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# Representation of Maps

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Causal maps may be represented in two main forms: via diagram or matrix. With the

diagram method the concepts (constructs) are usually represented as a word or words

enclosed in a box. The linkages are represented as lines with arrowheads. The lines

originate from the cause concept (construct) with the arrowhead pointing to the effect

concept (construct). Whenever possible, the map should be drawn so that the arrows

flow from left to right with little or no crossing of the lines (Axelrod, 1976). In some

instances there are mutually connected concepts. When two concepts are mutually

connected the concepts are causally connected in both directions (the two concepts are

both causes and effects of each other) (Knoke & Kuklinski, 1982). Mutually connected

concepts are represented as a two-headed arrow.

With the matrix representation the two primary matrices utilized are the adjacency and

reachability. An adjacency matrix is a matrix representing the association of direct

linkages between two constructs (Knoke & Kuklinski, 1982). If you are interested in the

presence or absence of a causal relationship between concepts, the adjacency matrix

contains only “0’s” and “1’s” (Carley & Palmquist, 1992). In the matrix the in-degrees is

the sum of all of the linkages flowing into the concept. Stated another way, it is the number

of times that the concept is an effect concept in a causal statement. The out-degrees is

the sum of all of the linkages flowing out of the concept. Again, stated another way, it

is the number of times that the concept is a cause concept in a causal statement. Table

7 provides a sample adjacency matrix.

If you are interested in not only the presence or absence of a causal relationship between

concepts but also the strength of the relationships, then the adjacency matrix contains

“0” for no relationship and a whole number (e.g., “4”) for the number of times that

relationship is recorded (Carley & Palmquist, 1992). The method for calculating the

frequency of linkages between two constructs is a percentage of the total linkages

between all constructs (Ford & Hegarty, 1983).

The reachability matrix indicates both the direct and indirect effects of a variable on all

other variables (Nelson, et al., 2000) and is calculated by the formula:

R = A + A2 + A3 + … + A n-1

where R is the reachability matrix, A is the adjacency matrix and n is the number of

variables. Table 8 provides a sample reachability matrix.

It is important to note that while the diagram and matrix methods are both appropriate for

causal mapping representation, as the maps become more complex researchers should

carefully consider their choice. For example, in Figure 6 you can see that this is an

extremely complex causal map (many concepts with many linkages). While possible, it

may be easier to derive the structural properties using the matrix method (aided by

computer analysis).

Table 8. Sample reachability matrix from Figure 4 map

1 2 3 4

Object - 1 1 1

Method 0 - 0 0

OO Development 0 0 - 1

Identifying Objects 0 0 0 -

Figure 6. Complex causal map

Analysis of Causal Maps

There are two aspects of causal mapping that have been consistently addressed in the

literature on analysis: content and structure (Nadkarni & Narayanan, in press). The

content refers to the meaning of specific concepts embedded in a causal map, and the

structure reflects the organization of the concepts in a map. In addition to these two

1 2 3 4

Object - 1 1 0

Method 0 - 0 0

OO Development 0 0 - 1

Identifying Objects 0 0 0 -

Table 7. Sample adjacency matrix from Figure 4 map

aspects, some researchers have begun to address the behavioral aspects of causal maps.

Behavior (as defined in Chapter I) asks the question, once we understand what the map

is telling us, can we use the map to make predictions? Toward the end of the book we

propose approaches to study the behavior of causal maps. In this chapter I focus on the

content and structural aspects only.