Genetic instrumental variable regression: Explaining socioeconomic and health outcomes in nonexperimental data

Thomas A. DiPretea,1,2, Casper A. P. Burikb,1, and Philipp D. Koellingerb,1,2

aDepartment of Sociology, Columbia University, New York, NY 10027; and bDepartment of Economics, Vrije Universiteit Amsterdam, 1081 HV Amsterdam, The Netherlands

Edited by Kenneth W. Wachter, University of California, Berkeley, CA, and approved March 21, 2018 (received for review May 3, 2017)

Identifying causal effects in nonexperimental data is an enduring complex,” which means that the observed heritability is due to challenge. One proposed solution that recently gained popu- the accumulation of effects from a very large number of genes larity is the idea to use genes as instrumental variables [i.e., that each have a small, often statistically insignificant, influ- Mendelian randomization (MR)]. However, this approach is prob- ence (10). lematic because many variables of interest are genetically corre- Furthermore, genes often influence several seemingly unre- lated, which implies the possibility that many genes could affect lated traits, a phenomenon called direct or vertical pleiotropy both the exposure and the outcome directly or via unobserved (11, 12). For example, a mutation of a single gene that causes confounding factors. Thus, pleiotropic effects of genes are them- the disease phenylketonuria is responsible for mental retardation selves a source of bias in nonexperimental data that would also and also for abnormally light hair and skin color (13). Pleiotropy undermine the ability of MR to correct for endogeneity bias from is not restricted to diseases. All genes involved in healthy cell nongenetic sources. Here, we propose an alternative approach, metabolism and cell division can be expected to directly influ- genetic instrumental variable (GIV) regression, that provides esti- ence a broad range of traits such as body height, cognitive ability, mates for the effect of an exposure on an outcome in the presence and longevity, even if the effect on each of these traits may be of pleiotropy. As a valuable byproduct, GIV regression also pro- tiny. Similarly, any gene involved in neurodevelopment and brain vides accurate estimates of the chip heritability of the outcome function is likely to contribute to human behavior and mental variable. GIV regression uses polygenic scores (PGSs) for the out- health in some way (14). come of interest which can be constructed from genome-wide In addition to direct pleiotropic effects, genes can also have association study (GWAS) results. By splitting the GWAS sam- indirect or horizontal pleiotropic effects, where a genetic vari- ple for the outcome into nonoverlapping subsamples, we obtain ant influences one trait, which in turn influences another trait multiple indicators of the outcome PGSs that can be used as (11). The similarity of the genetic architecture of two traits is instruments for each other and, in combination with other meth- estimated by their genetic correlation (i.e., the correlation of the ods such as sibling fixed effects, can address endogeneity bias “true” effect sizes of all genetic variants on both traits) (15), from both pleiotropy and the environment. In two empirical appli- which captures both direct and indirect pleiotropic effects (16– cations, we demonstrate that our approach produces reasonable 18). Genetic correlations exist between many traits and often estimates of the chip heritability of educational attainment (EA) and show that standard regression and MR provide upwardly Significance biased estimates of the effect of body height on EA. We propose genetic instrumental variable (GIV) regression—a causal effects | genetic instrumental variables | polygenic scores | method that controls for pleiotropic effects of genes on two genome-wide association studies | pleiotropy variables. GIV regression is broadly applicable to study out- comes for which polygenic scores from large-scale genome- major challenge in the social sciences and in epidemiol- wide association studies are available. We explore the perfor- Aogy is the identification of causal effects in nonexperimental mance of GIV regression in the presence of pleiotropy across data. In these disciplines, ethical and legal considerations along a range of scenarios and find that it yields more accurate with practical constraints often preclude the use of experiments estimates than alternative approaches such as ordinary least- to randomize the assignment of observations between treatment squares regression or Mendelian randomization. When GIV and control groups or to carry out such experiments in sam- regression is combined with proper controls for purely envi- ples that represent the relevant population (1). Instead, many ronmental sources of bias (e.g., using control variables and important questions are studied in field data which make it diffi- sibling fixed effects), it improves our understanding of the cult to discern between causal effects and (spurious) correlations causal relationships between genetically correlated variables. that are induced by unobserved factors (2). Obviously, confusing correlation with causation is not only a conceptual error; it can Author contributions: T.A.D., C.A.P.B., and P.D.K. designed research, performed research, also lead to ineffective or even harmful recommendations, treat- contributed new analytic tools, analyzed data, and wrote the paper. ments, and policies, as well as a significant waste of resources The authors declare no conflict of interest. (e.g., as in ref. 3). This article is a PNAS Direct Submission. One important source of bias in field data stems from genetic This open access article is distributed under Creative Commons Attribution- effects: Twin studies (4) as well as methods based on molecular NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND). genetic data (5, 6) allow estimation of the proportion of vari- See Commentary on page 5626. ance in a trait that is due to linear genetic effects (so-called 1 T.A.D., C.A.P.B., and P.D.K. contributed equally to this work. narrow-sense heritability). Using these and related methods, 2 To whom correspondence may be addressed. Email: [email protected] or p.d. an overwhelming body of literature demonstrates that almost [email protected]. all important human characteristics, behaviors, and health out- This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. comes are influenced both by genetic predisposition and by 1073/pnas.1707388115/-/DCSupplemental. environmental factors (7–9). Most of these traits are “genetically Published online April 23, 2018.

