~ Asym/(1 + exp((xmid - log(conc))/scal)), 
             data = DNase1, start = list(Asym = 3, xmid = 0, scal = 1))
logLik_o(mod1)

## compare AIC of vertical versus orthogonal residuals.
AIC(mod1)
AIC(logLik_o(mod1))

## Comparing an unweighted and a weighted onls() fit on the same data:
## logLik_o() includes the precision Jacobian correction, so AIC/BIC
## from the two fits are validly comparable.
mod2 <- onls(density ~ Asym/(1 + exp((xmid - log(conc))/scal)), 
              data = DNase1, start = list(Asym = 3, xmid = 0, scal = 1),
              sigma_x = 0.05, sigma_y = 0.1)
AIC(logLik_o(mod1))
AIC(logLik_o(mod2))

onls documentation built on Sept. 21, 2026, 9:06 a.m.