BINARY and CATEGORICAL TRAITS

BINARY and CATEGORICAL TRAITS
GENETIC ANALYSIS OF BINARY and CATEGORICAL TRAITS PART ONE

TABLE 1. Twin Pair Concordances for Major Depression
(Virginia Twin Study data, adapted from Neale and Cardon, 1992) MZ FEMALE PAIRS DZ FEMALE PAIRS Twin B Unaffected Affected Twin A - Unaffected 329 83 201 94 - Affected 95 82 63

2 x concordant affected pairs + discordant pairs
2 x Total Pairs Prevalence = e.g. for MZ pairs = e.g. for DZ pairs = 1180 = % 880 = % Prevalance = proportion of affected (alcoholic) twins in the general population.

2 x concordant affected pairs
2 x concordant affected pairs + discordant pairs Probandwise concordance rate = e.g. for MZ pairs = e.g. for DZ pairs = 166 = % 126 = % Probandwise concordance rate = probability that cotwin of a depressed twin will also have a history of depression. Recurrence Risk-ratio Probandwise concordance rate Prevalence =

Why do we have (2 x number of concordant affected pairs) in the numerator and denominator of the expression for the probandwise concordance rate? Consider a simple example where there are 4 affected individuals, who came from 3 twin pairs, ie, 1 — 0 1 — 0 1 — 1 There are 4 potential probands, so if we randomly select an affected individual, the probability that the cotwin of that individual is also affected will be 50%

TABLE 1a. Twin Pair Concordances for Alcohol Dependence (DSM-IIIR)
(Virginia Twin Study data, from Kendler et al., 1992) MZ Female Pairs DZ N pairs 590 440 Population prevalence 8.1% 10.2% Probandwise concordance 31.6% 24.4%

Number of concordant alcoholic pairs
= N pairs x prevalence x probandwise concordance MZ: 15 pairs DZ: 11 pairs Number of discordant pairs = 2 x N pairs x prevalence x (1 - probandwise concordance) MZ: 65 pairs DZ: 68 pairs Number of concordant unaffected pairs MZ: 510 pairs DZ: 361 pairs

The “PAIRWISE” concordance rate is redundant --
Some investigators also report a “PAIRWISE” CONCORDANCE RATE - the proportion of pairs with at least one twin affected who are concordant. The “PAIRWISE” concordance rate is redundant -- CR 2-CR PAIRWISE CONCORDANCE RATE = where CR is the probandwise concordance rate

a) Normal Liability Threshold Model b) Multiple-threshold Model
1 t 1 t 2 UNAFFECTED AFFECTED UNAFFECTED MILD SEVERE CASES CASES Alcoholism Risk Alcoholism Risk

CUMULATIVE NORMAL FREQUENCY DISTRIBUTION Threshold value (t)
Prevalence (area under the standard normal curve) 0.0 50% 0.25 40% 0.53 30% 0.84 20% 1.04 15% 1.28 10% 1.64 5% 1.95 2.5% 2.33 1% 3.08 0.1% -0.25 60%

Relatives’ Recurrence Risk Ratio
Table 3. Population distribution of pairs of relatives with both alcoholic, neither alcoholic, or only one relative alcoholic, as a function of (i) lifetime prevalence of alcoholism, and (ii) liability correlation for alcoholism in relatives PREVALENCE Relative A Relative B Liability correlation Both affected Discordant A affected B affected unaffected Risk to relative of an alcoholica Relatives’ Recurrence Risk Ratio (%) 30% 0.6 17.3 12.7 57.3 57.6 1.9 0.3 12.8 17.2 52.8 42.7 1.4 0.15 10.9 19.1 50.9 36.2 1.2 20% 9.9 10.1 69.9 49.6 2.5 6.6 13.4 66.6 33.1 1.7 5.2 14.8 65.2 26.2 1.3 10% 3.9 6.1 83.9 39.0 2.2 7.8 82.2 21.6 1.5 8.5 81.5 15.2 ai.e. Probandwise concordance rate

EXAMPLE DATA-FILE FOR MX RAW ORDINAL DATA:
MZF DEPRESSION DATA (depmzf.dat) 0 1 83 1 0 95 1 1 83

0 0 310 0 1 32.5 1 0 32.5 1 1 15 EXAMPLE DATA-FILE (II):
DERIVED FROM PUBLISHED SOURCES MZF ALCOHOL DEPENDENCE DATA (alcmzf.dat) 1 1 15

! tetrachoric.mx ! estimating tetrachoric correlations #define nvar 1 #define maxthresf 1 ! number of thresholds Analysis of depression data: estimating tetrachorics & confidence intervals data NI=3 NG=4 LAbels twina twinb countmz Ordinal fi=depmzf.rec ! Count is a definition variable that we use to tell MX the frequency count ! for each element of the 2x2 table ! Definition_variables countmz / Begin matrices; W LO nvar nvar fr ! w*w' is the tetrachoric correlation Y LO nvar nvar fr ! y*y' is 1-tetrachoric correlation M FU maxthresf nvar fi S DI nvar nvar ! Matrix that will store weight variable end matrices; SP M 3 MATRIX M ! This tells MX to store the definition variable count in S SP S -1 mat w 0.7 mat y 0.7

Begin algebra; R=W. W'; E=Y. Y'; V=R+E; end algebra; FREQ S;
Begin algebra; R=W*W'; E=Y*Y'; V=R+E; end algebra; FREQ S; ! tells MX that S contains the weight (frequency) variable TH M|M; ! tells MX that row and column thresholds contained in M|M CO V|R_ R'|V; ! formula for correlation matrix! bo y(1,1) bo w(1,1) bo m(1,1) interval r(1,1) ! compute 95% confidence interval for correlation OPT func=1.E-12 OPT RS END

