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Conversion of Revealed Causal Map into Evidential Diagram and Belief

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Propagation

In this section we first discuss how a revealed causal map can be converted to a belief

function evidential diagram and then discuss how beliefs can be propagated through this

evidential diagram. Our example is displayed in Figure 3.

Conversion of Revealed Causal Map into Evidential

Diagram

The conversion process of revealed causal map into evidential diagram can be described

in the following five steps:

1. Identify the main variables (i.e., constructs) in the revealed causal map.

2. Determine the possible values of these variables (such as, ‘true/false’, or ‘high/

medium/low’).

3. Determine the relationships among the variables (see the details below).

4. Connect the variables through the corresponding relationships.

5. Identify potential items of evidence pertaining to the variables in the diagram and

connect these items of evidence to the relevant variables.

The above approach yields the desired evidential diagram for belief-function analysis.

In Steps 1 and 2, we have identified nine variables (i.e., constructs; see Figure 3) and their

corresponding categorical values (Table 4).

Step 3 (determining the relationships among various variables) is a somewhat difficult

process. Expert judgments about these relationships must be rendered. For example, the

relationship R1 defining the relationship between ‘Role Recognition (RR)’ and ‘Job

Security (JT)’ was extremely difficult to model. In this case, the survey data provided only

information on whether the subjects recognize their changing role on the job and did not

specify any details on how this knowledge might influence ‘Job Security’. For IT

personnel, ‘Role Recognition’ might mean that ‘yes’ there is ‘Job Security’, but it also

may mean that there is no ‘Job Security’. Thus, lacking any other information, we assume,

for the present discussion, that when ‘Role Recognition’ is yes, ‘Job Security’ is 50%

‘yes’, and 50% ‘no’. However, when there is no knowledge about ‘Role Recognition’,

there is no knowledge about ‘Job Security’. Such a relationship can be expressed in terms

of m-values as given below.

m-values for R1:

mR1({(yes RR, yes JT), (no RR, yes JT), (no RR, no JT)}) = 0.5,

mR1({(yes RR, no JT), (no RR, yes JT), (no RR, no JT)}) = 0.5.

The above relationship propagates9 50% of mE1(yes RR), the belief on ‘Role Recognition’

being ‘yes’ from evidence E1 (Figure 4), to ‘yesJT’ 50% of mE1(yes RR) to ‘noJT’, and 100%

of mE1(noRR) and mE1({yesRR, noRR}) to ({yesJT, noJT}), as described in the assumed

relationship. In other words, the m-values propagated from variable ‘Role Recognition

(RR)’ to variable ‘Job security (JT)’ are given as:

mJT←RR(yesJT) = 0.5mE1(yesRR), mJT¬RR(noJT) = 0.5mE1(yesRR), and

mJT←RR({yesJT, noJT}) = mE1(noRR) + mE1({yesRR, yesRR})

For the relationship R2 we assume the following. On average, a person with the

knowledge that there is no job security will sign up for job training with 90% belief and

a person with the knowledge that there is no problem with the job security will not sign

up for job training with 90% belief. This relationship can be modeled in the following way:

m-values for R2:

mR2({(yes JT, no ST), (no JT, yes ST)}) = 0.9, and

mR2({(yes JT, yes ST), (yes JT, no ST), (no JT, yes ST), (no JT, no ST)}) = 0.1.

Similar to endnote 9, one can easily show the following m-values to be the result of mvalues

propagated from variable ‘Job Security (JT)’ to variable ‘Sign up for Job Training

(ST)’ through the relationship R2:

mST←JT(yesST) = 0.9mJT(noJT), mST¬JT(noST) = 0.9mJT(yesJT), and

mST←JT({yesST, noST}) = 0.1 + 0.9mJT({yesRR, yesRR}).

We assumed the following m-values for R3 (see Table 4 for the definitions of the

symbols):

m-values for R3:

mR3({(yes ST, yesUS), (no ST, no US)}) = 0.75, and

mR3({(yes ST, yes US), (yes ST, no US), (no ST, yes US), (no ST, no US)}) = 0.25.

The relationship previously discussed implies that if variable ‘ST’ is ‘yes’, i.e., a person

sings up for training, then variable ‘US’ will be ‘yes’, i.e., the person will have the

opportunity to use the new skill with 0.75 belief, and the remaining 0.25 belief is assigned

to ignorance. Similarly, the relationship implies that if ‘ST’ is ‘no’ then ‘US’ is ‘no’ with

belief 0.75, i.e., if one does not sign up for job training then he/she will not have the use

of new skill with belief 0.75. The remaining 0.25 belief represents ignorance.

For the relationship R4, we assume the following m-values:

m-values for R4:

mR4({(yesGS, yesUS), (noGS, noUS)}) = 1.0.

