https://orcid.org/0000-0002-3809-2129
This is an uncorrected proof.
Figures
Abstract
We recently developed a causal inference framework and test—the Transmission Mean Test (TMT)—to identify causal genotype–phenotype relationships in population-sampled parent–child trios, where one child per family is observed. Here, we establish the generalized TMT (gTMT) for population-sampled nuclear families, allowing multiple offspring per family. This extension focuses on detecting genetic loci with non-zero average causal effects (ACE) on child phenotypes, taking into account that siblings share similar random family-specific effects. We construct a potential outcomes trait model that considers both individual-level and family-level heterogeneity, captures additive and non-additive genetic effects, and accommodates both quantitative (continuous or count) and dichotomous traits. We design an unbiased estimate of the ACE and develop a sampling variance estimate
to form a statistic testing the null hypothesis of no causal effect. We provide both theory and empirical evidence demonstrating that gTMT is robust to confounding factors such as the population structure and family-specific effects. We analyze nuclear families in the UK Biobank as an illustrative example of the gTMT in action. When parental genotypes are missing, we propose to further extend gTMT by using Bayesian calculations on child genotypes to model parental genotypes as intermediate random variables.
Author summary
There has been a recent resurgence of interest in family designs, as genome-wide association studies have reached sizes at which subtle biases can overcome statistical control in population-based studies. At the same time, methodological advances in causal inference have created new opportunities for causal discoveries. In this work, we develop the generalized Transmission Mean Test (gTMT), a causal inference framework for identifying causal genotype–phenotype relationships in population-sampled nuclear families with multiple offspring per family. Building on our previous work on the Transmission Mean Test that focuses on parent–child trio data, this extension allows for shared family-specific effects among siblings and accommodates both quantitative and dichotomous traits. We formulate a potential outcomes model that captures individual- and family-level heterogeneity, construct unbiased estimators of average causal effects, and derive valid inference procedures. Through theoretical analysis and empirical studies, including an application to nuclear families in the UK Biobank, we demonstrate that gTMT is robust to a broad class of confounding effects, including population structure and family-level confounding. We further propose a Bayesian extension to address missing parental genotypes.
Citation: Tang Y, Storey JD (2026) A generalized test of genotype–phenotype causality in population-sampled nuclear families. PLoS Genet 22(7): e1012231. https://doi.org/10.1371/journal.pgen.1012231
Editor: Wei Pan, University of Minnesota School of Public Health, UNITED STATES OF AMERICA
Received: March 3, 2026; Accepted: June 29, 2026; Published: July 15, 2026
Copyright: © 2026 Tang, Storey. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The ‘geneticTMT’ software package is available at https://github.com/StoreyLab/geneticTMT. The results in this paper can be reproduced with code available at https://github.com/StoreyLab/causal-nuclear-family.
Funding: This work was supported in part by United States National Institutes of Health (HG006448 to JDS). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
Genome-wide association studies (GWAS) are typically conducted on population-sampled individuals. Although GWAS identify statistically significant associations between genetic variants and phenotypes, such associations are not interpreted as causal findings due to possible confounding factors that are associated with both genotypes and phenotypes. This is not a weakness of GWAS, but rather it is a general limitation of observational studies where associations are identified.
In contrast, family-based studies sample closely related individuals based on pedigrees to estimate linkage between genetic variants and phenotypes of interest, while accounting for common confounding factors such as population structure and non-random mating [1–15]. However, formally establishing causal inference in family-based designs has received less attention. We have recently proposed a framework and robust test of causality in studies involving population-sampled parent–child trios, called the Transmission Mean Test (TMT) [16].
The TMT builds on the Neyman-Rubin potential outcomes framework—a gold standard statistical framework for causal inference [17–19]—based on the randomization of alleles transmitted during the meiosis process. The TMT is designed to have a general applicability to a broad class of phenotypes (continuous, count, or dichotomous traits). The TMT demonstrates robustness to a wide-range of confounding factors including population structure, parent effects, and non-genetic variation.
The original TMT is designed for trio studies that sample two parents and one child per family. However, many nuclear family datasets include multiple offspring per family. For example, the Norwegian Mother and Child Cohort Study (MoBa) includes approximately 16,400 families with two or more offspring in addition to around 44,000 parent–child trios [15,20]. Families with multiple offspring are commonly observed among first-degree relatives in biobanks that originally intend to sample unrelated individuals. For example, previous work identified around 1,000 trios and 37 quartets (two parents and two offspring) in the UK Biobank cohort [21]. Additionally, the Finnish biobank FinnGen contains more than 12,000 nuclear families for which both parents and one or more offspring were directly genotyped [22,23].
Here, we extend the TMT and its underlying causal inference framework to population-sampled nuclear families, where each family contains two parents and any number of offspring. The primary objective is to infer causal effects from genotype to phenotype when family genotypes are available and the phenotype has been measured on the offspring. We consider a sample of J nuclear families, each having two parents. Within family j (), there are
offspring,
. A special case of this is the trio study that samples two parents and one child per family so that
for all
.
