An experimental comparison of fuzzy logic and analytic hierarchy process for medical decision support systems

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journal homepage: www.intl.elsevierhealth.com/journals/cmpb

An experimental comparison of fuzzy logic and analytic hierarchy process for medical decision support systems Faith-Michael Emeka Uzoka a,∗ , Okure Obot b , Ken Barker c , J. Osuji d a

Department of Computer Science and Information Systems, Mount Royal University, 4825 Mount Royal Gate SW, Calgary, Canada T3E6K6 b Department of Mathematics, Statistics, and Computer Science, University of Uyo, Nigeria c Department of Computer Science, University of Calgary, Calgary, Canada d School of Nursing, Mount Royal University, Calgary, Canada

a r t i c l e

i n f o

a b s t r a c t

Article history:

The task of medical diagnosis is a complex one, considering the level vagueness and uncer-

Received 15 August 2009

tainty management, especially when the disease has multiple symptoms. A number of

Received in revised form 8 June 2010

researchers have utilized the fuzzy-analytic hierarchy process (fuzzy-AHP) methodology in

Accepted 8 June 2010

handling imprecise data in medical diagnosis and therapy. The fuzzy logic is able to handle vagueness and unstructuredness in decision making, while the AHP has the ability to carry

Keywords:

out pairwise comparison of decision elements in order to determine their importance in the

Fuzzy

decision process. This study attempts to do a case comparison of the fuzzy and AHP methods

Analytical hierarchy process

in the development of medical diagnosis system, which involves basic symptoms elicita-

Malaria

tion and analysis. The results of the study indicate a non-statistically significant relative

Medical Diagnosis

superiority of the fuzzy technology over the AHP technology. Data collected from 30 malaria

Medical decision support system

patients were used to diagnose using AHP and fuzzy logic independent of one another. The results were compared and found to covary strongly. It was also discovered from the results of fuzzy logic diagnosis covary a little bit more strongly to the conventional diagnosis results than that of AHP. © 2010 Elsevier Ireland Ltd. All rights reserved.

1.

Introduction

The task of carrying out an effective and efficient differential medical diagnosis is a complex one. It involves a state space search of medical knowledge, which could become unwieldy, especially when the variables involved are numerous [1]. It is recognized [2] that a very important task in achieving hospital efficiency is to optimize the diagnostic process in terms of the number and duration of patient examinations, with accompanying accuracy, sensitivity, and specificity. The task of medical diagnosis, unlike other diagnostic processes, is made more complex because a lot of imprecision is involved. Patients cannot describe exactly what has happened to them or how they

∗

feel; doctors and nurses cannot tell exactly what they observe; laboratories report results with some degree of error; medical researchers cannot precisely characterize how diseases alter the normal functioning of the body [3]. Because of the complexity and vagueness involved in the diagnosis process, medical doctors need some level of decision support. It has been proven by [4] that the human mind can handle a combination of 7 ± 2 pieces of information at a time. Clinical decision support systems (DSSs) utilize two or more items of patient data, and some inference procedure to generate case-specific advice [5]. Four key functions of electronic clinical decision support systems are outlined in [6]: (a) “Administrative: supporting clinical coding and documentation, authorization of procedures, and referrals”; (b) “Managing clinical complexity

Corresponding author. Tel.: +14034406674. E-mail address: [email protected] (F.-M.E. Uzoka). 0169-2607/$ – see front matter © 2010 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.cmpb.2010.06.003