E4970–E4979 | PNAS | vol. 115 | no. 22 www.pnas.org/cgi/doi/10.1073/pnas.1707388115 Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 irt tal. et DiPrete schizophrenia (36), (BMI) such index traits mass body of the (35), heritability on height the light body driving shed as is to that begin architecture results association have traits genetic These genome-wide associations human (34). large-scale genetic (GWASs) many in studies replicable avail- of reported now robust, been architecture are of recently datasets genetic plethora large a the data very and genetic study result, of to a geno- collection As Array-based able cheap. the MR. and of made fast challenges have two technologies first typing the address 33). 32, by to (27, hindered genes most still of are function influ- the attempts that of such genes knowledge of limited but subset our exposure, be a not the isolate to obviously only try could ence could genes One these IVs. as are confound, used genes the of of effects pleiotropic source direct the personality, if Fourth, environment, (31). parental habits) the and (e.g., of not with genotypes is the correlate for offspring that control parents the everything of to with genotype correlates possible the and is parents, random it the of Unless genotype parents. trait, the the on the of conditional only in genotype assigned variance (29, the randomly instruments the are weak genotypes of of itself Third, problem part by 30). well-known gene small the to any very leads trait, a which complex only genetically capture envi- a will is Second, from ancestry. exposure effects with the genetic correlated if are genes true which that isolate know confounds and to ronmental need exposure are we there the First, However, idea. affect of interest. this influence to of may causal challenges outcome the it four some identify exposure, to on the IVs exposure affect as the genes them use which to known draw. possible be random be a could of it result the if is par- So offspring the the the of of genotype by genotype the the gametes on because ents, conditional of appealing Thus, production meiosis. principle of the in process in is randomized idea are The genotypes randomiza- Mendelian (25–28). approach (MR) this tion termed and IVs construct for variable outcome the affect direction) The units.] not same (SUTVA): other does the assumption unit in one value of affected treatment “treatment” is unit IV stable the the by and (every- affected monotonicity is are who satisfy to one particu- other need In IVs [Two valid challenging. difficult. that is conditions is restriction exclusion requirements the both satisfying lar, satisfy the that their regression on IVs finding practice, effect produce valid the In their restriction). and exclusion of through so-called solely (the variables term exposure outcome control error the with other the correlation the they of Second, on independent “relevant”). conditional be be to to need corre- IVs need be (i.e., to variables regression to need control the other need IVs the in on they Valid conditional exposure First, outcome. the with conditions. an lated on important exposure two isolate to that satisfy interest of of pro- exposure effect They the Valid the on experiments: (24). “shock” natural exogenous (IVs) to an variables vide similar instrumental conceptually use are to IVs is data this imental in conclusions wrong to lead still design. Further- study can nonexistent. twins causation differences or the reverse because between small or differences be pairs environmental to require twin unobserved tend more, studies MZ pairs twin of such MZ sizes that within sample is also large challenge approach very practical environment. this parental the shared addition, from However, arise genet- In that definition effects (23). for by controls for identical are outcomes who (almost) compare 22), ically (21, to twins is (MZ) confounds monozygotic genetic of presence the the to (20). effects genetic rise that for studies control giving bias not substantially do (19), may pleiotropy direct correlations that concern phenotypic their exceed eetavne ncmlxtatgntc aei possible it make genetics trait complex in advances Recent to information genetic use to proposed have Epidemiologists nonexper- in effects causal isolate to strategy popular Another in standard gold the possible, not is design experimental an If etos1–5 sections Appendix, here.) naturally exploit and we individuals, possi- of data which endowment not observational genetic between the on is like relationship experiments rely EA occurring causal must the or variables study two height to the body attempt any on randomizes Thus, based which ble. groups design should (it experimental into height clean body people whether a on check that EA of we (Note effect of control,” not). causal effects “negative a a the finds method As estimate our EA. we can on Then, EA height method. of body our (47) with heritability chip obtained so-called con- be a the that of demonstrate estimate we sistent and First, (51). Health (HRS) available Study publicly Retirement the using of applications version empirical enhanced in an smaller term generally we under what is or bias (EMR). regression MR, MR the OLS, GIV that evidence with simulations with with than of show assumptions we set these and comprehensive pleiotropy, a of from because inter- of term regressor a error when of case complex We est more SNPs. sam- of the GWAS number to error the the turn to then the when relative large and with sufficiently covariates, are are uncorrelated sizes included model ple is the the of PGS the net in true on term covariates the PGSs regression other the when GIV the of exogenous, that when effect show variable the to for outcome on estimates go accurate We produces approach. our direct of substantial tions of presence the expo- in an of even effect outcome pleiotropy. causal practically an the on for estimates bounds sure avail- regression lower and widely variables GIV upper using useful instrumental software. implemented pro- genetic statistical be we call can able particular, that we In regression that pleiotropy. (GIV) approach interest from ordinary an of arising outcome pose in bias the bias for reduce PGSs of to use source that alternative strategies propose serious We estimation MR. a and demon- (OLS) is regression We least-squares pleiotropy data. that nonexperimental strate using relationships causal to (48–50). efforts MR in recent assumptions despite exogeneity