Abstract
Twin data can provide valuable insight into the relationship between the stages of phenomena such as disease or substance abuse. Initiation of substance use may be caused by factors that are the same as, partially shared with, or completely independent of those that cause progression from use to abuse. Comparison of rates of progression among the cotwins of twins who do vs. do not initiate provides indirect information about the relationship between initiation and progression. Existing models for this relationship have been difficult to extend because they are usually expressed in terms of explicit integrals. In this paper, the problem is overcome by regarding the analysis of twin data on initiation and progression as a special case of missing data, in which individuals who do not initiate are regarded as having missing data on progression measures. Using the general framework for the analysis of ordinal data with missing values available in Mx makes extensions that include other variables much easier. The effects of continuous covariates such as age on initiation and progression becomes simple. Also facilitated are the examination of initiation and progression in two or more substances, and transition models with two or more steps. The methods are illustrated with data on the effects of cohort on liability to cannabis use and abuse, bivariate analysis of tobacco use and dependence and cannabis use and abuse, and the relationships between initiation of smoking, regular smoking and nicotine dependence. Other suitable applications include the relationship between symptoms and diagnosis, such as fears and the progression to phobia.
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Notes
This assumption is a consequence of partitioning the variation in progression into components due to liability to initiate, and residual components.
References
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Acknowledgements
Michael Neale is grateful for support from PHS grants RR08123, MH01458, DA-18673. Eric Harvey was supported by NIMH training grant MH-20030, Hermine Maes by HL-60688, Patrick Sullivan by MH-59160, and Kenneth Kendler by AA-09095, MH/AA-49492 and DA-11287.
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Edited by Michael Stallings
Appendices
Appendix A
Mx script for fitting univariate model to data collected from twins using explicit calls to the multivariate normal distribution integration routine mnor
!
! \(\tt{Mx\ script\ for\ causal\ conditional\ model}\)
!
# \(\tt{ngroup}\) 6
\(\tt{Group\ 1\ Compute\ MZ\ Correlations}\)
\(\quad\tt{Calculation}\)
\(\tt{Begin\ Matrices;}\)
\(\quad\tt{A\ Di\ 2\ 2\ Free}\)
\(\quad\tt{C\ Di\ 2\ 2\ Free}\)
\(\quad\tt{E\ Di\ 2\ 2\ Free}\)
\(\quad\tt{B\ Fu\ 2\ 2}\)
\(\quad\tt{I\ Id\ 2\ 2}\)
