Comparing Cohort Allocations With Approximate Information

Why Compare Allocations Through Information?

When a cohort can be divided among covariate strata in more than one way, the total sample size alone does not tell us which allocation will estimate population parameters more precisely. Subjects in different strata can contribute different amounts and patterns of information. A stratum may be especially informative about one parameter but relatively uninformative about another. Comparing the resulting population Fisher matrices makes those differences explicit.

The comparison in this tutorial is between prespecified allocations with the same total cohort size. The stratum-specific sampling schedules, model, nominal parameter values, and information approximation are held fixed. Only the number of subjects assigned to each stratum changes. This isolates the allocation’s contribution to approximate information; it does not establish which allocation is best for every possible scientific objective.

Definition: Let $K$ denote the number of covariate strata, and let $n_k$ be the number of subjects allocated to stratum $k$. For a fixed within-subject design and a fixed nominal population model, let $J_k$ be the approximate population Fisher information matrix contributed by one subject in stratum $k$. The approximate population Fisher matrix for allocation $n=(n_1,\ldots,n_K)$ is $$ I(n)=\sum_{k=1}^{K} n_k J_k. $$ The total cohort size is $N=\sum_{k=1}^{K}n_k$. When comparing allocations here, $N$ is the same for every candidate.

A matrix $J_k$ summarizes how a subject in stratum $k$ is expected to inform the population parameters under the chosen approximation. Its diagonal entries represent information about individual parameters, and its off-diagonal entries represent information coupling between parameters. The matrix can be calculated using the population Fisher information methods introduced in the earlier material on numerically computing a population Fisher matrix. In this tutorial, the $J_k$ matrices are treated as inputs to the allocation comparison.

The notation $J_k$ is per subject. If a calculation instead supplies information for a group of $m_k$ subjects in stratum $k$, that group contribution must be divided by $m_k$ before using it as $J_k$. Otherwise, multiplying by $n_k$ would count the group size twice.

Assumptions Behind the Comparison

The additive formula requires subjects to contribute independently to the likelihood, conditional on the model and their assigned strata. It also assumes that the same per-subject design and model apply to every subject within a stratum, so subjects in that stratum have the same $J_k$. If information differs within a stratum because of different sampling schedules, the relevant contributions must instead be summed subject by subject or by design subgroup.

The matrices must be calculated on a common parameterization and at the same nominal population parameter values. In a local first-order calculation, the information is an approximation around those nominal values; changing the allocation does not remove that locality. The comparison also presumes that the proposed counts are feasible and that the model, observation process, and approximation are applied consistently across candidates.

For an inverse-information interpretation of uncertainty, the total matrix $I(n)$ should be nonsingular and sufficiently well-conditioned for the intended calculation. If it is singular, some parameter direction is not identified by the approximate information, and $I(n)^{-1}$ does not exist. A nearly singular matrix can yield very large, unstable uncertainty estimates even when its entries look substantial.

Worked Example: Comparing Information for One Parameter

Suppose a hypothetical model has one target parameter, the log of clearance, and two covariate strata. A subject in stratum 1 contributes approximate information $J_1=0.80$, while a subject in stratum 2 contributes $J_2=0.30$. Compare allocations of 40 subjects as $(20,20)$ and $(30,10)$.

For the equal allocation, the total information is $$ I(20,20)=20(0.80)+20(0.30)=16+6=22. $$ For the second allocation, $$ I(30,10)=30(0.80)+10(0.30)=24+3=27. $$ Both designs have 40 subjects, but the second has greater approximate information for this scalar parameter.

For a regular scalar estimation problem, the inverse information approximates the variance of the estimator. Thus the approximate variances are $1/22=0.04545$ and $1/27=0.03704$, respectively, in squared log-clearance units. Their approximate standard errors are $\sqrt{1/22}=0.2132$ and $\sqrt{1/27}=0.1925$ log-clearance units. Under the stated model and approximation, the $(30,10)$ allocation is more informative for this parameter. This conclusion does not address whether other scientific or operational objectives favor a different allocation.

From Independent Subjects to a Cohort Matrix

The additive form follows from the definition of Fisher information. Let $Y_i$ denote the observations from subject $i$, and let $\ell_i(\theta)$ be that subject’s log-likelihood for population parameter vector $\theta$. For independent subjects, the cohort log-likelihood is the sum of subject log-likelihoods. Therefore its score, the derivative of the log-likelihood with respect to $\theta$, is the sum of the subject scores.

