Path analysis is a generalization of multiple regression that allows one to estimate the strength and sign of directional relationships for complicated causal schemes with multiple dependent variables ( Wright, 1920; Li, 1975). The causal scheme is usually considered an a priori hypothesis of potential effects (but see Shipley, 1997), and alternative hypotheses can be proposed and tested against each other ( Mitchell, 1993). Conversely, the a priori causal scheme can be taken as a given and used to make predictions about patterns of evolution. It is in this latter sense that we use our method, where the pieces of the causal scheme are built either from first principles or previous experimental and observational studies.
A path diagram ( Fig. 1) is a scheme of causal relationships. Consider an annual plant that grows vegetatively for some period of time, then ceases growth, flowers, sets seed and dies. More complex life cycles can be accommodated with this method; the example developed here is for simplicity of presentation. Five traits are measured: cotyledon size (z1), time of inflorescence initiation (bolting time; z2), number of rosette leaves at flowering initiation (z3), inflorescence height (z4) and number of fruits (z5). In our causal scheme, cotyledon size affects both time of inflorescence initiation and number of leaves, and both of them affect inflorescence height. Inflorescence height in turn influences fruit production. In this formulation, only first-order effects are included. That is, inflorescence height at the end of the season depends only on timing of inflorescence initiation and number of rosette leaves, not additionally on cotyledon size. Such second-order effects could be added by including additional paths.
A path diagram, besides showing the nature and direction of causal relationships, also includes estimates of the strength of those relationships, the path coefficients (p). A path coefficient is the standardized slope of the regression of the dependent variable on the independent variable in the context of the other independent variables. For example, inflorescence height (z4) is regressed on bolting time (z2). The slope (b42) is then standardized (p42) by multiplying it by the ratio of the standard deviations of the independent and dependent variables, respectively. If there is only a single independent variable, this standardized coefficient is a Pearson product-moment correlation; if there are additional independent variables, it is a standardized partial regression coefficient. The standardization acts to remove differences in scale among variables. Typically these relationships are assumed to be monotonic and linear, possibly after transformation. However, nonmonotonic (e.g. quadratic) relationships can be included by adding squared traits (see below). If a trait is transformed, the same transformation must be used for that trait as both a dependent and an independent variable.
When moving from the multiple regression formulation of Lande & Arnold (1983) to a path-analytical framework, we run into a terminology discrepancy between statisticians and evolutionary biologists. To avoid confusion it is necessary to refine previous word usage. In doing so we bring precision while avoiding unnecessary jargon.
In Lande & Arnold’s multiple regression framework there are only two ways a trait can affect fitness, a direct connection between the two traits, and a connection that proceeds backwards through a correlation with another trait and then forward to fitness ( Fig. 1A). Lande & Arnold refer to these two possible pathways as direct and indirect effects, respectively. A path-analytical framework adds a third possibility. A trait appearing early in the model may have its effect through an intermediate trait or traits and ultimately influence fitness. In standard path analysis parlance ( Pedhazur, 1982; see also Mitchell-Olds & Bergelson, 1990), a forward connection through an intermediate trait is referred to as an indirect effect (or a mediated effect), while a backward connection is a noncausal effect that includes both spurious effects due to shared causes and correlated effects due to correlated causes (e.g. in Fig. 1B, cotyledon size is a shared cause for leaf number and bolt date, while in Fig. 1A fitness is affected by four correlated causes). To add to the confusion, in selection analysis, correlational selection refers to simultaneous selection on a combination of traits ( Phillips & Arnold, 1989).
In order to avoid misunderstanding while not straying too far from previous usage, we propose the following terminology which replaces ‘effect’ with ‘selection’ when referring to the measures described by Lande & Arnold, and reserves the term ‘effect’ for describing paths or connections between traits. Direct selection is any forward connection between a trait and fitness, whether through an intermediate trait or not. In path analysis terminology this is the sum of all direct and indirect effects associated with a given trait. Indirect selection is any backward connection between a trait and fitness. In path analysis this is often called the noncausal effect. Total selection is the sum of direct and indirect selection. Correlational selection is defined as above ( Phillips & Arnold, 1989). A direct effect is any direct connection between traits. An indirect effect is any forward connection between two traits that goes through an intermediate trait. The causal effect is the sum of the direct and indirect effects (also known as the total effect or the total causal effect of Mitchell-Olds & Bergelson, 1990). A noncausal effect is any backward connection between two traits, including both spurious and correlated effects.
Partitioning the selection differential
The covariance between a trait and fitness is the selection differential (s). Such a covariance might develop for several reasons ( Arnold, 1983). In a multiple regression framework, there might be direct or indirect selection on the trait, i.e. s describes total selection. In a path-analytical framework, it might develop because of direct effects, indirect effects or noncausal effects ( Pedhazur, 1982; Schemske & Horvitz, 1988). Given the definitions mentioned earlier, it is therefore possible to use path analysis to partition the selection differential into direct selection (direct + indirect effects) and indirect selection (noncausal effects).
