Heterogeneity
HGEN619 class 2007
Heterogeneity Questions I
Univariate Analysis: What are the contributions
of additive genetic, dominance/shared
environmental and unique environmental factors
to the variance?
Are the contributions of genetic and
environmental factors equal for different groups,
such as sex, race, ethnicity, SES, environmental
exposure, etc.?
Heterogeneity Questions II
Are these differences due to differences in the
magnitude of the effects (quantitative)?
e.g.
Is the contribution of genetic/environmental
factors greater/smaller in males than in females?
Are the differences due to differences in the
nature of the effects (qualitative)?
e.g. Are
there different genetic/environmental factors
influencing the trait in males and females?
Groups
Comparison
gender
age
nationality
environment
Concordant for group
membership
MZ & DZ:
MM & FF pairs
MZ & DZ:
young & old pairs
MZ & DZ:
OZ & US pairs
MZ & DZ:
urban & rural pairs
Discordant for group
membership
DZ:
opposite sex pairs
MZ & DZ:
urban/ rural pairs
Heterogeneity Model I
same sex pairs
.5
1
1
A
1
1
1
C
1
A
E
1
C
E
am cm em
am cm em
PM
PM
.5
1
1
A
C
E
Male
Male
am2 + cm2 + em2 .5ama'm +cmc'm
Male
.5ama'm +cmc'm am2 + cm2 + em2
1
1
1
Male
1
A
1
C
af cf ef
af cf ef
PF
PF
E
Female
Female
Female
af2 + cf2 + ef2
.5afa'f + cfc'f
Female
.5afa'f + cfc'f
af2 + cf2 + ef2
Homogeneity Model I
same sex pairs
.5
1
1
A
1
1
C
1
C
E
a c e
a c e
PM
PM
1
A
1
A
E
.5
1
1
C
E
Male
Male
a2 + c2 + e2
Male
.5a2 + c2
a2 + c2 + e2
Female
Female
1
1
1
Male
1
A
1
C
a c e
a c e
PF
PF
E
Female
a2 + c2 + e2
Female
.5a2 + c2
a2 + c2 + e2
Scalar Model I
same sex pairs
.5
1
1
A
1
1
C
1
C
E
a c e
a c e
PM
PM
1
A
1
A
E
.5
1
1
C
E
Male
Male
a2 + c2 + e2
.5a2 +c2
Male
.5a2 +c2
a2 + c2 + e2
Female
Female
Female
k(a2 + c2 + e2)
Vk(.5a2 + c2)
Female
Vk(.5a2 + c2)
k(a2 + c2 + e2)
1
1
1
Male
1
A
1
C
a c e
a c e
PF
PF
Vk
Vk
PF
PF
E
Models for Concordant Pairs
OS
G1
G2
EP
Heterogeneity
12
a1 c 1 e1
a2 c 2 e2
6
Homogeneity
12
a1 c 1 e1
a1 c 1 e1
3
Scalar
12
a1 c 1 e1
a1 c 1 e1 k
4
Homogeneity Model II
opposite sex pairs
.5
1
1
A
1
1
1
C
1
A
E
1
C
a c e
a c e
PM
PF
Male
Male
a2 + c2 + e2
Female
.5a2 + c2
Female
a2 + c2 + e2
E
Heterogeneity Model II
opposite sex pairs
.5
1
1
A
1
1
1
C
1
A
E
1
C
am cm em
af cf ef
PM
PF
Male
Male
Female
E
Female
am2 + cm2 + em2 .5ama'f + cmc'f
.5afa'm + cfc'm
af2 + cf2 + ef2
Homogeneity Model III
.5
1