Fisher information can be expressed as the expected outer product of the score, under the usual regularity conditions that permit differentiation and expectation to be interchanged and give a mean-zero score. Independence makes the expected cross-products between distinct subjects’ scores zero. The cohort information is consequently the sum of the subject information matrices. Grouping subjects with identical stratum-specific information gives:

I(n)=\sum_{i=1}^{N} J_{s_i} =\sum_{k=1}^{K} n_kJ_k,

where $s_i$ is the stratum assigned to subject $i$. This derivation explains why the allocation counts weight the stratum matrices linearly. It does not imply that every design criterion is linear: for example, determinants and inverse-matrix variances are nonlinear functions of the resulting sum.

Comparing Matrix Information

For a parameter vector with several components, comparing only diagonal entries can be misleading. Off-diagonal entries describe information coupling, and the uncertainty of a parameter estimate depends on the inverse of the full matrix, not generally on the reciprocal of its diagonal information alone. Common comparisons include:

D-type comparison: compare $\det I(n)$, or equivalently $\log\det I(n)$ when the matrices are positive definite. A larger determinant corresponds to a smaller volume for the local joint uncertainty ellipsoid, under the same parameterization.
A-type comparison: compare $\operatorname{tr}\{I(n)^{-1}\}$. A smaller trace corresponds to a smaller sum of approximate variances across the parameters as represented on the chosen scale.
Targeted comparison: for a specified linear combination $c^\mathsf{T}\theta$, compare $c^\mathsf{T}I(n)^{-1}c$. This is the approximate variance for that contrast. The vector $c$ defines the scientific target.

These are distinct criteria, not interchangeable labels for “more information.” Their rankings can differ because one allocation can improve precision for a particular parameter direction while another improves joint precision overall. If parameters have substantially different scales, criteria involving determinants, eigenvalues, or traces can be difficult to interpret without a deliberate parameter scaling; the comparison must use the same scaling for every candidate.

Worked Example: Comparing Two-Parameter Population Matrices

Consider a hypothetical population model with two dimensionless parameters, log clearance and log volume, and two strata. Suppose the per-subject approximate matrices are $$ J_1= \begin{pmatrix} 1.2 & 0.2\\ 0.2 & 0.5 \end{pmatrix}, \qquad J_2= \begin{pmatrix} 0.4 & -0.1\\ -0.1 & 1.1 \end{pmatrix}. $$ Both are positive definite: their determinants are $1.2(0.5)-0.2^2=0.56$ and $0.4(1.1)-(-0.1)^2=0.43$, both positive, and their first diagonal entries are positive. Compare allocations A: $(30,30)$ and B: $(45,15)$, each with 60 subjects.

For allocation A, $$ I_A=30J_1+30J_2 =\begin{pmatrix}36&6\\6&15\end{pmatrix} +\begin{pmatrix}12&-3\\-3&33\end{pmatrix} =\begin{pmatrix}48&3\\3&48\end{pmatrix}. $$ For allocation B, $$ I_B=45J_1+15J_2 =\begin{pmatrix}54&9\\9&22.5\end{pmatrix} +\begin{pmatrix}6&-1.5\\-1.5&16.5\end{pmatrix} =\begin{pmatrix}60&7.5\\7.5&39\end{pmatrix}. $$

The determinants are $$ \det(I_A)=48(48)-3^2=2295,\qquad \det(I_B)=60(39)-7.5^2=2283.75. $$ By the D-type criterion, A has the slightly larger determinant. For the A-type criterion, the trace of the inverse of a two-by-two matrix $\begin{pmatrix}a&b\\b&d\end{pmatrix}$ is $(a+d)/(ad-b^2)$. Hence $$ \operatorname{tr}(I_A^{-1})=\frac{96}{2295}=0.04183,\qquad \operatorname{tr}(I_B^{-1})=\frac{99}{2283.75}=0.04335. $$ The A-type criterion also favors A, since its approximate sum of parameter variances is smaller.

The matrices are positive definite because their diagonal entries and determinants are positive. Their inverse-based interpretations are therefore available under the stated regularity and local approximation assumptions. These results support a narrow conclusion: for these hypothetical matrices and these two criteria, allocation A is preferred. The difference is small, so rounding, model uncertainty, or changes in nominal assumptions could matter in an applied design assessment.

Targeted Precision Can Change the Ranking

The preceding comparison does not make allocation A universally preferable. A prespecified scientific target may be one parameter or a contrast rather than the joint parameter vector. For allocation A above, the approximate variance of the first parameter is the first diagonal entry of $I_A^{-1}$, equal to $48/2295=0.02092$. For allocation B it is $39/2283.75=0.01708$. Thus B is better for the first parameter alone, although A performs slightly better under the D- and A-type criteria.