Partitioning an observed selection differential into these components depends on the causal model applied to the system. Consider the multiple regression model used in a typical selection analysis ( Fig. 1A). In this model there is no hierarchy of relationships among traits; all four of the observed traits influence fitness directly, and are correlated with one another. This model therefore only allows direct and noncausal effects on fitness, since there are no intermediate traits through which indirect effects might arise. Contrast the path model ( Fig. 1B). In this model only one trait (height) has a path leading directly to fitness with no intermediate steps, but all other traits may have indirect (mediated) or noncausal effects on fitness (Table 1). In this example, direct selection on inflorescence height is straightforward, p54 ≡ β*4, the regression of fruit production on inflorescence height. We use β to symbolize the selection coefficient to maintain consistency with the conventions established by Lande & Arnold (1983), but add the asterisk to indicate that the value may differ from that estimated using multiple regression (see below).
Table 1. Decomposition of the correlation between different traits and fitness under multiple regression and path analysis models (Fig. 1). Direct selection includes both direct and indirect effects, and indirect selection includes noncausal (spurious and correlational) effects. The sum of direct and indirect selection is the total selection accounted for by the model.
Direct selection coefficients (β*i ) are calculated as the sum of the direct and indirect effects for a trait; while the noncausal effects estimate indirect selection. Consider selection on leaf number. In this model, leaf number has no direct effect on fitness, but the indirect effect of leaf number on fruit production through inflorescence height estimates the strength of direct selection, and is calculated as β*3 ≡ p43p54. Indirect selection goes along a second pathway through a shared cause, seedling size. It is calculated by following the path backwards from leaf number to seedling size, then forward through bolting time to fitness (p31p21 p42p54). Total selection on leaf number can be calculated by adding together direct and indirect selection: s*3 ≡ p43p54 + p31p21 p42p54 (the asterisk indicates that this estimate relates to a path diagram, as above). Along a single path, coefficients are multiplied, while separate paths are summed. See Li (1975) for details on combining path coefficients. In complicated path diagrams there may be many pathways; however, the rules for calculating path coefficients are straightforward ( Li, 1975; Pedhazur, 1982). Several computer programs calculate path coefficients automatically [e.g. Procedure CALIS ( SAS Institute 1989a, b), LISREL ( Jöreskog & Sörbom, 1988 ), EQS ( Bentler, 1993), RAMONA ( SYSTAT for Windows, SPSS, Inc.)], and the more sophisticated versions offer a variety of estimation options, including ordinary least squares and maximum likelihood. Most major software packages calculate both standardized and unstandardized coefficients, so one must take care when reading the output to make sure which coefficients are being reported where.
We notate the model-implied covariance between a trait and fitness as s* (the ‘predicted’ covariance; Cohen & Cohen, 1983), to distinguish it from the observed selection differential (s) which does not depend on the causal model. Likewise, we distinguish between selection gradients estimated from a path model (β*i ; see also Koenig et al., 1991 ) and those estimated from a linear regression (βi). Structural equation modelling programs readily calculate s* and β*i values as part of the model-dependent predicted covariance matrix.
The sum of direct and indirect selection estimates the selection differential (s*) for a trait. The model-implied selection differential (s*) need not equal the observed selection differential (s) when the path diagram is ‘over-identified’ ( Pedhazur, 1982; Hayduk, 1987; see below). For example, our model ( Fig. 1B) implies a covariance between leaf number and fitness of s*3 (Table 1), a value which may deviate from the observed covariance (s3) for a variety of reasons, including misspecification of the causal model and sampling error.
In this fashion a path-analytical framework differs from a multiple regression framework, which will of necessity exactly recreate the original covariance matrix from the selection coefficients because it is ‘just-identified’ ( Pedhazur, 1982; Hayduk, 1987), having just enough information to estimate the regression coefficients. In contrast, many path models are over-identified, in that they contain more information than is needed to estimate the path coefficients. For example, in the linear path model ( Fig. 1B) there are 10 correlations among the five traits, but the path model only estimates five path coefficients.
This redundancy of information due to over-identification has two consequences. First, it means that the observed covariance between variables may differ from that implied by the model (Table 2). These deviations are an indication of the extent to which the data are consistent with the path diagram. Second, these deviations can be used to test the goodness-of-fit between the model and the data ( Pedhazur, 1982; Loehlin, 1987; Hayduk, 1988; Mitchell, 1993).
Table 2. Linear selection analyses showing direct and indirect selection coefficients for multiple regression and path analysis models. Coefficients in bold type are statistically significant at P < 0.0001. Statistical significance can be assessed for all traits for total selection and direct selection in the multiple regression model. In the path analysis model, significance can be assessed only for direct selection on height because only this trait is directly connected to fitness. Indirect selection in the path analysis model is calculated by subtracting direct selection (total effects) from s* (the model-implied correlation between the trait and fitness). R2 for the multiple regression model = 0.761. R2 values for the path model are: Bolting time = 0.045, Leaf number = 0.011, Height = 0.359, Fitness = 0.749.
