Mx Practical TC18, 2005 Dorret Boomsma, Nick Martin, Hermine H. Maes.
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Transcript of Mx Practical TC18, 2005 Dorret Boomsma, Nick Martin, Hermine H. Maes.
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Mx Practical
TC18, 2005Dorret Boomsma, Nick Martin,Hermine H. Maes
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Basic Genetic Epidemiology
Is the trait genetic?Collect phenotypic data on large samples of
MZ & DZ twinsCompare MZ & DZ correlationsPartition/ Quantify the variance in genetic and
environmental componentsTest significance of genetic variance
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Practical Example
Dataset: Dutch Adult Twins Cardiovascular Study: Fasting blood samples Variables: LDL, ApoB, ApoE(ln) Age range: 35-60 years Sample Sizes: MZ: 91 pairs, DZ: 116 pairs
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MZ & DZ correlations
*
*
MZDZ
0.8
a2
1.00.90.80.7 0.60.60.50.40.30.20.10.0
e2
c2
UN
0.4
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Raw Dataset: DutchMZ.rec
3 1 44.5 2.35 1.01 12.92 1 44.5 3.21 1.31 12.213 1 40.5 2.70 0.91 13.77 1 40.5 2.93 1.04 14.533 1 44.5 3.68 1.10 11.41 1 44.5 3.71 1.23 12.693 1 37.5 2.34 0.88 6.84 1 37.5 1.73 0.80 6.843 1 55.5 2.46 1.17 10.19 1 55.5 3.88 1.46 12.921 0 40.5 4.86 1.61 13.14 0 40.5 5.03 1.67 14.353 1 52.5 5.74 1.99 16.27 1 52.5 5.96 1.67 17.66....
MZ twins Data NInput=11 Rectangular File=DutchMZ.rec Labels zyg sex1 age1 ldl1 apob1 lnapoe1 sex2 age2 ldl2 apob2 lnapoe2
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Univariate Genetic Analysis
Saturated ModelsFree variances, covariances > correlationsFree means
Univariate ModelsVariances partitioned in a, c/d and eFree means (or not)
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Free means, (co)variances
MZ twins DZ twins
10 parametersCorrelation = covariance / square root of (variance1 * variance2)Covariance = correlation * square root of (variance1 * variance2)
T2T1
v1MZ cov
v2
1
m1 m2
T2T1
v3DZ cov
v4
1
m3 m4
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Mx Group Structure
Title Group type: data, calculation, constraint
[Read observed data, Labels, Select] Matrices declaration
Begin Matrices; End Matrices; [Specify numbers, parameters, etc.]
Algebra section and/or Model statement Begin Algebra; End Algebra; Means Covariances
[Options] End
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! Estimate means and correlations! Dutch Adult Twins: Lipid levels
#define nvar 1 #define nvarx2 2 #NGroups 2
G1: MZ twins Data NInput=11 Rectangular File=DutchMZ.rec Labels .. Select ldl1 ldl2 ; Begin Matrices; M Full nvar nvarx2 Free S Diag nvarx2 nvarx2 Free R Stnd nvarx2 nvarx2 Free End Matrices; ! Starting values Means M; Covariance S*R*S'; End
G2: DZ twins Data NInput=18 Rectangular File=DutchDZ.rec Labels .. Select ldl1 ldl2 ; Begin Matrices; M Full nvar nvarx2 Free S Diag nvarx2 nvarx2 Free R Stnd nvarx2 nvarx2 Free End Matrices; ! Starting values Means M; Covariance S*R*S'; End
Correlations_MZ+DZ.mx
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Correlations + Test MZ=DZcor
MZ DZ Chi (df=1)
p
LDL .78 .45 16.32 .000
ApoB .79 .46 16.67 .000
lnApoE .89 .51 33.31 .000
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Means, ACE
T2
AEA C
a ac e
T1
1 111
E
e
1
C
c
1
1
1
1
m1 m2
T2
AEA C
a ac e
T1
1 111
E
e
1
C
c
1
.5
1
1
m3 m4
MZ twins DZ twins
7 parameters
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Expected CovariancesObserved Cov Variance Twin 1 Covariance T1T2
Covariance T1T2 Variance Twin 2
MZ Expected Cov a2+c2+e2+d2 a2+c2+d2
a2+c2+d2 a2+c2+e2+d2
DZ Expected Cov a2+c2+e2+d2 .5a2+c2+.25d2
.5a2+c2+.25d2 a2+c2+e2+d2
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! Estimate variance components - ACED model! Dutch Adult Twins: Lipid levels
#define nvar 1 #define nvar2 2 #NGroups 4
Title 1: Model Parameters Calculation Begin Matrices; X Lower nvar nvar Free ! a Y Lower nvar nvar Free ! c Z Lower nvar nvar Free ! e W Lower nvar nvar ! d H Full 1 1 ! 0.5 Q Full 1 1 ! 0.25 End Matrices; Matrix H .5 Matrix Q .25
