Information Under Missing Observations

Why Missing Components Change Information

A sampling schedule describes measurements that are intended to be collected. The likelihood for a particular subject, however, is built from the measurements that are actually observed. If a planned concentration is unavailable, its value cannot contribute a residual, and its covariance with the available measurements cannot be used as though the missing value had been recorded. The subject’s information must be recalculated for the observed components.

This distinction matters in population design because information depends on both the sensitivity of the model and the covariance among observations. Losing one component can change the information about several parameters, not only the parameter most directly associated with that component. The calculation below applies to the first-order approximate Gaussian model: it is an information calculation for that approximation, not a claim that the approximation is exact.

The selection matrix has one row for each observed component and one column for each planned component. Each row contains a 1 in the column for the retained component and 0 elsewhere. It performs the same selection on the mean and on both dimensions of the covariance matrix. If the first and third of four planned samples are observed, for example, \(S_i\) keeps rows and columns 1 and 3. The resulting covariance is the covariance submatrix for those two samples.

Information for the Observed Subvector

For a multivariate normal model whose mean and covariance may depend on \(\psi\), the expected Fisher information for the observed vector has a mean-sensitivity contribution and a covariance-sensitivity contribution. Write \(m_{i,O,a}=\partial m_{i,O}/\partial\psi_a\) and \(V_{i,O,a}=\partial V_{i,O}/\partial\psi_a\). Then the subject-level matrix has entries

The first term measures how changes in the parameters move the expected observations relative to their variability. The second term measures information carried by parameter-dependent covariance. If a parameter affects only the mean, its covariance derivative is zero; if it affects only the covariance, its mean derivative is zero. For a population parameter affecting both, both terms may contribute. The derivatives and covariance in this formula must all refer to the observed subvector.

A short justification follows from the normal log-likelihood. For residual \(r=Y_{i,O}-m_{i,O}\), the score for \(\psi_a\) is

Under the stated normal model, \(E(r)=0\) and \(E(rr^\mathsf{T})=V_{i,O}\). The expected product of the two linear mean-score terms gives \(m_{i,O,a}^{\mathsf{T}}V_{i,O}^{-1}m_{i,O,b}\). The expected product of the centered quadratic covariance-score terms gives the trace term, using the fourth-moment identity for a multivariate normal vector. The cross-products have expectation zero because centered normal third moments are zero. Thus \(E(U_aU_b)\) gives the displayed information formula. This derivation also shows why substituting the observed covariance is essential: it is the distribution of the data entering the score.

Covariance Parameters Must Also Be Recalculated

When a parameter controls variability, missingness changes information through the observed covariance and its derivative. A simple example makes this explicit. For a scalar normal observation with mean \(\alpha\) and variance \(v\), the information for \((\alpha,v)\) is diagonal: the mean contributes \(1/v\), and the variance contributes \(1/(2v^2)\). If multiple independent observations share these parameters, their information adds.

Combining Subjects With Different Missingness Patterns

For independent subjects, the total approximate information is the sum of their subject-level information matrices. Each subject contributes the matrix computed from that subject’s own observed components. A planned design can be summarized by its intended schedule, but realized-data information depends on which measurements are actually available. The two quantities answer different questions: one concerns the design under its assumed observation process, and the other concerns the data collected.

Assumptions, Diagnostics, and Alternatives

The subvector calculation is appropriate when the observed values follow the marginal distribution implied by the approximate model and the missingness mechanism can be ignored for inference about \(\psi\). A standard sufficient condition is missing at random together with distinct parameters for the observation process and the data model: conditional on observed information, the probability of the missingness pattern does not depend on the unobserved values, and the missingness parameters are separate from \(\psi\). Under this condition, the likelihood for \(\psi\) is based on the observed-data distribution. If missingness depends on an unobserved concentration even after conditioning on observed information, the mechanism is not ignorable in general; a joint model for outcomes and missingness, or an explicit sensitivity analysis, may be needed.

The formula also presumes that \(V_{i,O}\) is nonsingular and that the approximate distribution and its parameter derivatives are defined at the nominal parameter values. A singular observed covariance or a non-finite derivative prevents the ordinary inverse-based calculation. Very small eigenvalues can make the information numerically unstable, so the observed covariance should be checked for symmetry, positive definiteness, and conditioning. The resulting information should also be checked for symmetry and positive semidefiniteness up to numerical error.

A useful diagnostic is to list, for every subject, the planned components, the observed components, and the corresponding selection matrix. Then verify that the dimension of \(V_{i,O}\) equals the number of observed components and that its entries are exactly the matching rows and columns of \(V_i\). Compare planned and realized information only after stating which missingness pattern each represents. Unexpected changes in rank or very large changes in information can reveal a coding error, a poorly supported parameter direction, or a genuine loss of information.

The main alternative to direct subject-level calculation is to average over possible missingness patterns before data are observed. If the pattern probabilities are specified and independent of \(\psi\), expected design information is the probability-weighted sum of the information for each pattern. This is useful for comparing prospective designs with expected dropout or missed visits. It is not the same as using the realized pattern counts after data collection. If the observation process depends on outcomes or unknown model parameters, pattern probabilities and outcome information may need to be modeled jointly; simply weighting the ordinary observed-subvector matrices can then be inadequate.

A frequent mistake is to compute information for the full vector and delete the rows and columns corresponding to missing observations. That operation does not generally recover observed-data information, because the full inverse covariance already uses the dependence structure of all planned components. The correction is to select the observed covariance submatrix first, differentiate the observed mean and covariance, and then apply the information formula. Likewise, replacing a missing value with zero or with a typical value falsely treats an imputation as a measurement unless the analysis explicitly models that imputation and its uncertainty.

Finally, first-order approximate information is local to the nominal population parameters and to the chosen approximation. Correct handling of missing components does not eliminate approximation error in the model itself. It ensures only that the information calculation corresponds to the data that are assumed to enter the likelihood. Comparing this approximation with other information calculations is a separate question.