This relationship implies that if ‘GS’ is ‘yes’ then ‘US’ is ‘yes’ with 1.0 belief. Also, if ‘GS’

is ‘no’ then ‘US’ is ‘no’ with 1.0 belief. In other words, if one has the opportunity to gain

new skills on the job then there is 1.0 belief that there is opportunity to use the new skills.

Similarly, if there is no opportunity to gain new skills on the job then there is no

opportunity to use the new skills.

The relationship R5 relates variables ‘US’, ‘FS’, ‘AW’, and ‘CW’ to the variable ‘Job

Satisfaction (JS)’. We have assumed the following relative weights, 0.125, 0.125, 0.25, and

0.5, respectively, for ‘US’, ‘FS’, ‘AW’, and ‘CW’ when propagating information (mvalues)

to the variable ‘JS’.

Step 4 simply represents a diagram with all the variables interconnected through the

assumed relationships (see Figure 4). In Step 5, we identify various items of evidence

pertaining to different variables and connect them to the corresponding variables. Table

4 provides a list of evidence pertaining to the nine variables in Figure 4. Once these items

of evidence are connected to the corresponding variables, we develop the evidential

diagram shown in Figure 4 for the analysis.

Propagation of Beliefs through Evidential Diagram

In order to propagate information in terms of m-values from all the variables to the variable

of interest, say, ‘Job Satisfaction (JS)’ in Figure 4, we need to follow the following steps.

First, gather all the information (m-values) at ‘Role Recognition (RR)’, propagate that

information (m-values) to variable ‘Job Security (JT)’ through the relationship R1 by first

vacuously extending to the space of R1, combining it with the m-values at R1 using

Dempster’s rule, and then marginalizing the resulting m-values to the space of ‘JT’.

Combine this information (m-values) with the m-values at ‘JT’ obtained from evidence

E2.1 and E2.2, again using Dempster’s rule. Next step is to propagate the resulting mvalues

at ‘JT’ through R2 to the variable ‘Sign up for Training to Gain New Skills (ST)’.

This is achieved again by vacuously extending the total m-values at ‘JT’ to the space of

R2, combining them with the m-values at R2 using Dempster’s rule, and then marginalizing

them to the space of variable ‘ST’. Combine this information (m-values) with the m-values

obtained from evidence E3 for ‘ST’. The resulting m-values are then propagated to the

variable ‘Opportunity to use New Skills (US)’. Combine these m-values with the m-values

obtained from the variable ‘Opportunity to Gain New Skills on the Job (GS)’ and the mvalues

from evidence E5 for ‘US’.

In the final step, we need to propagate all the m-values from the four variables, ‘US’, ‘FS’,

‘AW’, and ‘CW’ through the relationship R5 to the variable ‘Job Satisfaction (JS)’ by

vacuously extending the respective m-values to the space of R5, combine these m-values

with the m-values defining R5 and then marginalize them to the space of ‘Job Satisfaction’.

The marginalized m-values on ‘Job Satisfaction (JS)’ can be written as:

mJS(yesJS) = 0.125mUS(yesUS) + 0.125mFS(yesFS) + 0.25mAW(yesAW) + 0.5mCW(yesCW).

mJS(noJS) = 0.125mUS(noUS) + 0.125mFS(noFS) + 0.25mAW(noAW) + 0.5mCW(noCW).

mJS({yesJS, noJS}) = 1 - mJS(yesJS) - mJS(noJS).

These m-values provide the impact of all the variables in the evidential diagram in Figure 4.

Given that the evidential diagram in Figure 4 is a tree, the propagation of m-values from

various variables to the variable of interest, ‘Job Satisfaction’ is much easier than

propagation in a network of variables. We programmed the logic of vacuous extension,

marginalization, and Dempster’s rule of combination in a spreadsheet program in MS

Excel, which then was used to perform various analyses as discussed in the next section.

Decision Analysis of Causal Map Using

Belief Functions

In this section, we discuss how one can analyze the impact of one variable on the other

variables in the network given in Figure 4. Such an analysis allows the decision maker

to isolate an independent variable while holding the rest of the variables in the model

constant. In this example, the overall belief in job satisfaction is 0.803 given the inputs

from the Survey Results and industry data. The above value implies that based on the

subjects responses, on average, employees are satisfied with their jobs in the environment

surveyed with 0.803 level of belief. In order to investigate the impact of a number

of variables on the level of job satisfaction, we use a range of possible responses (0 to

1.0) for the variables while keeping the inputs from other items of evidence fixed at values

obtained from the survey as given in the respective figures.

First, we investigate the impact of ‘Job Security’ on ‘Job Satisfaction’. We vary the input

belief from evidence E2.2 for the negation of ‘Job Security’ from 0 to 1.0, keeping the rest

of inputs fixed. As seen in Figure 5, the impact of ‘No Job Security’ is pretty severe. As

the belief in no job security increases the belief in job satisfaction decreases with

increasing rate. In other words, if an employee sees strong evidence in support of ‘no

job security’ then he/she will have very low job satisfaction.

