2 Conventions Used in Path Analysis

A path diagram usually consists of boxes and circles, which are connected by arrows. Consider the diagram in Figure 5.1 for example.

Figure 5.1: Path diagram for three latent ($A, B$ and $C$) and two observed ($D$ and $E$) variables, illustrating correlations ($p$ and $q$) and path coefficients ($r,s, w, x, y$ and $z$).

Squares or rectangles are used to enclose observed (manifest or measured) variables, and circles or ellipses surround latent (unmeasured) variables. Single-headed arrows (`paths') are used to define causal relationships in the model, with the variable at the tail of the arrow causing the variable at the head. Omission of a path from one variable to another implies that there is no direct causal influence of the former variable on the latter. In the path diagram in (Figure 5.1) D is determined by $A$ and $B$, while $E$ is determined by $B$ and $C$. When two variables cause each other, we say that there is a feedback-loop, or `reciprocal causation' between them. Such a feedback-loop is shown between variables D and E in our example. Double-headed arrows are used to represent a covariance between two variables, which might arise through a common cause or their reciprocal causation or both. In many treatments of path analysis, double-headed arrows may be placed only between variables that do not have causal arrows pointing at them. This convention allows us to discriminate between dependent/endogenous variables and independent/ultimate/exogenous variables. Dependent variables are those variables we are trying to predict (in a regression model) or whose intercorrelations we are trying to explain (in a factor model). Dependent variables may be determined or caused by either independent variables or other dependent variables or both. In Figure 5.1, $D$ and $E$ are the dependent variables. Independent variables are the variables that explain the intercorrelations between the dependent variables or, in the case of the simplest regression models, predict the dependent variables. The causes of independent variables are not represented in the model. $A, B$ and $C$ are the independent variables in Figure 5.1. Omission of a double-headed arrow reflects the hypothesis that two independent variables are uncorrelated. In Figure 5.1 the independent variables $B$ and $C$ correlate, $C$ also correlates with $A$, but $A$ does not correlate with $B$. This illustrates (i) that two variables which correlate with a third do not necessarily correlate with each other, and (ii) that when two factors cause the same dependent variable, it does not imply that they correlate. In some treatments of path analysis, a double-headed arrow from an independent variable to itself is used to represent its variance, but this is often omitted if the variable is standardized to unit variance. However, for completeness and mathematical correctness, we do recommend to always include the standardized variance arrows. By convention, lower-case letters (or numeric values, if these can be specified) are used to represent the values of paths or double-headed arrows, in contrast to the use of upper-case for variables. We call the values corresponding to causal paths path coefficients, and those of the double-headed arrows simply correlation coefficients (see Figure 5.1 for examples). In some applications, subscripts identify the origin and destination of a path. The first subscript refers to the variable being caused, and the second subscript tells which variable is doing the causing. In most genetic applications we assume that the variables are scaled as deviations from the means, in which case the constant intercept terms in equations will be zero and can be omitted from the structural equations. Each dependent variable usually has a residual, unless it is fixed to zero ex-hypothesi. The residual variable does not correlate with any other determinants of its dependent variable, and will usually (but not always) be uncorrelated with other independent variables. In summary therefore, the conventions used in path analysis:

• Observed variables are enclosed in squares or rectangles. Latent variables are enclosed in circles or ellipses. Error variables are included in the path diagram, and may be enclosed by circles or ellipses or (occasionally) not enclosed at all.
• Upper-case letters are used to denote observed or latent variables, and lower-case letters or numeric values represent the values of paths or two-way arrows, respectively called path coefficients and correlation coefficients.
• A one-way arrow between two variables indicates a postulated direct influence of one variable on another. A two-way arrow between two variables indicates that these variables may be correlated without any assumed direct relationship.
• There is a fundamental distinction between independent variables and dependent variables. Independent variables are not caused by any other variables in the system.
• Coefficients may have two subscripts, the first indicating the variable to which arrow points, the second showing its origin.