We consider the setting where unphased biallelic single nucleotide polymorphism (SNP) data have been collected on I SNPs per individual. For a SNP, let a and b be the two alleles that generate three possible genotypes {aa,ab,bb}. We numerically encode these three genotypes by {0,1,2}, respectively. In family j, let be genotype i of the mother,
genotype i of the father, and
genotype i of offspring k in family j,
,
. Section A.1 in S1 Appendix depicts a full set of genotypes in a nuclear family, as well as the inheritance process of randomized allele transmissions. We additionally denote
for each child, where
is the allele transmitted to offspring k in family j from the maternal side and
the allele from the paternal side (where 0 is a transmission and 1 is b transmission). Let
be the phenotype of interest of offspring k in family j, where this phenotype can be real values, whole numbers, or dichotomous indicators {0, 1}. Writing
and
, the goal is to infer the causal effect from
to
for
, denoted
. We quantify this causal effect as the expected difference between potential outcomes of the traits corresponding to different alleles transmitted from parents to their offspring, i.e., what is typically called the average causal effect (ACE) from
to
. We develop an unbiased estimator of the ACE, together with its sampling variance estimate, to derive a test-statistic for testing the hypothesis of zero ACE versus non-zero ACE.
This paper introduces theoretical and technical innovations, building on previous work. First, to construct trait-level potential outcomes, we detail a model of the phenotype that accounts for within-family effects shared by siblings, forming the basis to quantify within-family covariances (among siblings) and between-family covariances (among offspring from different families). Second, we show how our proposed unbiased ACE estimate incorporates the phenotype values so that the estimate is robust to model misspecification and unmodeled polygenic background effects. Third, when estimating the sampling variance of the unbiased ACE estimator, we take into account covariances among all offspring to derive an estimate for the variance that shows the desired operating characteristics leading to a valid test-statistic. Along with our earlier work [16], we develop theory and methods for an atypical scenario in causal inference where two parallel randomized experiments of the same “treatment”—one per parent—are considered simultaneously on a set of subjects.
Our work has both connections to and differences with other methods in family-based studies. We previously showed the transmission disequilibrium test (TDT) [1–4] is a test of causality and a special case of our framework in ref. [16]. Nuclear family data have been utilized for association analyses [5–7,12–14], but these existing approaches do not test for causality. Another line of research utilizes phased genomes of parent–child trios and requires assumptions about recombination rates to test for probabilistic independence based causality [24,25], which is a different approach than that taken here.
The remainder of this paper starts by revisiting the TMT method for parent–child trios in Section 2.1. Then we establish the generalized TMT (gTMT) framework for nuclear families by developing an unbiased ACE estimator in Section 2.2, deriving its sampling variance estimate in Section 2.3, and proposing a hypothesis test in Section 2.4. We simulate nuclear family data to validate the performance of the gTMT in Section 3.1, contrast the gTMT with family-based association methods in Section 3.2, analyze UK Biobank data as an illustrative example in Section 3.3, and propose an approach to handle missing parental genotypes in Section 3.4. In Section 4, we develop and detail the underlying models and theory of the proposed framework to demonstrate that the gTMT is a valid causal inference method.
2 Proposed method
2.1 The TMT statistic for parent–child trios
We first review the previously proposed TMT for parent–child trios. The TMT is conducted on a per locus basis, so we drop the genetic locus index i. For parent–child trios, one child is observed per family so we drop the sibling subscript k. Let be an indicator function. Let N be the total number of randomized allele transmissions; in trio studies, N equals the total number of heterozygous parents in that
In ref. [16], we defined the TMT parameter as
where Y(1) and Y(0) are the potential outcomes of the two possible transmitted alleles. We fully detail these quantities in Eqs 9–12 for the generalized setting considered here. We showed that is non-zero if and only if the ACE from the child genotype G to the child phenotype Y is non-zero (i.e., there exists a non-zero causal effect from G to Y). We constructed a hypothesis test where the null hypothesis is
and the alternative is
. We defined the TMT statistic for the hypothesis test as follows.
Definition 1 (TMT statistic for parent–child trios). Define the assignment indicators
and define the population mean estimate
The original TMT statistic is
The assignment indicators capture whether a or b alleles were transmitted for each heterozygous parent of a child. In ref. [16], we proved that is an unbiased estimator of
in that
. We developed a sampling variance estimate
for
when H0 is true and is a lower bound for
when H1 is true. We defined the test statistic
with the p-value calculated either by
where
or by permutation.
In ref. [16], we extended the causal framework of the TMT to settings in which causal and noncausal loci may be linked through population-level linkage disequilibrium (LD), within-family meiotic genetic linkage, or both. Although this is a limitation of population-sampled studies, we demonstrated, both theoretically and empirically, that the statistical power of the test is generally higher at the direct causal loci than at the noncausal loci linked to them. We extended our framework to include this phenomenon, which we called “causal linkage” and denoted it by rather than true causality denoted by →. The same conclusions apply in the following sections for nuclear families.
2.2 The gTMT statistic for nuclear families
Here, we allow each family to have an arbitrary number of offspring. Let be the total number of offspring in family j,
,
. Note that
in ref. [16] for parent–child trios. The total number of randomized allele transmissions N is now calculated as
The gTMT parameter is the same as the original TMT parameter in Eq 1. In Theorem 1, we show that
is zero if and only if the ACE from the child genotype G to the child phenotype Y is zero, so that a non-zero
indicates the existence of a causal effect. A hypothesis test can be conducted by testing the null hypothesis
versus the alternative
. To do so, we define the gTMT statistic
as follows.
Definition 2 (gTMT statistic). Define the assignment indicators
and the population mean estimate
The gTMT statistic is
The factor 2/N comes from the fact that there is a 1/2 probability of ending of in the “control” or “treatment“ groups for each transmission, so we divide the sum in Eq 5 by N/2. In Theorem 2, we show that is an unbiased estimator for
in that
. Fig 1 is a schematic showing how the randomized transmission of alleles leads to randomized assignments to the “control” and “treatment” groups, and the statistic
measuring their difference.
