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Revealed Causal Mapping Technique

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Revealed causal mapping is a form of content analysis that attempts to discern the mental

models of individuals based on their verbal or text-based communications (Carley, 1997;

Darais et al., 2003; Narayanan & Fahey, 1990; Nelson et al., 2000). The general structure

of the causal map can reveal a wealth of information about cognitive associations,

explaining idiosyncratic behaviors and reasoning.

The actual steps used to develop the IT Job Satisfaction revealed causal map in the

present paper are outlined in Table 1. The research constructs were not determined a

priori, but were derived from the assertions in the data. The sequence of steps directly

develops the structure of the model from the data sample.

First, a key consideration in using RCM is the determination of source data (Narayanan

& Fahey, 1990). Since this study assessed the job satisfaction of IT professionals, it was

logical to gather data from IT workers in a variety of industries. Interviews were

conducted with employees of IT departments, and responses were analyzed to produce

the model presented later in the chapter.

Second, the researchers identified causal statements from the original transcripts or

documents. The third step in the procedure is to combine concepts based on coding rules

Table 1. Steps for revealed causal mapping technique

Step Description

1 Identify source data

2 Identify causal statements

3 Create concept dictionary

4 Aggregate maps

5 Produce RCM and analyze maps

 (Axelrod, 1976; Wrightson, 1976), producing a concept dictionary (see Appendix A).

Synonyms are grouped to enable interpretation and comparison of the resultant causal

maps. Care must be taken to ensure that synonyms are true to the original conveyance

of the participant. For example, two interviewees might use different words that hold

identical or very similar meanings such as “computer application” and “computer

program.” In mapping these terms, the links are not identical until the concepts are coded

by the researcher. It is preferable for investigators to err on the side of too many concepts,

rather than inadvertently combine terms inappropriately for the sake of parsimony.

Next, the maps of the individual participants were aggregated by combining the linkages

between the relevant concepts. The result of this step is a representative causal map for

the sample of participants (Markoczy & Goldberg, 1995).

RCM produces dependent maps, meaning that the links between nodes indicate the

presence of an association explicitly revealed in the data (Nadkarni & Shenoy, 2001). The

absence of a line does not imply independence between the nodes, however. It simply

means that a particular link was not stated by the participants. This characteristic of RCM

demonstrates the close relationship of the graphical result (map) to the data set.

Therefore, it is vital that the sample be representative of the population of interest. The

following section introduces belief functions and the importance of evidential reasoning

in managerial decision making.

Dempster-Shafer Theory of Belief

Functions

Dempster-Shafer (D-S) theory of belief functions, which is also known as the belieffunction

framework, is a broader framework than probability theory (Shafer and Srivastava,

1990). Actually, Bayesian framework is a special case of belief-function framework. The

basic difference between the belief-function framework and probability theory or

Bayesian framework is in the assignment of uncertainties to a mutually exclusive and

collectively exhaustive set of elements, say ,with elements {a1, a2, a3,...an}. This set of

elements, = {a1, a2, a3,...an}, is known as a frame of discernment in belief-function

framework. In probability theory, probabilities are assigned to individual elements, i.e.,

to the singletons, and they all add to one. For example, for the frame, ={a1, a2, a3, …

an}, with n mutually exclusive and collectively exhaustive set of elements, ai’s, with i =

1, 2, 3, … n, one assigns a probability measure to each element, 1.0 ≥P(ai) ≥0, such that

n

i=1

P(ai ) 1.

Under belief functions, however, the probablity mass is distributed over the super set

of the elements instead of just the singletons. Shafer (1976) calls this probability mass

distribution the basic probability assignment function, whereas Smets calls it belief

masses (Smets 1998, 1990a, 1990b). We will use Shafer’s terminology of probability mass

distribution over the superset of .