obstacle the twins relax serious (i.e., dizygotic a challenge remains fourth or the pleiotropy) However, siblings genotypes. draws of parent’s random the are samples from siblings between large differences genetic the using the where and by in parents parent–offspring of or could the sample large trios) list of a in above genotypes (e.g., observed the the are offspring in if MR addressed to be principle challenge of genetic share third individual larger than The (an much traits markers. (47) complex a the genetically capture heritability of of PGSs variance their below), the less by to substantially return suggested we capture than issue traits still (35–37, in EA PGSs variation and Although schizophrenia, 46). this complex BMI, genetically 39, height, that predict as to demonstrate such begin outcomes exclude studies that PGSs that Recent yields GWASs approach sample. large-scale prediction in on estimated the SNP are each outcome estimated of effects an the The aggregate (SNPs)]. single- polymorphisms [typically that variants nucleotide genetic indexes measured currently linear all of are effects summarizing 39, PGSs (15, for 45). traits tool complex 44, genetically favored for the predispositions genetic become Polygenic the (40–43). have ancestry associations by (PGSs) genetic confounded reported scores not the is traits of many majority sug- for vast evidence the empirical that and population, gests the in structure genetic educational and (15). (39), (EA) attainment depression (38), disease Alzheimer’s (37), omldrvtosadtcncldtisaecnandin contained are details technical and derivations Formal approach our of usefulness practical the demonstrate we Next, assump- the out laying and intuition providing by begin We modeling for pleiotropy of implications the address we Here, for control to strategies several use GWASs High-quality (T ) sptnilycreae ihuosre aibe nthe in variables unobserved with correlated potentially is . PNAS | o.115 vol. | o 22 no. | E4971 SI

SOCIAL SCIENCES SEE COMMENTARY Theory The GIV strategy starts to break down when bias arises from Intuition. To build intuition for our approach, we introduce the unobserved nongenetic factors as well as from pleiotropic effects. concept of the true PGS for y which would be constructed using We show below that both GIV and MR produce biased estimates the true effects of each SNP on y. In theory, the true SNP effects in this case. However, we demonstrate that the combination could be estimated in a GWAS on y in an infinitely large sample of GIV-C and GIV-U still outperforms OLS and MR. Fur- that is drawn from the same population as the prediction sam- thermore, the GIV approach has additional utility because it ple. The true PGS would capture the narrow-sense heritability of can be combined with other strategies to reduce the effects y. Of course, the true PGS is unknown. All one can practically of environmental endogeneity (e.g., additional control variables obtain is a PGS from a finite GWAS sample that will capture a or family fixed effects). We demonstrate that these combined part, but not all, of the genetic influence on y because the effect strategies can potentially provide accurate information about the of each SNP is estimated with noise. The attenuated predictive effects of an exposure in situations with both genetic and non- accuracy of practically available PGSs (45, 47, 52) is conceptu- genetic sources of endogeneity. In contrast, the problems for ally similar to the well-known problem of measurement error MR that are produced by pleiotropy bias are not fixable in a in regression analysis. It has long been understood that multi- similar manner. ple indicators can, under certain conditions, provide a strategy to correct regression estimates for attenuation from measurement Assumptions. GIV regression builds on the standard identify- error (53–55). We show below that by splitting the GWAS sam- ing assumptions of IV regression (24). In the context of our ple into independent subsamples, one can obtain several PGSs approach, this implies six specific conditions: (i.e., multiple indicators) in the prediction sample. Each will have i) Polygenicity: The outcome is a genetically complex trait that even lower predictive accuracy than the original score due to is influenced by many genetic variants, each with a very small the smaller GWAS subsamples used in their construction, but effect. these multiple indicators can be used as instrumental variables ii) Complete genetic information: The available genetic data for each other, and the instruments will satisfy the assumptions include all variants that influence the variable(s) of interest. of IV regression to the extent that the measurement errors (the iii) Genetic effects are linear: All genetic variants influence difference between the true and calculated PGSs) are uncorre- the variable(s) of interest via additive linear effects. Thus, lated. Standard two-stage least-squares (2SLS) regression (24) there are no genetic interactions (i.e., epistasis) or dominant (readily available in statistics software packages) using at least alleles. one valid IV for the PGS of y can then be used to back out an iv) Unbiased GWAS results: The available GWAS results are unbiased estimate of the heritability of y. not systematically biased by omitted environmental variables. Next, presume the matter of interest is not heritability, but the For example, failure to control for population structure can causal effect of some treatment T on y, where T is also her- lead to spurious genetic associations (31). itable and some genes have direct pleiotropic effects on both. v) Nonoverlapping samples: It is possible to divide GWAS sam- If these genes are not known and not controlled for, regressing ples into nonoverlapping subsamples drawn from the same y on T would result in omitted variable bias. (Unfortunately, population. that is the reality in most social scientific and epidemiological vi) The genetic effects on y are the same in the GWAS and studies that use nonexperimental data.) Suppose the effects of the prediction samples; i.e., the genetic correlation between all genes that influence y through channels other than T could samples is one. be known. Theoretically, one could estimate these effects in a GWAS on y that controls for T in an infinitely large sample. Estimating Narrow-Sense SNP Heritability from Polygenic Scores. That information could be used to construct a “true conditional” Under these assumptions, consistent estimates