Analysis of depression data: DZm data NI=3 LAbels twina twinb countdz OR fi=depdzf.rec Definition_variables countdz / Begin matrices; W LO nvar nvar fr ! w*w' is the tetrachoric correlation for DZ group Y LO nvar nvar fr ! y*y' is 1-tetrachoric correlation for DZ group N FU maxthresf nvar fr S DI nvar nvar ! Matrix that will store weight variable end matrices; SP N 6 MATRIX N SP S -1 mat w 0.6 mat y 0.8 Begin algebra; R=W*W'; E=Y*Y'; V=R+E; end algebra; FREQ S; TH N|N; CO V|R_R'|V; bo y(1,1) bo w(1,1) bo n(1,1) interval r(1,1) ! compute 95% confidence interval for correlation OPT RS END

Constraint function - constrain variances to unity for MZ group CO NI=1 Begin matrices = group 1; U unit 1 nvar end matrices; CO \d2v(V) = u; end Constraint function - constrain variances to unity for DZ group CO NI=1 Begin matrices = group 2; U unit 1 nvar end matrices; CO \d2v(V) = u; end

Summary of VL file data for group 1
COUNTMZ TWINA TWINB Code E E E+00 Number E E E+00 Mean E E E-01 Variance E E E-01 Minimum E E E+00 Maximum E E E+00 Summary of VL file data for group 2 COUNTDZ TWINA TWINB Code Number Mean Variance Minimum Maximum

PARAMETER SPECIFICATIONS
GROUP NUMBER: 1 Analysis of depression data: estimating tetrachorics & confidence intervals MATRIX E This is a computed FULL matrix of order 1 by 1 It has no free parameters specified MATRIX M This is a FULL matrix of order 1 by 1 1 1 3 MATRIX R MATRIX S This is a DIAGONAL matrix of order 1 by 1 1 -1 MATRIX V MATRIX W This is a LOWER TRIANGULAR matrix of order 1 by 1 1 1 MATRIX Y 1 2

GROUP NUMBER: 2 Analysis of ordinal alcohol tolerance and dependence data: DZm MATRIX E This is a computed FULL matrix of order 1 by 1 It has no free parameters specified MATRIX N This is a FULL matrix of order 1 by 1 1 1 6 MATRIX R MATRIX S This is a DIAGONAL matrix of order 1 by 1 1 -1 MATRIX V MATRIX W This is a LOWER TRIANGULAR matrix of order 1 by 1 1 4 MATRIX Y 1 5

MX PARAMETER ESTIMATES
GROUP NUMBER: 1 Analysis of depression data: estimating tetrachorics & confidence intervals MATRIX E This is a computed FULL matrix of order 1 by 1 [=Y*Y'] 1 MATRIX M This is a FULL matrix of order 1 by 1 MATRIX R [=W*W'] MATRIX S This is a DIAGONAL matrix of order 1 by 1 MATRIX V [=R+E]

MATRIX W This is a LOWER TRIANGULAR matrix of order 1 by 1 1 MATRIX Y Matrix of EXPECTED thresholds TWINA TWINB Threshold Threshold (OBSERVED MATRIX is nonexistent for raw data) EXPECTED COVARIANCE MATRIX TWINA TWINB Function value of this group: Where the fit function is -2 * Log-likelihood of raw ordinal

GROUP NUMBER: 2 Analysis of ordinal alcohol tolerance and dependence data: DZm MATRIX E This is a computed FULL matrix of order 1 by 1 [=Y*Y'] 1 MATRIX N This is a FULL matrix of order 1 by 1 MATRIX R [=W*W'] MATRIX S This is a DIAGONAL matrix of order 1 by 1 MATRIX V [=R+E]

Your model has 6 estimated parameters and 18 Observed statistics
Observed statistics include 2 constraints. -2 times log-likelihood of data >>> Degrees of freedom >>>>>>>>>>>>>>>> 1 Confidence intervals requested in group 1 Matrix Element Int Estimate Lower Upper Lfail Ufail R 1 Confidence intervals requested in group 2 R This problem used 0.2% of my workspace Task Time elapsed (DD:HH:MM:SS) Reading script & data : 0: 0: 0.11 Execution : 0: 0: 2.85 TOTAL : 0: 0: 2.96 Total number of warnings issued: 0 ______________________________________________________________________________

Mx startup successful. MX-Sunos version 1. 49. tetrachoric. mx
** Mx startup successful ** **MX-Sunos version 1.49** ! tetrachoric.mx ! estimating tetrachoric correlations The following MX script lines were read for group #DEFINE NVAR 1 #DEFINE MAXTHRESF 1 ! NUMBER OF THRESHOLDS ANALYSIS OF ALCOHOLISM DATA: ESTIMATING TETRACHORICS & CONFIDENCE INTERVALS DATA NI=3 NO=2 NG=4 LABELS TWINA TWINB COUNTMZ ORDINAL FI=ALCMZF.REC Ordinal data read initiated NOTE: Rectangular file contained records with data ! Count is a definition variable that we use to tell MX the frequency count ! for each element of the 2x2 table ! DEFINITION_VARIABLES COUNTMZ / NOTE: Definition yields 4 data vectors for analysis NOTE: Vectors contain a total of observations

Summary of VL file data for group 1 COUNTMZ TWINA TWINB Code -1
Summary of VL file data for group COUNTMZ TWINA TWINB Code E E E Number E E E Mean E E E-01 Variance E E E-01 Minimum E E E+00 Maximum E E E Summary of VL file data for group COUNTDZ TWINA TWINB Code E E E Number E E E Mean E E E-01 Variance E E E-01 Minimum E E E+00 Maximum E E E+00