This contrast illustrates why the design objective should be stated before selecting an allocation. A determinant summarizes joint uncertainty volume; a trace summarizes a scale-dependent sum of variances; and a targeted contrast evaluates a particular direction. Reporting more than one criterion can help make a trade-off visible, but it does not replace identifying which parameter combinations matter scientifically.

Worked Example: A Contrast-Specific Allocation Choice

In a separate hypothetical two-parameter problem, suppose the stratum-specific information is diagonal: $$ J_1=\begin{pmatrix}1.0&0\\0&0.2\end{pmatrix}, \qquad J_2=\begin{pmatrix}0.2&0\\0&1.0\end{pmatrix}. $$ Compare allocation A, $(30,10)$, with allocation B, $(20,20)$. Each has 40 subjects. The scientific target is the sum of the parameters, $c^\mathsf{T}\theta=\theta_1+\theta_2$, so $c=(1,1)^\mathsf{T}$.

The total matrices are $$ I_A=30J_1+10J_2=\begin{pmatrix}32&0\\0&16\end{pmatrix}, \qquad I_B=20J_1+20J_2=\begin{pmatrix}24&0\\0&24\end{pmatrix}. $$ The approximate variance for the target sum is $c^\mathsf{T}I^{-1}c$. For A, $$ c^\mathsf{T}I_A^{-1}c=\frac{1}{32}+\frac{1}{16}=0.09375. $$ For B, $$ c^\mathsf{T}I_B^{-1}c=\frac{1}{24}+\frac{1}{24}=0.08333. $$ Thus B gives lower approximate variance for the sum. If the target were $\theta_1$ alone, however, the variances would be $1/32=0.03125$ for A and $1/24=0.04167$ for B, favoring A. The preferred allocation depends on the target, not just the total sample size.

Diagnostics, Alternatives, and Limitations

Before accepting a comparison, verify that each candidate uses the same total $N$, the same stratum-specific sampling schedules, the same parameterization, and the same nominal model values. Reconstruct each total matrix from the stated counts and per-subject matrices. Check symmetry and positive definiteness when inverse-based criteria are used. Small negative eigenvalues can signal numerical error; a genuinely singular or indefinite result requires investigating the underlying calculation rather than proceeding as though a stable covariance approximation exists.

An allocation can be compared with a simple equal allocation or with a prespecified proportional allocation, but these are reference choices, not universal optima. A formal constrained search over allowable integer counts is another alternative: calculate $I(n)$ for each feasible allocation and select according to a declared objective. Such a search remains tied to that objective and to the assumed $J_k$ matrices. It does not automatically account for uncertainty in nominal parameters, operational feasibility beyond stated constraints, or departures from the information approximation.

The approximation is local: the matrices describe information at specified nominal values and under the adopted approximation. A ranking that is close may not be robust to plausible changes in those values or assumptions. In addition, a matrix criterion can conceal a poor result for a scientifically important parameter if the criterion does not directly emphasize that direction. Examine target-specific variances or parameter-wise uncertainty alongside any overall criterion when those quantities matter.

Common mistake: “Allocation B has a larger diagonal entry for clearance information, so B must be the better design.” This reasoning ignores the other parameters and the off-diagonal terms. The correction is to form the complete cohort matrices, then evaluate the stated objective—such as a determinant, inverse-information trace, or target-specific contrast variance—on the same parameter scale. No single ranking should be described as universally best unless the objective and assumptions justify it.

Key takeaway: With a fixed cohort size and fixed stratum-specific designs, approximate population information is the allocation-weighted sum of per-subject stratum matrices. Compare complete matrices using an explicitly chosen scientific criterion, and interpret the result as conditional on the model, nominal values, and approximation used.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

Two strata have scalar per-subject information 0.5 and 1.0. Compare allocations $(20,20)$ and $(30,10)$ for total information.
Why must per-subject information be multiplied by the stratum count only once?
For $I=\begin{pmatrix}10&2\\2&5\end{pmatrix}$, calculate the determinant and the approximate variance of the first parameter.
Can a design have a smaller approximate variance for one parameter but a worse D-type criterion than another design? Explain.

Solutions. 1. The totals are $20(0.5)+20(1.0)=30$ and $30(0.5)+10(1.0)=25$; the equal allocation has more information. 2. $J_k$ is defined per subject, so $n_kJ_k$ is the total contribution. Multiplying a group-level matrix by the subject count again would double-count. 3. The determinant is $10(5)-2^2=46$. The first diagonal element of $I^{-1}$ is $5/46=0.1087$, in squared units of the first parameter. 4. Yes. A D-type criterion summarizes joint uncertainty volume, while a single-parameter variance evaluates only one direction; the criteria can therefore rank designs differently.