1
A
1
1
C
1
E
1
1
A
C
E
.8 .5 .3
.8 .5 .3
PM
PF
Variance
Covariance
MZm
.98
.89
MZf
.98
.89
DZm
.98
.57
DZf
.98
.57
DZmf
.98
.57
Correlation
.91
.91
.58
.58
.58
Heterogeneity Model III
.5
1
1
1
1
A
C
1
E
1
A
1
C
E
.5 .5 .3
.8 .5 .3
PM
PF
Variance
Covariance
MZm
.59
.50
MZf
.98
.89
DZm
.59
.38
DZf
.98
.57
DZmf
.59/.98
.45
Correlation
.85
.91
.64
.58
.59
Scalar Model II
opposite sex pairs
.5
1
1
A
1
1
C
1
1
A
E
1
C
a c e
a c e
PM
PF
Vk
Male
Male
Female
Female
a2 + c2 + e2
Vk(.5a 2 + c2) k(a2 + c2 + e2)
PF
E
Scalar Model III
1
1
A
.5
1
1
1
C
E
1
A
1
C
E
.8 .5 .3
.8 .5 .3
PM
PF
1.5
PF
Variance
Covariance
MZm
.98
.89
MZf
2.20
2.00
DZm
.98
.57
DZf
2.20
1.28
DZmf
.98/2.2
.85
Correlation
.91
.91
.58
.58
.58
Models for Concordant and
Discordant Pairs
OS
G1
G2
EP
Heterogeneity
15
a1 c 1 e1
a2 c 2 e2
6
Homogeneity
15
a1 c 1 e1
a1 c 1 e1
3
Scalar
15
a1 c 1 e1
a1 c 1 e1 k
4
General I
II
15
15
a1 c 1 e1
a1 c 1 e1
a2 c 2 e2 rg 7
a2 c 2 e2 rc 7
General Model I
same sex pairs
.5
1
1
A
m
.5
1
A
1
1
C
1
1
1
A
E
A
m
h am cm em
A
E
Male
Male
Male
am2+cm2+em2+h2 .5am2 +cm2+.5h2
Male
.5am2 +cm2+.5h2 am2+cm2+em2+h2
PM
.5
1
C
h am cm em
PM
1
1
1
1
C
1
E
1
A
1
C
af cf ef
af cf ef
PF
PF
E
Female
Female
Female
af2 + cf2 + ef2
.5afa'f + cfc'f
Female
.5afa'f + cfc'f
af2 + cf2 + ef2
General Model II
opposite sex pairs
.5
1
1
A
m
1
A
1
1
1
C
E
h am cm em
PM
Female
A
1
C
af cf ef
PF
Male
Male
1
Female
am2+cm2+em2+h2 .5ama'f + cmc'f
.5afa'm + cfc'm
af2 + cf2 + ef2
E
General Model III
.5
1
1
A
m
1
A
1
1
C
1
1
A
E
.8 0 .5 .3
1
C
E
.8 .5 .3
PM
PF
Variance
Covariance
MZm
.98
.89
MZf
.98
.89
DZm
.98
.57
DZf
.98
.57
DZmf
.98
.25
Correlation
.91
.91
.58
.58
.26
Male-female genetic correlation (rg)
amaf
rg=
If am2~=0 and af2~=0
and h2=0 then rg=1
If am2=0 and h2~=0
then rg=0
If am2~=0 and h2~=0
then 0<rg<1
Vaf2 (am2+h2)
General Model IV
.5rg
1
1
1
1
1
A
C
1
A
E
1
C
am cm em
af cf ef
PM
PF
Male
Male
Female
E
Female
am2 + cm2 + em2 .5rgama'f +cmc'f
.5rgafa'm +cfc'm
af2 + cf2 + ef2
Summary of Models
Homogeneity Model:
no
quantitative, no qualitative differences
Heterogeneity Model:
quantitative
but no qualitative differences
Scalar Model:
special
case of heterogeneity model with proportional
quantitative differences
General Model:
quantitative
and qualitative differences
Heterogeneity
differences in variance components
NOT mean differences
group membership
continuous heterogeneity
#NGroups 8
!