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and details: keeping patients on research and chemotherapy protocols; tracking orders, referrals follow-up, and preventive care; (c) “Cost control: monitoring medication orders; avoiding duplicate or unnecessary tests”; (d) “Decision support: supporting clinical diagnosis and treatment plan processes; and promoting use of best practices, condition-specific guidelines, and population-based management.” A number of expert systems have attempted to address the subject of knowledge acquisition, representation, and utilization in medical diagnosis. However, the problem of managing imprecise knowledge still exists. The first attempts at creating decision support tools for medical diagnosis began with the application of statistical methods for medical diagnosis, initiated by the pioneering efforts of Lipkin, Hardy, and Engle in the 1950s at the Cornell Medical School [7]. Logical and probabilistic approaches were explored to the diagnosis of hematological disorders. This era saw the applicability of Bayesian inference, utility theory, Boolean logic and discriminant analysis to medical diagnostic problems [8]. Bayesian inference is a popular statistical decision making process, which provides a paradigm for updating information by using Baye’s theorem, a statement of conditional probabilities relating causes (states of nature) to outcomes. Utility theory allows decision makers to give formalized preference to a space defined by the alternatives and criteria. The scores for each alternative are combined with measures of each criterion’s importance (i.e. weight) to give a total utility for the alternative. Boolean logic is a form of algebra in which all values are reduced to either true or false, while discriminant analysis is a mathematical approach which tries to differentiate between classes, categories or clusters of groups. It partitions a sample into Yes or No groups, Positive and Negative values. By the early 1970s, it became evident that statistical tools were unable to deal with most complex clinical problems [9]. The first attempt at applying artificial intelligence (AI) principles in medical diagnosis started with the efforts made by Kulikowski in 1970, aimed at moving away from purely engineering approaches toward a deeper consideration of the ‘cognitive model’ that the human physician uses in diagnosis [7]. Pattern recognition methods were the focus of AI application in medical diagnosis until 1974 when Shortliffe published the first rule based approach for therapy advice in infectious diseases [10]. Rule based programs use the “if-then-rules” in chains of deductions to reach a conclusion. Szolovits observed [3] that rule based systems are good for narrow domains of medicine, but most serious diagnostic problems are so broad and complex that straightforward attempts to chain together larger sets of rules encounter major difficulties. Such systems lacked the model of the disease or clinical reasoning. In the absence of such models, the addition of new rules leads to unanticipated interactions between rules, resulting in serious degradation of program performance [11]. Furthermore, rule based systems attempt to represent different kinds of information (defining terms, expressing domain facts, supporting inference, and problem solving) in the same formalism. This compounding of different kinds of knowledge results in poorly structured systems that are difficult to understand and maintain [12]. As research in medical diagnosis deepened, emphasis shifted to the representation and utilization of unstructu-

11

red, imprecise, and dynamic knowledge. Szolovits recognized [13] that uncertainty is the central and critical fact about medical reasoning. Uncertainty and imprecision characterizes the sources of information available to medical expert systems. Such sources include the patient, physician, laboratory, technical methods of evaluation, and mathematical models that simulate the diagnostic process [14]. Researchers in medical diagnostic systems in the past decade have attempted to find ways to manage uncertainty in medical diagnosis using soft computing methods [13]. One of the earliest efforts in this direction attempted to develop heuristic methods for imposing structure on ill-structured components of medical diagnosis, resulting in the INTERNIST-1 diagnostic program [15]. Evolutionary algorithms [2], case-based reasoning [16], and hypertext-based [17] systems, and knowledge base technology [18] have been applied in the management of imprecise and unstructured medical knowledge [19] proposed a neuro-case rule based hybridization in medical diagnosis. The utilization of fuzzy logic and AHP became very popular in attempting to resolve the problems of imprecision and uncertainty in medical diagnosis. This is because of the ability of fuzzy logic to handle vague information [20] and the ability of AHP to mathematically model unstructured information [21]. The AHP has been proposed for the building of the kernel of medical decision support system in [22], while a framework for utilizing AHP in the diagnosis of fever has been reported as well [23]. The AHP is a multi-criteria decision analysis [MCDA] method that uses mathematical algorithms to transform qualitative subjective judgments into quantitative data, which produces a computational model that serves as input into the evaluation of decision alternatives. It uses judgments from a group of decision makers along with hierarchical decomposition of a problem to derive a set of ratio-scaled measures for decision alternatives. With the AHP the analyst structures a problem hierarchically and then, through an associated measurement-and-decomposition process, determines the relative priorities consistent with overall objectives [24]. The AHP is based on four axioms: reciprocal judgments, homogeneous elements, hierarchic or feedback dependent structure, and rank order expectations [25]. Fuzzy logic [26] is a generalization of the conventional set theory as a mathematical way to represent vagueness of parameters. The basic idea in fuzzy logic is that statements are not just ‘true’ or ‘false’, but partial truth is also accepted. Fuzzy logic exhibits complementary characteristics by offering a very powerful framework for approximate reasoning. Fuzzy systems are capable of acquiring knowledge from domain experts, and attempt to model the human reasoning process at a cognitive level [27]. This study seeks to compare the fuzzy methodology with the AHP methodology in order to experimentally ascertain their levels of effectiveness/ utility in medical diagnosis. This would assist medical decision system builders in deciding on which of the tools should form the backbone in a fuzzy-AHP (or AHP-fuzzy) system. Malaria is utilized as an experimental case study for three reasons: (1) Malaria has symptoms most of which are highly vague; (2) malaria is a major killer disease in the tropics that requires a lot of attention especially in the field of medical decision making; (3) The data for this study were collected from a tropical country (Nigeria) where data

