The Inclusion of Measured Covariates in the Neale and Kendler Model Fitting Approach

Abstract

Neale and Kendler (1995, Am J Hum Genet 57:935–953) described 13 comorbidity models, providing the most comprehensive set of alternative hypotheses regarding the possible causes of comorbidity to date. This research note describes an extension of the Neale and Kendler model fitting approach that permits the inclusion of measured covariates. An example of 13 models examining the comorbidity between alcohol dependence and illicit drug dependence is presented.

Appendix

Mx script for alternate forms model

G1 Alternate forms model

!

! If you are above threshold on L, you get disorder A with prob p

! If you are above threshold on L, you get disorder B with prob r

! Note: (q = 1-p and s = 1-r)

! If you are below threshold on L you are immune to A&B

! control group correlation matrix

DA CALC NG = 4

MATRICES

A Lo 1 1 Fixed

C Lo 1 1 Free

H fu 1 1

I id 1 1

begin algebra;

X = A*A′;

Y = C*C′;

end algebra;

COMPUTE I|H@X + Y_

H@X + Y|I;

Matrix H .5

Matrix A 0

Option RS SE

End

G2 CALCULATION GROUP clinical group correlation matrix

DA CALC

MATRICES

A Lo 1 1 = A1

C Lo 1 1 = C1

H fu 1 1

I id 1 1

begin algebra;

X = A*A′;

Y = C*C′;

end algebra;

COMPUTE I|H@X + Y_

H@X + Y|I;

Matrix H 0.5

Option RS

End

G3 Evaluate model control group

Data NI = 6

ordinal fi = control.rec

Labels age1 age2 sex1 sex2 type dummy

Definition_variables age1 age2 sex1 sex2 type/

Matrices

A Full 2 2 = %E1 ! corr between A factors

I Iden 1 1

P Full 1 1 free! probability of getting A if above threshold (p in Table 3)

R Full 1 1 free! probability of getting B if above threshold (r in Table 3)

H Full 1 2 ! matrix for age definition variable

Y Full 1 2 ! matrix for sex definition variable

L Full 1 2 ! estimated threshold for A

O Full 1 2 ! age difference in threshold

N Full 1 2 ! sex difference in threshold

M Full 1 1 ! mean of dummy variable

V Full 1 1 ! variance of dummy variable

J Full 4 1 ! for extracting right part of K

W Full 1 1 ! 0 + + These are to control integral type

Z Full 1 1 ! 1 − −

begin algebra;

T = L + (H.O) + (Y.N); ! Threshold = estimated threshold + (age × age difference in threshold) + (sex × sex difference in threshold)

D = \muln((A_T_T_Z|Z)) ; !lower/lower (siblings 1 and 2 are below ! threshold); (LL in Table 3)

E = \muln((A_T_T_Z|W)); !lower/upper (sibling 1 is below threshold and ! sibling 2 is above threshold) (LU in Table 3)

F = \muln((A_T_T_W|Z)); !upper/lower (sibling 1 is above threshold and ! sibling 2 is below threshold) (UL in Table 3)

G = \muln((A_T_T_W|W)); !upper/upper (siblings 1 and 2 are above ! threshold) (UU in Table 3)

Q = I-P; ! ((1-p) in Table 3)

S = I-R; ! ((1-r) in Table 3)

K = D + F.Q.S + E.Q.S + G.Q.Q.S.S_ ! 1 (expected probability of 0000)

F.R.Q + G.R.Q.Q.S_ ! 2 (expected probability of 0100)

E.R.Q + G.R.Q.Q.S_ ! 3 (expected probability of 0001)

F.P.S + G.P.Q.S.S_ ! 4 (expected probability of 1000)

E.P.S + G.P.Q.S.S_ ! 5 (expected probability of 0010)

F.P.R + G.P.R.Q.S_ ! 6 (expected probability of 1100)

E.P.R + G.P.R.Q.S_ ! 7 (expected probability of 0011)

G.R.R.Q.Q_ ! 8 (expected probability of 0101)

G.P.R.Q.S_ ! 9 (expected probability of 1001)