The second sensitivity analysis is conducted on the impact of having an opportunity to

use new skills on the job. This analysis reveals that the opportunity to use new skills has

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Figure 4. Evidential diagram* of the causal map in Figure 3

New Skills (ST) {yesST, noST}

4. Opportunity to Gain New Skills

on the Job (GS) {yesGS, noGS}

1. Recognition of Tech. & Bus. Role

Change (RR) {yesRR, noRR}

2. Job Security (JT)

{yesJT, noJT}

[yes/no]

5. Opportunity to use New

Skills (US) {yesUS, noUS}

Job Satisfaction (JS)

{yesJS, noJS}

7. Challenging work (CW)

{yesCW, noCW}

6. Feedback from Superiors/

Co-workers (FS) {yesFS, noFS}

8. Autonomy of Work (AW)

{yesAW, noAW}

E1. Survey Results: Q51, Q52,

Q53, Q55 (0.63, 0.37)

E2.1: Layoffs reported – in the firm, R1

industry, general public (0.27, 0.73)

E3: Survey Results, Q34

(0.79, 0.0)

R4 R3

E4: Survey Results, Q30

(0.84, 0.0)

R2

R5

E2.2: Survey Results, Q28

(0.83, 0.0)

E6: Survey Results, Q6, Q14-Q22

(0.66, 0.34)

E5: Survey Results, Q35

(0.81, 0.0)

E8: Survey Results, Q2, Q16, Q20

(0.77, 0.23)

E7: Survey Results, Q26

(0.81, 0.0)

*The oval shaped boxes represent variables and the rectangular boxes represent items of evidence.

The numbers in a rectangular box represent the level of support for and against the variable it

is connected to. These numbers were determined from the Survey Results except for E2.1 which

was determined from the industry data.

Belief Function Approach to Evidential Reasoning in Causal Maps 129

Figure 5. Belief in job satisfaction versus belief in no job security*

*The following input m-values in Figure 4 were used for the graph: mE1(yesRR)=0, mE1(noRR)=0,

mE2.1(yesJT)=0.27, mE2.1(noJT)=0.73, mE2.2(yesJT)=0, mE2.2(noJT)=0, mE3(yesST)=0, mE3(noST)=0,

mE4(yesGS)=0, mE4(noGS)=0, mE5(yesUS) varied from 0 - 1, mE5(noUS)=0, mE6(yesFS)=0.66,

mE6(noFS)=0.34, mE7(yesCW)=0.81, mE7(noCW)=0, mE8(yesAW)=0.77, mE8(noAW)=0.23.

Impact of Belief in 'No' Job Security on Belief in Job Satisfaction

0.65

0.67

0.69

0.71

0.73

0.75

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Belief in 'No' Job Security

Belief in Job Satisfaction

Figure 6. Belief in job satisfaction versus belief in opportunity to use new skills*

Impact of Belief in Opportunity to use New Skills on Belief in Job

Satisfaction

0.72

0.74

0.76

0.78

0.80

0.82

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Belief in Opportunity to use New Skills

Belief in Job Satisfaction

*The following input m-values in Figure 4 were used for the graph: mE1(yesRR)=0, mE1(noRR)=0,

mE2.1(yesJT)=0.27, mE2.1(noJT)=0.73, mE2.2(yesJT)=0, mE2.2(noJT)=0, mE3(yesST)=0, mE3(noST)=0,

mE4(yesGS)=0, mE4(noGS)=0, mE5(yesUS) varied from 0 - 1, mE5(noUS)=0, mE6(yesFS)=0.66,

mE6(noFS)=0.34, mE7(yesCW)=0.81, mE7(noCW)=0, mE8(yesAW)=0.77, mE8(noAW)=0.23.

Figure 7. Belief in job satisfaction versus belief in feedback from supervisors and coworkers*

Impact of Belief in Feedback from Supervisors and Co-Workers

on Belief in Job Satisfaction

0.71

0.73

0.75

0.77

0.79

0.81

0.83

0.85

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Belief in Feedback from Supervisors and Co-Workers

Belief in Job Satisfaction

*The following input m-values in Figure 4 were used for the graph: mE1(yesRR)=0, mE1(noRR)=0,

mE2.1(yesJT)=0.27, mE2.1(noJT)=0.73, mE2.2(yesJT)=0, mE2.2(noJT)=0, mE3(yesST)=0, mE3(noST)=0,

mE4(yesGS)=0, mE4(noGS)=0, mE5(yesUS)=0.81, mE5(noUS)=0, mE6(yesFS) varied from 0 - 1, mE6(noFS)=0,

mE7(yesCW)=0.81, mE7(noCW)=0, mE8(yesAW)=0.77, mE8(noAW)=0.23.

a significant positive impact on ‘Job Satisfaction’ as seen from Figure 6. As the belief in

opportunity to use new skills increases, the belief in job satisfaction increases. We find

an 8.5% increase in job satisfaction over the range from 0 – 1.0 for belief in opportunity

to use new skills. This impact is linear, unlike the previous case.