Heterogeneity Model
Title G1: Model Parameters
Calculation
Begin Matrices;
X Lower nvar nvar Free
Y Lower nvar nvar
Z Lower nvar nvar Free
W Lo wer nvar nvar Free
H Full 1 1
Q Full 1 1
End Matrices;
Matrix H .5
Matrix Q .25
Begin Algebra;
A= X*X';
C= Y*Y';
E= Z*Z';
D= W*W';
End Algebra;
Title G2: MZ data
#include ozbmi.dat Select if zyg =1
Begin Matrices = Group 1;
Covariance
A+C+E+D | A+C+D
_
A+C+D
| A+C+E+D;
Title G3: DZ data
#include ozbmi.dat Select if zyg =3
Begin Matrices = Group 1;
Covariance
A+C+E+D
| H@A+C+Q@D _
H@A+C+Q@D | A+C+E+D;
Title G4: Standardization
Begin Matrices = Group 1;
Begin Algebra;
V=A+C+E+D; P=A|C|E|D; S=P@V~;
End Algebra;
Interval S 1 1 - S 1 4
Title G5: Model Parameters
Calculation
Begin Matrices;
X Lower nvar nvar Free
Y Lower nvar nvar
Z Lower nvar nvar Free
W Lower nvar nvar Free
H Full 1 1
Q Full 1 1
End Matrices;
Matrix H .5
Matrix Q .25
Begin Algebra;
A= X*X';
C= Y*Y';
E= Z*Z';
D= W*W';
End Algebra;
Title G6: MZ data
#include ozbmi.dat Select if zyg =2
Begin Matrices = Group 5;
Covariance
A+C+E+D | A+C+D
_
A+C+D
| A+C+E+D;
Title G7: DZ data
#include ozbmi.dat Select if zyg =4
Begin Matrices = Group 5;
Covariance
A+C+E+D
| H@A+C+Q@D _
H@A+C+Q@D | A+C+E+D;
Title G8: Standardization
Begin Matrices = Group 5;
Begin Algebra;
V=A+C+E+D; P=A|C|E|D; S=P@V~;
End Algebra;
Interval S 1 1 - S 1 4
Option Sat=5995.659,2667
ozbmifmyace4.mx
#NGroups 6
Title G1: Model Parameters
Calculation
Begin Matrices;
X Lower nvar nvar Free
Y Lower nvar nvar
Z Lower nvar nvar Free
W Lower nvar nvar Free
H Full 1 1
Q Full 1 1
End Matrices;
Matrix H .5
Matrix Q .25
Begin Algebra;
A= X*X';
C= Y*Y';
E= Z*Z';
D= W*W';
End Algebra;
!
Homogeneity Model
Title G2: MZ data
#include ozbmi.dat Select if zyg =1
Begin Matrices = Group 1;
Covariance
A+C+E+D | A+C+D
_
A+C+D
| A+C+E+D;
Title G3: DZ data
#include ozbmi.dat Select if zyg =3
Begin Matrices = Group 1;
Covariance
A+C+E+D
| H@A+C+Q@D _
H@A+C+Q@D | A+C+E+D;
Title G4: MZ data
#include ozbmi.dat Select if zyg =2
Begin Matrices = Group 1;
Covariance
A+C+E+D | A+C+D
_
A+C+D
| A+C+E+D;
Title G5: DZ data
#include ozbmi.dat Select if zyg =4
Begin Matrices = Group 1;
Covariance
A+C+E+D
| H@A+C+Q@D _
H@A+C+Q@D | A+C+E+D;
Title G6: Standardization
Begin Matrices = Group 1;
Begin Algebra;
V=A+C+E+D; P=A|C|E|D; S=P@V~;
End Algebra;
Interval S 1 1 - S 1 4
Option Sat=5995.659,2667
ozbmifmyace4eq.mx
! Estimate genetic and environmental components - ACED model
! OZ BMI data - younger females & males + opposite sex twins I
#NGroups 7
#define nvar 1
G1: Parameters
Calculation
Begin Matrices;
X Lower nvar nvar
Y Lower nvar nvar
Z Lower nvar nvar
W Lower nvar nvar
S Lower nvar nvar
T Lower nvar nvar
U Lower nvar nvar
V Lower nvar nvar
H Full 1 1
Q Full 1 1
F Full 1 1 Free
End Matrices;
Matrix H .5
Start .5 F 1 1 1
Begin Algebra;
A= X*X';
C= Y*Y';
E= Z*Z';
D= W*W';
K= S*S';
L= T*T';
N= U*U';
O= V*V';
End Algebra; End
#define nvar2 2
Free
Free
Free
Free
Free
Free
!
!
!
!
!
!
!
!
!
!
!
FEMALES additive genetic path, a
FEMALES common environmental path, c
FEMALES specific environmental path, e
FEMALES dominance genetic path, d
MALES additive genetic path, a
MALES common environmental path, c
MALES specific environmental path, e
MALES dominance genetic path, d
scalar, 0.5
scalar, 0.25
free for DZO
!