MX PARAMETER ESTIMATES GROUP NUMBER: Analysis of alcoholism data: estimating tetrachorics & confidence intervals MATRIX E This is a computed FULL matrix of order 1 by [=Y*Y'] MATRIX M This is a FULL matrix of order 1 by MATRIX R This is a computed FULL matrix of order 1 by [=W*W'] MATRIX S This is a DIAGONAL matrix of order 1 by

MATRIX V This is a computed FULL matrix of order 1 by 1 [=R+E] 1 1 1
MATRIX V This is a computed FULL matrix of order 1 by [=R+E] MATRIX W This is a LOWER TRIANGULAR matrix of order 1 by MATRIX Y This is a LOWER TRIANGULAR matrix of order 1 by Matrix of EXPECTED thresholds TWINA TWINB Threshold Threshold (OBSERVED MATRIX is nonexistent for raw data) EXPECTED COVARIANCE MATRIX TWINA TWINB TWINA TWINB Function value of this group: Where the fit function is -2 * Log-likelihood of raw ordinal

GROUP NUMBER: 2 Analysis of ordinal alcohol tolerance and dependence data: DZm MATRIX E This is a computed FULL matrix of order 1 by 1 [=Y*Y'] 1 MATRIX N This is a FULL matrix of order 1 by 1 MATRIX R [=W*W'] MATRIX S This is a DIAGONAL matrix of order 1 by 1 MATRIX V [=R+E]

MATRIX Y This is a LOWER TRIANGULAR matrix of order 1 by 1 1 Your model has 6 estimated parameters and Observed statistics Observed statistics include 2 constraints. -2 times log-likelihood of data >>> Degrees of freedom >>>>>>>>>>>>>>>> 1 Confidence intervals requested in group 1 Matrix Element Int Estimate Lower Upper Lfail Ufail R 1 Confidence intervals requested in group 2 R This problem used 0.2% of my workspace Task Time elapsed (DD:HH:MM:SS) Reading script & data : 0: 0: 0.11 Execution : 0: 0: 3.41 TOTAL : 0: 0: 3.52 Total number of warnings issued: 0

ESTIMATED TETRACHORIC CORRELATIONS
(estimating separate thresholds for each zygosity group) DEPRESSION ALCOHOL DEPENDENCE ρ 95% CI MZF 0.43 0.53 DZF 0.18 0.35 -2 log-likelihood

TEST FOR ZYGOSITY DIFFERENCE IN PREVALENCE
(takes into account non-independence!) DEPRESSION ALCOHOL DEPENDENCE -2 ln L (i) Separate thresholds model (ii) Equal thresholds Heterogeneity (i - ii) χ2 = 5.265, p=0.02 χ2 = 2.408, p=0.12 1 1

This approach extends naturally to fitting univariate genetic models.

! univariate.mx ! fitting a univariate genetic model to 2x2 data #define nvar 1 #define maxthresf 1 ! number of thresholds Analysis of depression data: fitting ACE model data NI=3 NG=3 LAbels twina twinb countmz Ordinal fi=depmzf.rec ! Count is a definition variable that we use to tell MX the frequency count ! for each element of the 2x2 table ! Definition_variables countmz / Begin matrices; W LO nvar nvar fr ! additive genetic path (A=w*w') X LO nvar nvar fr ! shared environmental path (C=x*x') Y LO nvar nvar fr ! non-shared environmental path (E=y*y') Z LO nvar nvar fi ! non-additive genetic path (D=z*z') M FU maxthresf nvar fi ! matrix of thresholds S DI nvar nvar ! Matrix that will store weight variable end matrices; SP M 4 MATRIX M ! This tells MX to store the definition variable count in S SP S -1 mat w 0.5 mat x 0.5 mat y 0.7

Begin algebra; A=W. W'; C=X. X'; E=Y. Y'; D=Z
Begin algebra; A=W*W'; C=X*X'; E=Y*Y'; D=Z*Z'; V=A+C+D+E; end algebra; FREQ S; ! tells MX that S contains the weight (frequency) variable TH M|M; ! tells MX that row and column thresholds contained in M|M CO V|A+D+C_ A'+D'+C'|V; ! formula for correlation matrix! bo y(1,1) bo w(1,1) x(1,1) bo m(1,1) interval a(1,1) c(1,1) e(1,1) ! compute 95% confidence interval for correlation OPT func=1.E-12 OPT RS END

Analysis of depression data: DZm data NI=3 NO=4 LAbels twina twinb countdz OR fi=depdzf.rec Definition_variables countdz / Begin matrices = group 1; S DI nvar nvar ! Matrix that will store weight variable g DI ! constant (=0.5) for coefficient of additive genetic component h DI ! constant (=0.25) for coefficient of dominance genetic component n FU maxthresf nvar fi ! matrix of thresholds end matrices; SP N 5 MATRIX N MAT g 0.5 MAT h 0.25 SP S -1 FREQ S; TH N|N; CO ! formula for correlation matrix! bo n(1,1) OPT RS END

Constraint function - constrain variance to unity
CO NI=1 Begin matrices = group 1; U unit 1 nvar end matrices; CO \d2v(V) = u; end