\(\tt{End\ Matrices;}\)
\(\tt{Specify\ B\ !\ causal\ parameter\ from\ initiation\ to\ progression}\)
\(\quad\tt{0\ 0}\)
\(\quad\tt{7\ 0}\)
\(\tt{!\ starting values}\)
\(\tt{Matrix\ A\ .7\ .7}\)
\(\tt{Matrix\ C\ .5\ .5}\)
\(\tt{Matrix\ E\ .5\ .5}\)
\(\tt{Matrix\ B\ 0\ 0\ .4\ 0}\)
\(\tt{!\ parameter\ bounds}\)
\(\tt{\hbox{Bound 0 1 A 1 1 A 2 2 C 1 1 C 2 2}}\)
\(\tt{\hbox{Begin algebra;}}\)
\(\tt{\hbox{X = A*A' ;}}\)
\(\tt{\hbox{Y = C*C' ;}}\)
\(\tt{\hbox{Z = E*E' ;}}\)
\(\tt{\hbox{R = (I@((I-B)\~{}))\& (X+Y+Z| X+Y \_}}\)
\(\tt{\hbox{X+Y| X+Y+Z);}}\)
\(\tt{\hbox{End algebra;}}\)
\(\tt{\hbox{End group}}\)
\(\tt{\hbox{Group 2 DZ Correlation matrix}}\)
\(\tt{\hbox{Calculation}}\)
\(\tt{\hbox{Begin Matrices=Group 1;}}\)
\(\tt{\hbox{H Fu 1 1 ! .5}}\)
\(\tt{\hbox{ End Matrices;}}\)
\(\tt{\hbox{ Matrix H .5}}\)
\(\tt{\hbox{ Begin algebra;}}\)
\(\tt{\hbox{ R = (I@((I-B)\~{})) \&(X+Y+Z| h@X+Y \_}}\)
\(\tt{\hbox{h@X+Y| X+Y+Z);}}\)
\(\tt{\hbox{End algebra;}}\)
\(\tt{\hbox{End}}\)
\(\tt{\hbox{Fit model to MZ data with user-defined fit function (ML)}}\)
\(\tt{\hbox{Data Ni=1 No=1}}\)
\(\tt{\hbox{Begin Matrices;}}\)
\(\tt{\hbox{d full 1 1 ! two}}\)
\(\tt{\hbox{i zero 1 1}}\)
\(\tt{\hbox{n full 1 1 ! scalar 2.0}}\)
\(\tt{\hbox{o full 9 1}}\)
\(\tt{\hbox{r computed =R1 ! correlation matrix A1B1A2B2}}\)
\(\tt{\hbox{t full 1 4 ! thresholds abab}}\)
\(\tt{\hbox{w zero 1 4 ! means}}\)
\(\tt{\hbox{z unit 1 1}}\)
\(\tt{\hbox{End matrices;}}\)
\(\tt{\hbox{matrix d 2}}\)
\(\tt{\hbox{matrix n 2}}\)
\(\tt{\hbox{matrix o ! non, ex, current cell frequencies}}\)
\(\tt{\hbox{214 53 5}}\)
\(\tt{\hbox{55 117 17}}\)
\(\tt{\hbox{1 20 18}}\)
\(\tt{\hbox{! mnor function takes matrices with 4 more rows than columns.}}\)
\(\tt{\hbox{! first n (=4) rows are correlation matrix}}\)
\(\tt{\hbox{! row n+1 is mean vector}}\)
\(\tt{\hbox{! row n+2 is upper thresholds}}\)
\(\tt{\hbox{! row n+3 is lower threshold}}\)
\(\tt{\hbox{! row n+4 is indicator, 0 = integrate from -infinity to upper threshold}}\)
\(\tt{\hbox{! 1 = integrate from lower threshold to +infinity}}\)
\(\tt{\hbox{!}}\)
\(\tt{\hbox{Begin algebra ;}}\)
\(\tt{\hbox{e = -o.}\ln}\)
\(\tt{(\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{i}|\hbox{i}|\hbox{i}|\hbox{i})) +}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{i}|\hbox{i}|\hbox{i}|\hbox{z})) +}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{i}|\hbox{z}|\hbox{i}|\hbox{i})) +}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{i}|\hbox{z}|\hbox{i}|\hbox{z})) \_}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{i}|\hbox{i}|\hbox{z}|\hbox{i})) +}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{i}|\hbox{z}|\hbox{z}|\hbox{i})) \_}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{i}|\hbox{i}|\hbox{z}|\hbox{z})) +}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{i}|