Basic Probability Assignment Function (m-Values)

In the present context, the basic probability assignment function represents the

strength of evidence. For example, suppose that we have received feedback from a survey

of the IT employees of a company about whether their work is challenging or not. On

average, the employees believe that their work is challenging but they do not say this

with certainty; they put a high level of comfort, say 0.85 (on a scale 0 – 1.0), that their work

is challenging. But, they do not say that their work is not challenging. This response can

be represented through the basic probability assignment function, m-values1, on the

frame, {‘yesCW’, ‘noCW’}, of the variable ‘Challenging Work (CW)’ as: m(yesCW) = 0.85,

m(noCW) = 0, and m({yesCW, noCW}) = 0.15. These values imply that the evidence suggests

that the work is challenging to a degree 0.85, it is not challenging to a degree zero (there

is no evidence in support of the negation), and it is undecided to a degree 0.15.

Mathematically, the basic probability assignment function represents the distribution

of probability masses over the superset of the frame, . In other words, probability

masses are assigned to all the singletons, all subsets of two elements, three elements,

and so on, to the entire frame. Traditionally, these probability masses are represented in

terms of m-values and the sum of all these m-values equals one, i.e.,

B

m(B) 1

⊆

, where

B represents a subset of elements of frame . The m-value for the empty set is zero, i.e.,

m() = 0.

In addition to the basic probability assignment function, i.e., m-values, we have one

other function, Belief function, represented by Bel(.), that is of interest in the present

discussion. As defined below, Bel(A), determines the degree to which we believe, based

on the evidence, that A is true. This function is discussed further below.

Belief Functions

The function, Bel(B), defines the belief in B, a subset of elements of frame , that is true,

and is equal to m (B) plus the sum of all the m-values for the set of elements contained

in B, i.e., =

CB

Bel(B) = m(C)

. Let us consider the example described earlier to illustrate

the definition. Based on the Survey Results, we have 0.85 level of belief that the

employees have challenging work, zero belief that the employees do not have challenging

work. This evidence can be mapped in the following belief functions by using the above

definition:

Bel(yesCW) = m(yesCW) = 0.85,

Bel(noCW) = m(noCW) = 0.0,

Bel({yesCW, noCW}) = m(yesCW) + m(noCW) + m({yesCW, noCW})

= 0.85 + 0.0 + 0.15 = 1.0.

The belief values discussed previously imply that we have direct evidence from

surveying the employees that the work is challenging to a degree 0.85, no belief that the

work is not challenging, and the belief that the work is either challenging or not

challenging is 1.0. Note that in our example there is no state or element contained in

‘yesCW’ or ‘noCW’. Thus, m-values and Bel(.) for these elements are the same.

Dempster’s Rule of Combination

Dempster’s rule of combination is similar to Bayes’ rule in probability theory. It is used

to combine various independent items of evidence pertaining to a variable or a frame of

discernment. As mentioned earlier, the strength of evidence is expressed in terms of mvalues.

Thus, if we have two independent items of evidence pertaining to a given variable,

i.e., we have two sets of m-values for the same variable then the combined m-values are

obtained by using Dempster’s rule. For a simple case2 of two items of evidence pertaining

to a frame Dempter's rule of combination is expressed as:

-1

1 i 2 j

i,j

Bi Bj=B

m(B) = K . m (B )m (B ),

where m(B) represents the strength of the combined evidence and m1 and m2 are the two

sets of m-values associated with the two independent items of evidence. K is the

renormalization constant given by:

1 i 2 j

i,j

Bi Bj=

K = 1 - m (B )m (B ).

∩

The second term in K represents the conflict between the two items of evidence. When

K = 0, i.e., when the two items of evidence totally conflict with each other, these two items

of evidence are not combinable.

A simple interpretation of Dempster’s rule is that the combined m-value for a set of

elements B is equal to the sum of the product of the two sets of m-values (from each item

of evidence), m1(B1) and m2(B2), such that the intersection of B1 and B2 is equal to B and

renormalize the m-values to add to one by eliminating the conflicts.

Let us consider an example to illustrate Dempster’s rule. Consider that we have the

following sets of m-values from two independent items of evidence pertaining to a

variable, say A, with two values, ‘a’, and ‘~a’, representing respectively, that A is true

and is not true:

m1(a) = 0.4, m1(~a) = 0.1, m1({a, ~a}) = 0.5,

m2(a) = 0.6, m2(~a) = 0.2, m2({a, ~a}) = 0.2.