Mx startup successful. MX-Sunos version 1. 49. univariate. mx
** Mx startup successful ** **MX-Sunos version 1.49** ! univariate.mx ! fitting a univariate genetic model to 2x2 data The following MX script lines were read for group #DEFINE NVAR 1 #DEFINE MAXTHRESF 1 ! NUMBER OF THRESHOLDS ANALYSIS OF DEPRESSION DATA: FITTING ACE MODEL DATA NI=3 NO=2 NG=3 LABELS TWINA TWINB COUNTMZ ORDINAL FI=DEPMZF.REC Ordinal data read initiated NOTE: Rectangular file contained records with data ! Count is a definition variable that we use to tell MX the frequency count ! for each element of the 2x2 table ! DEFINITION_VARIABLES COUNTMZ / NOTE: Definition yields 4 data vectors for analysis NOTE: Vectors contain a total of observations BEGIN MATRICES; W LO NVAR NVAR FR ! ADDITIVE GENETIC PATH (A=W*W') X LO NVAR NVAR FR ! SHARED ENVIRONMENTAL PATH (C=X*X') Y LO NVAR NVAR FR ! NON-SHARED ENVIRONMENTAL PATH (E=Y*Y') Z LO NVAR NVAR FI ! NON-ADDITIVE GENETIC PATH (D=Z*Z') M FU MAXTHRESF NVAR FI ! MATRIX OF THRESHOLDS S DI NVAR NVAR ! MATRIX THAT WILL STORE WEIGHT VARIABLE END MATRICES;

MX PARAMETER ESTIMATES GROUP NUMBER: Analysis of depression data: fitting ACE model MATRIX A This is a computed FULL matrix of order 1 by [=W*W'] MATRIX C This is a computed FULL matrix of order 1 by [=X*X'] E MATRIX D This is a computed FULL matrix of order 1 by [=Z*Z'] MATRIX E This is a computed FULL matrix of order 1 by [=Y*Y'] MATRIX M This is a FULL matrix of order 1 by

MATRIX S This is a DIAGONAL matrix of order 1 by 1 1 1 83.0000
MATRIX V This is a computed FULL matrix of order 1 by [=A+C+D+E] MATRIX W This is a LOWER TRIANGULAR matrix of order 1 by MATRIX X This is a LOWER TRIANGULAR matrix of order 1 by E MATRIX Y This is a LOWER TRIANGULAR matrix of order 1 by MATRIX Z This is a LOWER TRIANGULAR matrix of order 1 by

Matrix of EXPECTED thresholds TWINA TWINB Threshold 1 0. 5493 0
Matrix of EXPECTED thresholds TWINA TWINB Threshold Threshold (OBSERVED MATRIX is nonexistent for raw data) EXPECTED COVARIANCE MATRIX TWINA TWINB TWINA TWINB Function value of this group: Where the fit function is -2 * Log-likelihood of raw ordinal

Observed statistics include 1 constraints. -2 times log-likelihood of data >>> Degrees of freedom >>>>>>>>>>>>>>>> 3 Confidence intervals requested in group 1 Matrix Element Int Estimate Lower Upper Lfail Ufail A C E This problem used 0.1% of my workspace Task Time elapsed (DD:HH:MM:SS) Reading script & data : 0: 0: 0.10 Execution : 0: 0:15.76 TOTAL : 0: 0:15.86 Total number of warnings issued: 1 ______________________________________________________________________________

Mx startup successful. MX-Sunos version 1. 49. univar2. mx
** Mx startup successful ** **MX-Sunos version 1.49** ! univar2.mx ! fitting a univariate genetic model to 2x2 data The following MX script lines were read for group #DEFINE NVAR 1 #DEFINE MAXTHRESF 1 ! NUMBER OF THRESHOLDS ANALYSIS OF ALCOHOL DEPENDENCE DATA: FITTING ACE MODEL DATA NI=3 NO=2 NG=3 LABELS TWINA TWINB COUNTMZ ORDINAL FI=ALCMZF.REC Ordinal data read initiated NOTE: Rectangular file contained records with data ! Count is a definition variable that we use to tell MX the frequency count ! for each element of the 2x2 table ! DEFINITION_VARIABLES COUNTMZ / NOTE: Definition yields 4 data vectors for analysis NOTE: Vectors contain a total of observations

MX PARAMETER ESTIMATES GROUP NUMBER: Analysis of alcohol dependence data: fitting ACE model MATRIX A This is a computed FULL matrix of order 1 by [=W*W'] MATRIX C This is a computed FULL matrix of order 1 by [=X*X'] MATRIX D This is a computed FULL matrix of order 1 by [=Z*Z'] MATRIX E This is a computed FULL matrix of order 1 by [=Y*Y']

MATRIX M This is a FULL matrix of order 1 by 1 1 1 1
MATRIX M This is a FULL matrix of order 1 by MATRIX S This is a DIAGONAL matrix of order 1 by MATRIX V This is a computed FULL matrix of order 1 by [=A+C+D+E] MATRIX W This is a LOWER TRIANGULAR matrix of order 1 by MATRIX X This is a LOWER TRIANGULAR matrix of order 1 by MATRIX Y This is a LOWER TRIANGULAR matrix of order 1 by MATRIX Z This is a LOWER TRIANGULAR matrix of order 1 by

Matrix of EXPECTED thresholds TWINA TWINB Threshold 1 1. 4017 1
Matrix of EXPECTED thresholds TWINA TWINB Threshold Threshold (OBSERVED MATRIX is nonexistent for raw data) EXPECTED COVARIANCE MATRIX TWINA TWINB TWINA TWINB Function value of this group: Where the fit function is -2 * Log-likelihood of raw ordinal

Observed statistics include 1 constraints. -2 times log-likelihood of data >>> Degrees of freedom >>>>>>>>>>>>>>>> 3 Confidence intervals requested in group 1 Matrix Element Int Estimate Lower Upper Lfail Ufail A C E This problem used 0.1% of my workspace Task Time elapsed (DD:HH:MM:SS) Reading script & data : 0: 0: 0.10 Execution : 0: 0: 7.55 TOTAL : 0: 0: 7.65 Total number of warnings issued: 1 ______________________________________________________________________________