of the chip heri- PGS in a prediction sample. Adding the true conditional PGS tability of a trait (i.e., the proportion of variance in a trait that to a regression of y on T in the prediction sample would effec- is due to linear effects of currently measurable SNPs) can be tively eliminate bias arising from direct pleiotropy. However, the obtained from polygenic scores (for full details, see SI Appendix, true conditional PGS is also unknown and all we can practically section 2). If y is the outcome variable, X is a vector of exoge- obtain is a noisy proxy of it from a finite GWAS sample. While it nous control variables, and S ∗ is a summary measure of genetic is not guaranteed, the general conclusion of the literature is that y|X tendency for y in the presence of controls for X , then one the use of proxy variables is an improvement over omitting the variable being proxied (56, 57). Furthermore, having a valid IV can write for the conditional score would potentially correct for its noise ∗ y = α + X β + γS +  [1] and get us closer to estimating the true causal effect of T on y|X y. As before, a valid IV can be practically obtained by splitting = α + X β + γ(Gζy|X ) + , the GWAS sample into independent parts and standard IV esti- mation techniques such as two-stage least squares can be used. where G is an n × m matrix of genetic markers, and ζy|X is the We refer to this approach as conditional genetic IV regression m × 1 vector of SNP effect sizes, where the number of SNPs (GIV-C). is typically in the millions. If the true effects of each SNP on If conditional GWAS results are not available, one can still the outcome were known, the entire genetic tendency for y ∗ add the unconditional PGS for y as a control variable and use would be captured by the true unconditional score Sy|X , and 2 ∗ IV regression with multiple indicators for this score to correct the marginal R of Sy|X in Eq. 1 would be the chip heritabil- for measurement error. We refer to this as unconditional GIV ity of the trait. We refer to the estimate of the PGS from actually regression (GIV-U). GIV-U still corrects for bias arising from available GWAS data in the presence of controls for X as Sy|X , direct pleiotropy, but this strategy will overcontrol and result in where estimates for T that are biased toward zero because the uncondi- ∗ S = S + v1 = Gζ + Gu [2] tional PGS also includes indirect pleiotropic effects of genes that y|X y|X y|X y|X affect y only because they affect T . However, extensive simula- and uy|X is the estimation error in ζy|X and Sy|X is substituted ∗ tions show that the combination of GIV-C and GIV-U turns out for Sy|X in Eq. 1. The variance of a trait that is captured by its to produce reasonable upper and lower bounds for the effect of available PGS increases with the available GWAS sample size T on y across a broad range of scenarios if the only sources of to estimate ζy|X and converges to the SNP-based narrow-sense bias are pleiotropic genes. heritability of the trait at the limit if all relevant genetic markers

E4972 | www.pnas.org/cgi/doi/10.1073/pnas.1707388115 DiPrete et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 irt tal. et DiPrete This SNPs. of of estimates sets noisier overlapping produces partially splitting least by at PGS for with the subsamples sample of discovery indicators multiple GWAS obtain the to is bias tion currently datasets the with ignor- questions strate- to available. important alternative down for address calls attenuation to large therefore, gies this having situation, bring This from levels. away to able time sizes long a sample are enough we esti- size, sample the GWAS in of attenuation mate substantial of demonstrated estimates consistently obtain to assumptions use S statistical that of methods combination estimation a nonconventional requires cases to were size sample GWAS 52). the 47, if (45, and large sufficiently GWAS the in included were erse on regressed estimate heritability the of heritability that γ genotype chip assuming with the further correlated of and not one, are of or in SD of contained a level variables and sufficient zero a mean to means satisfied which be independent, can the mutually accuracy. in assumption of separation essentially this set spatial be that total iso- sufficient the to have the to from that on DNA possible SNPs markers other currently of thousand each millions is markers hundred to it close several genetic located late However, general, are chromosome. 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SOCIAL SCIENCES SEE COMMENTARY ˆ We refer to the combined use of Sy|XT as a control and ST|X we construct an estimate for Sy1|XT , namely Sy1|XT , by using as an IV for T as EMR. However, controlling for Sy|XT as a Sy2|X (the unconditional PGS from the second GWAS sam- S ∗ ˆ proxy for y|XT is not a perfect solution to pleiotropy because it ple) as its IV. We call this approach, where Sy1|XT is used as ∗ leaves a component of Sy|XT in the error term which is correlated the regressor in the second stage, GIV-C. We also use a third with ST due to pleiotropy. As a result, the EMR estimate for δ estimator that uses the same IV as in GIV-C (i.e., Sy2|X ), but will be biased. The practical question, then, is whether alternative that substitutes the unconditional PGS for y (i.e., substitutes strategies that split the GWAS sample for Y to obtain multi- Sy1|X for the conditional PGS Sy1|XT ) in the structural model S ˆ ple indicators of y|XT that can be used as IVs for each other in Eq. 3. We then use Sy2|X to predict Sy1|X , obtaining Sy1|X (e.g., Sy1|XT and Sy2|XT ) are sufficient to rescue ST as a prac- as the regressor in the second stage. We call this third approach tically useful IV for T .