G.P.R.Q.S_ ! 10 (expected probability of 0110)

G.P.R.R.Q_ ! 11 (expected probability of 1101)

G.P.R.R.Q_ ! 12 (expected probability of 0111)

G.P.P.S.S_ ! 13 (expected probability of 1010)

G.P.P.R.S_ ! 14 (expected probability of 1110)

G.P.P.R.S_ ! 15 (expected probability of 1011)

G.P.P.R.R; ! 16 (expected probability of 1111)

end algebra;

Matrix J 1 1 1 1

Matrix M 0

Matrix V .159154943

Matrix P .5

Matrix R .5

Matrix W 0

Matrix Z 1

Specify L 100 100

Specify O 101 101

Specify N 102 102

Matrix L 1.5 1.5

SP H age1 age2;

SP Y sex1 sex2;

SP J -5 0 -5 0;

Means M;

Covariance V;

Weight (\part(K,J)) /

!Option user-defined RS

Option RS

End

G4 Evaluate model clinical group

Data NI = 6

ordinal fi = clinical.rec

Labels age1 age2 sex1 sex2 type dummy

Definition_variables age1 age2 sex1 sex2 type /

Matrices

A Full 2 2 = %E2 ! corr between A factors

I Iden 1 1

P Full 1 1 = P3! probability of getting A if above thresh (p in Table 3)

R Full 1 1 = R3! probability of getting B if above thresh (p in Table 3)

H Full 1 2 ! matrix for age definition variable

Y Full 1 2 ! matrix for sex definition variable

L Full 1 2 = L3! estimated threshold for A

O Full 1 2 = O3! age difference in threshold

N Full 1 2 = N3! sex difference in threshold

M Full 1 1 ! mean of dummy variable

V Full 1 1 ! variance of dummy variable

J Full 4 1 ! for extracting right part of K

W Full 1 1 ! 0 + + These are to control integral type

Z Full 1 1 ! 1 − −

b full 1 1 ! ascertainment parameter for illicit drug dependence

c full 1 1 free ! ascertainment parameter for alcohol dependence

begin algebra;

T = L + (H.O) + (Y.N);

D = \muln((A_T_T_Z|Z)); !lower/lower (LL in Table 3)

E = \muln((A_T_T_Z|W)); !lower/upper (LU in Table 3)

F = \muln((A_T_T_W|Z)); !upper/lower (UL in Table 3)

G = \muln((A_T_T_W|W)); !upper/upper (UU in Table 3)

Q = I-P; ! ((1-p) in Table 3)

S = I-R; ! ((1-r) in Table 3)

K = b.(F.R.Q + G.R.Q.Q.S)_ ! 2 (expected probability of 0100)

c.(F.P.S + G.P.Q.S.S)_ ! 4 (expected probability of 1000)

(b + c).(F.P.R + G.P.R.Q.S)_ ! 6 (expected probability of 1100)

b.(G.R.R.Q.Q)_ ! 8 (expected probability of 0101)

c.(G.P.R.Q.S)_ ! 9 (expected probability of 1001)

b.(G.P.R.Q.S)_ ! 10 (expected probability of 0110)

(b + c).(G.P.R.R.Q)_ ! 11 (expected probability of 1101)

b.(G.P.R.R.Q)_ ! 12 (expected probability of 0111)

c.(G.P.P.S.S)_ ! 13 (expected probability of 1010)

(b + c).(G.P.P.R.S)_ ! 14 (expected probability of 1110)

c.(G.P.P.R.S)_ ! 15 (expected probability of 1011)

(b + c).(G.P.P.R.R); ! 16 (expected probability of 1111)

end algebra;

Matrix J 1 1 1 1

Matrix M 0

Matrix V .159154943

Matrix W 0

Matrix Z 1

Matrix b 1

Matrix c .5

SP H age1 age2;

SP Y sex1 sex2;

SP J -5 0 -5 0;

Means M;

Covariance V;

Weight (\part(K@\sum(K)∼),J)/

Option RS

Option Th = -3

End