Information in Stratified Population Designs

Why Stratum Counts Matter

A population design may deliberately recruit subjects from several prespecified strata, such as age groups or categories of a baseline covariate. Even when every subject follows the same measurement schedule, the expected information contributed by one subject can differ across strata. A design with many subjects from one stratum therefore need not provide the same information as a design with the same total number distributed differently.

The calculation starts with a subject-level Fisher information matrix for each stratum. The key operation is to multiply each stratum’s matrix by the number of subjects assigned to that stratum, then add the resulting matrices. This gives information for the full cohort, conditional on its stratum counts and under the independence assumptions stated below.

Definition: Let \(\theta\) be a common vector of population-model parameters and let \(h=1,\ldots,H\) index prespecified strata. Write \(I_h(\theta)\) for the Fisher information matrix contributed by one subject in stratum \(h\), under the design used for that stratum. If \(n_h\) independent subjects are recruited from stratum \(h\), the stratified cohort’s information matrix is \(I_{\mathrm{total}}(\theta)=\sum_{h=1}^{H}n_h I_h(\theta)\). If the total sample size \(N=\sum_h n_h\) is positive, its average information per subject is \(\bar I(\theta)=I_{\mathrm{total}}(\theta)/N\).

Each \(I_h(\theta)\) is a matrix for the same parameter vector, with parameters in the same order and on the same scale in every stratum. Its entries describe information about individual parameters and their joint estimation. The matrices may differ because covariate values, response variability, or other model features differ across strata. When subjects within a stratum use different designs, the more general total is \(\sum_h\sum_{i=1}^{n_h} I_{hi}(\theta)\), where \(I_{hi}(\theta)\) is the information for subject \(i\) in stratum \(h\).

Why the Matrices Add

The addition rule follows from the likelihood for independent subjects. Let \(Y_{hi}\) denote the observations from subject \(i\) in stratum \(h\), and let \(\ell_{hi}(\theta)\) be that subject’s log likelihood conditional on the stratum and design. Conditional independence gives a joint log likelihood that is the sum of the subject log likelihoods. Fisher information is the expected negative Hessian of the log likelihood; expectation and differentiation of this sum give the sum of the subject information matrices.

\ell_{\mathrm{total}}(\theta)=\sum_{h=1}^{H}\sum_{i=1}^{n_h}\ell_{hi}(\theta), \qquad I_{\mathrm{total}}(\theta) = -E\!\left[\frac{\partial^2\ell_{\mathrm{total}}(\theta)} {\partial\theta\,\partial\theta^\mathsf{T}}\right] = \sum_{h=1}^{H}\sum_{i=1}^{n_h} I_{hi}(\theta).

If every subject in stratum \(h\) has the same design and model conditions, then \(I_{hi}(\theta)=I_h(\theta)\), which yields the definition above. The same result follows from the score covariance formulation: independent subject scores have mean zero, so cross-covariances between scores from distinct subjects are zero. Within-subject correlation does not violate this rule: it must be represented within that subject’s \(I_{hi}(\theta)\). Dependence between subjects, such as a shared unmodelled cluster effect, can create nonzero cross-subject terms and invalidate simple addition.

This derivation assumes a common parameter vector, a correctly specified likelihood, and independent subjects conditional on the modeled stratum and design. It also treats stratum membership and the cohort counts as given. If stratum proportions are themselves unknown parameters of a sampling model, information about those proportions requires modeling the sampling process; it is not automatically included in the conditional information matrices above.

Worked Example: Total Information for a Scalar Parameter

Suppose a hypothetical design estimates one parameter, the population mean of log apparent clearance, \(\theta\). Each subject in stratum A contributes \(0.40\) units of information per squared log-clearance unit, and each subject in stratum B contributes \(1.20\) in the same units. The planned cohort has \(n_A=30\) and \(n_B=20\). Assume the subjects are independent and the stated information values apply to every subject within their respective strata.