VIRGINIA TWIN STUDY: Female Like-Sex Pairs
Summary Model-Fitting Results Additive Genetic Variance 95% CI Shared Environmental Variance Non-Shared Environmental Variance Major depression 42.5 0.0 57.5 Alcohol dependence 35.9 17.2 46.9

Model-fitting results: Depression in the Virginia Twin Study
Parameter Estimates (%) Likelihood-ratio versus full model Model A C E D d.f. χ2 p A D E 30 -- 57 13 A C E 43 A E 1 0.15 0.70 C E 33 67 6.44 0.01 100

We can easily handle data where only one twin has responded.
HOWEVER, we are assuming that missing data are MCAR - Missing Completely at Random. We can include twins with missing cotwins (indicated by .) in the same data-file as complete pairs. Alternatively, if we want to test for differences in prevalence for complete pairs versus singles (suggestive of an ascertainment bias), we can include singleton twins as separate groups, allowing a test of equality of thresholds.

EXAMPLE: Alcohol Dependence Data from
1992 Survey of the Australian Twin Panel (1981 cohort) MZ Male DZ Male 0 0 274 138 1 0 37.5 38.5 0 1 1 1 49 19 0 . 34 31.5 1 . 8 . 0 . 1

Table 2. Numbers of twin pairs concordant and discordant for smoking
status in the Australian twin panel 1981 survey. MZ Female (N=1232 pairs) DZ Female (N=747 pairs) I II III I Non-smoker 629 310 II Successful quitter 110 64 98 33 III Current smoker 124 115 190 146 61 99 MZ Male (N=567 pairs) DZ Male (N=350 pairs) 221 121 77 70 44 27 31 53

MULTIPLE THRESHOLD MODEL
For n categories, we need to estimate (n-1) thresholds. The safest way to estimate multiple thresholds is to estimate: t0 t1 = t0 + t1 (t1 > 0) t2 = t1 + t2 (t2 > 0) and so on. This is especially important when we estimate confidence intervals. Note that if L = and M = then LM = etc. Hence, we merely need to constrain M2 etc. > 0.

! univariate3x3.mx ! fitting a univariate genetic model to 3x3 data #define nvar 1 #define maxthresf 2 ! number of thresholds Analysis of smoking data: fitting ACE model data NI=3 NG=3 LAbels twina twinb countmz Ordinal fi=smkmzf.rec ! Count is a definition variable that we use to tell MX the frequency count ! for each element of the 3x3 table ! Definition_variables countmz / Begin matrices; W LO nvar nvar fr ! additive genetic path (A=w*w') X LO nvar nvar fr ! shared environmental path (C=x*x') Y LO nvar nvar fr ! non-shared environmental path (E=y*y') Z LO nvar nvar fi ! non-additive genetic path (D=z*z') M FU maxthresf nvar fi ! matrix of thresholds L LO maxthresf maxthresf ! used to ensure t1 < t2 S DI nvar nvar ! Matrix that will store weight variable end matrices; SP M 4 5 MATRIX M

MATRIX L ! This tells MX to store the definition variable count in S SP S -1 mat w 0.5 mat x 0.5 mat y 0.7 Begin algebra; A=W*W'; C=X*X'; E=Y*Y'; D=Z*Z'; V=A+C+D+E; T=L*M; end algebra; FREQ S; ! tells MX that S contains the weight (frequency) variable TH T|T; ! tells MX that row and column thresholds contained in T|T CO V|A+D+C_ A'+D'+C'|V; ! formula for correlation matrix! bo y(1,1) m(2,1) bo w(1,1) x(1,1) bo m(1,1) interval a(1,1) c(1,1) e(1,1) ! compute 95% confidence interval for correlation OPT func=1.E-12 OPT RS END

Analysis of ordinal smoking data: DZm data NI=3 X
LAbels twina twinb countdz OR fi=smkdzf.rec Definition_variables countdz / Begin matrices = group 1; S DI nvar nvar ! Matrix that will store weight variable g DI ! constant (=0.5) for coefficient of additive genetic component h DI ! constant (=0.25) for coefficient of dominance genetic component n FU maxthresf nvar fi ! matrix of thresholds end matrices; SP N 6 7 MATRIX N MAT g 0.5 MAT h 0.25 SP S -1 Begin algebra; T=L*N; end algebra; FREQ S; TH T|T; CO ! formula for correlation matrix! bo n(1,1) bo n(2,1) OPT RS END

Constraint function - constrain variances to unity CO NI=1 Begin matrices = group 1; U unit 1 nvar end matrices; CO \d2v(V) = u; end

Mx startup successful. MX-Sunos version 1. 49. univariate3x3. mx
** Mx startup successful ** **MX-Sunos version 1.49** ! univariate3x3.mx ! fitting a univariate genetic model to 3x3 data The following MX script lines were read for group #DEFINE NVAR 1 #DEFINE MAXTHRESF 2 ! NUMBER OF THRESHOLDS ANALYSIS OF SMOKING DATA: FITTING ACE MODEL DATA NI=3 NO=9 NG=3 LABELS TWINA TWINB COUNTMZ ORDINAL FI=SMKMZF.REC Ordinal data read initiated NOTE: Rectangular file contained records with data and records where all data were missing ! Count is a definition variable that we use to tell MX the frequency count ! for each element of the 3x3 table ! DEFINITION_VARIABLES COUNTMZ / NOTE: Definition yields 9 data vectors for analysis NOTE: Vectors contain a total of observations BEGIN MATRICES; W LO NVAR NVAR FR ! ADDITIVE GENETIC PATH (A=W*W') X LO NVAR NVAR FR ! SHARED ENVIRONMENTAL PATH (C=X*X') Y LO NVAR NVAR FR ! NON-SHARED ENVIRONMENTAL PATH (E=Y*Y') Z LO NVAR NVAR FI ! NON-ADDITIVE GENETIC PATH (D=Z*Z') M FU MAXTHRESF NVAR FI ! MATRIX OF THRESHOLDS L LO MAXTHRESF MAXTHRESF ! USED TO ENSURE T1 < T2 S DI NVAR NVAR ! MATRIX THAT WILL STORE WEIGHT VARIABLE END MATRICES;