!
!
!
!
!
!
!
FEMALES additive genetic variance
FEMALES common environmental variance
FEMALES unique environmental variance
FEMALES dominance variance
MALES additive genetic variance
MALES common environmental variance
MALES unique environmental variance
MALES dominance variance
ozbmifmyace5.mx
! Estimate genetic and environmental components - ACED model
! OZ BMI data - younger females & males + opposite sex twins II
Title G2: MZf data
#include ozbmi.dat
Select if zyg =1
Select bmi1 bmi2 ;
Begin Matrices = Group 1;
M Full 1 nvar2 Free
End Matrices;
Means M;
Covariance
A+C+E+D | A+C+D
_
A+C+D
| A+C+E+D;
Option RSiduals;
End
Title G3: DZf data
#include ozbmi.dat
Select if zyg =3
Select bmi1 bmi2 ;
Begin Matrices = Group 1;
M Full 1 nvar2 Free
End Matrices;
Means M;
Covariance
A+C+E+D
| H@A+C+Q@D _
H@A+C+Q@D | A+C+E+D;
Option RSiduals
End
Title G4: MZm data
#include ozbmi.dat
Select if zyg =2
Select bmi1 bmi2 ;
Begin Matrices = Group 1;
M Full 1 nvar2 Free
End Matrices;
Means M;
Covariance
K+L+N+O | K+L+O
_
K+L+O
| K+L+N+O ;
Option RSiduals;
End
Title G5: DZm data
#include ozbmi.dat
Select if zyg =4
Select bmi1 bmi2 ;
Begin Matrices = Group 1;
M Full 1 nvar2 Free
End Matrices;
Means M;
Covariance
K+L+N+O
| H@K+L+Q@O _
H@K+L+Q@O | K+L+N+O ;
Option RSiduals
ozbmifmyace5.mx
End
! Estimate genetic and environmental components - ACED model
! OZ BMI data - younger females & males + opposite sex twins III
Title G6: DZfm data
#include ozbmi.dat
Select if zyg =5
Select bmi1 bmi2 ;
Begin Matrices = Group 1;
M Full 1 nvar2 Free
End Matrices;
Means M;
Covariance
A+C+E+D
| F@(X*S')+(Y*T')+Q@(W*V') _
F@(S*X')+(T*Y')+Q@(V*W') | K+L+N+O ;
Option RSiduals
End
Title G7: Standardization
Calculation
Begin Matrices = Group 1;
End Matrices;
Start .5 all
Start 20 M 2 1 1 - M 2 1
Start 20 M 3 1 1 - M 3 1
Start 20 M 4 1 1 - M 4 1
Start 20 M 5 1 1 - M 5 1
Start 20 M 6 1 1 - M 6 1
nvar2
nvar2
nvar2
nvar2
nvar2
ozbmifmyace5.mx
! Estimate genetic and environmental components - ACED model
! OZ BMI data - younger females & males + opposite sex twins IV
Begin Algebra;
G= A+C+E+D;
! FEMALES total variance
J= K+L+N+O;
! MALES total variance
P= A%G| C%G| E%G| D%G_ ! FEMALES FIRST ROW standardized variance components
K%J| L%J| N%J| O%J; ! MALES SECOND ROW standardized variance components
End Algebra;
Option NDecimals=4
!ADE model
! Option Sat=8310.308, 3633
Option Multiple Issat
!Interval @95 P 1 1 1 P 1 1 2 P 1 1 3 P 1 1 4 P 1 2 1 P 1 2 2 P 1 2 3 P 1 2 4
End
! Test for qualitative sex differences (nature of effect difference by sex)
Drop @.5 F 1 1 1
End
Save ozbmi5.mxs
! Test for quantitative sex differences (magnitude of effect difference by sex)
Equate X 1 1 1 S 1 1 1
! additive genetic factor loadings equal for M&F
Equate Z 1 1 1 U 1 1 1
! unique environmental factor loadings equal for M&F
Equate W 1 1 1 V 1 1 1
! dominance factor loadings equal for M&F
End
ozbmifmyace5.mx