The third variable analyzed is ‘Feedback from Supervisors/Co-workers’. As shown in

Figure 7, the results demonstrate a substantial positive impact of feedback on the job

satisfaction. In particular, job satisfaction increases about 19% as we progress from the

lower to higher levels of perceived feedback. It is obvious that feedback is a powerful

variable in predicting job satisfaction.

Next, we conduct a sensitivity analysis with the independent variable, ‘Challenging

Work’. ‘Job Satisfaction’ was extremely sensitive to increases in the perceived level of

challenging work. From no belief that the job is challenging to the higher range of belief,

1.0, the model indicates that the belief in job satisfaction moves from 0.388 to 0.838; a 129%

increase as seen in Figure 8. These results indicate that challenging work is the most

powerful variable in the model in the prediction of job satisfaction.

Finally, a sensitivity analysis was conducted on ‘Autonomy of Work’. The results

indicate that ‘Autonomy of Work’ has a significant impact on the dependent variable,

‘Job Satisfaction’. Job satisfaction was found to be very sensitive to autonomy. As the

perceived autonomy increases from 0 to 1.0, job satisfaction improves from 60% to 85%,

an increase of 41.6%. These results are presented in Figure 9.

These sensitivity analyses have shown the impact on job satisfaction from a broad range

of variables and their corresponding beliefs. However, we do want to point out that the

Figure 8. Belief in job satisfaction versus belief in challenging work*

Impact of Belief in C hallenging W ork on Belief in Job Satisfaction

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Belief in Challenging Work

Belief in Job Satisfactio

*The following input m-values in Figure 4 were used for the graph: mE1(yesRR)=0, mE1(noRR)=0,

mE2.1(yesJT)=0.27, mE2.1(noJT)=0.73, mE2.2(yesJT)=0, mE2.2(noJT)=0, mE3(yesST)=0, mE3(noST)=0,

mE4(yesGS)=0, mE4(noGS)=0, mE5(yesUS)=0.81, mE5(noUS)=0, mE6(yesFS)=0.66, mE6(noFS)=0.34,

mE7(yesCW) varied from 0 - 1, mE7(noCW)=0, mE8(yesAW)=0.77, mE8(noAW)=0.23.

Figure 9. Belief in job satisfaction versus belief in autonomy of work*

Impact of Belief in Autonomy of Work on Belief in Job Satisfaction

0.50

0.60

0.70

0.80

0.90

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Belief in Autonomy of Work

Belief in Job Satisfaction

*The following input m-values in Figure 4 were used for the graph: mE1(yesRR)=0, mE1(noRR)=0,

mE2.1(yesJT)=0.27, mE2.1(noJT)=0.73, mE2.2(yesJT)=0, mE2.2(noJT)=0, mE3(yesST)=0, mE3(noST)=0,

mE4(yesGS)=0, mE4(noGS)=0, mE5(yesUS)=0.81, mE5(noUS)=0, mE6(yesFS)=0.66, mE6(noFS)=0.34,

mE7(yesCW)=0.81, mE7(noCW)=0, mE8(yesAW) varied from 0 - 1, mE8(noAW)=0.

interrelationships among the intermediate variables and the relative weights assigned to

‘Opportunity to use New Skills’, ‘Feedback from Supervisors and Co-Workers, ‘Challenging

Work’, and ‘Autonomy of Work’, have direct impact on the results for the

dependent variable, ‘Job Satisfaction’.

In summary, the above analysis provides an example of how an evidential reasoning

approach under Dempster-Shafer theory of belief functions can be used to determine the

impact on a given construct or constructs of other constructs in a revealed causal map.

It should be noted that a revealed causal map of a decision problem is only a static model

while an evidential diagram of a revealed causal map provides a dynamic model for

analyzing the behaviors of various constructs under different conditions.

Conclusions and Future Directions for

Research

In this chapter we have demonstrated the use of evidential reasoning approach under

Dempster-Shafer (D-S) theory of belief functions to analyze revealed causal maps. As an

example, we used a simplified causal map obtained through a Revealed Causal Mapping

(RCM) technique where the participants were from information technology (IT) organizations

who provided the concepts to describe the target phenomenon of ‘Job Satisfaction’.

They also identified the associations between the concepts. After creating the

causal map of the problem being investigated, we developed an evidential diagram. This

diagram consists of the variables or constructs of the causal map, interconnected to the

other variables with some relationships. These relationships were defined by the

decision maker based on experience. Various items of evidence were identified that

pertained to different variables. Estimates of the beliefs in terms of m-values in support

of, or negation of, the variables were made for each item of evidence using survey

questions (Buche, 2003, particularly Appendix C). These m-values were then propagated

through the evidential network to obtain the overall belief of ‘Job Satisfaction’.

