### Getting your Ideas Down on Paper: Diagrammatic Representations

Creating Path Diagrams1

It is useful to break down a large problem into a series of causal propositions. Each variable incorporated into your set of propositions must have a clear and specific operational meaning. Simple propositions suggest that variation in one variables (often referred to as the "antecedent" or "independent" variable) affects variation in the second variable (generally thought of as the "antecedent" or "dependent" variable). Thus, for example, incompatibility is proposed to be antecedent to marital distress (its consequence). It is useful to try to identify other variables that may also affect marital happiness/distress. One way to do this is to focus on the "dependent" variable and ask yourself what all the possible independent, or antecedent, variables might be.

These variables can be labeled, for purposes of discussion, as alternative variables. Alternative variables that may affect marital satisfaction include a person's aspirations about marriage, his or her self-esteem, and the partner's physical appearance. The proposition that the level of compatibility creates conditions conducive to the development of marital happiness is not made any less tenable because it does not cover all of the relevant causes. Alternative variables, nonetheless, need to be considered. Such variables may covary with the hypothesized antecedent, and in such cases it is possible to mistakenly attribute the effects due to one variable to another. Suppose, for example, that physically attractive people are more likely to find a mate who is highly compatible. In psyche a situation, it is possible that any association between compatibility and marital happiness may be due to the association of compatibility and physical attractiveness, with attractiveness being the effective cause of happiness. Statistical analyses are available that control for the association of physical attractiveness and compatibility, allowing one to estimate the independent effects of each on marital satisfaction.

Moderating variables are introduced to account for situations where the relationship between the antecedent and the consequent variable is presumed to depend on some third variable. Thus, for example, "living arrangement" might be introduces as a moderating variable that affects the relationship between compatibility and satisfaction. The relationship between the two focal variables is hypothesized to be stronger when partners live together rather than apart.

Mediating variables may be introduced to explain why an antecedent variable affects a consequent variable. Thus, for example, incompatibility may be proposed to create conflict, which in turn produces martial unhappiness. The introduction of an intervening variable transforms one proposition (incompatibility leads to unhappiness) into two linked propositions (incompatibility leads to conflict; conflict leads to unhappiness).

Baron and Kenny (1986) provide a clear explication of the meaning of mediating variables. The figure referred to in their explication is shown below:

A variable functions as a mediator when it meets the following conditions: (a) variations in levels of the independent variable significantly account for variations in the presumed mediator (i.e., Path a), (b) variations in the mediator significantly account for variatons in the dependent variable (i.e., Path b), and (c) when Paths a and b are controlled, a previously significant relation between the independent and dependent variables is no longer significant, with the strongest demonstration of mediation occurring when Path c is reduced to zero. If the residual Path c is not zero, this indicates the operation of multiple mediating factors. Because most areas of psychology, including social, treat phenomena that have multiple causes, a more realistic goal may be to seek mediators that significantly decrease Path c rather than eliminating the relations between the independent and dependent variables altogether. From a theoretical perspective, a significant reduction demonstrates that a given mediator is indeed potent, albeit not both a necessary and sufficient condition for an effect to occur (p. 1176).

Antecedent, consequent, moderating, and mediating variables represent the basic building blocks for diagrammatic representations of causal processes. In the next section, we will consider ways to create such representations.

A Method for Diagramming your Model2

To create a diagram, you must commit yourself to specific theoretical constructs and measured variables, as well as provide a rationale for linking the constructs. Path diagrams provide an easy and convenient way to represent linkages between and among constructs (cf., Loehlin, 1987).

Path diagrams distinguish theoretical constructs from measured variables. Theoretical constructs are abstract by nature; they are often referred to as latent variables because they are not measured directly. "Measured variables" writes Falk (1987), "are actual observations and are frequently called markers, indicants, or manifest variables" (p. 14). Markers of compatibility might be similarity in gender-role attitudes, the extent to which partners like the same leisure activities, and their level of agreement in regard to religious matters. Theoretical constructs are represented in path diagrams as spheres, measured variables by squares.

The distinction between theoretical constructs and measured variables is useful to keep in mind even when only one measure for each theoretical construct is used. The failure to find an empirical association between two measured variables, each standing for a different theoretical construct, does not necessarily mean the two theoretical constructs are unrelated. Other more valid and more reliable measures may demonstrate the hypothesized relationship. A path diagram can be used to show the linkages between theoretical constructs and between such constructs and measured variables. The connections can be shown using two kinds of arrows: (a) straight one-headed arrows representing unidirectional, or causal, relationships between variables; and (b) curved, two-headed arrows depicting covariation, or correlation. Straight arrows going both directions between variables can be used to shoe mutual influence. Numbered subscripts on variables may designate time periods (when the measures pertain to data gathered at different points in time). The phase during the PAIR Project when data were gathered can be shown by such subscripts. The figure below shows the three types of linkages between constructs.