For offspring k in family j, if a heterozygous parent transmits an allele 0 to the child, the corresponding phenotype is included in the “control” group. If a heterozygous parent transmits an allele 1,
is included in the “treatment” group. The top layer to the middle shows the allele transmission from parents to offspring. In the middle layer, the number of offspring per family varies. The bottom layer shows how the trait values are assigned to the control and treatment groups, the difference of which is measured by
.
2.3 Sampling variance of the gTMT statistic
As Fig 1 presents, one child can be assigned twice, once, or zero times to either the treatment group or the control group. The same child can also be assigned simultaneously once to each group, which will have zero contribution to based on Eq 5 and does not contribute to the variance of
. These properties show how our particular setting is an atypical instance of causality under randomization (see ref. [16] for a full discussion). This leads to four groups contributing to the overall sampling variance of
, each of which may have a different group-specific variance:
In , one heterozygous parent transmitted an allele 0 to the child, assigning the child phenotype to the “control”. In
, one heterozygous parent transmitted an allele 1 to the child, assigning the child phenotype to the “treatment”. In
, two heterozygous parents transmitted two alleles 0 to the child, assigning the child phenotype twice to the control. In
, two heterozygous parents transmitted two alleles 1 to the child, assigning the child phenotype twice to the treatment. For each of these four sets, we form the following estimates:
We form the sampling variance estimate for as:
We discuss the theoretical foundations of the sampling variance estimate in Theorem 3, where we disentangle the variances and covariances within and between families.
2.4 Proposed hypothesis test
We form the following test statistic
By the Central Limit Theorem, is approximately Normal(0,1) when the null hypothesis of no causality is true. In this case, the p-value is calculated by
where
. One can also use a permutation null distribution as an alternative to the Normal(0,1) distribution to calculate p-values (details in Section A.2 in S1 Appendix). We verify that the permutation null and the Normal(0,1) yield similar results in the following section.
3 Results
3.1 Performance of gTMT as a test of causality
We implemented an algorithm (detailed in Section B.1 in S1 Appendix) to simulate genotypes of 3,000 nuclear families including 1,000 trios (two parents and one child), 1,000 tetrads (two parents and two offspring) and 1,000 quintets (two parents and three offspring). This produced 6,000 offspring and 6,000 parents with 100,000 SNPs per individual. We simulated tractable population structure by generating genotypes via a standard admixture model [26–29]. We randomly assigned 100 SNPs to be causal and followed Section B.2 in S1 Appendix to generate child phenotypes across various levels of true heritability h2.
3.1.1 Unbiased estimation of the target parameter.
We confirmed the accuracy of as an unbiased estimate of the TMT parameter
among various levels of h2 (Fig 2A).
(A) A plot showing is an unbiased estimator of
. We generated 1,000 family trios, 1,000 tetrads and 1,000 quintets (in total 3,000 families with 6,000 offspring) from a structured population (FST = 0.2) with 100,000 SNPs per individual and we randomly chose 100 genetic loci to be causal. We simulated the offspring phenotypes for
. Shown are 1,000 randomly chosen instances of
for each h2. (B) gTMT controls FPR at the desired significance level per setting. We display numerical data for (B) in Section B.3 in S1 Appendix. (C) The ROC curve from a randomly chosen simulated data set. (D) The distribution of the test statistic
is well approximated by Normal(0,1) when the null hypothesis of no causality is true.
3.1.2 Causal FPR and statistical power.
We tested for non-zero ACE per locus and calculated p-values for both causal and non-causal SNPs by utilizing both the Normal(0,1) null distribution and the permutation null. We then computed the false positive rate (FPR) and the true positive rate (TPR) at various significance thresholds in the unit interval [0,1]. The gTMT test controlled the FPR at the desired significance level across all levels of h2 (Fig 2B, numerical data in Section B.3 in S1 Appendix). The receiver operating characteristic (ROC) curve demonstrated gTMT having statistical power for detecting direct causal effects for all levels of h2 (Fig 2C).
3.1.3 Accuracy of the null distribution.
We confirmed that the distribution of the test statistic is well approximated by the standard Normal (0,1) distribution when the null hypothesis of no causality is true (Fig 2D).
3.1.4 Performance of gTMT in the presence of linkage disequilibrium.
In Section B.4 in S1 Appendix, we employed the software msprime [30] to simulate nuclear family genotypes in LD with an admixed ancestral population structure. Based on the “American Admixture” model [31], we generated a sample of 5,000 families with 6,000 offspring, containing 4,000 trios (two parents and one child per family) and 1,000 tetrads (two offspring per family), with 100,000 SNPs across 22 chromosomes per individual. We utilized the polygenic trait model to simulate offspring phenotype values and implemented the gTMT across all simulated genotypes. We validated that the gTMT has accurate ACE estimation at the causal loci, and generally higher statistical power at the causal loci than at the noncausal loci linked to them.
3.2 The gTMT versus family-based association methods in the presence of confounding
We compared gTMT with association methods applicable to family data, noting that those methods are intended for associations. The goal of the comparison is to show that gTMT is accurate for identifying causal effects in contrast to these association methods when the causal effects are confounded. A popular association method is the family-based association test (FBAT) [5–8], which is a score test based on the FBAT statistic . Conditioning on the trait Y and parental genotypes, FBAT derives the variance of U by
and calculates the test statistic
, with p-value
where X has a
distribution.
Another approach called the family-based GWAS (FGWAS) builds on a regression model to estimate b1 as the association parameter between Y and G while controlling for parental genotypes
and
[12–14]. Distinctions between FGWAS and the TMT have been further detailed in ref. [16]; it has also been shown that a variety of confounding effects can impact the performance of FGWAS [32].