Label Row X add_gen Label Row Y com_env Label Row Z spec_env Label Row W dom_gen Begin Algebra; A= X*X'; ! a^2 C= Y*Y'; ! c^2 E= Z*Z'; ! e^2 D= W*W'; ! d^2 End Algebra; End
ACEmodel_MZ+DZ.mx
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! Estimate variance components - ACED model! Dutch Adult Twins: Lipid levels
Title G2: MZ data Data NInput=11 Rectangular File=DutchMZ.rec Labels .. Select ldl1 ldl2 ; Begin Matrices = Group 1; M Full 1 nvar2 Free End Matrices; Matrix M 4 4 Means M; Covariance A+C+E+D | A+C+D _ A+C+D | A+C+E+D; Option RSiduals End
Title 3: DZ data Data NInput=18 Rectangular File=DutchDZ.rec Labels .. Select ldl1 ldl2 ; Begin Matrices = Group 1; M Full 1 nvar2 Free End Matrices; Matrix M 4 4 Means M; Covariance A+C+E+D | H@A+C+Q@D _ H@A+C+Q@D | A+C+E+D; Option RSiduals End
ACEmodel_MZ+DZ.mx
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! Estimate variance components - ACED model! Dutch Adult Twins: Lipid levels
Title G4: Standardization Calculation Begin Matrices = Group 1; End Matrices; Start .6 all Begin Algebra; V=A+C+E+D; P=A|C|E|D; S=P@V~; End Algebra; Label Col P a^2 c^2 e^2 d^2 Label Col S a^2 c^2 e^2 d^2 !ACE model ! Interval S 1 1 - S 1 4 Option NDecimals=4 Option Multiple Issat End
Save ACE.mxs ! Test significance of
genetic .. Drop X 1 1 1 End
Get ACE.mxs ! Test significance of shared
env Drop Y 1 1 1 Exit
ACEmodel_MZ+DZ.mx
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Chi-square tests and probs
Significance of A (df=1)
Significance of C (df=1)
LDL 25.14 .000 0.30 .584
ApoB 28.58 .000 0.16 .693
lnApoE 37.01 .000 1.97 .161
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Variance Components ACE
a2 c2 e2
LDL .72 .08 .20
ApoB .75 .06 .19
lnApoE .67 .20 .13
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Using templates & headers in Mx
Correlations_MZ+DZ_template.mx #include DutchMZ.dat Select t1$var t2$var ;
Correlations_MZ+DZ_header.mx #define $var ldl #include Correlations_MZ+DZ_template.mx
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Linkage Analysis
Where are the genes?Collect genotypic data on large number of
markersCompare correlations by number of alleles
identical by descent at a particular markerPartition/ Quantify variance in genetic (QTL)
and environmental componentsTest significance of QTL effect
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A
Q?
B
Q?X
C
Q?
Fully informative mating
mother father
offspringIBD = 1Sib1
ACBD
Sib2ADBC
offspring IBD = 0Sib1AC
Sib2BD
D
Q?
offspringIBD = 2Sib1
ACBD
Sib2ACBD
QTL
marker
dist
ance
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Identity by Descent (IBD) in sibs
Four parental marker alleles: A-B and C-D
Two siblings can inherit 0, 1 or 2 alleles IBD
IBD 0:1:2 = 25%:50%:25%
Derivation of IBD probabilities at one marker (Haseman & Elston 1972
Sib2
Sib1
AC AD BC BD
AC 2 1 1 0
AD 1 2 0 1
BC 1 0 2 1
BD 0 1 1 2
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DZ ibd0,1,2 correlations
*
*
DZibd2DZibd1
0.8
q2
1.00.90.80.7 0.60.60.50.40.30.20.10.0
e2
f2
DZibd0
0.4
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DZ ibd0,1,2 & MZ correlations
*
DZibd2DZibd1
1.00.90.80.7 0.60.60.50.40.30.20.10.0
DZibd0
0.4
a2
c2
e2
/MZ
0.8**
0.9
q2
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Raw Dataset: DutchDZ.rec
DZ twins Data NInput=18 Rectangular File= DutchDZ.rec Labels zyg sex1 age1 med1 ldl1 apob1 lnapoe1 sex2 age2
med2 ldl2 apob2 lnapoe2 ibd0_65 ibd1_65 ibd2_65 pihat65 pi65cat
position 65 on chromosome 19 ibd0_65 ibd1_65 ibd2_65: probabilities that sibling pair is
ibd 0, 1 or 2 pihat65: pihat estimated as ½(ibd1_65) + (ibd2_65) pi65cat: sample divided according to π<.25, π>.75 or
other
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Distribution of pi-hat
Adult Dutch DZ pairs: distribution of pi-hat (π) at 65 cM on chromosome 19
π = IBD/2 π < 0.25: IBD=0 group π > 0.75: IBD=2 group others: IBD=1 group pi65cat= (0,1,2)
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Can resemblance (e.g. correlations, covariances) between sib pairs, or DZ twins, be modeled as a function of DNA marker sharing at a particular chromosomal location?