As mentioned earlier, the general formula of Dempster’s rule yields the combines m-value

for an element or a set of elements of the frame of discernment by multiplying the two sets

of m-values such that the intersection of their respective arguments is equal to the

element or set of elements desired in the combined m-value, and by eliminating the

conflicts and renormalizing the resulting m-values such that the resulting m-values add

to one. This reasoning yields the following expressions as a result of Dempster’s rule for

binary variables:

m(a) = K-1[m1(a)m2(a) + m1(a)m2({a,~a}) + m1({a,~a})m2(a)],

m(~a) = K-1[m1(~a)m2(~a) + m1(~a)m2({a,~ a}) + m1({a,~ a})m2(~a)],

m({a,~ a}) = K-1m1({a,~ a})m2({a,~ a}),

and

K = 1 – [m1(a)m2(~a) + m1(~a)m2(a)].

As we can see above, m(a) is the result of the multiplication of the two sets of m-values

such that the intersection of their arguments is equal to ‘a’ and the renormalization

constant, K, is equal to one minus the conflict terms. Similarly m(~a) and m({a,~a}) are

the results of multiplying two sets of m-values such that the intersection of their

arguments is equal to ‘~a’ and ({a,~a}), respectively.

Substituting the values for the two m-values, we obtain:

K = 1 – [0.4x0.2 + 0.1x0.6] = 0.86,

m(a) = [0.4x0.6 + 0.4x0.2 + 0.5x0.6]/0.86 = 0.72093,

m(~a) = [0.1x0.2 + 0.1x0.2 + 0.5x0.2]/0.86 = 0.16279,

m({a,~a}) = 0.5x0.2/0.86 = 0.11628.

Thus, the total beliefs after combining both items of evidence are given by:

Bel(a) = m(a) = 0.72093, Bel(~a) = m(~a) = 0.16279,

and

Bel({a,~a}) = m(a) + m(~a) + m({a,~a}) = 0.72093 + 0.16279 + 0.11628 = 1.0.

The above values of beliefs in ‘a’ and ‘~a’ represent the combined beliefs from two items

of evidence. Belief that ‘a’ is true from the first item of evidence is 0.4; from the second

item of evidence it is 0.6, whereas the combined belief that ‘a’ is true based on the two

items of evidence is 0.72093; a stronger belief as a result of the combination. The

combined belief would have been much stronger if we did not have the conflict.

Evidential Reasoning Approach

Strat (1984) and Pearl (1990) have used the term “evidential reasoning” for decision

making under uncertainty. Under this approach one needs to develop an evidential

diagram (as shown in Figure 4 in the next section; see also Srivastava & Mock (2000) for

other examples) containing all the variables involved in the decision problem with their

interrelationships and the items of evidence pertaining to those variables. Once the

evidential diagram is completed, the decision maker can determine the impact of a given

variable on all other variables in the diagram by combining the knowledge about the

variables. In other words, under the evidential reasoning approach, if we have knowledge

about one or more variables in the evidential diagram, then we can make predictions

about the other variables in the diagram given that we know how these variables are

interrelated. Usually, the knowledge about the states of these variables is only partial,

i.e., there is uncertainty associated with what we know about these variables. As

mentioned earlier, we use Dempster-Shafer theory of belief functions to model these

uncertainties.

In the present case, variables in the evidential diagram represent the “constructs” of the

model obtained through the Revealed Causal Mapping (RCM) process, and the interrelationships

represent how one variable or a multiple of variables influence a given

variable. Such relationships among the variables can be defined either in terms of

categorical relationships such as, ‘AND’, and ‘OR’, or in terms of uncertain relationships,

such as a combination of ‘AND’ and ‘OR’, or some other relationships as discussed in

the next section.

In order to illustrate the evidential reasoning approach, let us first construct an evidential

diagram using a simple hypothetical decision problem involving three variables, X, Y, and

Z (see Figure 1). Let us assume for simplicity that these variables are binary, i.e., each

Figure 1: Example of an evidential nework*

*Rounded boxes represent variables (constructs), hexagonal box represents a relationship, and

rectangular boxes represent items of evidence pertinent to the variables they are connected

X: (x, ~x)

Y: (y, ~y)

Z: (z, ~z)

Evidence for Z

Evidence for Y

Evidence for X

A ND

variable has two values: either the variable is true (x, y, and z) or false (~x, ~y, and ~z).

