3 Building a Path Coefficients Model Mx Script

With the introduction from the previous sections and chapters, we are now in a position to set up a simple genetic model using Mx. The script in Appendix  fits a simple univariate genetic model, estimating path coefficients, to covariance matrices for two like-sex twin groups: MZ twin pairs reared together, and DZ twin pairs reared together. The script is written to ignore information on means. The full path diagram is given in Figure 6.1

Figure 6.1: Univariate genetic model for data from monozygotic (MZ) or dizygotic (DZ) twins reared together. Genetic and environmental latent variables cause the phenotypes $P_1$ and $P_2$

We have drawn this figure to correspond to the variables in the model. The latent genetic and environmental variables $A, C, E$ and $D$ cause the observed variables $P_1$ and $P_2$. The script is written to fit a model with free parameters $e, a$, and $d$, and fixing $c$ to zero -- implying that there are no effects of shared environment on BMI. The script is extensively documented using the comment facility in Mx: any line beginning with an exclamation mark is interpreted as a comment. We shall consider this first example Mx script in detail. Please note that reading this section is not a substitute for reading in detail the Mx manual [], but merely a quick introduction to the essentials of a Mx script for genetic applications. Each new statement in a script begins on a new line. For each group, we will have the following structure:

1. Title
2. Group type
3. Read and select any observed data, supply labels
4. Declare matrices to express the model
5. Specify parameters, (starting) values, equality constraints
6. Define matrix formulae for the model
7. Request fit functions, output and optimization options
8. End

We shall now examine the structure in greater detail, focusing on our BMI model. We plan to test hypotheses about the contributions of genetic and environmental factors to individual differences in BMI using data collected from MZ and DZ twins reared together. The Mx script therefore will have at least two groups. To simplify the structure of the script, we have added a calculation group at the beginning. We start the Mx script by indicating how many groups our job consists of with the #NGroups 3 statement.

• Title

  Calculate genetic and environmental variance components
  

  A new title must be given at the start of each group.
• Set the Group Type

  Calculation NGroups=3
  

  where NGroups is the number of groups. This parameter is specified for the first group only. Calculation groups allow the specification of matrix operations in an algebra section which can greatly simplify the structure of the script. Here, we use the calculation group to specify the free and fixed parameters in the model, $e, a, d$, and $c$, and calculate their squared quantities, to be used in the expectations of the variances and the MZ and DZ covariances of the model (see Section ).
• Matrices Declaration

  Begin Matrices;
   X Lower 1 1 Free
   Y Lower 1 1 Fixed
   Z Lower 1 1 Free
   W Lower 1 1 Free
   H Full 1 1
   Q Full 1 1
  End Matrices;
  

  The matrices declaration section begins with a Begin Matrices; line and ends with a End Matrices; line. Up to 26 matrices can be declared, each starting on a new line. Matrix names are restricted to one letter, from A to Z. The name is followed by the matrix type (see Mx manual for details on available matrix types), the number of rows and the number of colums. All matrix elements are fixed by default. If the keyword Free appears, each modifiable element has a free parameter specified to be estimated. In this example, four 1 $\times$ 1 matrices have been declared. Matrices X, Z and W represent free parameters a, e and d, respectively. The parameter c in matrix Y is fixed to zero (the word Fixed appears only for clarification). Two additional matrices, H and Q, are declared for fixed scalars to be used in the model specification.
• Labels, Numbers and Parameters

   Label Row X add_gen
   Label Row Y comm_env
   Label Row Z spec_env
   Label Row W dom_gen
   Matrix H .5
   Matrix Q .25
   Start .6 All
  

  Labels can be given for the row or column (or both) of any matrix. Values can be assigned to matrix elements using the Matrix command. If the matrix element is modifiable, the assigned value will be the starting value. The Start command here is used to assign the same starting value to All the free parameters in the model. In genetic problems, we must assign starting values to parameters. In the present case, the only parameters to be estimated are a, d and e. In choosing starting values for twin data, a useful rule of thumb is to assume that the total variance is divided equally between the parameters that are to be estimated. In this case the predicted total variance is $3\times .6^2 = 1.08$ which is close to the observed total variance in these data. For other data, other starting values may be required. Good starting values can save a significant amount of computer time, whereas bad starting values may cause any optimizer to fail to find a global minimum, or to hit a maximum number of iterations before converging.
• Algebra Section