Summary of VL file data for group 1 COUNTMZ TWINA TWINB Code -1
Summary of VL file data for group COUNTMZ TWINA TWINB Code E E E Number E E E Mean E E E+00 Variance E E E-01 Minimum E E E+00 Maximum E E E Summary of VL file data for group COUNTDZ TWINA TWINB Code Number Mean Variance Minimum Maximum

GROUP NUMBER: 1 Analysis of smoking data: fitting ACE model MATRIX A This is a computed FULL matrix of order 1 by 1 [=W*W'] 1 MATRIX C [=X*X'] MATRIX D [=Z*Z'] MATRIX E [=Y*Y'] MATRIX L This is a LOWER TRIANGULAR matrix of order 2 by 2

MATRIX M This is a FULL matrix of order 2 by 1 1 MATRIX S This is a DIAGONAL matrix of order 1 by 1 MATRIX T This is a computed FULL matrix of order 2 by 1 [=L*M] MATRIX V This is a computed FULL matrix of order 1 by 1 [=A+C+D+E] MATRIX W This is a LOWER TRIANGULAR matrix of order 1 by 1 MATRIX X

MATRIX Y This is a LOWER TRIANGULAR matrix of order 1 by 1 1 MATRIX Z Matrix of EXPECTED thresholds TWINA TWINB Threshold Threshold (OBSERVED MATRIX is nonexistent for raw data) EXPECTED COVARIANCE MATRIX TWINA TWINB TWINA TWINB Function value of this group: Where the fit function is -2 * Log-likelihood of raw ordinal

MATRIX W This is a LOWER TRIANGULAR matrix of order 1 by 1 1 MATRIX X MATRIX Y MATRIX Z Matrix of EXPECTED thresholds TWINA TWINB Threshold Threshold (OBSERVED MATRIX is nonexistent for raw data) EXPECTED COVARIANCE MATRIX TWINA TWINB TWINA TWINB Function value of this group: Where the fit function is -2 * Log-likelihood of raw ordinal

Observed statistics include 1 constraints. -2 times log-likelihood of data >>> Degrees of freedom >>>>>>>>>>>>>>>> 3 Confidence intervals requested in group 1 Matrix Element Int Estimate Lower Upper Lfail Ufail A C E This problem used 0.1% of my workspace Task Time elapsed (DD:HH:MM:SS) Reading script & data : 0: 0: 0.24 Execution : 0: 0:29.75 TOTAL : 0: 0:30.00 Total number of warnings issued: 2 ______________________________________________________________________________

SMOKING IN WOMEN % 95% CI Additive genetic variance 58.4 39.9-77.8
Shared environmental variance 18.2 Non-shared environmental variance 23.4 -2 log-likelihood

BIVARIATE GENETIC APPLICATIONS
It is a simple step to modify the univariate script to allow for bivariate (or even trivariate) genetic analyses. If the traits being analyzed have varying numbers of thresholds, maxthres will be the maximum number of thresholds, and we will have, say, MAT M 0 1.0 In the next example, we analyze Australian twin data on lifetime history of major depression and current smoking status. Here, the original raw data are given in depsmkmf.rec and depsmkdf.rec. Notice that the data have been sorted -- this will improve the efficiency of the MX run.

! ordinal_bivariate.mx #define nvar 2 #define nvar2 4 #define maxthres 2 Analysis of ordinal depression (0/1) and smoking ! initiation/persistence (0/1/2) data NI=nvar2 NG=3 Ordinal fi=depsmkmf.rec Begin matrices; M FU maxthres nvar fr L LO maxthres maxthres W LO nvar nvar fr X LO nvar nvar fr Y LO nvar nvar fr end matrices; MAT L MATRIX M SP M st 0.7 y(1,1) y(2,2) w(1,1) w(2,2) st 0.2 x(1,1) x(2,2) st 0.2 w(2,1) x(2,1) y(2,1)

Begin algebra; A=W. W'; O=\stnd(A); C=X. X'; r=\stnd(C); E=Y
Begin algebra; A=W*W'; O=\stnd(A); C=X*X'; r=\stnd(C); E=Y*Y'; q=\stnd(E); P=A+C+E; end algebra; TH L*M|L*M; CO P | A + C _ A' + C' | P ; bo y(1,1) y(2,2) bo x(1,1) x(2,2) w(1,1) w(2,2) bo x(2,1) y(2,1) w(2,1) bo m(2,2) bo m(1,1) ! interval a(1,1) a(2,2) c(1,1) c(2,2) e(1,1) e(2,2) o(1,2) r(1,2) q(1,2) OPT func=1.E-12 OPT RS END

Analysis of ordinal depression and smoking data: DZF data NI=nvar2 Ordinal fi=depsmkdf.rec Begin matrices = group 1; N FU maxthres nvar fr g fu 1 1 end matrices; MATRIX N SP N mat g 0.5 TH L*N | L*N ; CO P | + C _ + C' | P ; bo n(2,2) bo n(1,1) OPT RS END