Also, let us assume that variable Z is related to X and Y through the ‘AND’ relationship.

This relationship implies that Z is true (z) if and only if X is true (x) and Y is true (y), but

it is false (~z) when either X is true (x) and Y is false (~y), or X is false (~x) and Y is true

(y), or both X and Y are false (~x, ~y). Now we draw a diagram consisting of the three

variables, X, Y, and Z, represented by rounded boxes and connect them with a relational

node represented by the hexagonal box. Further, connect each variable with the

corresponding items of evidence represented by rectangular boxes. Figure 1 depicts the

evidential diagram for the above case.

As mentioned earlier, an evidential reasoning approach helps us infer about one variable

given what we know about the other variables in the evidential diagram. For example, in

Figure 1, we can predict about the state of Z given what we know about the states of X

and Y, and the relationship among them. Under the belief-function framework, this

knowledge is expressed in terms of m-values. For example, knowledge about X and Y,

based on the corresponding evidence, can be expressed in terms of m-values3, mX at X,

and mY at Y, as: mX(x) = 0.6, mX(~x) = 0.2, mX({x,~x}) = 0.2, and mY(y) = 0.7, mY(~y) = 0,

mY({y,~y}) = 0.3. The first set of m-values suggests that the evidence relevant to X

provides 0.6 level of support that X is true, i.e., mX(x) = 0.6, 0.2 level of support that X is

not true, i.e., mX(~x) = 0.2, and 0.2 level of support undecided, i.e., mX({x,~x}) = 0.2. One

can provide a similar interpretation of the m-values for Y. The ‘AND’ relationship

between X and Y, and Z can be expressed in terms of the following m-values: m({xyz,

x~y~z, ~xy~z, ~x~y~z}) = 1.0. This relationship implies that z is true if and only if x is true

and y is true, and it is false when either x is true and ~y is true, ~x is true and y is true,

or ~x and ~y are true.

Based on the knowledge about X and Y above and the relationship of Z with X and Y,

we can now make inferences about Z. This process consists of three steps which are

described in Appendix C in detail. Basically, Step 1 involves propagating4 beliefs or mvalues

from X and Y variables to the relational node ‘AND’ through vacuous5 extension.

This process yields two sets of m-values at ‘AND’, one from X and the other from Y:

Table 2: List of symbols related to m-values used in the propagation process in Figure 1

Symbol Description

x, y, and z These symbols, respectively, represent that the variables X, Y, and Z, are true.

~x, ~y, and ~z These symbols, respectively, represent that the variables X, Y, and Z, are not true.

X={x,~x} The frame of X which represents all the possible values of X.

Y={y,~y} The frame of Y which represents all the possible values of Y.

AND= {xyz, x~y~z,

~xy~z, ~x~y~z}

The frame of ‘AND’ relationship. The elements in the frame are the only possible

values under the logical ‘AND’ relationship between Z, and X and Y.

mX({.}) m-value for the element or the set of elements {x,~x} in the argument for variable X.

mY({.}) m-value for the element or the set of elements {y,~y} in the argument for variable Y.

mAND({.}) m-value for the elements in the argument for the ‘AND’ relationship.

mAND←X({.}) m-value for the element or elements in the argument propagated to ‘AND’

relationship from variable X.

mAND←Y({.}) m-value for the element or elements in the argument propagated to ‘AND’

relationship from variable Y.

mZ←AND({.}) m-values propagated from ‘AND’ to variable Z in Figure 1.

mAND←xand mAND←y (See Table 2 for definitions of symbols). Also, we already have one

set of m-values, mAND, at the relational node ‘AND’. Step 2 involves combining the three

sets of m-values at the ‘AND’ node using Dempster’s rule. Step 3 involves propagating

the resulting m-values from the ‘AND’ node to variable Z by marginalization6. This

process yields mz←AND.These m-values are then combined with the m-values at Z, mZ,

obtained from the evidence pertaining to Z. The resultant m-values will provide the belief

values whether Z is true or not true. As mentioned earlier, the details of the propagation

process7 are discussed in Appendix C through a numerical example.