† This is a practical question beyond the GIV-U. reach of formal mathematics and best answered by simulation We generally expect the use of GIV-C to perform better than analyses. Unfortunately, and as we show in SI Appendix, section the use of the proxy Sy|XT in simple OLS. If the true effect of T 2, the pleiotropy-induced violation of the exclusion restriction on y is positive and positive pleiotropy is present, the estimated when using genetic IVs for T is sufficient to produce serious bias effect of T on y will have positive bias. This follows from the ∗ in the estimated effect of T even if one attempts to control for positive correlation between Sy|XT and T and from the positive ∗ pleiotropy using such strategies. The magnitude of bias clearly effect of S on y. The presence of the proxy Sy|XT in the first ∗ y|XT depends on the quality of the proxies for Sy|XT . However, we approach (simple OLS) adds a partially offsetting negative bias, find that pleiotropy leads to considerable bias in MR in virtually because the correlation between Sy|XT and T is positive and the all scenarios we investigated (SI Appendix, sections 2, 3, and 4 effect γ˜OLS is also positive, but γˆOLS Sy|XT is being subtracted, and Tables S2–S15). which causes the offsetting bias to be negative. The net bias is The problems posed by pleiotropy cannot be completely elim- expected to be positive, but we would expect it to be smaller ∗ inated without knowledge of Sy|XT (SI Appendix, section 2). with the inclusion of the proxy than with no proxy at all, both However, this situation does not mean that the estimation prob- because the correlation between T and Sy|XT would be lower ∗ lems introduced by pleiotropy are intractable. When endogeneity than between T and Sy|XT and because we expect γˆOLS to be bias is driven by genetic correlation, the PGS for y can still attenuated relative to γ. When GIV-C is used instead, the term be used to obtain more accurate estimates of the effect of ∗ ˆ in the error becomes γSy|XT − γˆIVc Sy1|XT . The presence in the T than can be obtained with MR or, for that matter, with ∗ first stage of GIV-C of T , which is correlated with S , pre- OLS that lacks controls for the direct effect of genetic mark- y|XT γ ers on y. To gain insight into the best strategy, we consider the vents the IV strategy from obtaining a consistent estimate of . γˆ > γˆ reasons for the pleiotropy bias. Regardless of whether the esti- Nonetheless, we would generally expect IVc OLS , and there- δ mation strategy is OLS or the second stage of IV regression fore we expect the positive bias for the estimate of to be smaller δ involving OLS on predicted variables from the first stage, the when using GIV-C than when estimating using OLS and the S coefficient bias comes from the extent to which the expected esti- proxy y1|X . We confirm this in the simulations in SI Appendix, ‡ Tables S2–S7. mate of the OLS coefficients differs from the true coefficients; ∗ With GIV-U, the problem term in the error is γS − namely, y|XT 0 −1 0 γˆIVu Sˆ . As before, the presence of the first term produces E[βˆ|X ] = β + E[X X
X |X ]. [5] y1|X a positive bias in the estimate of δ, while the second term pro- In other words, the coefficient bias from OLS is the expected duces an offsetting negative bias. The offset will be stronger regression coefficient of the error on the included variables in when the unconditional PGS for y is the regressor in the struc- the regression. If  is the sum of an omitted variable, z, which tural model, because the coefficients of the genetic markers in is correlated with the regressors, and additional variables that Sy1 are δˆξˆ+ ζˆ, where ξ is the effect of the genetic marker on are uncorrelated with the regressors, then the bias for each   T . The presence of δˆ Gξˆ in the second endogenous term in coefficient βk in Eq. 5 becomes the product of the regression ∗ ˜ coefficient for xk in the regression of z on all of the omitted the error (i.e., the second term in γSy|XT − γ˜Sy|XT ) produces variables multiplied by the effect of z on the outcome. For sim- a stronger downward bias. This downward bias is made still plicity, we assume that the only variables in the regression are stronger by the use of γˆIVu instead of γˆOLS as the coefficient, ∗ ˜ T and a potential proxy for Sy|XT , which we call Sy|XT . For because we expect the first-stage regression to reduce the down- ˜ ˆ ward bias of γˆOLS . In other words, we expect these three proxies any given proxy, Sy|XT , the bias in the estimate of δ (the coef- ficient for T in Eq. 3) comes from the expected coefficient of to behave differently in the simulations, and, as we will see, this T from a regression of γS ∗ − γ˜S˜ on T and S˜ . We expectation is met in practice. We establish via a comprehen- y|XT y|XT y|XT sive set of simulations that GIV-C and GIV-U provide upper ˜ consider three alternative approaches for the proxy Sy|XT , which and lower bounds for the effect of T across a range of plau- we call simple OLS, GIV-C, and GIV-U. First, we use Sy|XT ∗ sible scenarios for pleiotropy and for heritability (SI Appendix, as a proxy for Sy|XT in a simple OLS regression. Second, we Tables S2–S7). We further establish through simulation analyses observe that Sy|XT is correlated with T ; its inclusion in the error that GIV-C and GIV-U perform similarly in the case when endo- (by virtue of its being controlled) may affect the bias in δˆ. So geneity arises from pleiotropy and when it arises from pleiotropy in combination with genetic confounds for reasons other than pleiotropy [epistasis, effects from rare alleles, or genetic nurtur- ing effects, where the environment of ego is shaped by genetically †An earlier version of our paper pursued this approach and called it GIV regression. However, we later found that controlling for T in a GWAS for y induces a correlation of related individuals to ego (63)] (SI Appendix, section 3 and Sy1|XT with ST that invalidates the latter as an IV. The version of GIV-U and GIV-C regres- Tables S8–S10). sion we describe below does not have this problem because it does not use a genetic instrument for T anymore. Instead, GIV-U and GIV-C both rely on a proxy-control strat- Simulations egy that uses only an instrument for Sy1|XT or Sy1|X to correct for measurement error ∗ ∗ in these proxies for Sy|XT and Sy|X . We explored the performance of GIV regression in finite sam- ple sizes using three sets of simulation scenarios (SI Appendix, ‡It is possible to have finite-sample bias that disappears asymptotically, in which case the estimator is consistent. We use the expectation formula instead because it is arguably sections 2–4). The simulations generated genetic and pheno- more straightforward to understand. typic data at the individual level from a set of known models in