The first step in creating a path diagram, regardless of whether multiple measures are used, involves the creation of a diagram showing the linkages between the latent, or theoretical, variables. Such a diagram is called a Latent Variable Path Model, or the "inner model" (Falk, 1987).

The preliminary model shown below portrays a rather common formulation suggesting that compatibility affects marital satisfaction. The arrow drawn from compatibility to marital satisfaction suggests that compatibility has a causal effect on satisfaction. It is incumbent upon the theoretician to create a plausible rationale for linking the variables.

Compatibility might be posited to account, at least in part, for the affective quality of the marriage. The affective quality, in turn, may be hypothesized to affect satisfaction. This formulation suggests that the impact of compatibility on marital satisfaction is mediated through marital interaction. This set of propositions puts together ideas concerning the causes of marital satisfaction based upon compatibility theories and social learning theory.

The next figure shows an even more elaborated formulation of the connection between compatibility and marital satisfaction. The line drawn from behavioral interdependence to the line connecting compatibility and affective quality of the marriage suggests that the relationship between the latter two constructs depends upon the extent of behavioral interdependence. In this case, it is suggested that compatibility has little or no connection with the affective quality of interaction when interdependence is low, but a strong connection when interdependence is high. Behavioral interdependence thus moderates the connection between compatibility and satisfaction.

(Click to see full-size)

The constructs should be ordered in the path diagram from left to right, with those at the left causally prior to, or predictive of, those on the right. The theoretical construct(s) to the far left are generally thought of as exogenous (outside) because the model takes their values as "given," rather than as something to be explained. Compatibility is an exogenous variable. The variables that are shown to be influenced, either directly or indirectly, by exogenous variables are said to be endogenous.  It should be clear that the causal chain could be extended farther back; another model, for example, might explain "compatibility" in terms of dating experience, the idea being that those who shop around will be more likely to select a more compatible partner.

The next step in constructing a path diagram involves showing the connection between the latent variables and the markers, or measured variables. It is frequently the case that each theoretical construct is measured with a single indicator. For purposes of illustration, however, we will develop a more complex model in the example shown below.

(Click to see full-size)

Two variables measuring compatibility are included - similarity in gender role attitudes and similarity in leisure interests. The curved two-headed arrow drawn between these two measures of compatibility portrays them as correlated, but not causally related. The rationale behind this depiction, as well as each of the other arrows in the model, needs to be articulated. It might be argued, for example, that couples who are "selected" with regard to gender role attitudes might also be selected in terms of leisure interests. If some people more than others select mates on the basis of compatibility, then it would not be surprising to find that couples who are compatible in one regard would also be compatible in others.

The arrows drawn to similarity in gender role attitudes and leisure interests identify them as indicants of compatibility. The frequency of affectional expression and the frequency of conflict and negativity are shown as indicants of the affective quality of marriage. You may have noticed that affectional expression and negativity are not shown as correlated. The absence of the arrow in this instance is based on a review of previous research that shows no correlation between affectional expression and conflict/negativity among couples who are happy with their marriage (note that the PAIR Project partners are generally happy in the early years of marriage. If you were using data from the follow-up, however, you might carefully examine this correlation or lack thereof, since many partners are unhappy in the fourth phase). The spouses' living arrangement is used as an indicant of behavioral interdependence, suggesting that the connection between compatibility and the affective quality of marriage depends upon whether the spouses live together or have a commuting marriage. The right hand side of the model shows a single measure of satisfaction.

The logic behind the arrows drawn between variables in a diagram must be thought through carefully. You should have a clear idea as to what is implied by each arrow you draw. Your model should include only those variables that are necessary, either from a theoretical or practical standpoint. A good model is no more complex than it needs to be. It is important to recognize that this particular causal model represents only one of many possible ways of conceptualizing how these variables are linked to one another. For example, it could be argued that a more fine-grained measure of behavioral interdependence is necessary. An even more elaborate model might suggest that different kinds of compatibility are apt to be important depending upon the couples' living arrangements.

1. This discussion draws heavily from two sources: R. Frank Falk's (1987), A primer for soft modeling, Institute of Human Development, University of California, Berkeley; and John C. Loehlin's (1987), Latent variable models: An introduction to factor, path, and structural analysis. Hillsdale, NJ: Erlbaum. These sources use path diagrams ("nomograms") to introduce readers to causal modeling.

2. The purpose of this section is to help you show your ideas in diagrammatic form rather than to sketch out diagrams amenable to a particular kind of statistical analysis. Some of the terms and methods adapted here have slightly different meanings in other contexts.