12

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i. Microbiological examination of body fluids such as cerebrospinal fluid (CSF), urine, faeces, sputum, microscopy, culture and sensitivity. ii. Immunodiagnosis, which includes serology and antigen detection. iii. Tissue diagnosis (aspirate and biopsy), which could include microscopy culture, deoxyribonucleic acid (DNA) test, cytology and histopathology. iv. Other tests such as computerized tomography (CT) and magnetic resonance imaging (MRI).

Fig. 1 – Model of medical diagnosis. Adapted from Uzoka and Famuyiwa [18].

is easily collected on malaria cases. Data for 30 patients were collected with the help of six medical doctors from four hospitals (two public and two private) for the purpose of conducting the experiment. Each doctor was requested to obtain diagnosis for five malaria patients. With the consent of the patient, the doctor utilized a ‘diagnosis sheet’, with which he rated each patient based on the 22 identified malaria symptoms. At the end of the all necessary investigations, the doctor provided a diagnosis, indicating the intensity of malaria attack. The symptoms evaluations on the diagnosis sheet for each patient were then processed using the AHP and fuzzy methodologies independently to obtain diagnoses, which were then compared with the final diagnoses by the medical doctors in order to evaluate the quality of the diagnoses by the fuzzy and AHP methodologies. In Section 2, a brief overview of the traditional method of malaria diagnosis is presented. Section 3 presents the AHP methodology for malaria diagnosis, while Section 4 shows the fuzzy methodology. In Section 5 the results are compared and discussed, while some conclusions are drawn in Section 6.

2. Conventional method of malaria diagnosis The diagnosis of malaria follows the conventional process of medical diagnosis modeled in Fig. 1. It involves the following steps [1,20]: a. Patient interrogations, which seeks an understanding of the following: the complaint, history of the complaint, patient’s previous history, patient’s social history, and patient’s family history. b. Basic laboratory investigations such as full blood count, blood biochemistry, erythrocycle sedimentations, and Xray. c. Special investigations, which could include the following:

It is noted from the studies carried out in [28] that most cases of malaria diagnosis identify the incidence of early stage malaria after patient interrogation and clinical examinations. However in cases of severe malaria, or complicated symptoms, further investigations are carried out. This study addresses diagnosis of malaria, based on patient interrogation and clinical examinations. Symptoms of malaria that have been identified in the literature include the following: fever, rigors, aches, sweating, tiredness, abdominal pain, diarrhea, loss of appetite, nausea, cough, as well as vomiting [29]. Other symptoms of malaria identified in [30] include: shaking, chills, dizziness, pains, delirium, fall in temperature, and shortness of breath. Complications of malaria are secondary conditions, symptoms, or other disorders that are caused by malaria. In many cases the distinction between symptoms of malaria and complications of malaria is unclear or arbitrary. The following complications of malaria are identified [30]: cerebral malaria, mother–infant transmission, low birth weight, anemia, jaundice, kidney http://www.wrongdiagnosis.com/k/kidney failure/intro.htm failure, fluid imbalance, enlarged spleen, enlarged http://www.wrongdiagnosis.com/sym/enlarged liver.htmliver, backwater fever, hematuria, liver complications, brain complications, and death. Other complications include [31]: acute renal failure (ARF), disseminated intravascular coagulation (DIC), and acute respiratory distress syndrome (ARDS).

