What Identifiability Means
A population design is useful only if its observations can distinguish the population parameters of interest. A design may produce many measurements and still fail to distinguish two parameters if changing them in certain combinations leaves the modeled distribution unchanged. The approximate population Fisher matrix provides a local, first-order way to detect such failures.
It is important to separate local identifiability from nonsingular Fisher information. Local identifiability asks whether different parameter values sufficiently close to a specified value imply different distributions. Nonsingular Fisher information asks whether every parameter direction changes the distribution to first order, under the model and approximation used to calculate that information. The latter is a useful sufficient condition for regular local information, but it is not necessary for local identifiability.
The distinction matters in practice. A singular matrix may indicate redundant parameters, insufficiently varied sampling times, or an approximation that misses higher-order changes. It does not, by itself, prove that the exact model is locally unidentifiable.
Approximate Population Information and Its Null Directions
Suppose subjects are independent. For subject \(i\), let the observation vector be approximated by a multivariate normal distribution with mean \(\mu_i(\theta)\) and covariance matrix \(\Sigma_i(\theta)\), where \(\theta\) is a \(p\)-dimensional population parameter vector. The approximation assumes differentiable means and covariances and positive-definite \(\Sigma_i\). It is a first-order population information calculation, not necessarily the exact information of a nonlinear mixed-effects model's marginal likelihood.
Write \(\mu_{i,a}=\partial\mu_i/\partial\theta_a\) and \(\Sigma_{i,a}=\partial\Sigma_i/\partial\theta_a\). Under this Gaussian approximation, the Fisher information element for parameters \(\theta_a\) and \(\theta_b\) is
The first term records information from changes in the mean; the second records information from changes in the covariance. A subject's repeated observations can therefore contribute information through both their expected values and their within-subject covariance pattern.
To see the identifiability connection, consider any direction \(v\) in parameter space. Its directional derivatives for subject \(i\) are \(D\mu_i[v]=\sum_a v_a\mu_{i,a}\) and \(D\Sigma_i[v]=\sum_a v_a\Sigma_{i,a}\). In the following expression, the square root of \(\Sigma_i^{-1}\) is well-defined because \(\Sigma_i\) is positive definite, and the matrix inside the trace is symmetric.
Each term is nonnegative. The mean term is a squared length after weighting by a positive-definite matrix. For a symmetric matrix \(A\), \(\operatorname{tr}(A^2)\) is the sum of the squares of its entries, so the covariance term is also nonnegative and is zero only when \(A=0\). Since \(\Sigma_i^{-1/2}\) is invertible, that occurs exactly when \(D\Sigma_i[v]=0\). Thus \(v^{\mathsf T}Fv=0\) exactly when, for every subject, both \(D\mu_i[v]=0\) and \(D\Sigma_i[v]=0\).
For a positive-semidefinite Fisher matrix, singularity means there is a nonzero \(v\) with \(v^{\mathsf T}Fv=0\). The derivation shows that, under the stated Gaussian approximation, this is equivalent to a direction that changes neither the modeled means nor covariances to first order across the design. Equivalently, the combined sensitivity columns for the parameters are linearly dependent. Dependence among mean sensitivities alone is not enough if covariance sensitivities distinguish the same parameter combination.
Worked Examples
Worked Example: Two Parameters With the Same Mean Effect
Suppose each of four hypothetical subjects contributes one observation, with \(Y_i\sim N(\theta_1+\theta_2,4)\). The variance is known and fixed, and the observations are independent. Assess first-order identifiability of \(\theta_1\) and \(\theta_2\).
The mean derivatives are both 1, and the covariance derivatives are zero. Each subject's information contribution is therefore \[ \frac{1}{4} \begin{pmatrix} 1&1\\ 1&1 \end{pmatrix}. \] Summing over four subjects gives \[ F= \begin{pmatrix} 1&1\\ 1&1 \end{pmatrix}. \]
For \(v=(1,-1)^{\mathsf T}\), \(Fv=(0,0)^{\mathsf T}\). Moving a small amount in the direction that increases \(\theta_1\) and decreases \(\theta_2\) by the same amount does not change the mean or variance. The modeled distribution depends on the parameters only through \(\theta_1+\theta_2\), so they cannot be separately identified from these observations. Four subjects improve information about the sum, but do not resolve this confounding.
Worked Example: Covariance Sensitivity Separates Parameters
Consider three independent hypothetical subjects, each with one observation from \(Y_i\sim N(\alpha+\beta,\exp(\alpha-\beta))\). At the nominal value \(\alpha=\beta=0\), determine whether the approximate information is singular. Assume the Gaussian model is exact for this example.
At the nominal value, the variance is \(1\). The mean derivative vector is \((1,1)^{\mathsf T}\), and the variance derivatives with respect to \(\alpha\) and \(\beta\) are \(1\) and \(-1\). For one observation, the mean contribution is \[ \begin{pmatrix}1&1\\1&1\end{pmatrix}, \] and the covariance contribution is \[ \frac{1}{2} \begin{pmatrix}1&-1\\-1&1\end{pmatrix}. \]
Thus, the information per subject is \[ \begin{pmatrix}1.5&0.5\\0.5&1.5\end{pmatrix}, \] and for three subjects it is \[ F= \begin{pmatrix}4.5&1.5\\1.5&4.5\end{pmatrix}. \] Its determinant is \(4.5(4.5)-1.5(1.5)=18\), which is positive; the matrix is nonsingular. The mean alone responds only to \(\alpha+\beta\), but the variance responds to \(\alpha-\beta\). Together, these two distributional features distinguish the parameters locally.
