Candidate Populations With Stratified Sampling

Why Stratify Candidate Generation?

A simple random draw from a target population can be appropriate when the aim is to generate independent population-representative candidates. But a finite draw may cluster in some parts of parameter space and leave others sparsely represented. In virtual patient generation, this can make an initial candidate set a poor map of the range of plausible parameter vectors, even when each parameter has a specified marginal distribution.

Stratified sampling addresses this coverage issue by dividing a range into strata and deliberately placing candidates in each stratum. Latin hypercube sampling (LHS) is a particular randomized design that stratifies each parameter separately. It improves marginal coverage by construction; it does not guarantee that every combination of parameter values is represented, nor does it by itself encode dependence between patient traits.

Definition: A stratified sample divides a target range or population into non-overlapping strata and draws a specified number of candidates from each. In Latin hypercube sampling with \(n\) candidates and \(d\) parameters, the probability scale of each parameter is divided into \(n\) equal-probability strata, and exactly one candidate is assigned to each stratum for that parameter. The assignments can differ across parameters.

Randomized Latin Hypercube Construction

Suppose parameter \(j\) has a continuous target cumulative distribution function \(F_j\), for \(j=1,\ldots,d\). To make an LHS of size \(n\), independently randomize a permutation \(\pi_j\) of \(1,\ldots,n\) for each parameter. For candidate \(i\), draw \(V_{ij}\sim\mathrm{Uniform}(0,1)\), independently across candidates and parameters, and set

U_{ij}=\frac{\pi_j(i)-1+V_{ij}}{n}, \qquad \theta_{ij}=F_j^{-1}(U_{ij}).

Here \(U_{ij}\) is the probability-scale value, and \(\theta_{ij}\) is the resulting parameter value. The \(V_{ij}\) draws randomize positions within their assigned strata; they are not arbitrary constants. Since \(V_{ij}\) lies strictly between 0 and 1 with probability one, \(U_{ij}\) lies strictly inside its stratum and avoids the upper endpoint \(U_{ij}=1\), where an inverse cumulative distribution function can be infinite or undefined. In numerical implementations, the random number generator and any endpoint handling should be checked explicitly.

This randomization is important for the marginal distribution claim. For a fixed parameter, the randomly assigned stratum is equally likely to be any of the \(n\) strata. Conditional on a stratum, \(U\) is uniform within that stratum. Therefore, for \(u\) in stratum \(k\), the probability of falling at or below \(u\) is

P(U\leq u) =\frac{k-1}{n}+\frac{1}{n}\frac{u-(k-1)/n}{1/n} =u.

Thus each randomized coordinate \(U_{ij}\) is marginally uniform on \((0,1)\), and applying \(F_j^{-1}\) gives the specified marginal distribution \(F_j\). This is a statement about the design randomization, not a claim that every finite realization has exactly the target distribution. Nor does it mean that fixed, hand-picked within-stratum values preserve that distribution.

Worked Example: Transforming Strata to a Parameter Range

Suppose a hypothetical QSP model uses a clearance parameter \(CL\) uniformly distributed from 2 to 10 L/h. Construct four probability-scale strata and use the probability values \(0.10,\ 0.40,\ 0.60,\ 0.90\), one in each stratum.

The four strata are \([0,0.25), [0.25,0.50), [0.50,0.75), [0.75,1)\). The listed values fall in these strata in order. For a uniform distribution from 2 to 10 L/h, \(F^{-1}(u)=2+8u\). Thus the four candidate clearances are \(2+8(0.10)=2.8\), \(2+8(0.40)=5.2\), \(2+8(0.60)=6.8\), and \(2+8(0.90)=9.2\) L/h.

Each candidate lies in a different probability stratum, and the inverse transformation maps the probability scale to the stated clearance range. The particular four values are one possible realization, not four equally spaced quantiles required by LHS. In randomized LHS, each position within its stratum is randomly drawn.

Worked Coverage Comparison

A useful comparison examines both marginal strata and joint cells. The following small example uses two hypothetical parameters \(x\) and \(y\), each uniform from 0 to 1, and \(n=4\) candidates. For LHS, choose the following stratum assignments and within-stratum random values. The values of \(V\) are illustrative realizations from the specified uniform randomization.

Candidate	\(x\) stratum	\(V_x\)	\(x\)	\(y\) stratum	\(V_y\)	\(y\)
1	1	0.20	0.05	4	0.70	0.925
2	3	0.60	0.65	1	0.30	0.075
3	2	0.40	0.35	3	0.80	0.700
4	4	0.80	0.95	2	0.10	0.275

For example, candidate 1 has \(x=(1-1+0.20)/4=0.05\) and \(y=(4-1+0.70)/4=0.925\). The \(x\) stratum numbers are 1, 3, 2, 4, so each of the four strata is used once; the \(y\) stratum numbers are 4, 1, 3, 2, also using each once. The rows pair the parameter values according to the independently randomized assignments.

Worked Example: Comparing LHS With a Simple Random Draw

For comparison, consider one hypothetical simple random draw of four points from the same independent uniform distributions. Suppose its \(x\) values are \(0.08,\ 0.22,\ 0.27,\ 0.91\), and its \(y\) values are \(0.12,\ 0.18,\ 0.44,\ 0.53\). Use four equal-width bins on each axis.