3. The AHP methodology for malaria diagnosis The analytic hierarchy process, attempts to support multicriteria analysis of decision variables in order to determine the relative importance of each variable in the decision matrix on a pairwise basis [21]. The variables involved in malaria diagnosis are numerous; as such, their combinatorial analysis may become explosive, and lead to a decay of the medical expert’s preference. This is further complicated by the inability of the human mind to handle more than 7 ± 2 pieces of information at the same time [4]. The AHP deals with dependence among variables or clusters of decision structure to combine statistical and judgmental information. The analytical hierarchy process is a very popular and classical method of evaluation where priorities are derived from eigenvalues of the pairwise comparison matrix of a set of elements expressed on ratio scales. AHP falls into a class of techniques known under the name Multiple-Criteria Decision Aid (MCDA). An evaluation problem solved by MCDA can be modeled as a 7-ple {A, T, D, M, E, G, R} where A is the set

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Fig. 2 – Hierarchy of basic malaria diagnosis criteria.

of alternatives under evaluation in the model, T is the type of evaluation, D is the tree of the evaluation attributes, M is a set of associated measures, E is the set of scales associated to the attributes, G is the set of criteria constructed in order to represent the decision maker’s preference and R is the preference aggregation procedure. Multi-criteria decision methods can be classified into two: discrete and continuous methods based on the nature of alternatives to be considered in the decision process [32]. Continuous methods deal with quantities which vary continuously in the decision problem. Such methods include linear/goal programming and aspiration based models. Discrete methods have finite number of alternatives, a set of objectives, criteria for evaluating alternatives, and a method for ranking the alternatives [33]. Discrete methods are further divided into: (1) Weighting techniques such as Simple Additive Weighting [SAW] [34]; (2) ranking techniques, such as Preference Ranking Organization METHod for Enrichment Evaluations [PROMETHEE] [35], Technique for Order Preference by Similarity to Ideal Solution [TOPSIS] [34], and Ordered Weighted Averaging [OWA] [36]; and (3) mixed techniques, such as the ELECTRE [37], the analytic hierarchy process [AHP] [38], the Multi-Attribute Value Theory [39], and Value Focused Thinking [VFT] [40]. Discrete MCDA methods are most suitable in medical diagnosis because symptoms are discrete in nature with varying degrees of intensity. Prominent discrete methods that have been used in decision situations include the MAVT, VFT, and AHP. The MAVT uses a utility function that applies to outcomes of decision options which are uncertain. The decision maker makes a subjective judgment of evaluating trade-offs among alternatives. This judgment is obtained by formalizing a value structure. The VFT is fairly similar to the MAVT, except for the fact that the MAVT is used to solve decision problems based on alternatives (alternative-focused thinking), while the VFT is a way to identify desirable decision situations and reap the benefits of such situations by solving them. Value Focused Thinking is different from alternative-focused thinking in three ways [40]: (1) Significant effort is allocated to the articulation of values in VFT; (2) value articulation precedes other activities; and (3) articulated values are explicitly utilized in the identification of decision opportunities and creation of alternatives [32]. The AHP technique uses the same paradigm as the VFT. However, the AHP adopts a different approach in relative value estimation (weights) and scoring of alternatives [33]. The AHP methodology utilizes the pairwise comparison of