\hbox{z}|\hbox{z}|\hbox{z})) \_}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{z}|\hbox{i}|\hbox{i}|\hbox{i})) +}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{z}|\hbox{i}|\hbox{i}|\hbox{z})) \_}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{z}|\hbox{i}|\hbox{z}|\hbox{i})) \_}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{z}|\hbox{i}|\hbox{z}|\hbox{z})) \_}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{z}|\hbox{z}|\hbox{i}|\hbox{i})) +}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{z}|\hbox{z}|\hbox{i}|\hbox{z})) \_}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{z}|\hbox{z}|\hbox{z}|\hbox{i})) \_}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{z}|\hbox{z}|\hbox{z}|\hbox{z})));}\)
\(\tt{\hbox{End algebra ;}}\)
\(\tt{\hbox{Compute d.} \backslash \hbox{sum(e) ;}}\)
\(\tt{\hbox{Option user rs}}\)
\(\tt{\hbox{End}}\)
\(\tt{\hbox{Fit model to DZ data with user-defined fit function (ML)}}\)
\(\tt{\hbox{Data Ni=1 No=1}}\)
\(\tt{\hbox{Begin Matrices = Group 3;}}\)
\(\tt{\hbox{! re-declare o and r as they are different for DZ's}}\)
\(\tt{\hbox{o full 9 1}}\)
\(\tt{\hbox{r comp =R2 ! correlation matrix A1B1A2B2}}\)
\(\tt{\hbox{End matrices;}}\)
\(\tt{\hbox{Specify t 10 11 10 11 ! equate thresholds for twin 1 \& twin 2 (and MZ/DZ)}}\)
\(\tt{\hbox{Matrix t .2 .0 .2 .0}}\)
\(\tt{\hbox{Matrix o}}\)
\(\tt{\hbox{100 52 3}}\)
\(\tt{\hbox{45 82 16}}\)
\(\tt{\hbox{6 18 4}}\)
\(\tt{\hbox{Bound -2 3 10 11}}\)
\(\tt{\hbox{Begin algebra ;}}\)
\(\tt{\hbox{e = -(o).} \ln}\)
\(\tt{(\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{i}|\hbox{i}|\hbox{i}|\hbox{i})) +}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{i}|\hbox{i}|\hbox{i}|\hbox{z})) +}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{i}|\hbox{z}|\hbox{i}|\hbox{i})) +}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{i}|\hbox{z}|\hbox{i}|\hbox{z})) \_}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{i}|\hbox{i}|\hbox{z}|\hbox{i})) +}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{i}|\hbox{z}|\hbox{z}|\hbox{i})) \_}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{i}|\hbox{i}|\hbox{z}|\hbox{z})) +}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{i}|\hbox{z}|\hbox{z}|\hbox{z})) \_}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{z}|\hbox{i}|\hbox{i}|\hbox{i})) +}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{z}|\hbox{i}|\hbox{i}|\hbox{z})) \_}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{z}|\hbox{i}|\hbox{z}|\hbox{i})) \_}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{z}|\hbox{i}|\hbox{z}|\hbox{z})) \_}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{z}|\hbox{z}|\hbox{i}|\hbox{i})) +}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{z}|\hbox{z}|\hbox{i}|\hbox{z})) \_}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{z}|\hbox{z}|\hbox{z}|\hbox{i})) \_}\)