To illustrate the usefulness of the evidential reasoning approach under Dempster-Shafer

theory of belief functions, we performed various sensitivity analyses to determine the

impact of different variables on ‘Job Satisfaction’. This technique enables researchers

to predict the level of job satisfaction when given evidence for the other variables in the

model. As further validation for our findings, our results are directly in line with previous

literature on job satisfaction for workers in general. IT personnel are very similar to other

professions and vocations. An evidential diagram similar to the one discussed here

would be useful in predicting whether a specific work environment would be more or less

satisfactory to an employee before joining the job.

In this chapter we have explained the steps necessary to convert revealed causal maps

into evidential diagrams. The analysis of the transformed diagram is useful in forming

predictions about human behavior. This technique incorporates the existence of uncertainty

in the level of belief associated with the evidence. Therefore, the researcher is able

to include in the diagram personal intuition and confidence based on direct experience.

Another advantage of the evidential reasoning approach over a revealed causal map is

that the former provides a dynamic model of a decision problem while the later provides

only a static model. As a limitation, the evidential reasoning approach may become quite

complex especially when variables or constructs in the diagram are highly integrated. For

ease of instruction, the example discussed herein was fairly simplistic, with primarily

linear associations.

References

Ang, S., & Slaughter, S.A. (2001). Work outcomes and job design for contract versus

permanent information systems professionals on software development teams.

MIS Quarterly, 25, 321-350.

Axelrod, R. (1976). Structure of decisions: The cognitive maps of political elites.

Princeton, NJ: Princeton University Press.

Bougon, M.G., Weick, K., & Binkhorst, D. (1977). Cognition in organizations: An analysis

of the Utrecht Jazz Orchestra. Administrative Science Quarterly, 22, 606-639.

Bovee, M, Srivastava, R. P., & Mak, B. (2003, January). A conceptual framework and

belief-function approach to assessing overall information quality. International

Journal of Intelligent Systems, 18(1), 51-74.

Buche, M.W. (2003). IT professional work identity: Constructs and outcomes. Unpublished

dissertation, University of Kansas, Lawrence, KS.

Carley, K. (1997). Extracting team mental models through textural analysis. Journal of

Organizational Behavior, 18, 533-558.

Curley, S. P., & Golden, J.I. (1994). Using belief functions to represent degrees of belief.

Organization Behavior and Human Decision Processes, 271-303.

Darais, K.M., Nelson, K.M., Rice, S.C., & Buche, M.W. (2003). Identifying the enablers

and barriers of IT personnel transition. In C. Shayo & M. Igbaria (Eds.), Strategies

for managing IS/IT personnel (pp. 92-112). Hershey, PA: Idea Group.

Gupta, Y.P., Guimaraes, T., & Raghunathan, T.S. (1992). Attitudes and intentions of

information center personnel. Information & Management, 22, 151-160.

Hackman, J.R., & Oldham, G. (1976). Motivation through the design of work: Test of a

theory. Organizational Behavior and Human Performance, 16, 250-279.

Harrison, K., Srivastava, R.P., & Plumlee, R.D. (2002). Auditors’ Evaluations of Uncertain

Audit Evidence: Belief Functions versus Probabilities. In R. P. Srivastava & T.

Mock (Eds.), Belief functions in business decisions, (pp. 161-183). Heidelberg:

Springer-Verlag.

Huff, A.S. (1990). Mapping strategic thought. Chichester, UK: Wiley.

Igbaria, M., & Guimaraes, R. (1993). Antecedents and consequences of job satisfaction

among information center employees. Journal of Management Information Systems,

9, 145-174.

Markoczy, L., & Goldberg, J. (1995). A method for eliciting and comparing causal maps.

Journal of Management, 21, 305-333.

Nadkarni, S., & Shenoy, P.P. (2001). A Bayesian Network approach to making inferences

in causal maps. European Journal of Operational Research, 128, 479-498.

Narayanan, V.K., & Fahey, L. (1990). Evolution of revealed causal maps during decline:

A case study of Admiral. In A. Huff (Ed.). Mapping strategic thought (pp. 109-133).

London: John Wiley & Sons.

Nelson, K.M. (2000). IT personnel transition and organization transition strategy.

National Science Foundation Grant.

Nelson, K.M., Nadkarni, S., Narayanan, V.K., & Ghods, M. (2000). Understanding

software operations support expertise: A causal mapping approach. MIS Quarterly,

24, 475-507.

Pearl, J. (1990). Bayesian and belief-functions formalism for evidential reasoning: a

conceptual analysis. In Readings in uncertain reasoning (pp. 540-574). San

Mateo, CA: Morgan Kaufmann.

Radding, A. (1997). Rock-solid incentives. Network World, 14, 31-34.