Here, we considered family-specific confounding effects that are correlated with the parental genotypes. We evaluated the statistical power of gTMT, FBAT and FGWAS in detecting causal genetic variants. Note that associations can be affected by confounding, whereas a causal inference method should be robust to such confounding. We implemented an algorithm (detailed in Section B.1 in S1 Appendix) to simulate genotypes of 1,500 nuclear families (500 trios, 500 tetrads and 500 quintets). This produced 3,000 offspring and 3,000 parents with 100,000 SNPs per individual. We randomly chose 100 causal SNPs and generated child traits by
where is the set of causal SNPs and
is the family effect correlated with parental genotypes at a causal locus
randomly chosen from
(details in Section B.5 in S1 Appendix). We applied gTMT, FBAT and FGWAS to compute p-values at the causal loci. Specifically, for 3,000 iterations, we generated child genotypes at the causal loci, simulated child traits by Eq 8, and applied the three methods to derive p-values. This produced 3,000 p-values per causal locus per method. We calculated empirically the type II error (
) as the proportion of p-value greater or equal to a significance-level
among 3,000 p-values for each method, i.e.,
. Then we used
to estimate the statistical power.
Fig 3 shows the effect of confounding from the causal locus . We considered a range of confounding effect sizes s (in Eq 8) from s = 0 (no confounding) to s = 5 and computed the statistical power for each method. The gTMT maintained the same power across all levels of the confounding effect size, while both FBAT and FGWAS had decreasing power when the confounding effect size increased. We conducted the same power analysis for larger samples: (i) 6,000 offspring from 3,000 families (1,000 trios, 1,000 tetrads and 1,000 quintets); and (ii) 12,000 offspring from 6,000 families (2,000 trios, 2,000 tetrads and 2,000 quintets). Both scenarios showed higher power for the gTMT than the other two methods in the presence of confounding effects (Fig 3).
Notice that the confounding effect in Eq 8 is correlated with homozygous parental genotypes, which has biological implications. As a hypothesized example, when studying genetic effects on blood pressure, if both parents are homozygous for alleles in genes such as SLC4A5 or AGT that predispose one to salt-sensitive hypertension [33–36], parents are more likely to have elevated blood pressure and adopt salt-restricted diets or anti-hypertensive behaviors. This, in turn, creates a shared low-sodium environment within a family, which directly influences offspring blood pressure and is independent of the transmitted alleles. Consequently, these biologically mediated environmental factors can confound the relationship between offspring genotypes and phenotypes.
In Section B.5 in S1 Appendix, we explore the performances of gTMT, FBAT, and FGWAS when the confounding effect, in Eq 8, also depends on heterozygous parents. We validated the robustness of the gTMT to confounding effects in various scenarios. We also showed that without confounding effects, all three methods have similar statistical power because association signals without confounding are similar to causal findings.
3.3 UK Biobank data analysis
3.3.1 Identifying nuclear families.
We used kinship estimates from KING [37], age, and gender to identify 990 trios (two parents and one child) and 37 quartets (two parents and two offspring) with 1,064 distinct offspring and 2,054 parents in total. We detail the procedure for identifying nuclear families in Section C.1 in S1 Appendix.
3.3.2 The gTMT for blood pressure.
We analyzed three blood pressure measures, including systolic blood pressure (SBP), diastolic blood pressure (DBP), and pulse pressure (PP) where PP = SBP - DBP. We detail the quality control pipeline for these blood pressure measures in Section C.2 in S1 Appendix. We implemented gTMT across 9,563,967 autosomal SNPs that have minor allele frequency (MAF) greater than 1% and imputation information scores [38] greater than 0.8. We calculated one p-value per SNP and display as the genome-wide gTMT profile. For all three blood pressure measures, we successfully detected several genome-wide significant signals that are supported by previous GWAS results. See Section C.3 in S1 Appendix for the genome-wide gTMT profile, gTMT p-values, and supporting literature. Some of these significant variants are located within genes that have well-supported biological mechanisms that may influence blood pressure. For example, we identified a locus associated with SBP located within the gene PAM that activates critical hormones required by many cardiovascular peptides central to blood pressure regulation [39,40]. For DBP, we identified associations within the gene CAMKID, which has been shown to involve in aldosterone synthesis, fitting into a canonical blood pressure control pathway [41]. Another example is a PP corresponding locus within the gene SLC20A2, which has emerged as PP-related gene in several large-scale GWAS [42–47] and has been prioritized as a functional target for hypertension [48].
3.3.3 Comparing to family-based association methods.
We conducted FBAT and FGWAS to detect associations between autosomal SNPs and blood pressure measures. Fig 4 displays quantile-quantile plots comparing p-values from gTMT, FBAT, and FGWAS versus p-values under the null hypothesis of no causality that are Uniform(0,1) distributed. In general, both FBAT and FGWAS were more conservative than gTMT. For the pulse pressure (PP) measure, only gTMT successfully detected genome-wide significant signals while both FBAT and FGWAS did not. One possible reason is that intra-individual random effects acting as confounders are driving the FBAT and FGWAS statistical significance in SBP and DBP. When analyzing PP, since PP = SBP − DBP, intra-individual random effects are likely eliminated. For PP, therefore, statistical significance was absent for FBAT and FGWAS, while gTMT maintained notable statistical significance.