\(\tt{\backslash\hbox{mnor }(\hbox{r}\_\hbox{w}\_\hbox{t}\_\hbox{t}\_(\hbox{z}|\hbox{z}|\hbox{z}|\hbox{z})));}\)
\(\tt{\hbox{End algebra ;}}\)
\(\tt{\hbox{Compute d.} \backslash\hbox{sum(e);}}\)
\(\tt{\hbox{Option user}}\)
\(\tt{\hbox{End}}\)
\(\tt{\hbox{Group 5 constrain variances to 1}}\)
\(\tt{\hbox{Constraint NI=1}}\)
\(\tt{\hbox{Begin Matrices = Group 1;}}\)
\(\tt{\hbox{U Unit 1 2}}\)
\(\tt{\hbox{z izero 4 2}}\)
\(\tt{\hbox{End matrices;}}\)
\(\tt{\hbox{Constraint U} = \backslash\hbox{d2v(r)*z;}}\)
\(\tt{\hbox{End}}\)
\(\tt{\hbox{Group 6 - standardize estimates}}\)
\(\tt{\hbox{Calculation}}\)
\(\tt{\hbox{Begin Matrices = Group 1;}}\)
\(\tt{\hbox{I Id 2 2}}\)
\(\tt{\hbox{J iden 4 4}}\)
\(\tt{\hbox{End matrices;}}\)
\(\tt{\hbox{Begin algebra;}}\)
\(\tt{\hbox{K = I@((I-B)\~{});}}\)
\(\tt{\hbox{L} = \backslash\hbox{v2d}(\backslash\hbox{sqrt}(\backslash\hbox{d2v((I@((I-B)\~{}))*(X+Y+Z}| \hbox{X+Y \_}}\)
\(\tt{\hbox{X+Y}|\hbox{ X+Y+Z)* (I@((I-B)\~{})'))));}}\)
\(\tt{\hbox{M = L\~{}*K*L;}}\)
\(\tt{\hbox{R} = \backslash\hbox{d2v(X+Y+Z);}}\)
\(\tt{\hbox{S} = ((\backslash\hbox{d2v(X))\%R)}\_((\backslash\hbox{d2v(Y))\%R)}\_((\backslash\hbox{d2v(Z))\%R);}}\)
\(\tt{\hbox{End algebra;}}\)
\(\tt{\hbox{Labels row S}}\)
\(\tt{\hbox{A\_A A\_B C\_A C\_B E\_A E\_B}}\)
\(\tt{\hbox{Labels row L}}\)
\(\tt{\hbox{MZT1Abeta MZT1Bbeta}}\)
\(\tt{\hbox{MZT1Abeta MZT1Bbeta}}\)
\(\tt{\hbox{option func=1.e-10}}\) | \(\tt{\hbox{! function precision for optimization}}\) |
\(\tt{\hbox{option df=18}}\) | \(\tt{\hbox{! adjust df}}\) |
\(\tt{\hbox{option nd=4}}\) | \(\tt{\hbox{! 4 decimal places}}\) |
\(\tt{\hbox{option eps=.00000001}}\) | \(\tt{\hbox{! integration precision for mnor}}\) |
\(\tt{\hbox{option th=-2}}\) | \(\tt{\hbox{! retry optimization from final point twice}}\) |
\(\tt{\hbox{End}}\)
Appendix B
Mx script for fitting conditional causal model including cohort/age effects
\(\tt{\hbox{! CCC with age}}\)
\(\tt{\hbox{\#ngroup 6}}\)
\(\tt{\hbox{Group 1 Compute MZ Correlations}}\)
\(\tt{\hbox{Calculation}}\)
\(\tt{\hbox{Begin Matrices;}}\)
\(\tt{\hbox{A Di 2 2 Free}}\)
\(\tt{\hbox{C Di 2 2 Free}}\)
\(\tt{\hbox{E Di 2 2 Free}}\)
\(\tt{\hbox{B Fu 2 2}}\)
\(\tt{\hbox{I Id 2 2}}\)
\(\tt{\hbox{End matrices;}}\)
\(\tt{\hbox{Specify B ! causal parameter from initiation to progression}}\)
\(\tt{\hbox{0 0}}\)
\(\tt{\hbox{7 0}}\)
\(\tt{\hbox{! starting values}}\)
\(\tt{\hbox{Matrix A .7 .7}}\)
\(\tt{\hbox{Matrix C .5 .5}}\)
\(\tt{\hbox{Matrix E .5 .5}}\)
\(\tt{\hbox{Matrix B 0 0 .4 0}}\)
\(\tt{\hbox{! parameter bounds}}\)
\(\tt{\hbox{Bound 0 1 A 1 1 A 2 2 C 1 1 C 2 2}}\)
\(\tt{\hbox{Begin algebra;}}\)
\(\tt{\hbox{X = A*A' ;}}\)
\(\tt{\hbox{Y = C*C' ;}}\)
\(\tt{\hbox{Z = E*E' ;}}\)
\(\tt{\hbox{R = (I@((I-B)\~{})) \& (X+Y+Z}|\hbox{ X+Y \_}}\)
