Valid Permutations Under Family Ascertainment

Why Ascertainment Changes the Randomization Question

Nonparametric linkage (NPL) statistics assess whether affected relatives share marker alleles or chromosomal segments more often than expected under a null of no linkage. A permutation calibration estimates that null distribution by repeatedly randomizing some part of the data and recalculating the statistic. The crucial question is not simply which values can be shuffled. It is which randomizations have the stated null distribution after accounting for how the families entered the study?

Family ascertainment is the rule used to recruit or select pedigrees. For example, a study may recruit a pedigree only if at least two siblings are affected, or may require a particular affected relative to be the proband. Those rules shape the observed affection patterns. A null replicate that ignores them may create families that could not have been sampled, or give too much probability to patterns that the study design excluded. In either case, the simulated tail probability need not calibrate the NPL statistic.

Here, the design features can include the pedigree relationships, covariates, affection labels, observed or simulated marker data, and the recruitment rule. Which of these are held fixed depends on the randomization method. There is no single universally valid operation called “the NPL permutation”: a conditional affection-label permutation and a Mendelian gene-drop answer different conditional questions.

Two Conditional Routes to a Null Distribution

Let \(T\) denote the NPL statistic, \(Y\) the affection labels, \(M\) the marker information, \(D\) the pedigree and relevant covariates, and \(A\) the ascertainment event. A valid null replicate must reflect the distribution of the statistic under the null and under the conditions used by the analysis. In particular, if recruitment was based on \(Y\), then the selected data are conditional on \(A\), which is a property of \(Y\).

One route is to hold the marker data and pedigree fixed and randomize affection labels. This is appropriate only when the labels are exchangeable among the individuals being permuted after conditioning on the design and ascertainment rule. The permissible assignments may need to preserve the number affected, a fixed affected proband, covariate strata, or other recruitment features. If the phenotype model gives different probabilities to individuals, an unrestricted shuffle is not justified merely because everyone belongs to one family.

A second route is to hold the pedigree and observed affection pattern fixed and simulate marker inheritance under the no-linkage null, often by gene-dropping. Founders’ marker alleles or haplotypes, the marker map and allele model, the pedigree relationships, and the study’s missingness pattern must be treated consistently with the chosen design. Descendant marker genotypes are then generated through Mendelian transmissions under the null. If founders are untyped, their marker alleles must be handled by a stated model rather than silently treated as known.

These routes should not be conflated. In a label permutation, observed marker information is retained and phenotype labels move only among exchangeable positions. In a gene-drop, the affection labels remain fixed while marker inheritance is simulated. A method may also use a carefully defined conditional distribution of inheritance given marker observations, but it must specify what is being conditioned on and how the statistic is recalculated. In particular, one cannot claim to retain all observed child genotypes exactly while freely replacing the inheritance pattern that those genotypes constrain.

Ascertainment and Exchangeability Conditions

The key issue is whether the selection rule is invariant over the proposed randomizations. If a study enrolls families because a designated proband is affected, a permutation that makes that proband unaffected does not preserve the sampled design. If recruitment requires at least two affected siblings, an assignment with fewer than two affected siblings is not a possible selected family. Sometimes a constraint such as a fixed affection count makes every proposed assignment satisfy the rule, but that alone does not establish exchangeability: individuals may still differ in age, sex, covariates, or probability of selection.

Under the no-linkage null, marker inheritance is independent of affection status at the trait locus, conditional on the relevant pedigree and design, when the marker region was not itself used to select families and the null model correctly represents the transmissions. This provides the rationale for holding selected affection labels fixed while gene-dropping marker inheritance. It does not make arbitrary permutations of affection labels valid. Nor does it protect a simulation that ignores marker-dependent recruitment, genotyping-based exclusions, or other selection features.

A practical audit asks four questions before any randomization is run:

• What was selected? State the recruitment or exclusion rule, including any designated proband or affected-relative requirement.
• What is held fixed? Specify pedigree structure, affection labels, covariates, founder information, marker model, and missingness as appropriate.
• What is randomized? Name the precise labels, transmissions, or other quantities that change in each replicate.
• Why are those outcomes exchangeable? State the null and the conditions under which every allowed replicate has the required probability.

Constraining Affection-Label Permutations

When an analysis does permute affection labels, the allowed assignments must reflect both exchangeability and ascertainment. Conditioning on the number affected is a common constraint, but it is not automatically sufficient. A fixed proband status, covariate strata, or family-specific recruitment requirement may impose further restrictions. Even a constrained set of assignments is valid only if the remaining individuals are exchangeable under the null, conditional on the variables used to define the constraints.

Gene-Dropping While Preserving Family and Marker Structure

In gene-dropping replicates, pedigree relationships are not shuffled: parents remain parents and siblings remain siblings. A marker allele is passed from each parent according to the specified Mendelian transmission model. For a multipoint analysis, transmissions across markers must also respect the map and recombination model; independently drawing a fresh inheritance state at every marker would generally erase the dependence along a chromosome. The statistic is recomputed from the simulated marker data using the same analysis rule as for the observed data.

“Preserve marker information” therefore means preserving the relevant marker model and design, not necessarily copying every observed descendant genotype into every replicate. A simulation might condition on typed founder haplotypes, use allele frequencies to model untyped founders, retain the marker map, and keep the locations of missing genotypes fixed while generating the values that would be observed. Alternatively, a specifically conditional procedure may condition on observed marker genotypes and sample compatible inheritance states. Those are different null distributions and must not be described as if they were the same procedure.

Likewise, missingness should not be handled by silently changing which relatives contribute from replicate to replicate unless that change is part of the specified model. A common design-preserving choice is to hold the observed typing pattern fixed and apply it to each replicate. This keeps the statistic’s information pattern comparable while allowing simulated marker inheritance to vary. If missingness depends on genotype or affection status, simply fixing its pattern may not suffice; that dependence needs to be represented in the null mechanism.

Why the Conditioning Must Be Explicit

A permutation p-value is a tail probability under a particular conditional null, not a general probability that the null hypothesis is true. If the randomization fails to preserve the selection process, its reference distribution may not describe the NPL statistic for the families actually observed. This is especially consequential in linkage studies because family recruitment often deliberately enriches the sample for affected relatives.

The short justification for holding affection labels fixed in a gene-drop is a conditional-independence argument. Under the no-linkage null, and assuming marker data did not determine recruitment, marker transmissions have the same null distribution among pedigrees selected by affection status as among otherwise comparable pedigrees with those affection labels. Symbolically, the intended implication is

for the specified transmission model and ascertainment event \(A\). This equality explains why a gene-drop can keep \(Y\) and \(A\) fixed. It is not automatic: it can fail if the marker region affects the trait, if the marker was used in recruitment, or if the simulation omits relevant selection or dependence. Those possibilities require a different null model or additional conditioning.

Common Mistake

Wrong reasoning: “All siblings are relatives in the same family, so their affection labels can be permuted freely; the permutation automatically accounts for family ascertainment.”

Correction: Relatedness alone does not imply exchangeability. A designated affected proband, a minimum affected count, differences among siblings, or a marker-based selection rule can restrict or change the probabilities of assignments. State the recruitment event, condition on the appropriate features, and justify equal probabilities only for assignments that are genuinely exchangeable under the null. If that justification is unavailable, use a null simulation that represents the ascertainment and transmission process rather than an unrestricted shuffle.