Data constraint CO NI=1 Begin matrices = group 1; U unit 1 nvar end matrices; CO \d2v(P) = u; end

Summary of VL file data for group 1 Code 1. 0000 2. 0000 3. 0000 4
Summary of VL file data for group Code Number Mean Variance Minimum Maximum Summary of VL file data for group Code Number Mean Variance Minimum Maximum

I am not sure I have found a solution that satisfies
*** WARNING! *** I am not sure I have found a solution that satisfies Kuhn-Tucker conditions for a minimum. NAG's IFAIL parameter is 6 Looks like I got stuck here. Check the following: 1. The model is correctly specified 2. Starting values are good 3. You are not already at the solution The error can arise if the Hessian is ill-conditioned You can try resetting it to an identity matrix and fit from the solution by putting TH=-n on the OU line where n is the number of refits that you want to do If all else fails try putting NAG=30 on the OU line and examine the file NAGDUMP.OUT and the NAG manual

GROUP NUMBER: 1 Analysis of ordinal depression (0/1) and smoking initiation/persistence (0/1/2) MATRIX A This is a computed FULL matrix of order 2 by 2 [=W*W'] MATRIX C [=X*X'] E E-05 E E-01 MATRIX E [=Y*Y'] MATRIX L This is a LOWER TRIANGULAR matrix of order 2 by 2

MATRIX M This is a FULL matrix of order 2 by 2 MATRIX O This is a computed FULL matrix of order 2 by 2 [=\STND(A)] MATRIX P [=A+C+E] MATRIX Q [=\STND(E)]

MATRIX R This is a computed FULL matrix of order 2 by 2 [=\STND(C)] MATRIX W This is a LOWER TRIANGULAR matrix of order 2 by 2 MATRIX X E-04 E E-01 MATRIX Y

Matrix of EXPECTED thresholds
Threshold Threshold (OBSERVED MATRIX is nonexistent for raw data) EXPECTED COVARIANCE MATRIX Function value of this group: Where the fit function is -2 * Log-likelihood of raw ordinal

*** WARNING! *** Minimization may not be successful. See above CODE RED - Hessian/precision problem Your model has 15 estimated parameters and Observed statistics Observed statistics include 2 constraints. -2 times log-likelihood of data >>> Degrees of freedom >>>>>>>>>>>>>>>> This problem used 1.2% of my workspace Task Time elapsed (DD:HH:MM:SS) Reading script & data : 0: 0: 6.63 Execution : 0: 5:18.42 TOTAL : 0: 5:25.04 Total number of warnings issued: 2 ______________________________________________________________________________

BINARY and CATEGORICAL TRAITS
GENETIC ANALYSIS OF BINARY and CATEGORICAL TRAITS PART TWO

HIGH-RISK SAMPLING SCHEMES
The Ordinal data-option in MX allows us to analyze twin or family data collected under a two-stage sampling scheme, where in the first stage we study a random sample of families, but in the second stage the probability that a family will be assigned for interview is a function of phenotypic values observed at the first stage. For example, we may decide that we will do follow-up assessments with all pairs where at least one twin is affected at stage one, but only 10% of pairs where neither twin was affected at stage one.

To illustrate this, we have created a simulated data-set, with the following parameters, using multsim2_2mz.mx and multsim2_2dz.mx. WAVE 1 WAVE 2 VA 50% rG = 1.00 VC 9% rC = 1.00 VE 41% rE = 0.71 Prevalence 25% First, we analyze this data assuming that all twin pairs (1000 MZ, 1000 DZ pairs) are assessed at both waves (ordinal_bivariate_simulated.mx).

SIMULATED TWO-WAVE DATA
TWIN A TWIN B Wave 1 Wave 2 MZ_FULL DZ_FULL 560 519.5 1 33.6 37.8 60.35 92.4 5.8 4.7 17.25 15.3 92.7 64.6 TOTAL PAIRS

Estimated parameters: WAVE 1 WAVE 2 % 95% CI r VA 49.7 ( ) 49.5 ( ) rG = 1.00 VC 9.3 ( ) 9.5 ( ) rC = 1.00 VE 41.1 ( ) ( ) rE = 0.71 Prevalence 25.0 (t1=0.6748) (t2=0.6747)

HIGH-RISK SAMPLING SCHEMES
(II) Next, we analyze the data-set that would arise under our two-stage sampling scheme, using ordinal_bivariate_hirisk.mx. This is exactly the same program as in the previous case, except that we have changed file names! In 90% of cases where neither twin was affected at stage one, the stage two phenotypic values are set to missing. What parameter estimates do we recover in this case?

Two-Wave data simulating high-risk sampling
MZHIRISK DZHIRISK 56 51.95 . 504 467.55 1 3.36 3.78 30.24 34.02 33.6 37.8 60.35 92.4 0.58 0.47 5.22 4.23 5.8 4.7 17.25 15.3 92.7 64.6 TOTAL PAIRS - Wave 1: 1000 1000.1 - Wave 2: 430.30

Estimated parameters: WAVE 1 WAVE 2 % 95% CI r VA 50.2 ( ) 50.3 ( ) rG = 1.00 VC 8.8 8.6 rC = 1.00 VE 41.0 41.1 rE = 0.71 Prevalence 25.0 (t1=0.6745) (t2=0.6737)

HIGH-RISK SAMPLING SCHEMES (III)
Notice that we included all pairs who were assessed at stage one. What happens if we focus on the stage two phenotype and include only those pairs who have data at stage two? Data are in mzlistwise.rec and dzlistwise.rec; the program is univariate_listwise.mx. Estimated parameters: WAVE 2 % 95% CI VA 23.0 VC 0.0 VE 77.0 Prevalence 50.0 (t1=0.004) ( ) When we ignore the wave one data, our estimates of population prevalence (not unexpectedly) and genetic and environmental parameters, are seriously biased!