Saffiotti, A., & Umkehrer, E. (1991). Pulcinella: A general tool for propagating uncertainty

in valuation networks. Proceedings of the Seventh National Conference on

Artificial Intelligence (pp. 323-331), University of California, Los Angeles.

Shafer, G. (1976). A mathematical theory of evidence. Princeton University Press.

Shafer, G., Shenoy, P.P., & Srivastava, R.P. (1988, May). Auditor’s Assistant: A

knowledge engineering tool for audit decisions. Proceedings of the 1988 Touche

Ross/University of Kansas Symposium on Auditing Problems (pp. 61-79).

Shafer, G., & Srivastava, R.P. (1990). The Bayesian and belief-function formalisms: A

general perspective for auditing. Auditing: A Journal of Practice and Theory,

(Supplement), 110-148.

Shenoy, P.P. (1991). Valuation-based system for discrete optimization. In P.P. Bonissone,

M. Henrion, L. N. Kanal, and J. Lemmer (Eds.), Uncertainty in artificial intelligence,

(Vol. 6, pp. 385-400). Amsterdam: North-Holland.,

Shenoy, P.P., & Shafer, G. (1990). Axioms for probability and belief-function propagation.

In Uncertainty in artificial intelligence. Elsevier Science Publishers.

Smets, P. (1998). The transferable belief model for quantified belief representation. In P.

Smets (Ed.), Quantified representation for uncertainty and imprecision, (Vol. 1).

Smets, P. (1990a, May). The combination of evidence in the transferable belief model.

IEEE Transactions on Pattern Analysis and Machine Intelligence, 12, 5.

Smets, P. (1990b). Constructing the pignistic probability function in a context of

uncertainty. In M. Henrion, R.D. Shachter, L.N. Kanal & J F. Lemmer (Eds.),

Uncertainty in artificial intelligence 5. North-Holland: Elsevier Science Publishers

B.V.

Srivastava, R.P. (1995, March). The belief-function approach to aggregating audit

evidence. International Journal of Intelligent Systems, 10(3), 329-356.

Srivastava, R.P. (1993, Fall). Belief functions and audit decisions. Auditors Report, 17(1),

8-12.

Srivastava, R.P., & Datta, D. (2002). Belief-function approach to evidential reasoning for

acquisition and merger decisions. In R. P. Srivastava & T. Mock (Eds.), Belief

functions in business decisions (pp. 220-248). Heidelberg: Springer-Verlag.

Srivastava, R.P., & Liu, L. (2003). Applications of belief functions in business decisions:

A review. Information Systems Frontiers (forthcoming).

Srivastava, R.P., & Lu, H. (2002, October). Structural analysis of audit evidence using

belief functions. Fuzzy Sets and Systems, 131(1), 107-120.

Srivastava, R.P., & Mock, T.J. (2002). Belief functions in business decisions. Heidelberg:

Springer-Verlag.

Srivastava, R.P., & Mock, T.J. (2000, Winter). Evidential reasoning for Webtrust assurance

services. Journal of Management Information Systems, 10(3), 11-32.

Strat, T.M. (1984). Continuous Belief functions for evidential reasoning. Proceedings of

the National Conference on Artificial Intelligence, Austin, Texas (pp. 308-313).

Thatcher, J.B., Stepina, L.P., & Boyle, R.J. (2002-2003). Turnover of information technology

workers: Examining empirically the influence of attitudes, job characteristics,

and external markets. Journal of Management Information Systems, 19, 231-261.

Wrightson, M.T. (1976). Coding rules. In R. Axelrod (Ed.), Structure of decisions: The

cognitive maps of political elites (pp. 291-332). Princeton, NJ: Princeton University

Press.

Zarley, D. Y. T., & Shafer, G. (1988). Evidential Reasoning using DELIEF. Proceedings

of the National Conference of Artificial Intelligence.

Endnotes

1 See the following references for more discussion on belief functions and their

applications: Bovee et al. (2003), Srivastava (1993), Srivastava and Datta (2002),

Srivastava and Liu (2003), and Srivastava and Mock (2000).

2 For three independent items of evidence, Dempster’s rules can be written as:

-1

1 i 2 j 3 k 1 i 2 j 3 k

i,j,k i,j,k

Bi Bj Bk =B Bi Bj Bk =

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

∩∩∩∩

.

One can easily generalize the above formula for n independent items of evidence

(see Shafer, 1976, for details).

3 The argument of m-function represents the state for which the value is assigned

and the subscript describes the evidence from which the value is derived. For

example, mX(x) = 0.6 represents 0.6 level of support for ‘x’ from an item of evidence

pertaining to the variable X.

4 Propagation is the process by which m-values on a variable or a set of variables

are moved (mapped) to another variable or a set of variables. For example, m-values

from variable X in Figure 1 can be propagated to the relational variable ‘AND’ that

consist of three variables, X, Y, and Z.