The simple random \(x\) values fall in bins 1, 1, 2, and 4, so three of four \(x\) bins are occupied. The \(y\) values fall in bins 1, 1, 2, and 3, so three of four \(y\) bins are occupied. In the LHS table, all four bins are occupied for each parameter. For this particular realization, LHS therefore has more complete marginal-bin coverage.

Joint coverage is less complete. The LHS rows occupy four of the 16 possible \(x\)-by-\(y\) bin combinations. The simple random rows occupy three combinations: \((1,1)\) twice, \((2,2)\), and \((4,3)\). Neither design covers all joint cells, and this one small comparison does not establish that LHS always covers more joint cells. Its guaranteed property here is one observation per marginal stratum, not comprehensive joint-space coverage.

The result illustrates the intended role of LHS: more deliberate coverage of each parameter’s marginal range in a fixed-size candidate set. It is not a guarantee of better representation for every function of the parameters, every rare combination, or every realized sample.

Strata for Known Groups

Stratification can also be defined using known groups rather than equal-probability intervals of continuous parameters. If a target population has groups with known proportions, the allocation of candidates to those groups should be stated explicitly. Proportional allocation preserves the target group proportions in the candidate count when the arithmetic permits it. Equal allocation instead deliberately gives each group the same number of candidates; unless corrected in a later analysis, that candidate set does not have the target group proportions.

Worked Example: Proportional Versus Equal Allocation

Suppose a hypothetical target population has two phenotype strata: 60% of patients are in group A and 40% in group B. Generate 100 candidates. Under proportional allocation, assign \(100(0.60)=60\) candidates to A and \(100(0.40)=40\) to B. The candidate proportions are \(60/100=0.60\) and \(40/100=0.40\), matching the stated target proportions.

Under equal allocation, assign 50 candidates to each group. The resulting unweighted candidate proportions are \(50/100=0.50\) and \(50/100=0.50\), not 0.60 and 0.40. Equal allocation may be useful when deliberate representation of each group is the design goal, but it must not be described as a proportionally representative sample. The allocation choice answers a design question; it does not change the target proportions.

Assumptions, Diagnostics, and Limitations

For the marginal-distribution result, the design must use a specified marginal \(F_j\), a valid inverse transformation, a random permutation of the strata, and randomized within-stratum values with the stated distribution. Independent permutations across parameters are a simple construction for marginal coverage. The parameter marginals alone do not specify the joint distribution of a virtual patient. In particular, independently constructed LHS coordinates represent independent parameter combinations at the row level; they do not reproduce correlations or other dependence merely because the marginal distributions are correct. When dependence is part of the target, the dependence construction must be incorporated and the resulting joint candidates checked.

Inspect candidate coordinates on both probability and parameter scales. A basic LHS check confirms that each probability-scale stratum contains exactly one value for every parameter. Also check transformed values for finite numbers, allowed ranges, and consistency with the intended marginal distributions. In a QSP model, assess whether the resulting parameter vectors satisfy the physiological feasibility constraints established for the model. Any subsequent restriction or modification can alter the original stratum counts, so the coverage claim applies to the design before such changes unless balance is verified again.

LHS is not automatically preferable in every setting. A simple random draw is often natural when independent random population draws are the objective, and its independent sampling structure can simplify some uncertainty calculations. LHS is attractive when a fixed candidate budget should cover each marginal range systematically. General stratified designs can target meaningful groups or intervals, but their performance depends on the chosen strata and allocation. LHS can still miss important joint regions, and increasing the number of strata does not by itself solve dependence, feasibility, or tail-coverage problems.

A further caution concerns uncertainty estimates. Points within an LHS are not independent in the same way as points from simple random sampling: each parameter’s stratum assignments are constrained to form a permutation. Therefore, uncertainty calculations that assume independent draws should not be applied automatically to LHS output. The design and the purpose of the calculation must be considered together.

Common pitfall: It is incorrect to choose any one value inside each stratum and then claim that the inverse transformation necessarily preserves the target marginal distribution. For example, choosing the lower boundary in every stratum does not produce randomized uniform probability-scale values, and an upper endpoint can produce an infinite transformed value. Use randomized within-stratum values such as independent \(V_{ij}\sim\mathrm{Uniform}(0,1)\); then the marginal-distribution statement follows under the design randomization.

Key takeaway: Randomized LHS guarantees one candidate in each probability stratum of each parameter and preserves the target marginal distribution under its randomization. It does not guarantee full joint coverage, target dependence, or continued balance after candidates are altered or excluded.

Check Your Understanding

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

With \(n=5\), what is the probability-scale interval for stratum 3?
Why does one value in each stratum not, by itself, prove that the transformed values have the target marginal distribution?
For two parameters and \(n=6\), how many marginal strata are occupied for each parameter in an LHS? Does that imply all 36 joint cells are occupied?
A target population has 30% of group A and 70% of group B. In 50 candidates, what are the proportional allocations, and what proportions result from equal allocation?

Solutions. (1) Stratum 3 is \([2/5,3/5)=[0.4,0.6)\). (2) The within-stratum values must also be randomized as specified; arbitrary fixed positions can distort the probability-scale distribution, and endpoint choices can cause transformation problems. (3) All 6 marginal strata are occupied for each parameter, but this does not imply full joint coverage; there are 36 possible joint cells, while LHS assigns only 6 rows. (4) Proportional allocation gives \(50(0.30)=15\) in A and \(50(0.70)=35\) in B, or 30% and 70%. Equal allocation gives 25 in each, or 50% and 50%, so it does not preserve the target group proportions.