\(\tt{\hbox{X+Y}|\hbox{ X+Y+Z);}}\)
\(\tt{\hbox{End algebra;}}\)
\(\tt{\hbox{End group}}\)
\(\tt{\hbox{Group 2 Compute DZ Correlation matrix}}\)
\(\tt{\hbox{Calculation}}\)
\(\tt{\hbox{Begin Matrices=Group 1;}}\)
\(\tt{\hbox{H Fu 1 1 ! .5}}\)
\(\tt{\hbox{End matrices;}}\)
\(\tt{\hbox{Matrix H .5}}\)
\(\tt{\hbox{Begin algebra;}}\)
\(\tt{\hbox{R = (I@((I-B)\~{})) \&(X+Y+Z}|\hbox{ h@X+Y \_}}\)
\(\tt{\hbox{h@X+Y}|\hbox{ X+Y+Z);}}\)
\(\tt{\hbox{End algebra;}}\)
\(\tt{\hbox{End}}\)
\(\tt{\hbox{Fit model to MZ data}}\)
\(\tt{\hbox{Data Ninput=10}}\)
\(\tt{\hbox{Labels zyg agea nicusea nicpc2a canusea canabua}}\)
\(\tt{\hbox{nicuseb nicpc2b canuseb canabub}}\)
\(\tt{\hbox{Ordinal file=ffpair2.rec}}\)
\(\tt{\hbox{Select if zyg = 1}}\)
\(\tt{\hbox{Select agea canusea canabua canuseb canabub ;}}\)
\(\tt{\hbox{Definition agea ;}}\)
\(\tt{\hbox{Begin Matrices;}}\)
\(\tt{\hbox{a full 1 1}}\)
\(\tt{\hbox{i zero 1 1}}\)
\(\tt{\hbox{n full 1 1 ! scalar 2.0}}\)
\(\tt{\hbox{o full 9 1}}\)
\(\tt{\hbox{r full 4 4 =R1 ! correlation matrix A1B1A2B2}}\)
\(\tt{\hbox{t full 1 4 ! thresholds abab}}\)
\(\tt{\hbox{u full 1 4 ! thresholds abab}}\)
\(\tt{\hbox{w zero 1 4 ! means}}\)
\(\tt{\hbox{z unit 1 1}}\)
\(\tt{\hbox{End matrices;}}\)
\(\tt{\hbox{Specify a agea ! A gets updated with age for each case}}\)
\(\tt{\hbox{! during calculation of covariances and thresholds}}\)
\(\tt{\hbox{matrix n 2}}\)
\(\tt{\hbox{covariance r ;}}\)
\(\tt{\hbox{thresholds t+u@a ;}}\)
\(\tt{\hbox{option rs}}\)
\(\tt{\hbox{End}}\)
\(\tt{\hbox{Fit model to DZ data}}\)
\(\tt{\hbox{Data Ninput=10}}\)
\(\tt{\hbox{Labels zyg agea nicusea nicpc2a canusea canabua}}\)
\(\tt{\hbox{nicuseb nicpc2b canuseb canabub}}\)
\(\tt{\hbox{Ordinal file=ffpair2.rec}}\)
\(\tt{\hbox{Select if zyg = 2}}\)
\(\tt{\hbox{Select agea canusea canabua canuseb canabub ;}}\)
\(\tt{\hbox{Definition agea ;}}\)
\(\tt{\hbox{Begin matrices =Group 3;}}\)
\(\tt{\hbox{o full 9 1}}\)
\(\tt{\hbox{r full 4 4 =R2 ! correlation matrix A1B1A2B2}}\)
\(\tt{\hbox{End matrices;}}\)
\(\tt{\hbox{specify t 10 11 10 11}}\)
\(\tt{\hbox{Matrix t .2 .0 .2 .0}}\)
\(\tt{\hbox{specify a agea}}\)
\(\tt{\hbox{specify u 12 13 12 13}}\)
\(\tt{\hbox{matrix u .01 .01 .01 .01}}\)
\(\tt{\hbox{bound -.05 .05 u 1 1 U 1 2}}\)
\(\tt{\hbox{Bound -2 3 10 11}}\)
\(\tt{\hbox{covariance r ;}}\)
\(\tt{\hbox{Thresholds t + u@a;}}\)
\(\tt{\hbox{End}}\)
\(\tt{\hbox{Group 5 - constrain variances = 1}}\)
\(\tt{\hbox{Constraint NI=1}}\)
\(\tt{\hbox{Begin Matrices = Group 1}}\)
\(\tt{\hbox{U Unit 1 2}}\)
\(\tt{\hbox{V iz 4 2}}\)
\(\tt{\hbox{End matrices;}}\)
\(\tt{\hbox{Constraint U} = \backslash\hbox{d2v(R)*V ;}}\)
\(\tt{\hbox{End}}\)
\(\tt{\hbox{Group 6 - standardize estimates}}\)
\(\tt{\hbox{Data calc}}\)
\(\tt{\hbox{Begin Matrices;}}\)
\(\tt{\hbox{A Di 2 2 = A1}}\)