  Begin Algebra;
   A= X*X';
   C= Y*Y';
   E= Z*Z';
   D= W*W';
  End Algebra;
  

  The algebra section begins with a Begin Algebra; statement and ends with a End Algebra; statement. Each algebra operation starts on a new line and ends with a semi-colon (it may run over several lines so a ; is essential to mark the end of a formula). The matrix on the left side of the = sign is newly defined as the result of the matrix operation on the right side of the = sign. Matrices on the right have to be declared in the matrices declaration section or defined in a previous algebra statement. In this example the quantities $a^{2}$, $c^{2}$, $e^{2}$ and $d^{2}$ are calculated in matrices A, C, E and D, respectively.
• End
  
  Every group ends with an End statement.

The structure for the data groups for MZ and DZ twins is very similar. We will only discuss the first data group in detail. The first line gives the title for this group.

• Data Section

  Data NInput_vars=2 NObservations=534
   Labels bmi_t1 bmi_t2
   CMatrix Symmetric File=ozbmimzf.cov
  

  where:
  1. NInputvars is number of input variables, i.e., $2n$, if there are $n$ variables assessed for each member of a twin pair
  2. NObservations is number of observations or sample size, i.e., number of pairs used to compute the data matrix in this group.
  Mx allows the user the option of reading a list of names for the observed variables (Labels). This is very useful for clarification of the Mx output. Mx will read a covariance matrix (CMatrix), a correlation matrix (KMatrix), or a matrix of polychoric and polyserial correlations (PMatrix). The matrix may be read as a lower triangle in free format (the default, the keyword Symmetric is optional), or as a full matrix if the keyword Full is specified. It will also read means (Means) when these are needed. Summary statistics can be read from within the Mx script, for example,

  CMatrix Symmetric 
  0.7247
  0.5891 0.7915
  

  Alternatively, the data matrices can be read from separate files, e.g.,

  CMatrix Symmetric File=ozbmimzf.cov
  

  The lines referring to the actual data, Data, Labels and CMatrix can be saved in a dat file (e.g. ozbmimzf.dat which can then be included in the Mx script with the following statement:

  #include ozbmimzf.dat
  

• Matrices declaration

  Matrices= Group 1
  

  The = Group 1 command includes all the declared and defined matrices from the group 1 into the current group.
• Model specification

  Covariances A+C+D+E | A+C+D_
              A+C+D   | A+C+D+E;
  

  The expected covariance matrix is specified using a matrix formulation with the expected variances for twin 1 and twin 2 on the diagonal and the expected covariance between twins, in this case for MZ's, as the off-diagonal element. The expectation for the variance, $a^2+c^2+d^2+e^2$, is translated into A+C+D+E; that for the MZ covariance, $a^2+c^2+d^2$, into A+C+D. The resulting four 1 $\times$ 1 matrices are concatenated using the horizontal bar | and the vertical bar _ operators to form the 2 $\times$ 2 expected covariance matrix, corresponding to the 2 $\times$ 2 observed covariance matrix. Note that the covariance statement needs to end with a semicolon.
• Options

  Option RSiduals
  

  Various options for statistical output and optimization can be specified. Usually, the choice of estimation procedure will be either maximum likelihood if covariance matrices are being analyzed, or weighted least squares if matrices of polychoric, polyserial, or product-moment correlations are being analyzed. The RSiduals option is very useful as it results in the printing of the observed, expected and residual matrices in the output.

The specification for the DZ group is very similar to that of the MZ group. Note the different number of observations, the new filename containing the DZ observed covariance matrix and the expected covariance matrix to match the expectation of the DZ covariance, $.5a^2+c^2+.25d^2$. A special form of matrix multiplication, the Kronecker product, represented by the symbol @, is used to premultiply the matrix A by the scalar .5 and the matrix D by the scalar .25. The specification extends easily to the multivariate case (see Section ?).

After successfully running the Mx input script, by default, Mx prints the

1. User's input script
2. Parameter Specifications
3. Parameter Estimates
4. Measures of overall goodness-of-fit.

Other useful output can be requested by additional options, including:

• NDecimals=$x$ - set number of decimals in printed output ($0<x<8$, default: $x$ =4) -- useful for simulation work.
• Iterations=$xx$ - set maximum number of iterations (default: 1000).

The Mx manual should be consulted for a full description of the options.