decision variables in order to arrive at a priority model. Such comparisons can either be done by using the eigenvalue technique [38] or regression analysis [42]. The AHP is preferred to most of the MCDA methods for the following reasons [33,42,52]: (1) AHP is structured, as such can be documented and replicated; (2) AHP is applicable to decision situations involving subjective judgments, and used both qualitative and quantitative data; (3) it provides measures of consistency of preferences; (4) AHP is suitable for group decision making; (5) eliciting required information as adopted by AHP is easy because of the fact that pairwise comparisons involve only two attributes at a time; (6) AHP meets the MCDA method properties of interaction, weighting, dominance, and scaling. The superiority of the AHP to other MCDA methods was also demonstrated in [44]. The AHP is based on four axioms: reciprocal judgments, homogeneous elements, hierarchic or feedback dependent structure, and rank order expectations [25]. The AHP methodology has undergone several modifications to incorporate risk and fuzzy elements in decision making. The pseudo code of AHP is presented in Appendix A. It is built on three basic principles, namely; decomposition, measurement of principles, and synthesis. Decomposition breaks down a problem into manageable elements that are treated individually. It begins with implicit description of the problems (the goal) and proceeds logically to the criteria (or states of nature) in terms of which outcomes are evaluated. The result of this phase is a hierarchical structure consisting of levels for grouping issues together as to their importance or influence with repect to the elements in the adjacent levels above. The decomposition of the malaria diagnostic variables into hierarchy is presented in Fig. 2. Measurement of preferences involves a pairwise comparison of evaluation variables, which are verbal statements about the strength of importance of a variable over another, represented numerically on an absolute scale. The comparison is done from the top level of the hierarchy to the bottom level in order to establish the overall priority index. Let P(i,j) be a pairwise comparison of two elements i and j; where {i,j} ∈ nk (nk = node k of the AHP tree). The larger the value of P(i,j), the more i is preferred to j in the priority rating. The following rules govern the entries in the PWC. Rule 1: P(j,i) = [p(i,j)]−1 Rule 2: If element i is judged to be of equal importance with element j, then: P(i,j) = P(j,i) = 1; in particular, P(i,i) = 1 for all i.

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Table 1 – Level 2 PWC matrix. FVR FVR ACH CNS GIT RSS GML

1.00

Table 3 – Aches PWC matrix.

ACH

CNS

GIT

RSS

GML

5.00 1.00

7.00 7.00 1.00

7.00 6.00 1.00 1.00

8.00 7.00 3.00 6.00 1.00

6.00 3.00 0.20 0.50 0.20 1.00

HDC MSH BKH JTP

HDC

MSH

BKH

JTP

1.00

6.00 1.00

6.00 1.00 1.00

6.00 1.00 1.00 1.00

Inconsistency: 0.00.

Inconsistency: 0.11.

Table 4 – Central nervous system PWC matrix. The pairwise comparison (PWC) matrices were obtained with the help of medical doctors (values averages and rounded off to whole numbers). The ability of AHP to test for consistency is one of the method’s greatest strengths. The AHP view of consistency is based on the idea of cardinal transitivity. For example, if criteria A is considered to be twice as important as criteria B, and criteria B is considered to be three times as important as criteria C, then it would imply (by perfect cardinal consistency) that criteria A be considered six times more important than Requirement C. if the experts (participants) judge Requirement A to be less important than Requirement C, it implies that a judgmental error exists and the prioritization matrix is inconsistent. To check for consistency of experts’ ratings, the principal eigenvalue max is calculated via matrix mathematics. The closer the max is to the number of criteria n, the more consistent is the result. Deviations from the consistency are represented by the consistency index (CI). Related to the CI is the consistency ratio (CR), which is the ratio of the CI to the average CI (ACI). A consistency ratio of 0.1 is acceptable [38]. However, [45] found that it is possible to answer rationally and consistently and obtain a consistency ratio above 0.1. The important issue is that respondents understand what they are doing [46], so as not to discard rational responses, which could lead to a loss of valuable information. To this end some studies have used consistency ratio of 0.2 as cut off [47]. In this study, a consistency cut off of 0.2 is adopted because of the number of comparisons and variations existing in the judgment of the medical experts due to the unstructured and vague nature of symptoms elicitation and analysis in malaria diagnosis. The pairwise comparison (PWC) matrix for the level 2 criteria of the AHP is represented in Table 1, while the PWC matrices for level 3 variables are presented in Tables 2–7. Synthesis involves the computation of eigenvalues and the eigenvector. The eigenvalues and eigenvector present a means of obtaining linear relationships among the evaluation variables. This initial table of eigenvalues and eigenvector helps to establish priority model. It is important to note that the pairwise comparisons are also carried out for elements of

CHL CHL NSA DRM TRD EXP DZN

FVR EXT SHV Inconsistency: 0.06.

FV

EXT

SHV

1.00

7.00 1.00

5.00 3.00 1.00

NSA

DRM

TRD

EXP

DZN

0.50 1.00

5.00 6.00 1.00

2.00 2.00 0.14 1.00

4.00 5.00 0.50 7.00 1.00

5.00 2.00 0.20 5.00 0.17 1.00

Inconsistency: 0.11.