E4974 | www.pnas.org/cgi/doi/10.1073/pnas.1707388115 DiPrete et al. 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Table litoyonly Pleiotropy only Pleiotropy litoyonly Pleiotropy only Pleiotropy y 2 h = = h h h h = T 2 y y Parameter y 2 y 2 y 2 0.5, T 2 0.5, . r h eiaiiyprmtr of parameters heritability the are = = = 0.8, = ρ n h fetof effect the and ν h h h y 0.5, T ρ .) ρ T 2 T 2 T 2 e h ν stecreainbetween correlation the is eeicue in included were = = = T 2 = y ρ = = 0.2 0.6 0.4 0.4, e ρ 0.2 = o l eut.A, results. all for ν T 0.4 s = = y 000)(.02 000)(0.0002) (0.0001) (0.0002) (0.0001) 0.5 000)(.02 000)(0.0001) (0.0001) (0.0002) (0.0001) 0.5 and T y T y and and n Ewti aetee f2 iuain sn different using simulations 20 of parentheses within SE and to T T y . s B, S2; Table stesaeo h ofudta scnrle o in for controlled is that confound the of share the is ieyt be. to likely of on whether levels effect about causal higher information GIV- useful and moderate at GIV-C provides to of Even U combination low heritability. the of heritability, and and conditions pleiotropy pleiotropy under of answer true levels the to close the underestimates of generally of with- GIV-U levels OLS effect EMR. moderate than or accurate MR, to more proxies, low is out and at of heritability modest effect and is pleiotropy the overestimation overestimates the generally of but GIV-C heritability variants, extreme. the genetic not and unobserved is nurturing) genetic (i.e., or confounds epistasis, genetic effect true other the or of bounds regression lower GIV of and contrast, In upper pleiotropy. results reasonable with provides endogene- Our along nongenetic IVs. present when is even genetic ity poorly for performs MR restriction vio- that the exclusion find by the undermined is of strategy lation MR When the provide MR. present, from results is estimates pleiotropy of The skeptical be from to sources. endogeneity reason include considerable environmental) also (i.e., G and bias, nongenetic con- F of genetic scenarios of source while sources only other founds, the plus is pleiotropy contains pleiotropy Table E where in scenario situations A–D to Scenarios refer size). effect 1 standardized the on details of effect the on environ- produces effect genetics some to an where unrelated both is situation that a factor behavioral in or occur mental Appendix, would (SI This variables for genetic 4). with models section correlated the itself in not was terms that error the between correlation genetic by i.e., (like members, variables sample (63). of with genes nurturing the also by genes affected therefore the are and with that correlated members are sample which of genes, for parental term by disturbance selected the in factors mental affect that factors genetic the for PGS conditional the 000)(.03 000)(0.0002) (0.0002) (0.0093) (0.0005) 000)(.02 000)(0.0001) (0.0001) (0.0012) (0.0001) 000)(.02 000)(0.0002) (0.0001) (0.0001) (0.0001) (0.0002) (0.0004) (0.0001) (0.0001) 000)(.02 000)(0.0001) (0.0001) (0.0002) (0.0001) .04154 .110.9419 1.0131 1.5040 1.1004 .04342 .760.8689 0.7263 1.0776 0.8575 1.1573 3.4922 1.0604 1.5020 1.5004 1.5024 1.3016 1.2011 .50153 .291.1193 1.4259 1.5032 1.4520 .16140 .920.6422 1.1922 1.4609 1.3106 L RGVCGIV-U GIV-C MR OLS umr fteerslsi nTbe1frtecs where case the for 1 Table in is results these of summary A a of presence the specified we simulations, of set third the In T T n h eei ofudfor confound genetic the and y on al S6 Table Appendix, SI and y C, S3; Table T T. ftesuc fedgniyi nyfo pleiotropy from only is endogeneity of source the if alsS–4 7 9 S11, S9, S7, S2–S4, Tables Appendix, SI ρ on ν T y y stecreainbetween correlation the is sstt . 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SOCIAL SCIENCES SEE COMMENTARY In the case where the true value of T is zero (i.e., where Table 2. Effects of the polygenic score for educational the true model is Eq. 1), we expect that GIV-U will produce attainment (PGS EA) on (residualized) educational attainment in an estimate that is close to zero as long as the endogeneity the Health and Retirement Study (HRS) comes either from pleiotropy or from other genetic confounds Variable OLS IV1 IV2 (see SI Appendix, sections 2–4, for details). The simulations in SI Appendix section 2 show that when the endogeneity is only from PGS EA UKB 0.259∗∗∗ 0.523∗∗∗ genetic sources, GIV-U estimates the effect of T to be close to (0.0183) (0.0385) zero regardless of the level of pleiotropy or inheritance that is PGS EA SSGAC 0.530∗∗∗ specified in the simulations. (0.0389) When the source of endogeneity is nongenetic in origin, we hˆ2 NA 0.134 0.138 find that neither MR nor proxy controls for pleiotropy provide NA (0.0197) (0.0202) a satisfactory method for determining the effect of T on y. In N 2,751 2,751 2,751 this scenario, the pleiotropy creates endogeneity bias for genetic *P < 0.05, **P < 0.01, ***P < 0.001. We regress the residual of EA on IVs that defeats the ability of MR to solve the problem of non- the different PGSs and calculate the implied heritability estimates. SEs are in genetic endogeneity via an IV strategy. Nongenetic endogeneity parentheses. All variables have been standardized. EA is measured in years can cause even GIV-U to overpredict the effect of T on y, of schooling needed to obtain the highest achieved educational degree although in our simulations it is clearly the most accurate of all according to International Standard Classification of Education (ISCED) clas- of the estimators that we have surveyed when the effect of T is sifications. We use the residual of EA after a regression on birth year, birth zero. Indeed, GIV-U always provides the most conservative esti- year squared, gender, and the first 20 principal components in the genetic mate across the entire range of scenarios that we have surveyed, data. PGS EA SSGAC: PGS for EA using meta-analysis from ref. 15, excluding both for the case where an effect of T on y exists and when the data from 23andMe, UK Biobank (UKB), and HRS; PGS EA UKB, PGS for EA effect of T is zero. using UKB data. IV1 uses PGS EA SSGAC as instrument and IV2 uses PGS EA UKB as instrument. NA, not applicable. Inference with the GIV can be further strengthened in cases where nongenetic endogeneity can be controlled either through observable variables or through strategies such as family fixed effects that reduce or eliminate the impact of nongenetic forms mate and the two GIV estimates overlap, demonstrating that of environmental endogeneity. Indeed, environmental endo- GIV regression can recover the chip heritability of EA from geneity is a concern in most applied-research questions that use polygenic scores. nonexperimental data. Reassuringly, our simulations show that GIV regression is a good estimation strategy in the presence The Relationship Between Body Height and Educational Attain- of both direct pleiotropy and environmental endogeneity if con- ment. Previous studies using both OLS and sibling or twin trol variables are