... meioses

meiosis is the process of gametogenesis in which either sperm or ova are formed

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...$bb$

This notation is described more fully in Chapter 3.

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... environment

Twins born serially by embryo implantation are currently far too rare for the purposes of statistical distinction between pre- and post-natal effects!

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... twins

These data are for illustration only; they would normally be treated as ordinal, not continuous, and would be summarized differently, as described in Section 2.3. Note also that we do not need to have equal numbers of pairs in the two groups.

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... pair

It is possible to use data files that contain both types of twins and some code to discriminate between them, but it is less efficient.

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... table

Mathematically these expected proportions can be written as double integrals. We do not explicitly define them here, but return to the subject in the context of ascertainment discussed in Chapter 

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... inheritance"

In fact quite a small number of genetic factors may give rise to a distribution which is for almost all practical purposes indistinguishable from a normal distribution [Kendler and Kidd, 1986].

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... matrices

The number of elements in a weight matrix for a covariance matrix is greater than that for a correlation matrix. For this reason, it is necessary to specify Matrix=PMatrix on the Data line of a Mx job that is to read a weight matrix.

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... inches

Note: 1 inch = 2.54cm; 1 foot = 12 inches.

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... mean

This is an application of the method described in Section 2.2.1. It looks a bit more intimidating here because of (a) the multiplication by the frequency, and (b) the use of letters not numbers. To gain confidence in this method, the reader may wish to choose values for $d$ and $h$ and work through an example.

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... transpose

Transposition is defined in Section 4.3.2 below. Essentially the rows become columns and vice versa.

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... above

Those readers wishing to know more about the uses of eigenvalues and eigenvectors may consult Searle (1982) or any general text on matrix algebra.

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... matrices

N.B. For a diagonal matrix one takes the reciprocal of only the diagonal elements!

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... variables,

Multivariate path diagrams, including delta path [van Eerdewegh, 1982], copath [Cloninger, 1980], and conditional path diagrams [Carey, 1986a] employ slightly different rules, but are outside the scope of this book. See Vogler (1985) for a general description.

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... relationship

i.e. we do not expect different heritabilities for twin 1 and twin 2; however for other relationships such as parents and children, the assumption may not be valid, as could be established empirically if we had genetically informative data in both generations.

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... exists

The reader may like to verify this by calculating the determinant according to the method laid out in Section 4.3.2 or with the aid of a computer.

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... sum

i.e. the unsigned difference between twin 1 and twin 2 of each pair, $\vert\mbox{BMI}_{\mbox{twin 1}} - \mbox{BMI}_{\mbox{twin 2}}\vert$ with $\mbox{BMI}_{\mbox{twin 1}} + \mbox{BMI}_{\mbox{twin 2}}$

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...gill81

small observed variances ($<.5$) can be problematic as the predicted covariance matrix may become non-positive definite.

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... matrix

A singular matrix cannot be inverted (see Chapter 4) and, therefore, the maximum likelihood fit function (see Chapter ) cannot be computed.

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... freedom

The degrees of freedom associated with this test are calculated as the difference between the number of observed statistics ($n_s$) and the number of estimated parameters ($n_p$) in the model. Our data consist of two variances and a covariance for each of the MZ and DZ groups, giving $n_s=6$ in total. The CE model has two parameters $c$ and $e$, so $n_s - n_p = 6-2=4$df.

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... phenotypes)

Except where explicitly noted, all models presented in this text treat observed variables as deviation phenotypes.

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... category

Excessive contributions to the $\chi^2$ by a small number of outliers could also be detected by fitting models directly to the raw data using Mx. Though a more powerful method of assessing the impact of outliers, it is beyond the scope of this volume.

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... assessments

Such independent assessments would risk retest effects if they were close together in time. Conversely, assessments separated by a long interval would risk actual phenotypic change from one occasion to the next. For a methodological review of this area, see Helzer (1977)

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... variance

The reader might like to consider what the components of this shared variance might include in these data obtained from the mothers of the twins and think forward to our treatment of rating data in Chapter 11.

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... effects

The reasoning goes like this: (e.g.) males have a elevated level of a chemical that prevents any gene expression from certain loci, at random with respect to the phenotype under study. Thus, both additive and dominant genetic effects would be reduced in males vs females, and hence the same genetic correlation between the sexes would apply to both.

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... data

In exploratory factor analysis the term ``factor structure'' is used to describe the correlations between variables and factors, but in confirmatory analysis, as described here, the term often describes the characteristics of a hypothesized factor model.

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... 1989)

We are grateful to Dr. Richard Schieken for making these data, gathered as part of a project supported by NHLBI award HL-31010, available prior to publication.

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...)

This problem is extreme when maximum likelihood is the fit function, because the inverse of $\Sigma$ is required (see Chapter ).

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... adequately

There are in fact some other special cases such as scalar sex-limitation - where identical genetic or environmental factors may have different factor loadings for males and females -- when the psychometric model may fit as well or better than the biometric model.

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... sex-limitation

This is to avoid estimated loadings of opposite sign in boys and girls - see Chapter 9.

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... children

However, if these effects were substantial and if MZ twins correlated more highly than DZ twins in their interactional style, the variance of parents' ratings should differ (Neale et al., 1992). Given sufficient sample size, these effects would lead to failure of these models.

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