Note that p-values above the identity line show signal for statistical significance. The genomic inflation factor values [49] are: 1.031 (gTMT), 1.007 (FGWAS), and 1.001 (FBAT) for the systolic blood pressure; 1.029 (gTMT), 1.004 (FGWAS), and 1.000 (FBAT) for the diastolic blood pressure; and 1.027 (gTMT), 1.005 (FGWAS), and 1.001 (FBAT) for the pulse pressure.
3.4 Missing parental gentoypes
When one or more parental genotype is missing, the following is a possible strategy to extend the gTMT by using Bayesian probabilities to model parental genotypes as intermediate random variables. Let be the genotypes of all offspring in family j. By using Bayes theorem, the conditional probability for observing the possible parental genotype pairs is
where . For the conditional probability
, we follow Mendel’s laws to calculate
among family trios and tetrads (Section A.3 in S1 Appendix). In an analogous way, one can enumerate
for families with an arbitrary number of offspring. The other two probabilities
and
can be calculated empirically based on the data in the study or other data sets sampled from the same population. This pipeline could be directly applied to sibling studies where parental genotypes are generally unavailable.
4 Theoretical foundations of the gTMT
We develop models and theory here to demonstrate that the gTMT is a valid test of causality. The following is an extension of the potential outcomes framework for trios in ref. [16] to the scenario we consider here where families may have more than one offspring.
4.1 The trait model
The trait for offspring k in family j is modeled as:
where is a random variable shared by siblings in family j, while
is a random variable specific to offspring k in family j. We do not make specific assumptions about the expected values of
and
nor do we require exogeneity. The dependence structure of these random variables is articulated in Assumption 2.
Here we retain Assumption 1 from ref. [16] that the genetic effects are either non-decreasing or non-increasing.
Assumption 1. The conditional expectation of offspring phenotype given parent-transmitted alleles, , is either a non-decreasing function of
such that
or is analogously a non-increasing function of
.
Note that under the trait model in Eq 9, if is non-decreasing, then
; if
is non-increasing, then
.
4.2 Potential outcomes and causal effects
In order to characterize the causal effect of the parent-transmitted alleles on the offspring phenotype Y, we previously [16] modeled potential outcomes for Y in terms of each parent individually. We formulated variables representing the phenotype that the offspring would have developed if receiving a particular allele from the parent. Let A be the parent-transmitted allele. The potential outcomes are Y(A = 0) and Y(A = 1), which we sometimes simplify as Y(1) and Y(0) when there is no ambiguity. Then the observed trait value Y can be written as a function of the potential outcomes,
where is the indicator function. When considering a particular parent’s contribution, let
and
be potential outcomes for the maternal side and
and
for the paternal side.
Here, we extend this to potential outcomes in the scenario of nuclear families with an arbitrary number of offspring per family. Under the trait model in Eq 9, for offspring k in family j, the potential outcomes are
leading to two pairs of causal effects per parent:
We use expectations of these causal effects to define the average causal effect (ACE) as follows; this definition applies to all offspring, so we omit the family and sibling indices j,k for the rest of this subsection.
Definition 3 (ACE for an allele). The ACE of a parent-transmitted allele A on offspring trait Y is the average difference between potential outcomes Y(A = 1) and Y(A = 0): . We say A is directly causal for Y, denoted
, if
.
We derive ACE per parent explicitly as:
Assuming and
in the sampled population, then
if and only if
. This leads to the following definition of the ACE of the child genotype
on the child phenotype Y, denoted as
.
Definition 4 (ACE for genotype). Under the trait model in Eq 9, if and only if
. Otherwise,
. The offspring genotype G is directly causal for Y, denoted by
, if
.
Based on Eq 12 and Definition 4, it is trivial that the following three properties are equivalent:
4.3 The gTMT parameter quantifies the causal effect
Recall that N is the total number of randomized allele transmissions from a parent to a child among all families, as previously defined in Eq 3. The TMT parameter in Eq 1,
is the same value regardless of whether we are considering trios or generalized nuclear families since it is defined in terms of one parent and one child. We show that a non-zero value of implies non-zero ACE through the following theorem.
Theorem 1. Under the trait model in Eq 9 and Assumption 1, the TMT parameter if and only if
.
We prove Theorem 1 in Section A.4 in S1 Appendix.
4.4 The gTMT statistic is an unbiased estimate
Based on Definition 2, we proposed as an unbiased estimate of
in that
, which we prove here. By Mendel’s Law of Segregation, the allele transmitted from a heterozygous parent to offspring is
for
. Since this randomization in meiosis precedes other factors such as the offspring individual effect
and the family effect
in Eq 9, we make the following assumption.
Assumption 2. Under the trait model in Eq 9,
When Assumption 2 is satisfied, the following lemma holds, which will later be used to show that is an unbiased estimate of
.
Lemma 1. Under the trait model in Eq 9 and given Assumption 2, the potential outcomes, and
, are conditionally independent of the parent-transmitted allele,
, given a heterozygous parental genotype,
, written as
We prove Lemma 1 in Section A.5 in S1 Appendix, and we later use Lemma 1 to prove Theorem 2. Lemma 1 satisfies the unconfoundedness assumption in “intention-to-treat analysis” [19], which is the basic assumption in classical causal inference analysis under the potential outcomes framework.
Here we prove that in Theorem 2. The proof is based on the trait model in Eq 9 and requires Assumption 1 and Assumption 2. We show the following lemma as an intermediate step.
Lemma 2. For the generalized TMT statistic and the population-mean estimate
,
so that
We prove Lemma 2 in Section A.6 in S1 Appendix. Lemma 2 leads to the following theorem that states is an unbiased estimate of
.