available that manage to absorb a substantial fixed-effects methods have found that taller people generally share of the nongenetic confounds (SI Appendix, Tables S13– have higher levels of EA (66–68). They are also more likely to S15). Therefore, we recommend using GIV regression always in perform well in various other life domains, including earnings, combination with control variables the capture possible environ- higher marriage rates for men (although with higher proba- mental confounds, ideally in datasets that allow controlling for bilities of divorce), and higher fertility (69–74). The question family fixed effects (e.g., using siblings or dizygotic twins). In SI is what drives these results. Can they be attributed to genetic Appendix, section 6, we provide additional practical guidelines effects that jointly influence these outcomes? Are there social for GIV regression. mechanisms that systematically favor taller or penalize shorter The simulations described in this paper certainly do not individuals? Or are there nongenetic factors (e.g., the uterine cover all conceivable data-generating processes, but they are and postbirth environments especially related to nutrition or dis- nonetheless of considerable utility. ease) that affect both height and these life-course outcomes? The literature on the relationship between height and EA has Empirical Applications found evidence that the association arises largely through the We illustrate the practical use of GIV regression in two empir- relationship between height and cognitive ability, which may sug- ical applications using data from the Health and Retirement gest that the height–EA association is driven largely by genetic Survey (HRS) for 2,751 unrelated individuals of northwest association between height and cognitive ability. We use GIV European descent who were born between 1935 and 1945 regression with individual-level data from the HRS to clarify (SI Appendix, section 5). the influence of height on EA, and we compare these results with those obtained from OLS and from MR. In addition, we The Narrow-Sense SNP Heritability of EA. First, we demon- conduct a negative control experiment that estimates the causal strate that GIV regression can recover the unbiased genomic– effect of EA on body height (which should be zero). A com- relatedness-matrix restricted maximum-likelihood (GREML) plete description of the materials and methods is available in estimate of the chip heritability of EA. Specifically, we follow the SI Appendix, section 5. common practice in GREML estimates of heritability and ana- GWAS summary statistics for height were obtained from the lyze the residual of EA from a regression of EA on birth year, Genetic Investigation of Anthropometric Traits (GIANT) con- birth year squared, gender, and the first 20 principal components sortium (35) and by running a GWAS on height (conditional on from the genetic data (64). Next, we standardize the residual and EA and unconditional on EA) in the UKB (75). The UKB was regress it on a standardized PGS for EA using OLS or GIV. not part of the GIANT sample. GWAS summary statistics for EA The results are displayed in Table 2. The OLS estimate of the were obtained from the Social Science Genetic Association Con- PGS accounts for 6.8% of the variance in EA (β2 = ∆R2 = sortium (SSGAC) for the unconditional PGSs. The most recent 6.8%), which is substantially lower than the 17.3% (95% CI ± study of the SSGAC on EA used a meta-analysis of 64 cohorts for 4%) estimate of chip heritability reported by ref. 64 in the same genetic discovery (15). We obtained meta-analysis results from data using GREML. [GREML yields unbiased estimates of SNP- this study with the HRS, UKB, and 23andMe cohorts excluded based heritability that are not affected by attenuation (65).] and we refer to the PGS constructed from these results as EA Instead, the GIV regression results in columns 2 and 3 of Table SSGAC. Furthermore, we obtained GWAS estimates for EA in 2 imply a chip heritability of 13.4% (CI ± 3.9%) and 13.8% the full UKB release (N = 442,183) from ref. 76. We refer to (±4.0%), respectively. Thus, the 95% CIs of the GREML esti- this PGS as PGS EA unc. UKB. We also created a PGS for EA

E4976 | www.pnas.org/cgi/doi/10.1073/pnas.1707388115 DiPrete et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 N UKB uncond. EA PGS UKB cond. EA PGS educational Variable on height of effect the (EA) of attainment Estimates 3. 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EA cond. Height on EA between GWAS (PGS a sample running UKB by height on conditional o egt o I- n I- G ASGCwsue sa instrument. an as instrument used as was used SSGAC was EA data PGS GIV-U (UKB) and Biobank For GIV-C UK For variables. from height. control height for as for included PGS are a components MR, father, principal EA 20 mother, first EA gender, the squared, and year birth year, Birth parentheses. in hs eut lopittwr rdciesrtg for strategy productive a toward point also results These 4 Table in results regression control negative the However, then pleiotropy, positive from came EA in bias the of all If of effect standardized estimated the report we 3 Table In *P < 5 **P 0.05, < IN n EA and GIANT 1 ***P 0.01, 0.136 006)(.41 006)(0.0262) (0.0264) (0.0481) (0.0262) ,5 ,5 ,5 2,751 2,751 2,751 2,751 L RGVCGIV-U GIV-C MR OLS < ∗∗∗ 0.Sadrie fetszsadSsare SEs and sizes effect Standardized 0.001. 0.160 SA.Teeoe whenever Therefore, SSGAC. ∗∗∗ 0.396 0.168 (0.0367) ∗∗∗ ∗∗∗ 0.110 0.384 (0.0354) ∗∗∗ ∗∗∗ eilg n h oilsine—w ed fiqiywhere inquiry of epi- in fields challenges sciences—two important social most the and and biggest observational the demiology with of one relationships is causal data of estimation Accurate Conclusion into genetic selection nonrandom of samples.) from dis- existence bias we the possible strategies address given estimation here EA cussed the of of none provided process (Obviously, the advantage confounds. in social the is the of toward height large progress artifact by how represent an understanding results of not these goal words, and other endogene- real In genetic-unrelated be from is ity. or would pleiotropy effect we from either the statisti- analysis, bias and that fixed-effects positive confident sibling be a rather to in continued weaker. significant GIV-U effect is relatively cally of endogeneity the the is estimate for if the endogeneity effect If estimate genetic-unrelated this underpredicts plausible if or EA a strong sug- on is therefore height