Table 5 – Gastro intestinal tract PWC. VMT VMT DRH DRT SMC

1.00

DRH

DRT

0.17 1.00

0.33 4.00 1.00

SMC 3.00 6.00 6.00 1.00

Inconsistency: 0.09.

Table 6 – Respiratory system PWC. ABT ABT CUG

CUG

1.00

0.17 1.00

Inconsistency: 0.00.

Table 7 – General malaise PWC. LOA LOA YOE SUW

1.00

YOE

SUW

6.00 1.00

5.00 0.33 1.00

Inconsistency: 0.09

the sub-criteria (variables) of all evaluation criteria (factors) [48]. This can be represented mathematically in the following [49]: The eigenvalue for cell {aij } is derived as: Eij =

Table 2 – Fever PWC matrix.

1.00

Vij T.j

where Eij is the eigenvalue of cell{aij }. Vij is the value of the PWC matrix for cell {aij }. T.j is the sum of the values on column j.

(1)

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Fig. 4 – Level 3 priorities (FVR). Fig. 3 – Level 2 priorities.

The eigenvector for variable k is a column vector given as: n 

k =

Ekj

j=1

(2)

n

where



k is the eigenvector corresponding to variable k ( Ekj is the eigenvalue of cell {akj } (j = 1,2,. . .,n) n is the number of evaluation variables.

k = 1). Fig. 5 – Level 3 priorities (ACH).

The next step is the evaluation of the patient’s state of health (with respect to malaria) based on the respective factors (or variables). The sum of the ratings on the factors are derived as a basis for determining the intensity of malaria attack. The final evaluation (diagnosis) is given as: wi =



Rij j

(3)

where wi is the weighted diagnosis of patient i. Rij is the rating of the patient on variable j. j is the eigenvector of variable j.

Fig. 6 – Level 3 priorities (CNS).

The tables of eigenvalues and eigenvector (priority vector) for level 2 criteria and level 3 variables respectively were computed. Fig. 3 shows the level 2 priorities, while Figs. 4–9 show the level 3 priorities. The level 2 diagnostic criteria evaluation gives an eigenvector, 1 , while the level 3 variables produce the eigenvector, 2 for each factor. 1 combines with the column vector of level 2 factors to give the Diagnostic Factor Index for level 2 criteria (DFI1 ), while 2 combines with the column vector of the level 3 variables to give the Diagnostic Factor Index for level 3 variables (DFI2 ), as shown in (4) and (5). Fig. 7 – Level 3 priorities (GIT).

DFI1 = 0.506FVR + 0.238ACH + 0.048CNS + 0.065GIT + 0.026RSS + 0.116GML

(4)

15

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2 eigenvector; namely, fever and aches which are very common symptoms of malaria. Evidence from the results shows that malaria has reasonably low effect on the respiratory and gastro intestinal tract systems. In Table 9, the case study of the 30 patients used for the system evaluation is presented. This is based on the diagnosis shown in Appendix B.

4. The fuzzy methodology for malaria diagnosis

Fig. 8 – Level 3 priorities (RSS).

The knowledge base for malaria contains both static and dynamic information. There are qualitative and quantitative variables, which are analyzed in order to arrive at a diagnostic conclusion. The fuzzy logic of the diagnosis of malaria involves fuzzification, inference, defuzzification.

4.1.

Fuzzification

The symptoms are assigned linguistic labels and some degrees of membership according to the following functions:

Fig. 9 – Level 3 priorities (GML). x= Table 8 – Malaria intensity scale. Uniform rating 1 2 3 4 5

ADFI range 0.000000–0.999048 0.999049–1.998096 1.998097–2.997144 2.997145–3.996192 3.996193–4.99524

Malaria intensity Very low Low Moderate High Very high

DFI2(FVR) = 0.731FVR + 0.188SHV + 0.081EXT DFI2(ACH) = 0.667HDC + 0.111MSH + 0.111BKH + 0.111JTP DFI2(CNS) = 0.272CHL + 0.293NSA + 0.032DRM + 0.244TRD +0.043EXP + 0.117DZN DFI2(GIT) = 0.106VMT + 0.596DRH + 0.244DRT + 0.054SMC DFI2(RSS) = 0.143ABT + 0.857CUG DFI2(GML) = 0.717LOA + 0.195YOE + 0.088SUW