Theorem 2. Under the trait model in Eq 9, when Assumption 1 and Assumption 2 are satisfied, the generalized TMT statistic is unbiased for the TMT parameter in that
We prove Theorem 2 in Section A.7 in S1 Appendix. To test the null hypothesis versus the alternative
, we propose a sampling variance estimate for
in Eq 6 in the next section.
4.5 Estimate of the sampling variance
Here, we derive from Eq 6 as an estimate of the sampling variance of
. Since the statistic
contains a population mean estimate
, we first calculate the variance of the following statistic with known
:
where
Theorem 3. When the null hypothesis that is true,
In general,
The proof of Theorem 3 is in Section A.8 in S1 Appendix. This proof shows how to disentangle the variances and covariances within and between families. Defining and referring back to the definition of
in Eq 13, we can see that
Notice that contains the covariance between siblings within the same family, i.e.,
where
. Also,
involves the covariance between offspring from different families, i.e.,
where
. In the proof of Theorem 3, we show that when the null hypothesis is true
and in general
. We also show that when the null hypothesis is true
and in general
. All these covariance properties are key to Theorem 3. Motivated by Eq 14 in Theorem 3, we form
in Eq 6 to estimate the variance of
under the null hypothesis. Our simulations in Fig 2 empirically validate
as an estimate of the variance of
under the null hypothesis.
5 Discussion
Building on our earlier work for parent–child trios [16], we have extended this framework to population-sampled nuclear families, establishing a robust inference method to detect causal effects of genotype on phenotype. We have presented theory and methods to rigorously prove the intuition that family-based designs enable causal inference, owing to their inherent randomization through allele transmission and their observed robustness to common confounding effects such as population structure and non-random mating.
For studies where individuals are randomly sampled from a population, association tests and downstream methods such as fine-mapping have been established as a viable approach to identifying associations between genotype and phenotype. GWAS usually implement a linear mixed model regression to identify associations. Such regression approaches make the potentially strong assumption of exogeneity in that the non-genetic variation has zero covariance with the genetic variation. This assumption is not always satisfied for population-sampled nuclear families because family specific effects— in Eq 9—can introduce non-genetic factors that induce confounding between the child’s genotypes and phenotype. The existence of such confounding factors can generate false positives for GWAS results when including nuclear families. The gTMT is robust to these confounding factors because randomization has been leveraged, captured by the potential outcomes in Eq 10 to derive the causal effects in Eq 11.
For very large sample size population-based studies, there are often first degree relatives present. For example, we identified 28,895 candidate parent–child relationships in the UK Biobank data. One could use the gTMT and future extensions in conjunction with existing GWAS methods in these studies. The full set of related individuals could be set aside, while an association analysis is then applied to the remaining unrelated individuals. A set of SNPs showing associations in this first stage analysis could then be tested for causality using the gTMT in the related individuals that have been set aside. This strategy would increase the power of the secondary gTMT stage because fewer SNPs would be tested, and it would also provide a valid test of causality of the associated SNPs from the first stage.
Going beyond nuclear families to more general closely related individuals, future work could develop a framework where one can first probabilistically model missing parent genotypes and then conduct the gTMT. Large-scale family-based genetic data can have first degree relatives across multiple generations, such as offspring, parents, and grandparents. An example is the Framingham Heart Study that has recruited 5,209, 5,124, and 4,095 participants from three generations of residents in Framingham, with each generation as the offspring of the previous generation [50–52]. For such data, the gTMT could be implemented to analyze these data across generations.
The gTMT is currently designed to analyze common genetic variants. For rare genetic variants, the proportion of heterozygous parents would be small, affecting the total number of randomized allele transmissions from heterozygous parents and the power of the gTMT. This limitation has also been recognized for family-based association tests. Previous work has proposed methods to detect associations between rare variants and traits [11,53–55]. An extension of the TDT for rare variant association tests has been developed for binary traits, but currently shows some inflated type I error rates [56]. An interesting future direction for the gTMT would be to build on these existing ideas to modify the gTMT to achieve reasonable power when analyzing rare variants.
Supporting information
S1 Appendix. This document presents comprehensive proofs for theorems and lemmas in the main text, extended discussions of methods, simulation details, and UK Biobank analysis.
https://doi.org/10.1371/journal.pgen.1012231.s001
(PDF)
Acknowledgments
This research has been conducted using the UK Biobank Resource under Application Number 104628.
References
- 1. Spielman RS, McGinnis RE, Ewens WJ. Transmission test for linkage disequilibrium: the insulin gene region and insulin-dependent diabetes mellitus (IDDM). Am J Hum Genet. 1993;52(3):506–16. pmid:8447318
- 2. Ewens WJ, Spielman RS. The transmission/disequilibrium test: history, subdivision, and admixture. Am J Hum Genet. 1995;57(2):455–64. pmid:7668272
- 3. Thomson G. Mapping disease genes: family-based association studies. Am J Hum Genet. 1995;57(2):487–98. pmid:7668275
- 4. Allison DB. Transmission-disequilibrium tests for quantitative traits. Am J Hum Genet. 1997;60(3):676–90. pmid:9042929
- 5. Laird NM, Horvath S, Xu X. Implementing a unified approach to family-based tests of association. Genet Epidemiol. 2000;19 Suppl 1:S36-42. pmid:11055368
- 6. Laird NM, Lange C. Family-based designs in the age of large-scale gene-association studies. Nat Rev Genet. 2006;7(5):385–94. pmid:16619052
- 7. Laird NM, Lange C. Family-based methods for linkage and association analysis. Adv Genet. 2008;60:219–52. pmid:18358323
- 8.