either results of GIV-U simulation that is Our height (18). gest and EA. high strategy EA affects extremely between effective height pleiotropy not an extent that not suggest what suffers studies to is Other estimate and MR MR whether that the determining and for that bias imply pleiotropy strongly from also height of They effect the EA. of on estimate that biased imply upwardly strongly an results provides impli- Our OLS estimates. the available the into the the of in insight time, cations same considerable inevitable the provides At is IV. example valid uncertainty empirical a or although data experimental desired, of absence be might as quite is endogeneity bias strong. genetic-unrelated positive this and unless in pleiotropy effect both true considered the from estimators underpredict will bias other more it the and be suggests paper, of will was regression all estimate height control than GIV-U on conservative negative the EA sources, the of environmental effect If from the zero. of to estimate as close GIV-U bounds the approximate estimates provide as the would long data, GIV-U such and GIV-C With environmental from fixed-effects. using shared family by for controlling via or allow variables directly that factors siblings environmental control- on data relevant by the either for confounds while pleiotropy ling environmental from for bias correcting address and to also GIV-U strategies use proxy to twin as arguably MZ is GIV-C from strategy results factors productive the most bias environmental The still data. could unobserved EA and but height effectively affecting pleiotropy, would for twins monozygotic control endogene- between environmental for Comparisons for adjust ity. fix to and used effective pleiotropy be by no cannot contaminated is therefore are there instruments that genetic implies its 3 affecting MR; is Table bias in environmental estimates as the well as pleiotropy that stration instrument instrument. an an as as used used was was GIANT height data PGS (UKB) GIV-U Biobank and For GIV-C UK For variables. from EA. control EA for as for included PGS are a components MR, father, principal EA 20 mother, first EA gender, the squared, and year birth year, Birth parentheses. in N UKB unc. height PGS UKB cond. height PGS EA Variable (EA) attainment educational of effect the height of on Estimates 4. Table hs eut ontpoiea la n etaconclusion a neat and clean as provide not do results These *P < 5 **P 0.05, < 1 ***P 0.01, 0.072 003)(.53 001)(0.0120) (0.0119) (0.0543) (0.0138) ,5 ,5 ,5 2,751 2,751 2,751 2,751 L RGVCGIV-U GIV-C MR OLS < 0.Sadrie fetszsadSsare SEs and sizes effect Standardized 0.001. ∗∗∗ PNAS 0.179∗∗ | o.115 vol. 0.050 0.448 (0.0174) | ∗∗∗ ∗∗∗ o 22 no. 0.040 0.446 (0.0175) | E4977 ∗∗∗ ∗∗∗

SOCIAL SCIENCES SEE COMMENTARY many questions of interest cannot be adequately addressed results for the outcome of interest from two nonoverlapping with properly designed experiments due to practical or ethical samples. Due to rapidly falling genotyping costs that enable a constraints. One important confound in nonexperimental data growing availability of genetic data and large GWAS samples comes from direct pleiotropic effects of genes on the exposure for many traits, these requirements have become increasingly and the outcome of interest. Both OLS and MR yield biased feasible for many applications. Indeed, the combination of new results in this case. We proposed GIV regression as an empirical estimation tools and continued rapid advancements in genetics strategy that controls for such pleiotropic effects using polygenic should provide a significant improvement in our understand- scores. GIV regression uses standard IV estimation algorithms ing of the effects of behavioral and environmental variables on such as two-stage least squares that are widely available in important socioeconomic and medical outcomes. existing statistical software packages. Our approach provides reasonable upper and lower bounds of causal effects in situa- ACKNOWLEDGMENTS. We are very grateful to Richard Karlsson Linner´ for tions when pleiotropic effects of genes are the only source of bias. help with the GWAS analyses in the UK Biobank, to Aysu Okbay for provid- ing us with subsets of the GWAS meta-analysis on educational attainment, We showed that OLS, MR, and GIV regression yield biased esti- and to S. Fleur Meddens for constructing genetic principal components mates if both genetic and environmental sources of endogeneity in the HRS data. We thank Daniel J. Benjamin, Jonathan P. Beauchamp, are present. However, GIV regression still outperforms OLS Benjamin Domingue, Lisbeth Trille Loft, Niels Rietveld, Eric Slob, Patrick and MR in this scenario. Furthermore, GIV regression can (and Turley, Hans van Kippersluis, and Tian Zheng for productive discussions and comments on earlier versions of the manuscript. Furthermore, we thank should) be combined with additional strategies that allow con- Elliot Tucker-Drob for directing us to the necessary correction of the heri- trolling for bias from purely environmental or behavioral factors, tability estimate in our model. This research was facilitated by the SSGAC such as using covariates or family-fixed effects. Together, these and by the research group on genetic and social causes of life chances at approaches can provide reasonable estimates of causal effects the Zentrum fur¨ interdisciplinare¨ Forschung Bielefeld. Data analyses made use of the UK Biobank resource under Application 11425. We acknowledge across a broad range of scenarios. data access from the GIANT Consortium. We used data from the HRS, which GIV regression is called for whenever an experimental design, is supported by the National Institute on Aging (NIA U01AG009740, RC2 a valid IV strategy, or a large-enough sample of MZ twins is AG036495, and RC4 AG039029). This study was supported by funding from not available and when pleiotropy is a potential problem—a a European Research Council Consolidator Grant (647648 EdGe) (to P.D.K.). HRS genotype data can be accessed via the database of Genotypes and Phe- situation that is frequently encountered in practice. The main notypes (dbGaP, accession no. phs000428.v1.p1). Researchers who wish to requirements for GIV regression are a prediction sample that link genetic data with other HRS measures that are not in dbGaP, such as has been comprehensively genotyped and large-scale GWAS educational attainment, must apply for access from HRS.

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