5 Vacuous extension is the process through which m-values on a smaller frame are

extended to a larger frame. For example, m(x) when vacuously extended to the joint

space of X and Y, i.e., the frame {xy, x~y, ~xy, ~x~y}, yields m(x) = m({xy, x~y}).

6 Marginalization of m-values is opposite to the vacuous extension. This process is

similar to marginalization in probability theory; it involves eliminating all the

unwanted variables by summing the m-values over the unwanted variables. For

example, assume that we have the following m-values on the joint space of X and

Y, X,Y = {xy, x~y, ~xy, ~x~y}: m({xy}) = 0.1, m({xy, x~y}) = 0.6, and m({xy, x~y, ~xy,

~x~y}) = 0.3. The marginalized m-values onto the space of X variable are: m({x})

= 0.1 + 0.6 = 0.7, and m({x, ~x}) = 0.3. Similarly, the marginalized m-values onto the

Y space are: m({y}) = 0.1, m({y, ~y}) = 0.9.

7 Through this example we are illustrating the details of the propagation process of

beliefs or m-values through a tree of variables as this is what is needed in our model

of IT job satisfaction obtained through the RCM process. A discussion on the

details of the propagation of beliefs through a network of variables is beyond the

scope of this chapter. Interested readers should see Srivastava (1995) and Shenoy

and Shafer (1990) for this kind of propagation.

8 A Markov tree is characterized by a set of nodes N and a set of edges E where each

edge is a two-element subset of N such that (Srivastava, 1995; see also, Shenoy,

1991):

• (N,E) is a tree.

• If N and N’ are two distinct nodes in N, and {N, N’} is an edge, i.e., {N,N'} ∈E ,

then N’≠

• If N and N’ are distinct nodes of N, and X is a variable in both N and N’, then

X is in every node on the path from N to N’.

9 As described in Section IV, in order to propagate m-values from ‘RR’ to ‘JT’ through

the relationship R1, one needs to vacuously extend the m-values from the space

of ‘RR’, {yesRR, noRR}, to the space of R1, which is the joint space of ‘RR’ and ‘JT’,

i.e., {(yesRR, yesJT), (yesRR, noJT), (noRR, yesJT), (noRR, noJT)}, combine the m-values

at R1, and then marginalize to the space of ‘JT’, {yesJT, noJT}.

10 This semi-structured interview guide was also part of NSF grant proposal and

Transition Study research project (Nelson, 2000; Buche, 2003).

Appendix A: Concept Dictionary with

Examples

Construct Description Example

Role not valued Company no longer needs certain skill sets

to support certain roles.

Generalists such as myself…don’t see that

role being valued much.

Role change Expectations of workers experience

transition.

I got into the analyst role, being the leader

and doing the coordination.

Fear of job loss Lack of job security.

Anyone would be worried about their

career.

workers to develop new skills.

them.

Opportunity to gain

new skills

Workers are taught new skills in classroom

or self-paced training.

Once you learn programming, and you

have that skill.

Opportunity to use

new skills

The job environment provides the

opportunities for workers to practice the

skills learned during training.

Using new skills to make the company

more competitive.

Feedback form

superiors and coworkers

Direct reaction obtained from supervisors

and co-workers that reduces ambiguity

The users let me know if the system meets

their needs.

Challenging projects Work assignments provide an intrinsic

motivation because the problem-solving

aspect takes effort.

Technical challenges of the job.

Autonomy of Work Workers have freedom and independence in

determining relevant job-related decisions.

Nobody really tells me what to do or how

to do it.

Job satisfaction Affective response to the current job

environment.

Pleasant work environment.

Appendix B: Interview Protocol10

1. What motivates you to come to work here every day?

4. What is the most important thing you contribute to this organization?

5. What could you contribute to your organization that you currently are unable to

contribute?

6. What barriers keep you from making this contribution?

7. Where do you realistically see yourself professionally in five years?

8. Where would you ideally like to see yourself professionally in five years?

9. What barriers might keep you from your ideal situation?

10. How much do you like change?

11. How much do you think the IT field, in general, is changing?

12. How much do you think the IT field at your company is changing?

14. How is your organization supporting you in personally making these changes?

15. What barriers do you see in making these changes?

16. What is your primary, one year professional goal?

18. In summary, how do you see yourself fitting into the organization’s “big picture”?

Appendix C: Propagation Illustration in

Figure 1

In this appendix we describe in detail the three steps involved in the propagation of mvalues

from variables X and Y in Figure 1 to variable Z.