As a result, statistical significance testing plays a different role in a path-analytical framework. When a direct selection coefficient comes from combining direct and indirect effects, its statistical significance is not directly tested. One can only test the significance of individual path coefficients ( Jöreskog & Sörbom, 1988). Instead, one tests the statistical significance of the entire model and compares alternative models ( Loehlin, 1987; Hayduk, 1988; Mitchell, 1993). After a model is chosen, the selection coefficients are calculated and their relative magnitudes compared. There are no unambiguous guidelines about how to interpret such selection coefficients. Magnitudes of <|0.2| are typically considered small, those from |0.2| to |0.4| are considered moderate and greater values are considered large.
Nonlinear and stabilizing/disruptive selection
Above we presented a path-analytical framework for directional selection, representing selection which changes population means. Selection can also be nonlinear, representing selection which changes population variances. Such nonlinear selection coefficients are typically symbolized as γ. To incorporate selection on variances into a path analysis, one creates new traits, which are squares of the original traits ( Fig. 2). Such analysis requires that the original trait be centred to a mean of 0 by subtracting the population mean from each observation. In the analyses below we used z-scores, calculated by subtracting the mean and then dividing by the standard deviation, in order to simplify interpretation ( Pedhazur, 1982; Lande & Arnold, 1983). Our use of squared traits to indicate quadratic paths is actually equivalent to what is done in a multiple regression analysis. Such an analysis, however, is seldom illustrated, only shown in equation form. Thus, the relationship is not immediately apparent.
Figure 2 The same path diagram as in Fig. 1(B), but with nonlinear path coefficients (q) now shown. These nonlinear coefficients are regressions on squares of the trait values and are indicated by squared traits, z2 terms. To distinguish paths through traits and squared traits, the latter are indicated with a prime. For example, the path from z1 to z2. 3 is q3′1. Variation due to error (U) is not included for simplicity. Conventions follow Fig. 1. Grey is used to denote elements that are also in the model in Fig. 1(B).
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The path diagram indicates that these squared traits are correlated with (rather than caused by) the original trait from which they are calculated, since neither is conceptually nor biologically antecedent to the other. The paths from and to the squared traits are designated as q, to differentiate them from linear paths (p) and correlations (r). To distinguish paths through traits from those through squared traits, we indicate the latter with a prime. For example, the path from z1 to z23 is q3′1. While we do not deal with the calculation of the path coefficients here, we note that the linear and nonlinear models must be estimated separately ( Cohen & Cohen, 1983; Lande & Arnold, 1983; Brodie et al., 1995 ). In a nonlinear (quadratic) model, it is the combination of the variable and its square that is of interest, not either one in isolation. The linear coefficient in a quadratic analysis does not necessarily equal the coefficient in a linear analysis because of the shared variance between the trait and its squared trait. The linear analysis provides the best estimate of directional selection, so in a nonlinear analysis the linear coefficients are typically ignored.
These new path coefficients through the squared traits measure nonlinear selection, selection on the variance ( Lande & Arnold, 1983; Mitchell-Olds & Shaw, 1987). If there is an internal maximum (minimum), then they measure stabilizing (disruptive) selection in the classic sense. Otherwise they represent selection which decreases (increases) the phenotypic variance. In standard linear path analysis these terms would not exist. If there were no internal maximum or minimum, a suitable transformation would be found to linearize the relationship. The case of an internal maximum or minimum is simply ignored in most cases. Linearization is typically justified on the grounds that the researcher is simply trying to create the most useful, predictive model with no epistemological import being given to the exact form of the equations. However in the case of natural selection, the nonlinear terms have real meaning. Thus, we need a method for incorporating them into the analysis.
Path and selection coefficients in this model are calculated as before, but involve a slightly more complicated decomposition of the selection differential because of the more complicated model. Selection directly on quadratic components is assessed simply by the direct and indirect effect coefficients (direct selection) for the squared traits as described above. Thus, direct nonlinear selection on leaf number is γ*3=q4′3′q54′ + q43′p54. Because there are no nonlinear (or linear) terms connecting leaf number with bolting time, this latter trait does not contribute to direct nonlinear selection on leaf number. Direct nonlinear selection is quantified by the total effect of each of the squared traits. Indirect selection involves the noncausal effect and can be complicated and tedious to calculate. For example, as part of the calculation of indirect nonlinear selection on seedling size, you would include the path from z12 through z1, z2 and z4 to z5 (r1p21 p42 p54). In practice, it is easiest to calculate the indirect selection coefficient by subtracting the direct selection coefficient of the trait (β*) from the model-implied covariance between the trait and fitness (s*).
In a similar fashion, correlational selection can be incorporated into a path-analytical framework. Estimating the interaction terms for such an analysis requires construction of new traits by multiplying together any pair of traits that have a direct effect on a third trait. The analysis is analogous to that described above, but is beyond the scope of this paper.