\(\tt{\hbox{C Di 2 2 = C1}}\)
\(\tt{\hbox{E Di 2 2 = E1}}\)
\(\tt{\hbox{B Fu 2 2 = B1}}\)
\(\tt{\hbox{I Id 2 2}}\)
\(\tt{\hbox{H Fu 1 1 ! .5}}\)
\(\tt{\hbox{J iden 4 4}}\)
\(\tt{\hbox{End matrices;}}\)
\(\tt{\hbox{Begin algebra;}}\)
\(\tt{\hbox{X = A*A' ;}}\)
\(\tt{\hbox{Y = C*C' ;}}\)
\(\tt{\hbox{Z = E*E' ;}}\)
\(\tt{\hbox{K = I@((I-B)\~{});}}\)
\(\tt{\hbox{L} = \backslash\hbox{v2d}(\backslash\hbox{sqrt}(\backslash\hbox{d2v((I@((I-B)\~{}))*(X+Y+Z}|\hbox{ X+Y \_}}\)
\(\tt{\hbox{X+Y}|\hbox{ X+Y+Z)* (I@((I-B)\~{})'))));}}\)
\(\tt{\hbox{M = L\~{}*K*L;}}\)
\(\tt{\hbox{R} = \backslash\hbox{d2v(X+Y+Z);}}\)
\(\tt{\hbox{S} = ((\backslash\hbox{d2v(X))\%R})\_((\backslash\hbox{d2v(Y))\%R})\_((\backslash\hbox{d2v(Z))\%R);}}\)
\(\tt{\hbox{End algebra;}}\)
\(\tt{\hbox{Labels row S}}\)
\(\tt{\hbox{A\_A A\_B C\_A C\_B E\_A E\_B}}\)
\(\tt{\hbox{Labels row L}}\)
\(\tt{\hbox{MZT1Abeta MZT1Bbeta}}\)
\(\tt{\hbox{MZT1Abeta MZT1Bbeta}}\)
\(\tt{\hbox{Interval B 1 2 1}}\)
\(\tt{\hbox{option mu nd=4}}\)
\(\tt{\hbox{option nag=10 db=1}}\)
\(\tt{\hbox{option func=1.e-8}}\)
\(\tt{\hbox{option th=-2}}\)
\(\tt{\hbox{option multiple issat}}\)
\(\tt{\hbox{End}}\)
\(\tt{\hbox{!fit submodel without age effect}}\)
\(\tt{\hbox{save cccage.mxs}}\)
\(\tt{\hbox{drop 12 13}}\)
\(\tt{\hbox{End}}\)
Appendix C
Mx script for fitting bivariate conditional causal model for initiation and progression in pairs of twins
\(\tt{\hbox{! Bivariate Genetic Cholesky Model CCC}}\)
\(\tt{\hbox{! Simulated ordinal data}}\)
\(\tt{\hbox{\#ngroups 4}}\)
\(\tt{\hbox{\#define nthresh1 1}}\)
\(\tt{\hbox{\#define nthresh2 1}}\)
\(\tt{\hbox{\#define nvar 4}}\)
\(\tt{\hbox{Group 1: set up model}}\)
\(\tt{\hbox{Calculation}}\)
\(\tt{\hbox{Begin Matrices;}}\)
\(\tt{\hbox{B Full nvar nvar Free}}\) | \(\tt{\hbox{! causal pathways}}\) |
\(\tt{\hbox{J Iden 2 2}}\) | |
\(\tt{\hbox{K Iden 4 4}}\) | |
\(\tt{\hbox{X diag nvar nvar Free}}\) | \(\tt{\hbox{! genetic structure}}\) |
\(\tt{\hbox{Y diag nvar nvar Free}}\) | \(\tt{\hbox{! common environmental structure}}\) |
\(\tt{\hbox{Z diag nvar nvar Free}}\) | \(\tt{\hbox{! specific environmental structure}}\) |
\(\tt{\hbox{W Lower nvar nvar}}\) | \(\tt{\hbox{! dominance structure (set to zero)}}\) |
\(\tt{\hbox{T Full nthresh2 8 Free}}\) |
\(\tt{\hbox{End matrices;}}\)
\(\tt{\hbox{Bound .0 2 X 1 1 X 2 2 X 3 3 X 4 4}}\)
\(\tt{\hbox{Bound .0 2 Y 1 1 Y 2 2 Y 3 3 Y 4 4}}\)
\(\tt{\hbox{Bound .1 2 Z 1 1 Z 2 2 Z 3 3 Z 4 4}}\)
\(\tt{\hbox{Matrix T}}\)
\(\tt{\hbox{0 0 0 0 0 0 0 0}}\)
\(\tt{\hbox{Bound -3 3 T 1 1 - T 1 8}}\)
\(\tt{\hbox{Specify B}}\)
\(\tt{\hbox{0 0 104 0}}\)
\(\tt{\hbox{101 0 105 0}}\)
\(\tt{\hbox{102 0 0 0}}\)
\(\tt{\hbox{103 0 106 0}}\)
\(\tt{\hbox{Bound -.99 .99 B 1 1 to B 4 4}}\)
\(\tt{\hbox{Labels Col B IS DS IC DC}}\)
\(\tt{\hbox{Labels Row B IS DS IC DC}}\)