Step 1: Propagation of m-values from X and Y to ‘AND’ node:

In order to propagate m-values from variable X, a smaller node with one variable and the

frame X={x,~x}, to the ‘AND’ node, a larger node consisting of three variable X, Y, and

Z with the frame AND= {xyz, x~y~z, ~xy~z, ~x~y~z}, we vacuously extend the m-values

at X to the space {xyz, x~y~z, ~xy~z, ~x~y~z} defined by the ‘AND’ node. This process

yields the following non-zero m-values from X to the ‘AND’ node:

mAND←X({xyz, x~y~z}) = mX(x) = 0.6,

mAND←X({~xy~z, ~x~y~z}) = mX(~x) = 0.2,

mAND←X({xyz, x~y~z, ~xy~z, ~x~y~z}) = mX({x, ~x}) = 0.2.

Similarly, we obtain the following non-zero m-values at the ‘AND’ node when the mvalues

from Y are propagated to the ‘AND’ node:

mAND←Y({xyz, ~xy~z}) = mY(y) = 0.7

mAND←Y({xyz, x~y~z, ~xy~z, ~x~y~z}) = mY({y,~y}) = 0.3

Step 2: Combine m-values from X and Y with the m-values at ‘AND’

We have the following set of m-values at the ‘AND’ node; one from X, one from Y, and

one at the ‘AND’ node defining the relationship.

m-values from X:

mAND←X({xyz, x~y~z}) = 0.6, mAND←X({~xy~z, ~x~y~z}) = 0.2, and

mAND←X({xyz, x~y~z, ~xy~z, ~x~y~z}) = 0.2.

m-values from Y:

mAND←Y({xyz, ~xy~z}) = 0.7,

mAND←Y({xyz, x~y~z, ~xy~z, ~x~y~z}) = 0.3.

m-values at the ‘AND’ node:

mAND({xyz, x~y~z, ~xy~z, ~x~y~z}) = 1.0.

After we combine the above m-values using Dempster’s rule, we obtain the following mvalues:

m({xyz}) = mAND←X({xyz, x~y~z}).mAND←Y({xyz, ~xy~z}).

mAND({xyz, x~y~z, ~xy~z, ~x~y~z}) = 0.6x0.7x1.0 = 0.42,

m({xyz, x~y~z}) = mAND←X({xyz, x~y~z}).

mAND←Y({xyz, x~y~z, ~xy~z, ~x~y~z}).mAND({xyz, x~y~z, ~xy~z, ~x~y~z})

= 0.6x0.3x1.0 = 0.18,

m({~xy~z}) = mAND←X({~xy~z, ~x~y~z}).mAND←Y({xyz, ~xy~z}).

mAND({xyz, x~y~z, ~xy~z, ~x~y~z})

0.2x0.7x1.0 = 0.14,

m({~xy~z, ~x~y~z}) = mAND←X({~xy~z, ~x~y~z}).

mAND←Y({xyz, x~y~z, ~xy~z, ~x~y~z }).

mAND({xyz, x~y~z, ~xy~z, ~x~y~z}) = 0.2x0.3x1.0 = 0.06,

m({xyz, ~xy~z}) = mAND←X({xyz, x~y~z, ~xy~z, ~x~y~z }).

mAND←Y({xyz, ~xy~z}).mAND({xyz, x~y~z, ~xy~z, ~x~y~z}) = 0.2x0.7x1.0 = 0.14,

m({xyz, x~y~z, ~xy~z, ~x~y~z})

= mAND←X({xyz, x~y~z, ~xy~z, ~x~y~z }).

mAND←Y({xyz, x~y~z, ~xy~z, ~x~y~z }).

mAND({xyz, x~y~z, ~xy~z, ~x~y~z }) = 0.2x0.3x1.0 = 0.06.

The above m-values are propagated to variable Z by marginalizing them to Z as described

next.

Step 3: Propagate m-values from ‘AND’ node to Z

The third step deals with propagating beliefs or m-values from ‘AND’ node to variable

Z. Since the “AND’ is a bigger node consisting of three variables, X, Y, and Z, the mvalues

have to be marginalized to variable Z. As discussed in endnote 6, marginalization

of belief functions or m-values is similar to marginalization of probabilities. The unwanted

variables are eliminated by summing the m-values over the variables. We obtain the

following m-values on variable Z as a result of propagation of m-values from X and Y

through the relationship ‘AND’ by marginalization of m-values at the ‘AND’ node:

mZ←AND({z}) = m({xyz}) = 0.42,

mZ←AND({~z}) = m({~xy~z}) + m({~xy~z, ~x~y~z}) = 0.14 + 0.06 = 0.20,

mZ←AND({z,~z}) = m({xyz, x~y~z}) + m({xyz, ~xy~z}) + m({xyz, x~y~z, ~xy~z, ~x~y~z})

= 0.18+ 0.14 + 0.06 = 0.38.

This completes the process. We now know that belief that Z is true is 0.42 (i.e., Bel(z) =

0.42), given that we know that X is true with belief 0.6 and Y is true with belief 0.7. Similarly,

we know that Z is not true with belief 0.20, i.e., Bel(~z) = 0.20, given the knowledge about

X and Y expressed in terms of the following m-values: 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.