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Compare correlations by IBD
DZ pairs (3 groups according to IBD) onlyEstimate correlations as function of IBD
(pi65cat)Test if correlations are equal
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Add MZ twins
DZ + MZ pairsEstimate correlations as function of IBD +
zygosityTest if DZibd2 correlation is equal to MZ
correlation
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Correlations
DZibd2 DZibd1 DZibd0 MZ
LDL .81 .49 -.21 .78
ApoB .64 .50 .02 .79
lnApoE .83 .55 .14 .89
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Tests .... Option Multiple Issat End
Save lipidcor.mxs ! Test for linkage ! Set 3 DZ IBD correlations equal Equate R 1 2 1 R 2 2 1 R 3 2 1 End
Get lipidcor.mxs ! Test for residual polygenic variance ! Set DZ IBD2 correlation equal to MZ correlation Equate R 1 2 1 R 4 2 1 Exit
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Chi-square tests and probs
All DZ equal (df=2) DZibd2 = MZ (df=1)
LDL 21.77 .000 0.09 .757
ApoB 7.98 .019 1.53 .216
lnApoE 12.45 .002 0.58 .448
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Compare correlations
DZ pairs (3 groups according to IBD) onlyEstimate correlations as function of IBDTest if correlations are equal
Correlations_DZibd.mx
DZ + MZ pairsEstimate correlations as function of IBD + zygTest if DZibd2 correlation is equal to MZ cor
Correlations_DZibd+MZ.mx
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DZ by IBD status
T2
QEQ F
q qf e
T1
1 111
E
e
1
F
f
1
.5
1
1m3 m4
T2
QEQ F
q qf e
T1
1 111
E
e
1
F
f
1
1
1m5 m6
Variance = Q + F + E Covariance = πQ + F + E
T2
QEQ F
q qf e
T1
1 111
E
e
1
F
f
1
1
1
1m1 m2
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DZ by IBD status + MZ
T2
AEA C
a ac e
T1
1 111
E
e
1
C
c
1
.51
1m1 m2
Q1
Q1
q q
1
T2
AEA C
a ac e
T1
1 111
E
e
1
C
c
1
.51
1m5 m6
Q1
Q1
q q
T2
AEA C
a ac e
T1
1 111
E
e
1
C
c
1
11
1m7 m8
Q1
Q1
q q
1
T2
AEA C
a ac e
T1
1 111
E
e
1
C
c
1
.51
1m3 m4
Q1
Q1
q q
.5
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Partition Variance
DZ pairs (3 groups according to IBD) onlyEstimate FEQTest if QTL effect is significant
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Covariance Statements G2: DZ IBD2 twins Matrix K 1 Covariance F+Q+E | F+K@Q _ F+K@Q | F+Q+E;
G3: DZ IBD1 twins Matrix K .5 Covariance F+Q+E | F+K@Q _ F+K@Q | F+Q+E;
G4: DZ IBD0 twins Covariance F+Q+E | F_ F | F+Q+E;
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Partition Variance
DZ + MZ pairsEstimate ACEQTest if QTL estimate/significance is different
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Covariance Statements +MZ G2: DZ IBD2 twins Matrix K 1 Covariance A+C+Q+E | H@A+C+K@Q _ H@A+C+K@Q | A+C+Q+E;
G3: DZ IBD1 twins Matrix K .5 Covariance A+C+Q+E | H@A+C+K@Q _ H@A+C+K@Q | A+C+Q+E;
G4: DZ IBD0 twins Covariance A+C+Q+E | H@A+C_ H@A+C | A+C+Q+E;
G5: MZ twins Covariance A+C+Q+E | A+C+Q _ A+C+Q | A+C+Q+E;
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Chi-square Tests for QTL
DZ pairs (df=1) DZ+MZ pairs (df=1)
LDL 12.25 .000 12.56 .000
ApoB 1.95 .163 2.13 .145
lnApoE 12.45 .000 12.29 .000
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Variance Components FEQ
f2 e2 q2
LDL .00 (.00-.32) .23 (.13-.40) .77 (.36-.87)
ApoB .27 (.00-.54) .41 (.24-.66) .32 (.00-.73)
lnApoE .19 (.00-.43) .16 (.09-.32) .65 (.33-.90)
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Variance Components ACEQ
a2 c2 e2 q2
LDL .04 (.00-.39) .00 (.00-.27) .21 (.15-.29) .75 (.37-.84)
ApoB .46 (.11-.84) .02 (.00-.29) .19 (.24-.27) .33 (.00-.67)
lnApoE .02 (.00-.33) .22 (.00-.45) .13 (.10-.18) .63 (.32-.89)
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Partition Variance
DZ pairs (3 groups according to IBD) onlyEstimate QFETest if QTL effect is significant
FEQmodel_DZibd.mx
DZ + MZ pairsEstimate Test if QTL estimate/significance is different
ACEQmodel_DZibd+MZ.mx