\(\tt{\hbox{Matrix X .8 .7071 .6 .5}}\)
\(\tt{\hbox{Matrix Z .6 .5 .8 .7071}}\)
\(\tt{\hbox{Matrix B}}\)
\(\tt{\hbox{0 0 .3 0}}\)
\(\tt{\hbox{.5 0 .3 0}}\)
\(\tt{\hbox{.3 0 0 0}}\)
\(\tt{\hbox{.3 0 .5 0}}\)
\(\tt{\hbox{Begin algebra;}}\)
\(\tt{\hbox{A= X*X' ;}}\)
\(\tt{\hbox{C= Y*Y' ;}}\)
\(\tt{\hbox{E= Z*Z' ;}}\)
\(\tt{\hbox{D= W*W' ;}}\)
\(\tt{\hbox{F= (J@(K-B))\~{} ;}}\)
\(\tt{\hbox{End algebra;}}\)
\(\tt{\hbox{End}}\)
\(\tt{\hbox{Group 2: Fit model to MZ twin pairs}}\)
\(\tt{\hbox{Data Ninput=10}}\)
\(\tt{\hbox{Labels zyg agea nicusea nicpc2a canusea canabua}}\)
\(\tt{\hbox{nicuseb nicpc2b canuseb canabub}}\)
\(\tt{\hbox{Ordinal file=ffinits.rec}}\)
\(\tt{\hbox{Select if zyg = 1}}\)
\(\tt{\hbox{select nicusea nicpc2a canusea canabua nicuseb nicpc2b canuseb canabub ;}}\)
\(\tt{\hbox{Begin Matrices= Group 1;}}\)
\(\tt{\hbox{Covariances F\&(A+C+E+D}|\hbox{ A+C+D \_}}\)
\(\tt{\hbox{A+C+D} |\hbox{ A+C+E+D) /}}\)
\(\tt{\hbox{Thresholds T ;}}\)
\(\tt{\hbox{Option RSidual}}\)
\(\tt{\hbox{End}}\)
\(\tt{\hbox{Group 3: Fit model to DZ twin pairs}}\)
\(\tt{\hbox{Data Ninput=10}}\)
\(\tt{\hbox{Labels zyg agea nicusea nicpc2a canusea canabua}}\)
\(\tt{\hbox{nicuseb nicpc2b canuseb canabub}}\)
\(\tt{\hbox{Ordinal file=ffinits.rec}}\)
\(\tt{\hbox{Select if zyg = 2}}\)
\(\tt{\hbox{select nicusea nicpc2a canusea canabua nicuseb nicpc2b canuseb canabub ;}}\)
\(\tt{\hbox{Begin Matrices= Group 1;}}\)
\(\tt{\hbox{H Full 1 1}}\)
\(\tt{\hbox{Q Full 1 1}}\)
\(\tt{\hbox{End matrices;}}\)
\(\tt{\hbox{Matrix H .5}}\)
\(\tt{\hbox{Matrix Q .25}}\)
\(\tt{\hbox{Covariances F\&(A+C+E+D} |\hbox{ H@A+C+Q@D \_}}\)
\(\tt{\hbox{H@A+C+Q@D} |\hbox{ A+C+E+D) /}}\)
\(\tt{\hbox{Thresholds T ;}}\)
\(\tt{\hbox{Option RSidual}}\)
\(\tt{\hbox{Options NDecimals=4}}\)
\(\tt{\hbox{option func=1.e-8}}\)
\(\tt{\hbox{End}}\)
\(\tt{\hbox{G4: Constrain variances}}\)
\(\tt{\hbox{Constraint NI=1}}\)
\(\tt{\hbox{Begin Matrices;}}\)
\(\tt{\hbox{U unit 1 4}}\)
\(\tt{\hbox{E symm 8 8 = \%e2}}\)
\(\tt{\hbox{Z iz 8 4}}\)
\(\tt{\hbox{End matrices;}}\)
\(\tt{\hbox{Constraint U} = \backslash\hbox{d2v(E) * Z ;}}\)
\(\tt{\hbox{Option}}\)
\(\tt{\hbox{End}}\)
Appendix D
Mx script for fitting three stage/two transition causal model for initiation and two progressions in pairs of twins
\(\tt{\hbox{! Bivariate Genetic Cholesky Model CCC}}\)
\(\tt{\hbox{! Simulated ordinal data}}\)
\(\tt{\hbox{!}}\)
\(\tt{\hbox{\#ngroups 4}}\)
\(\tt{\hbox{\#define nthresh1 1}}\)
\(\tt{\hbox{\#define nthresh2 1}}\)
\(\tt{\hbox{\#define nvar 3}}\)
\(\tt{\hbox{G1: set up model}}\)
\(\tt{\hbox{Calculation}}\)
\(\tt{\hbox{Begin Matrices;}}\)
\(\tt{\hbox{B Full nvar nvar Free}}\) | \(\tt{\hbox{! causal pathways}}\) |
\(\tt{\hbox{J Iden 2 2}}\) | |
\(\tt{\hbox{K Iden nvar nvar}}\) | |
\(\tt{\hbox{I Lower nthresh2 nthresh2}}\) | |