For one parameter, the information matrices reduce to scalar values. Thus,

I_{\mathrm{total}}=n_A I_A+n_B I_B =(30)(0.40)+(20)(1.20) =12+24=36.

The cohort has \(N=30+20=50\) subjects, so its average information per subject is \(36/50=0.7200\) in the same information units. Under a local large-sample information approximation, the variance is approximately \(1/I_{\mathrm{total}}=1/36=0.02778\) squared log-clearance units, and the standard error is approximately \(\sqrt{1/36}=0.1667\) log-clearance units.

These are local approximations, not guarantees of realized estimator precision. The calculation shows why the stratum counts matter: although stratum B contains only 40% of subjects, it contributes \(24/36=66.67\%\) of the total information in this example.

Matrix Information and Parameter Directions

For multiple parameters, cohort counts weight whole matrices rather than separate parameter-wise information values. Off-diagonal entries matter because the precision of a parameter estimate can depend on its joint estimation with other parameters. The matrix should be combined first; its inverse, when it exists, is the local covariance approximation for an efficient estimator under regularity conditions.

Worked Example: Combining Two-Parameter Information Matrices

Consider a hypothetical two-parameter model with parameters \(\theta_1\) and \(\theta_2\). For subjects in stratum A, suppose the per-subject information matrix is \(I_A=\begin{pmatrix}4&1\\1&1\end{pmatrix}\). For stratum B, suppose it is \(I_B=\begin{pmatrix}1&-0.5\\-0.5&3\end{pmatrix}\). A planned cohort has \(n_A=8\) and \(n_B=12\). Both matrices are symmetric and positive definite: their determinants are \(4(1)-1=3>0\) and \(1(3)-(-0.5)^2=2.75>0\), respectively. Assume independent subjects and the same parameter order in both strata.

The total information matrix is

I_{\mathrm{total}} =8I_A+12I_B = \begin{pmatrix}32&8\\8&8\end{pmatrix} + \begin{pmatrix}12&-6\\-6&36\end{pmatrix} = \begin{pmatrix}44&2\\2&44\end{pmatrix}.

Its determinant is \(44(44)-2^2=1932>0\), so it is invertible. The local covariance approximation is

I_{\mathrm{total}}^{-1} = \frac{1}{1932} \begin{pmatrix}44&-2\\-2&44\end{pmatrix} \approx \begin{pmatrix}0.02277&-0.001035\\-0.001035&0.02277\end{pmatrix}.

The approximate standard errors are the square roots of the diagonal entries: \(\sqrt{0.02277}\approx0.1509\) parameter units for each parameter. The negative off-diagonal covariance indicates a small negative local association between estimation errors in the two parameters. It is not a causal relationship. The example also shows why summing only the diagonal entries would be insufficient: the off-diagonal terms are part of the joint information calculation.

Changing the Allocation

At fixed total sample size, changing \(n_h\) changes the weighted sum of the stratum matrices. It can improve information in one parameter direction while reducing it in another. Consequently, “more information” is not always a single scalar statement: the interpretation depends on the parameter, contrast, or matrix summary of interest. For a specified parameter direction \(a\), the quantity \(a^\mathsf{T}I_{\mathrm{total}}a\) describes information in that direction; different allocations can rank differently for different choices of \(a\).

Worked Example: Same Total, Different Information Directions

Suppose two hypothetical strata contribute \(I_A=\begin{pmatrix}2&0\\0&0.5\end{pmatrix}\) and \(I_B=\begin{pmatrix}0.5&0\\0&2\end{pmatrix}\) per subject for parameters \(\theta_1,\theta_2\). Each matrix is positive definite. Compare two allocations of the same \(N=20\) independent subjects: allocation 1 has \(n_A=5,n_B=15\), and allocation 2 has \(n_A=10,n_B=10\).

For allocation 1,

I_{\mathrm{total},1} =5I_A+15I_B = \begin{pmatrix}10&0\\0&2.5\end{pmatrix} + \begin{pmatrix}7.5&0\\0&30\end{pmatrix} = \begin{pmatrix}17.5&0\\0&32.5\end{pmatrix}.