HIGH-RISK SAMPLING SCHEMES (IV)
Suppose that instead we acknowledge that our population is drawn from a population where the prevalence of the observed trait is 25%, and fix our estimate of the threshold value, t= As in the previous example, we limit ourselves to twin pairs where wave two assessments occurred. WAVE 2 % 95% CI VA 45.8 VC 0.0 VE 54.2 The bias to parameter estimates is substantially reduced! (There is still a bias, however: in particular, our estimate of the shared environmental variance is now zero.)

HIGH-RISK SAMPLING SCHEMES (V)
How do we explain these results? “Missing data theory” is an active area of research in statistics which is concerned with how we should adjust for missing observations -- which may be missing because of subject non-response, or because of sampling design (e.g. our two-stage sampling design). Missing data theory distinguishes between data that are (i) MCAR -- missing completely at random, i.e. non-response is completely unrelated to the variable we are studying (plausible for variables such as finger ridge count). (ii) MAR -- missing at random, i.e. non-response is random, but the probability of non-response may vary as a function of observed trait values (or underlying latent variables).

Probability of missing data
Suppose we have a 5-level variable with the following probabilities of missing data at subsequent follow-up: Trait Value Probability of missing data 1 60% 2 20% 3 35% 4 13% 5 50% These data are certainly not MCAR, but they do meet the definition of MAR.

HIGH-RISK SAMPLING SCHEMES (VI)
Under certain conditions, missingness is said to be ignorable, i.e. we can recover estimates of the underlying population parameters without needing to adjust for differential rates of non-response. For our two-stage high-risk sampling scheme, where we assumed random sampling at the first stage, but that only 10% of concordant unaffected pairs are assessed at the second stage, the stage-two data are MAR. Provided that we use the Ordinal option in MX (or the Raw data option, for continuous variables), and analyze all pairs observed at stage one, we can recover correct estimates of population prevalence and of genetic and environmental parameters.

If probability of non-response is (i) determined by one or more correlated phenotypes that are not included in the analyses; or (ii) partly a function of the stage-two phenotype (such as would be the case if individuals who were unaffected at wave one but had become affected by the time wave two were more likely not to agree to be assessed at wave two than individuals who remained unaffected throughout), missing data will be non-ignorable. In the case where we analyzed only the wave two data, but fixed the prevalence at 25% (i.e. assuming that missingness is determined by the stage two phenotype), missingness was still strictly non-ignorable, since it was determined by wave one and not wave two phenotypic values. However, since we simulated a very high test-retest correlation between wave one and wave two data, analyzing the data as though they were MAR greatly reduced biases to estimates of genetic and environmental parameters.

HIGH-RISK SAMPLING SCHEMES (VII)
Consider the following cases: Example 1: We are conducting a twin study of smoking. Probability of non-response is much higher for smokers (30%) than for non-smokers (15%). We include data from singleton twins as well as complete pairs. Are missing data ignorable? NO, since the probability of non-response is determined by the phenotype we wish to study. Example 2: We are conducting a twin study of smoking. We know that the prevalence of smoking is 25%, both in MZ and in DZ twin pairs. We can identify every twin pair where at least one twin is a smoker, and determine the smoking status of each twin. However, we don’t have data on concordant non-smoking pairs. Are missing data ignorable?

Missing data theory provides a framework for thinking about several important classes of problems in behavior genetics: (i) clinically ascertained samples; (ii) cooperation or retention bias; (iii) hierarchical or stage-dependent models of genetic and environmental influences on risk of psychopathology.

ASCERTAINED SAMPLES Suppose we identify twins from a treatment series, and then assess the twins and their cotwins. Typically, we will ascertain no more than one twin from each pair (‘single ascertainment’). If we have an estimate of the prevalence of the disorder we are studying in the population from which they are drawn (e.g. treated alcohol dependence), provided that that prevalence estimate applies equally to MZ and to DZ pairs, then we can derive estimates of genetic and environmental parameters. If we have “double proband” pairs, where each twin independently has come to clinical attention, we will count the pair twice, once with twin A and once with twin B as the proband.

For example, Gurling et al
For example, Gurling et al. (1984) reported that they failed to find significant heritability of alcohol dependence based on data from a small twin series, in which the number of unaffected and affected cotwins of alcohol dependent probands was as follows: Suppose the prevalence of treated alcohol dependence is 10%. We can fit an ACE model, fixing the threshold for MZ and DZ pairs to see univariate_ascertained.mx. Unaffected Affected MZ pairs: 10 5 (mzmasc.rec) DZ pairs: 14 6 (dzmasc.rec)

ASCERTAINED SAMPLES (II)
For the Gurling data, assuming 10% prevalence, we estimate: VA = 11.0% (95%CI %) VC = 40.4% (95%CI %) VE = 48.6% (95%CI %) Of course, the assumption that one in ten British males receives inpatient treatment for alcoholism may be a little strong. Suppose the prevalence of treated alcoholism is only 2.5% (corresponding to a threshold value of 1.96). Then we would obtain: VA = 7.5% (95%CI %) VC = 62.8% (95%CI %) VE = 29.8% (95%CI %) In other words, with these data we cannot reject the hypothesis of 75% heritability of alcohol dependence.

BINARY and CATEGORICAL TRAITS

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