\(\tt{\hbox{X diag nvar nvar Free}}\) | \(\tt{\hbox{! genetic structure}}\) |
\(\tt{\hbox{Y diag nvar nvar Free}}\) | \(\tt{\hbox{! common environmental structure}}\) |
\(\tt{\hbox{Z diag nvar nvar Free}}\) | \(\tt{\hbox{! specific environmental structure}}\) |
\(\tt{\hbox{W Lower nvar nvar}}\) | \(\tt{\hbox{! dominance structure}}\) |
\(\tt{\hbox{T Full nthresh2 6 Free}}\) |
\(\tt{\hbox{End matrices;}}\)
\(\tt{\hbox{Bound .0 1 X 1 1 X 2 2 X 3 3}}\)
\(\tt{\hbox{Bound .0 1 Y 1 1 Y 2 2 Y 3 3}}\)
\(\tt{\hbox{Bound .1 1 Z 1 1 Z 2 2 Z 3 3}}\)
\(\tt{\hbox{Bound -3 3 T 1 1 - T 1 6}}\)
\(\tt{\hbox{Specify B}}\)
\(\tt{\hbox{0 0 0}}\)
\(\tt{\hbox{101 0 0}}\)
\(\tt{\hbox{0 1020 0}}\)
\(\tt{\hbox{Bound -.99 .99 B 2 1 B 3 2}}\)
\(\tt{\hbox{Labels Col B IS RS ND}}\)
\(\tt{\hbox{Labels Row B IS RS ND}}\)
\(\tt{\hbox{Matrix X .8 .7071 .6}}\)
\(\tt{\hbox{Matrix Z .6 .5 .8}}\)
\(\tt{\hbox{Matrix B 0 0 0 .5 0 0 0 .5 0}}\)
\(\tt{\hbox{Begin algebra;}}\)
\(\tt{\hbox{A= X*X' ;}}\)
\(\tt{\hbox{C= Y*Y' ;}}\)
\(\tt{\hbox{E= Z*Z' ;}}\)
\(\tt{\hbox{D= W*W' ;}}\)
\(\tt{\hbox{F= (J@(K-B))\~{} ;}}\)
\(\tt{\hbox{End algebra;}}\)
\(\tt{\hbox{End}}\)
\(\tt{\hbox{G2: MZ twin pairs}}\)
\(\tt{\hbox{\#include patccc1.dat}}\)
\(\tt{\hbox{Select if zyg = 1}}\)
\(\tt{\hbox{select evera rega nda everb regb ndb ;}}\)
\(\tt{\hbox{Begin Matrices= Group 1;}}\)
\(\tt{\hbox{Covariances F\&(A+C+E+D} |\hbox{ A+C+D \_}}\)
\(\tt{\hbox{A+C+D} |\hbox{ A+C+E+D) /}}\)
\(\tt{\hbox{Thresholds T ;}}\)
\(\tt{\hbox{Option RSidual}}\)
\(\tt{\hbox{End}}\)
\(\tt{\hbox{G3: DZ twin pairs}}\)
\(\tt{\hbox{\#include patccc1.dat}}\)
\(\tt{\hbox{Select if zyg = 2}}\)
\(\tt{\hbox{select evera rega nda everb regb ndb ;}}\)
\(\tt{\hbox{Begin Matrices= Group 1;}}\)
\(\tt{\hbox{H Full 1 1}}\)
\(\tt{\hbox{Q Full 1 1}}\)
\(\tt{\hbox{End matrices;}}\)
\(\tt{\hbox{Matrix H .5}}\)
\(\tt{\hbox{Matrix Q .25}}\)
\(\tt{\hbox{Covariances F\&(A+C+E+D} |\hbox{ H@A+C+Q@D \_}}\)
\(\tt{\hbox{H@A+C+Q@D} |\hbox{ A+C+E+D) /}}\)
\(\tt{\hbox{Thresholds T ;}}\)
\(\tt{\hbox{Option RSidual}}\)
\(\tt{\hbox{Options NDecimals=4}}\)
\(\tt{\hbox{option func=1.e-8}}\)
\(\tt{\hbox{End}}\)
\(\tt{\hbox{G4: Constrain variances}}\)
\(\tt{\hbox{Constraint NI=1}}\)
\(\tt{\hbox{Begin Matrices;}}\)
\(\tt{\hbox{U unit 1 nvar}}\)
\(\tt{\hbox{E symm 6 6 = \%e2}}\)
\(\tt{\hbox{Z iz 6 nvar}}\)
\(\tt{\hbox{End matrices;}}\)
\(\tt{\hbox{Constraint U} = \backslash\hbox{d2v(E) * Z ;}}\)
\(\tt{\hbox{Option Multiple th=-2}}\)
\(\tt{\hbox{End}}\)
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Neale, M., Harvey, E., Maes, H. et al. Extensions to the Modeling of Initiation and Progression: Applications to Substance Use and Abuse. Behav Genet 36, 507–524 (2006). https://doi.org/10.1007/s10519-006-9063-x
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DOI: https://doi.org/10.1007/s10519-006-9063-x