Worked Example: Local Identifiability With Zero Fisher Information
Let a single hypothetical observation satisfy \(Y\sim N(\theta^3,1)\). Examine local identifiability and Fisher information at \(\theta=0\).
The normal distribution's mean is \(\theta^3\), so its derivative is \(3\theta^2\). At zero, that derivative is zero, and the variance is fixed. The Fisher information is therefore \(F(0)=(3\theta^2)^2|_{\theta=0}=0\), a singular (scalar) information value.
Nevertheless, the parameter is identifiable: if two values of \(\theta\) give the same normal distribution, their means must be equal, so \(\theta_1^3=\theta_2^3\), which implies \(\theta_1=\theta_2\). The map from \(\theta\) to the distribution is one-to-one, including in every neighborhood of zero. This example shows why zero first-order information at a point is not proof of local non-identifiability. The distribution changes at higher order in \(\theta\), which the first derivative does not capture.
How to Assess a Proposed Design
A practical assessment begins with the nominal parameter values and the actual subject-level design: sampling times, covariates, and observation patterns determine the sensitivities available to the design. State the approximation being used, calculate derivatives for each subject, and assess whether a nonzero parameter combination leaves all mean and covariance derivatives unchanged. The following sequence organizes that check.
Write each subject's approximate mean and covariance as functions of the population parameters. Confirm that the covariance matrices are positive definite and the derivatives exist at the nominal values.
For each parameter, calculate the mean and covariance derivatives for each subject. Do not assess mean sensitivities in isolation if the covariance also depends on the parameters.
Ask whether a nonzero vector \(v\) makes every subject's directional mean and covariance derivatives zero. If so, the approximate Fisher matrix is singular and the design lacks first-order information in that direction.
Identify which parameter combination is involved, then determine whether the issue reflects structural redundancy, the chosen design, a special nominal value, or a limitation of the approximation.
Exact linear dependence is not the only concern. Sensitivities can be nearly dependent, leaving a design with very weak information in some combination even if the calculated matrix is technically nonsingular. Numerical checks should therefore be repeated with careful parameter scaling and derivative verification; an apparent rank problem can arise from units or numerical error. Scaling helps interpretation and computation, but does not create information absent from the model and design.
A useful design diagnosis is to ask what change would make the implicated derivative patterns distinct. Different sampling times, additional within-subject observations, or a broader range of covariate values may help if they alter those patterns. More subjects with an unchanged design can increase the amount of information, but cannot generally remove an exact per-subject confounding such as the one in the first example.
Limits, Alternatives, and Common Pitfalls
The approximate Fisher assessment is local: its answer depends on the nominal parameter values, the design, and the approximation to the observation distribution. It also concerns first-order information. A full likelihood analysis or a direct check of whether distinct parameter values imply the same distribution can investigate identifiability beyond that first-order test, although those approaches may be more demanding. Comparing likelihood behavior or simulating data can help reveal weakly distinguished parameter combinations, but neither substitutes for stating the model and the question being assessed.
Reparameterizing may make a model easier to interpret or calculate with. It does not add information. If the transformation is smooth and has a nonsingular Jacobian at the nominal value (so it is a local diffeomorphism there), a genuine first-order information deficit remains a deficit under the transformed coordinates. Conversely, a transformation that is not one-to-one can introduce redundant representations, so its domain and mapping must be checked.
Common mistake: “The Fisher matrix is singular, so the parameters are locally unidentifiable.” This conclusion is too strong. Singularity says that some nonzero direction has no first-order change in the means or covariances under the stated approximation. It may reflect genuine redundancy, but—as \(N(\theta^3,1)\) at \(\theta=0\) demonstrates—the model can still be locally identifiable through higher-order changes. The correction is to report the first-order deficiency, identify its direction, and assess exact distributional identifiability separately when that distinction matters.
The converse also deserves care: a nonsingular approximate matrix is evidence of distinct first-order sensitivities under its assumptions, not a universal guarantee of practical identifiability across a broad parameter region. A local calculation cannot establish that parameter estimates will be precise under every sample size or model departure.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- If two parameters have identical mean sensitivities but different covariance sensitivities, must the approximate population Fisher matrix be singular?
- For \(Y\sim N(\theta^3,1)\) at \(\theta=0\), what does the zero Fisher information establish, and what does it not establish?
- In the first worked example, would doubling the number of subjects with the same design separate \(\theta_1\) from \(\theta_2\)? Explain.
- State the directional-derivative condition for a null direction of the Gaussian approximate Fisher matrix.
Solutions. (1) No. Covariance sensitivities may distinguish the parameters even when their mean sensitivities coincide, as in the second worked example. (2) It establishes the absence of first-order information at zero under this model; it does not establish local non-identifiability. The map \(\theta\mapsto\theta^3\) is one-to-one, so the parameter is locally identifiable. (3) No. Doubling replication increases information about \(\theta_1+\theta_2\), but the direction \((1,-1)^{\mathsf T}\) still changes neither the mean nor the variance. (4) For a nonzero \(v\), every subject must satisfy \(D\mu_i[v]=0\) and \(D\Sigma_i[v]=0\); under the stated positive-definite Gaussian approximation, this is equivalent to \(v^{\mathsf T}Fv=0\).