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A Note on the Parameterization of Purcell’s G × E Model for Ordinal and Binary Data

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Abstract

Following the publication of Purcell’s approach to the modeling of gene by environment interaction in 2002, the interest in G × E modeling in twin and family data increased dramatically. The analytic techniques described by Purcell were designed for use with continuous data. Here we explore the re-parameterization of these models for use with ordinal and binary outcome data. Analysis of binary and ordinal data within the context of a liability threshold model traditionally requires constraining the total variance to unity to ensure identification. Here, we demonstrate an alternative approach for use with ordinal data, in which the values of the first two thresholds are fixed, thus allowing the total variance to change as function of the moderator. We also demonstrate that when using binary data, constraining the total variance to unity for a given value of the moderator is sufficient to ensure identification. Simulation results indicate that analyses of ordinal and binary data can recover both the raw and standardized patterns of results. However, the scale of the results is dependent on the specification of (threshold or variance) constraints rather than the underlying distribution of liability. Example Mx scripts are provided.

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Notes

  1. Although it is not usually discussed explicitly, the genetic effect under discussion in G × E analyses is almost always additive in nature.

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Acknowledgments

The authors would like to thank Sophie van der Sluis, Dorret Boomsma, Lannie Ligthart and the reviewers for their helpful comments. SEM is supported by an Australian NHMRC Sidney Sax Fellowship (443036). MCN is supported in part by NIH grants DA-18673, MH-65322. BMN is supported in part by NIMH grants R01MH081803 and R01MH062873 to SV Faraone.

Author information

Authors and Affiliations

  1. Genetic Epidemiology Unit, Queensland Institute of Medical Research, Brisbane, QLD, Australia

    Sarah E. Medland

  2. Virginia Institute of Psychiatric and Behavioral Genetics, Virginia Commonwealth University, P.O. Box 980126 MCV, Richmond, VA, 23298-0126, USA

    Sarah E. Medland, Michael C. Neale & Lindon J. Eaves

  3. Department of Psychiatry, Virginia Commonwealth University, Richmond, VA, USA

    Michael C. Neale & Lindon J. Eaves

  4. Department of Human Genetics, Virginia Commonwealth University, Richmond, VA, USA

    Michael C. Neale & Lindon J. Eaves

  5. Department of Biological Psychology, Free University, Amsterdam, The Netherlands

    Michael C. Neale

  6. Social, Genetic, and Developmental Psychiatry Centre, Institute of Psychiatry, King’s College, London, UK

    Benjamin M. Neale

  7. Broad Institute of MIT and Harvard University, Cambridge, MA, USA

    Benjamin M. Neale

  8. Center for Human Genetic Research, Massachusetts General Hospital, Harvard Medical School, Boston, MA, USA

    Benjamin M. Neale

Authors
  1. Sarah E. Medland
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  2. Michael C. Neale
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  3. Lindon J. Eaves
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  4. Benjamin M. Neale
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Corresponding author

Correspondence to Sarah E. Medland.

Additional information

Edited by Dorret Boomsma.

Appendices

Appendix 1: Example Mx script for a generalized moderated regression model with the ordinal data using the interval liability model

  • G1: Parameters

  • #define ndef 1 ! n definition variables: sex moderator1

  • #define nmod 2 ! unmod, moderator1

  • #define nind 2 ! n individuals in largest sibship

  • #define nthr 2 ! number of thresholds 2 = 3 categories

  • Data Calc NGroups = 3

  • Begin Matrices;

  • A full 1 nmod free

  • C full 1 nmod free

  • E full 1 nmod free

  • M full 1 nind free ! mean

  • B full 1 ndef free ! effects of covariates on the mean

  • H unit nind nind ! mz constants

  • J stand nind nind ! dz constants

  • R ident nind nind

  • T Full nthr nind ! contains the new threshold matrix

  • U unit nind 1

  • X Unit nthr 1 ! contains a vector of 1s

  • End Matrices;

  • Value .5 J 2 1

  • eq m 1 1 1 m 1 1 2

  • ma t

  • 0 0

  • 1 1

  • !starting values for A C and E

  • !unmoderated and moderated parameter start values

  • MATRIX A -.5 .5

  • MATRIX C .2 .2

  • MATRIX E .3 .1

  • labels coloumn A unmod mod

  • labels coloumn C unmod mod

  • labels coloumn E unmod mod

  • labels coloumn b mod

  • Options RSiduals

  • End

  • G2: MZ

  • DATA NINPUT = 8

  • labels rep zyg mod1 mod2 c1 c2 tw1 tw2

  • Ord File = mz1

  • select mod1 mod2 tw1 tw2;

  • Definition_variables mod1 mod2;

  • Matrices = Group 1

  • V full nmod nind ! contains coefficients of the cov corrections

  • W full ndef nind ! contains covariates for means regression

  • End Matrices;

  • SP V 0 0 mod1 mod2;

  • VALUE 1 V 1 1 V 1 2

  • ! contains 1 s for unmoderated cov elements and covariates for moderated

  • ! cov elements

  • SP W mod1 mod2;

  • ! contains covariates for means regression

  • Begin Algebra;

  • S=

  • (H.((U@A)*V).(V′*(U@A)′)) + !variance/cov due to A

  • ((U@C)*V).(V′*(U@C)′) + !variance/cov due to C

  • (R.((U@E)*V).(V′*(U@E)′)); !variance/cov due to E

  • Z = (T − X@(M + (B*W)))%(X@\SQRT(\D2V(S)));

  • D = \V2D(U′%\SQRT(\D2V(S)));