For allocation 2,

I_{\mathrm{total},2} =10I_A+10I_B = \begin{pmatrix}20&0\\0&5\end{pmatrix} + \begin{pmatrix}5&0\\0&20\end{pmatrix} = \begin{pmatrix}25&0\\0&25\end{pmatrix}.

The first allocation has more information about \(\theta_2\) (32.5 versus 25), while the balanced allocation has more information about \(\theta_1\) (25 versus 17.5). Their determinants are \(17.5(32.5)=568.75\) and \(25(25)=625\), respectively. Thus, the balanced allocation has the larger determinant in this example, but that does not mean it is best for every objective: allocation 1 has greater information in the \(\theta_2\) direction. Allocation choices must be interpreted relative to the parameters and precision goal that matter.

Diagnostics, Comparisons, and Limits

The main alternative to a stratum-by-stratum calculation is to use one pooled per-subject matrix. This can be valid if it is formed as the weighted average \(\bar I=\sum_h(n_h/N)I_h\), since \(I_{\mathrm{total}}=N\bar I\). It is not valid to use an unweighted average when stratum counts differ, or to apply a matrix from one stratum to every subject when their information contributions differ. The weighted average is a convenient summary; the total matrix retains the scale needed for cohort-level precision.

Before interpreting a result, check that the matrices have matching dimensions, parameter order, and scaling; that cohort counts are nonnegative integers; and that the total count matches the planned sample size. Each valid Fisher information matrix should be symmetric and positive semidefinite, apart from small numerical asymmetries. If the combined matrix is singular, some parameter direction is not identified by the specified design. If it is nearly singular, the inverse may be numerically unstable and local standard errors may be large. Eigenvalue diagnostics and parameter scaling help assess these issues.

Common failure modes include treating repeated measurements as independent subjects, adding matrices computed for different parameterizations, and inverting a stratum matrix before adding it to the others. In general, information adds, but covariance matrices do not: \((\sum_h n_hI_h)^{-1}\) is not \(\sum_h(n_hI_h)^{-1}\). Also, a zero count for a stratum means its matrix contributes nothing to this conditional calculation; no evidence about that stratum’s responses is present in the cohort.

The calculation is local when the information matrices depend on nominal parameter values. It is only as reliable as those values and the model assumptions used to calculate each \(I_h\). It also presumes that the chosen stratum-specific information matrices already account for each subject’s measurement design and within-subject observation dependence. It does not by itself select an optimal allocation, account for recruitment costs, or quantify uncertainty in the information matrices.

Key takeaway: For independent subjects sharing a common parameter vector, multiply each stratum’s per-subject Fisher information matrix by its cohort count and add the matrices. The resulting total—not an unweighted average or a sum of covariances—is the basis for local cohort-level precision.

Check Your Understanding

Answer each question, then compare with the solutions below.

Two strata contribute scalar information \(0.30\) and \(0.90\) per subject. A cohort includes 10 subjects from the first stratum and 5 from the second. What is its total information and average information per subject?
Why can the total information matrix fail to equal the sum of subject-level matrices when subjects are dependent?
With a fixed sample size, can an allocation that increases information about one parameter reduce information about another? Explain using the matrix interpretation.
Does a conditional calculation based on fixed stratum counts automatically provide information about population stratum proportions?

Solutions. (1) Total information is \((10)(0.30)+(5)(0.90)=3+4.5=7.5\). There are 15 subjects, so average information is \(7.5/15=0.500\) per subject. (2) Dependence can make cross-covariances between distinct subjects’ scores nonzero, so the joint information includes cross-subject contributions rather than only the sum of separate subject matrices. (3) Yes. Different strata can contribute differently across parameter directions; increasing the count in a stratum may raise information in one direction while lowering it in another relative to a competing allocation. (4) No. The conditional matrices describe information about model parameters given stratum membership and counts; information about stratum proportions requires a model for the sampling process.