The Gaussian Approximation to the Subject-Level Likelihood

From a Linearized Population Model to a Subject Likelihood

For population design, the data from one subject are a vector of observations collected under that subject’s sampling schedule. A subject-level likelihood is the probability density assigned to that vector by a model, viewed as a function of the population parameters. In the approximation considered here, the vector is modeled as multivariate normal, with its mean and covariance obtained from the linearized population model. Earlier material on approximate population means and covariances supplies those model quantities; the task here is to assemble them into a likelihood and understand its two central terms.

This is an approximation to the marginal distribution of a subject’s observations. It is not generally the exact likelihood obtained by integrating a nonlinear mixed-effects model over its random effects. The approximation can be useful when the linearization is adequate, but the likelihood inherits any weaknesses in that approximation.

The positive-definiteness condition is essential: it ensures that the covariance has positive variances in every nonzero direction, that its determinant is positive, and that its inverse exists. The observation times and design are treated as fixed when evaluating this subject’s density. Within-subject observations need not be independent; their covariance is represented by the off-diagonal elements of \(\boldsymbol{\Sigma}_i\).

The Density and Its Logarithm

Under the stated approximation, the density for the observed vector is

Here \(L_i(\theta)\) denotes the subject-level likelihood contribution, evaluated at the observed \(\mathbf y_i\). Its logarithm is

The log form turns products and powers in the density into sums. It also separates the likelihood into a dimension-dependent normalizing constant, a covariance determinant term, and a residual quadratic form. The determinant term is not optional: evaluating only how close the observations are to the mean does not give the Gaussian likelihood.

The determinant and quadratic form must be considered together. A covariance matrix that is enlarged can reduce the quadratic penalty for a fixed residual, but it also increases the determinant penalty. Thus, simply increasing model variances cannot improve the log-likelihood without limit.

Why the Two Terms Appear

A short change-of-variables argument explains the structure. For a fixed positive-definite covariance matrix, write its Cholesky factorization as \(\boldsymbol{\Sigma}_i=LL^T\), where \(L\) is lower triangular with positive diagonal entries. Define the whitened residual \(\mathbf z=L^{-1}\mathbf r_i\). Then

The transformed residual has a standard multivariate normal density, proportional to \(\exp(-\mathbf z^T\mathbf z/2)\). Transforming that density back to the original observation coordinates multiplies it by the absolute Jacobian determinant \(\lvert L\rvert^{-1}\). Since \(\lvert\boldsymbol{\Sigma}_i\rvert=\lvert L\rvert^2\), this factor is \(\lvert\boldsymbol{\Sigma}_i\rvert^{-1/2}\). The standard \(m_i\)-dimensional normalizing constant is \((2\pi)^{-m_i/2}\). Taking logarithms yields the determinant and quadratic-form terms in \(\ell_i\). This derivation assumes \(L\) exists, which is guaranteed by positive definiteness.

Whitening is also a useful interpretation: the quadratic form is the ordinary squared Euclidean length of the residual after the covariance structure has been removed. It is not generally the sum of the squared raw residuals divided by one common variance.

How Covariance Changes the Likelihood

The next hypothetical comparison holds the residual vector fixed at \([0.3,-0.2]^T\) units and changes only the covariance. Compare the correlated matrix from the preceding example with the diagonal matrix having the same marginal variances. Both matrices are positive definite: their diagonal entries are positive, and their determinants are positive.

Assumptions, Diagnostics, and Limits

The calculation is valid as the stated Gaussian approximation when the model provides an appropriate approximate mean and covariance for the subject’s observation vector, and when the covariance at the parameter value is symmetric positive definite. If a population likelihood across subjects is formed by multiplying subject densities, this additionally assumes subjects are independent under the population model. Dependence among observations within a subject is allowed and must be represented in that subject’s covariance.

Several checks help detect numerical or modeling problems:

• Check symmetry and positive definiteness of \(\boldsymbol{\Sigma}_i\). A nonpositive determinant or failed Cholesky factorization prevents a valid nonsingular Gaussian density.
• Check the determinant and quadratic form separately. A negative computed quadratic form indicates a numerical or implementation error when the covariance is positive definite.
• Verify the residual order matches the row and column order of the covariance matrix. A correctly computed determinant cannot protect against pairing observations with the wrong covariance entries.
• Use a stable numerical method, such as Cholesky factorization, rather than explicitly forming a matrix inverse in software. The displayed inverse formula is useful for hand calculations, but direct inversion can be less stable computationally.
• Assess whether a Gaussian approximation is credible for the model and design. Strong nonlinearity, highly skewed marginal distributions, or behavior poorly captured by a first-order linearization can make the approximate likelihood misleading even when the covariance is mathematically valid.

The main alternative is the exact marginal likelihood, which integrates the conditional observation density over the random-effects distribution. In nonlinear mixed-effects models that integral may require numerical integration or approximation methods. The Gaussian approximation instead summarizes the subject’s marginal distribution through its approximate mean and covariance and inserts those quantities into a multivariate normal density. It is computationally direct, but it does not become exact merely because the mean and covariance have been calculated accurately.

Common Mistake and Practical Interpretation

A frequent error is to calculate only \(-\tfrac12\mathbf r_i^T\boldsymbol{\Sigma}_i^{-1}\mathbf r_i\) and call it the log-likelihood, or to omit the determinant because it does not involve the residual. That reasoning is incorrect whenever the covariance depends on model parameters: changing the covariance changes both terms. Even when comparing models with different covariance structures, the determinant remains part of the density normalization.

Another mistake is to interpret a likelihood density as a probability for an exact continuous observation. A point has probability zero under a continuous distribution; the density describes relative support per unit of measurement. Probabilities require integration over a region. Finally, \(\ell_i\) may be negative without indicating a poor or invalid model: continuous densities can be below one, and log densities can consequently be negative.