  • End Algebra;

  • Threshold Z;

  • Covariance D*S*D;

  • End

  • G3: DZ

  • DATA NINPUT = 8

  • labels rep zyg mod1 mod2 c1 c2 tw1 tw2

  • ord File = dz1

  • select mod1 mod2 tw1 tw2;

  • Definition_variables mod1 mod2;

  • Matrices = Group 1

  • V full nmod nind

  • W full ndef nind

  • End Matrices;

  • SP V 0 0 mod1 mod2;

  • SP W mod1 mod2;

  • VALUE 1 V 1 1 V 1 2

  • Begin Algebra;

  • S=

  • (J.((U@A)*V).(V′*(U@A)′)) + !variance/cov due to A

  • ((U@C)*V).(V′*(U@C)′) + ! variance/cov due to C

  • (R.((U@E)*V).(V′*(U@E)′)); ! variance/cov due to E

  • Z = (T-X@(M + (B*W)))%(X@\SQRT(\D2V(S)));

  • D = \V2D(U′%\SQRT(\D2V(S)));

  • End Algebra;

  • Threshold Z;

  • Covariance D*S*D;

  • End

Appendix 2: Example Mx script for a generalized moderated regression model with the binary data using a variance constraint at a specified value of the moderator

  • !Constrained binary script

  • #define ndef 1 ! n definition variables: sex moderator1

  • #define nmod 2 ! unmod, moderator1

  • #define nind 2 ! n individuals in largest sibship

  • G!: Parameters

  • Data Calc NGroups = 4

  • Begin Matrices;

  • A full 1 nmod free

  • C full 1 nmod free

  • E full 1 nmod free

  • M full 1 nind free ! threshold

  • B full 1 ndef free ! mean effects

  • H unit nind nind ! mz constants

  • J stand nind nind ! dz constants

  • U unit nind 1

  • D ident nind nind

  • End Matrices;

  • Value .5 J 2 1

  • !starting values for threshold

  • MATRIX M .8 .8

  • !starting values for beta

  • MATRIX B 0.05

  • !starting values for A C and E

  • !unmoderated and moderated parameter start values

  • MATRIX A 1.5 − .5

  • MATRIX C .7 .4

  • MATRIX E 2 1

  • labels coloumn A unmod mod

  • labels coloumn C unmod mod

  • labels coloumn E unmod mod

  • labels coloumn b mod

  • Options RSiduals

  • End

  • G2: MZ

  • DATA NINPUT = 10

  • labels loop zyg mod1 mod2 mod1b mod2b tw1 tw2 bintw1 bintw2

  • Ord File = mz1

  • select mod1 mod2 bintw1 bintw2;

  • Definition_variables mod1 mod2;

  • Matrices = Group 1

  • V full nmod nind ! contains coefficients of the cov corrections

  • W full ndef nind ! contains covariates for threshold regression

  • End Matrices;

  • SP V 0 0 mod1 mod2;

  • VALUE 1 V 1 1 V 1 2

  • ! contains 1 s for unmoderated cov elements and covariates for moderated

  • ! cov elements

  • SP W mod1 mod2;

  • ! contains covariates for thresholds regression

  • Thresholds M + B*W;

  • Covariance

  • (H.((U@A)*V).(V′*(U@A)′))+

  • ((U@C)*V).(V′*(U@C)′)+

  • (D.((U@E)*V).(V′*(U@E)′));

  • End

  • G3: DZ

  • DATA NINPUT = 10

  • labels loop zyg mod1 mod2 mod1b mod2b tw1 tw2 bintw1 bintw2

  • Ord File = dz1

  • select mod1 mod2 bintw1 bintw2;

  • Definition_variables mod1 mod2;

  • Matrices = Group 1

  • V full nmod nind

  • W full ndef nind

  • End Matrices;

  • SP V 0 0 mod1 mod2;

  • SP W mod1 mod2;

  • VALUE 1 V 1 1 V 1 2

  • Thresholds M + B*W;

  • Covariance

  • (J.((U@A)*V).(V′*(U@A)′))+

  • ((U@C)*V).(V′*(U@C)′)+

  • (D.((U@E)*V).(V′*(U@E)′));

  • End

  • Constraint group to force variance = 1 when moderator is zero

  • constraint

  • begin matrices;

  • A full 1 1 free

  • C full 1 1 free

  • E full 1 1 free

  • F full 1 1 free

  • G full 1 1 free

  • H full 1 1 free

  • M full 1 1 fixed !moderator

  • U unit 1 1

  • End matrices;

  • EQ A 1 1 1 A 4 1 1

  • EQ C 1 1 1 C 4 1 1

  • EQ E 1 1 1 E 4 1 1

  • EQ A 1 1 2 F 4 1 1

  • EQ C 1 1 2 G 4 1 1

  • EQ E 1 1 2 H 4 1 1

  • MATRIX m 0 !value of the moderator for constraint

  • begin Algebra;

  • T=

  • ((A + (F*M))*(A + (F*M)))+

  • ((C + (G*M))*(C + (G*M)))+

  • ((E + (H*M))*(E + (H*M)));

  • end algebra;

  • Contrain U = T;

  • option jiggle append

  • End

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Medland, S.E., Neale, M.C., Eaves, L.J. et al. A Note on the Parameterization of Purcell’s G × E Model for Ordinal and Binary Data. Behav Genet 39, 220–229 (2009). https://doi.org/10.1007/s10519-008-9247-7

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  • DOI: https://doi.org/10.1007/s10519-008-9247-7

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