Laird NM, Lange C. Family-based association test (FBAT). eLS. 2018;:1–8.
- 9. Chen W-M, Abecasis GR. Family-based association tests for genomewide association scans. Am J Hum Genet. 2007;81(5):913–26. pmid:17924335
- 10. Ott J, Kamatani Y, Lathrop M. Family-based designs for genome-wide association studies. Nat Rev Genet. 2011;12(7):465–74. pmid:21629274
- 11. Ionita-Laza I, Lee S, Makarov V, Buxbaum JD, Lin X. Family-based association tests for sequence data, and comparisons with population-based association tests. Eur J Hum Genet. 2013;21(10):1158–62. pmid:23386037
- 12. Kong A, Thorleifsson G, Frigge ML, Vilhjalmsson BJ, Young AI, Thorgeirsson TE, et al. The nature of nurture: Effects of parental genotypes. Science. 2018;359(6374):424–8. pmid:29371463
- 13. Young AI, Frigge ML, Gudbjartsson DF, Thorleifsson G, Bjornsdottir G, Sulem P, et al. Relatedness disequilibrium regression estimates heritability without environmental bias. Nat Genet. 2018;50(9):1304–10. pmid:30104764
- 14. Young AI, Nehzati SM, Benonisdottir S, Okbay A, Jayashankar H, Lee C, et al. Mendelian imputation of parental genotypes improves estimates of direct genetic effects. Nat Genet. 2022;54(6):897–905. pmid:35681053
- 15. Davies NM, Hemani G, Neiderhiser JM, Martin HC, Mills MC, Visscher PM, et al. The importance of family-based sampling for biobanks. Nature. 2024;634(8035):795–803. pmid:39443775
- 16. Tang Y, Cabreros I, Storey JD. Identifying Causal Genotype-Phenotype Relationships for Population-Sampled Parent-Child Trios. Genet Epidemiol. 2026;50(1):e70027. pmid:41521593
- 17. Neyman J. On the application of probability theory to agricultural experiments. Essay on principles. Statistical Science. 1923;5(4):465–72.
- 18. Rubin DB. Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of Educational Psychology. 1974;66(5):688–701.
- 19.
Imbens GW, Rubin DB. Causal Inference in Statistics, Social, and Biomedical Sciences. Cambridge University Press. 2015.
- 20. Magnus P, Birke C, Vejrup K, Haugan A, Alsaker E, Daltveit AK, et al. Cohort Profile Update: The Norwegian Mother and Child Cohort Study (MoBa). Int J Epidemiol. 2016;45(2):382–8. pmid:27063603
- 21. Bycroft C, Freeman C, Petkova D, Band G, Elliott LT, Sharp K, et al. The UK Biobank resource with deep phenotyping and genomic data. Nature. 2018;562(7726):203–9. pmid:30305743
- 22. Kurki MI, Karjalainen J, Palta P, Sipilä TP, Kristiansson K, Donner KM, et al. FinnGen provides genetic insights from a well-phenotyped isolated population. Nature. 2023;613(7944):508–18. pmid:36653562
- 23. Wang F, Liu A, Yang Z, Vartiainen P, Jukarainen S, Koskela S, et al. Effects of parental autoimmune diseases on type 1 diabetes in offspring can be partially explained by HLA and non-HLA polymorphisms. Cell Genom. 2025;5(6):100854. pmid:40286789
- 24. Bates S, Sesia M, Sabatti C, Candès E. Causal inference in genetic trio studies. Proc Natl Acad Sci U S A. 2020;117(39):24117–26. pmid:32948695
- 25. Yang Y, Wang C, Liu L, Buxbaum J, He Z, Ionita-Laza I. KnockoffTrio: A knockoff framework for the identification of putative causal variants in genome-wide association studies with trio design. Am J Hum Genet. 2022;109(10):1761–76. pmid:36150388
- 26. Pritchard JK, Stephens M, Donnelly P. Inference of population structure using multilocus genotype data. Genetics. 2000;155(2):945–59. pmid:10835412
- 27. Alexander DH, Novembre J, Lange K. Fast model-based estimation of ancestry in unrelated individuals. Genome Res. 2009;19(9):1655–64. pmid:19648217
- 28. Cabreros I, Storey JD. A Likelihood-Free Estimator of Population Structure Bridging Admixture Models and Principal Components Analysis. Genetics. 2019;212(4):1009–29. pmid:31028112
- 29. Ochoa A, Storey JD. Estimating FST and kinship for arbitrary population structures. PLoS Genet. 2021;17(1):e1009241. pmid:33465078
- 30. Baumdicker F, Bisschop G, Goldstein D, Gower G, Ragsdale AP, Tsambos G, et al. Efficient ancestry and mutation simulation with msprime 1.0. Genetics. 2022;220(3):iyab229. pmid:34897427
- 31. Browning SR, Browning BL, Daviglus ML, Durazo-Arvizu RA, Schneiderman N, Kaplan RC, et al. Ancestry-specific recent effective population size in the Americas. PLoS Genet. 2018;14(5):e1007385. pmid:29795556
- 32. Veller C, Coop GM. Interpreting population- and family-based genome-wide association studies in the presence of confounding. PLoS Biol. 2024;22(4):e3002511. pmid:38603516
- 33. Sanada H, Jones JE, Jose PA. Genetics of salt-sensitive hypertension. Curr Hypertens Rep. 2011;13(1):55–66. pmid:21058046
- 34. Carey RM, Schoeffel CD, Gildea JJ, Jones JE, McGrath HE, Gordon LN, et al. Salt sensitivity of blood pressure is associated with polymorphisms in the sodium-bicarbonate cotransporter. Hypertension. 2012;60(5):1359–66. pmid:22987918
- 35. Kelly TN, He J. Genomic epidemiology of blood pressure salt sensitivity. J Hypertens. 2012;30(5):861–73. pmid:22495127
- 36. Pilic L, Mavrommatis Y. Genetic predisposition to salt-sensitive normotension and its effects on salt taste perception and intake. Br J Nutr. 2018;120(7):721–31. pmid:30105963
- 37. Manichaikul A, Mychaleckyj JC, Rich SS, Daly K, Sale M, Chen W-M. Robust relationship inference in genome-wide association studies. Bioinformatics. 2010;26(22):2867–73. pmid:20926424
- 38. Howie B, Fuchsberger C, Stephens M, Marchini J, Abecasis GR. Fast and accurate genotype imputation in genome-wide association studies through pre-phasing. Nat Genet. 2012;44(8):955–9. pmid:22820512
- 39. Yoo HJ, Kim M, Kim M, Chae JS, Lee S-H, Lee JH. The peptidylglycine-α-amidating monooxygenase (PAM) gene rs13175330 A>G polymorphism is associated with hypertension in a Korean population. Hum Genomics. 2017;11(1):29. pmid:29162152
- 40. Powers KG, Ma X-M, Eipper BA, Mains RE. Identifying roles for peptidergic signaling in mice. Proc Natl Acad Sci U S A. 2019;116(40):20169–79. pmid:31455734
- 41. Frau F, Zaninello R, Salvi E, Ortu MF, Braga D, Velayutham D, et al. Genome-wide association study identifies CAMKID variants involved in blood pressure response to losartan: the SOPHIA study. Pharmacogenomics. 2014;15(13):1643–52. pmid:25410890
- 42. Hoffmann TJ, Ehret GB, Nandakumar P, Ranatunga D, Schaefer C, Kwok P-Y, et al. Genome-wide association analyses using electronic health records identify new loci influencing blood pressure variation. Nat Genet. 2017;49(1):54–64. pmid:27841878
- 43. Warren HR, Evangelou E, Cabrera CP, Gao H, Ren M, Mifsud B, et al. Genome-wide association analysis identifies novel blood pressure loci and offers biological insights into cardiovascular risk. Nat Genet. 2017;49(3):403–15. pmid:28135244
- 44. Giri A, Hellwege JN, Keaton JM, Park J, Qiu C, Warren HR, et al. Trans-ethnic association study of blood pressure determinants in over 750,000 individuals. Nat Genet. 2019;51(1):51–62. pmid:30578418
- 45. Sakaue S, Kanai M, Tanigawa Y, Karjalainen J, Kurki M, Koshiba S, et al. A cross-population atlas of genetic associations for 220 human phenotypes. Nat Genet. 2021;53(10):1415–24. pmid:34594039
- 46. Keaton JM, Kamali Z, Xie T, Vaez A, Williams A, Goleva SB, et al. Genome-wide analysis in over 1 million individuals of European ancestry yields improved polygenic risk scores for blood pressure traits. Nat Genet. 2024;56(5):778–91. pmid:38689001
- 47. Yang M-L, Xu C, Gupte T, Hoffmann TJ, Iribarren C, Zhou X, et al. Sex-specific genetic architecture of blood pressure. Nat Med. 2024;30(3):818–28. pmid:38459180
- 48. van Duijvenboden S, Ramírez J, Young WJ, Olczak KJ, Ahmed F, Alhammadi MJAY, et al. Integration of genetic fine-mapping and multi-omics data reveals candidate effector genes for hypertension. Am J Hum Genet. 2023;110(10):1718–34. pmid:37683633
- 49. Devlin B, Roeder K. Genomic control for association studies. Biometrics. 1999;55(4):997–1004. pmid:11315092
- 50. Dawber TR, Meadors GF, Moore FEJ. Epidemiological approaches to heart disease: the Framingham Study. American Journal of Public Health and the Nations Health. 1951;41(3):279–86.
- 51. Feinleib M, Kannel WB, Garrison RJ, McNamara PM, Castelli WP. The Framingham Offspring Study. Design and preliminary data. Prev Med. 1975;4(4):518–25. pmid:1208363
- 52. Splansky GL, Corey D, Yang Q, Atwood LD, Cupples LA, Benjamin EJ, et al. The Third Generation Cohort of the National Heart, Lung, and Blood Institute’s Framingham Heart Study: design, recruitment, and initial examination. Am J Epidemiol. 2007;165(11):1328–35. pmid:17372189
- 53. Chen H, Meigs JB, Dupuis J. Sequence kernel association test for quantitative traits in family samples. Genet Epidemiol. 2013;37(2):196–204. pmid:23280576
- 54. Jiang D, McPeek MS. Robust rare variant association testing for quantitative traits in samples with related individuals. Genet Epidemiol. 2014;38(1):10–20. pmid:24248908
- 55. Wang X, Zhang Z, Morris N, Cai T, Lee S, Wang C, et al. Rare variant association test in family-based sequencing studies. Brief Bioinform. 2017;18(6):954–61. pmid:27677958
- 56. He Z, O’Roak BJ, Smith JD, Wang G, Hooker S, Santos-Cortez RLP, et al. Rare-variant extensions of the transmission disequilibrium test: application to autism exome sequence data. Am J Hum Genet. 2014;94(1):33–46. pmid:24360806