(5)

⎧ 0 < x ≤ 1, very low ⎪ ⎪ ⎪ 1 < x ≤ 2, low ⎨ 2 < x ≤ 3,

moderate

4 < x ≤ 5,

very intense

⎪ ⎪ ⎪ ⎩ 3 < x ≤ 4, intense

(7)

Fuzzification begins with the transformation of the raw data using the functions that are expressed in Eqs. (8)–(12). Each of the labels is assigned a value to emphasize the degree of symptom presence as assigned in (7). In arriving at five linguistic variables; very low, low, moderate, intense and very intense, we consider what is currently practiced conventionally and the fact that further decomposition of the variables will be confusing and degrade the performance of the system. On the other hand, less membership degree is bound to leave some classes of diagnoses unattended to. Each linguistic fuzzy membership degree is classified into 3 classes for simplicity and to avoid unnecessary computational complications.

Combining (4) and (5) we have the Aggregate Diagnostic Factor Index (ADFI), given as: ADFI =

0.369886FVR + 0.095128SHV + 0.040986EXT + 0.158746HDC + 0.026418MSH + 0.026418BKH +0.026418JTP + 0.013056CHL + 0.014064NSA + 0.001536DRM + 0.011712TRD + 0.002064EXP +0.005616DZN + 0.00689VMT + 0.03874DRH + 0.01586DRT + 0.00351SMC + 0.003718ABT +0.022282CUG + 0.083172LOA + 0.02262YOE + 0.010208SUW

The ADFI forms the basis of diagnosing patients and determining the intensity of malaria attack. In order to determine the intensity of malaria attack, a scale of intensity is formed, based on state of ‘perfect information’, whereby each of the variables is considered to have uniform values drawing from a likert scale, to determine the intensity of malaria attack. The scale is depicted in Table 8. It is observed that the intensity of malaria attack is influenced more by the variables that have high values on the eigenvector; such as fever (0.3699), headache (0.1587), and shivering (0.0951). These variables also belong to the level 2 criteria that have high ratings on the Level

Very low (x) =

Low (x) =

⎧ ⎨

0, x − 0.9, ⎩ 1,

⎧ ⎪ ⎨ 0, ⎪ ⎩

(6)

4−x 5 1,

if x < 1 if 0 < x ≤ 1 if 1 < x < 2

(8)

if x < 2 if 1 < x ≤ 2, if 2 < x < 3

(9)

if2 < x ≤ 3

(10)

if3 < x < 4

⎧ ⎨ 0,

x−1

if 3 < x ≤ 4

⎩ 4

(11)

MOD MOD MOD MOD MOD INT INT INT L INT VINT MOD MOD MOD MOD L L L VL VL INT INT L L L L MOD MOD MOD INT INT L L VL L L L VL MOD INT L INT MOD INT L VL INT INT VL MOD VL VL VLMOD VINT MOD INT INT VL VL L L MOD MOD MOD MOD VINT L INT VL L INT VL L MOD INT L VL MOD MOD MOD MOD L MOD MOD INT VL VL INT MOD MOD MOD INT INT INT INT MOD L L MOD L VL MOD MOD MOD L L L VL MOD INT L L VL MOD INT INT INT VINT L VL MOD L L INT INT VL MOD MOD VL L MOD MOD

SUW YOE LOA CUG ABT SMC DRT DRH VMT

if x < 3

NAU

Intense (x) =

⎪ ⎩

6−x 5 1,

MOD MOD MOD INT INT L VL MOD VL L MOD

Moderate (x) =

⎧ ⎪ ⎨ 0,

High Moderate High High Moderate Moderate Moderate Moderate High Moderate High Moderate Low Moderate High Very high Moderate High Very high High High High Moderate Very high Moderate High Moderate Moderate High High

DZN

3.807656 2.084208 3.112266 3.72112 2.707014 2.905654 2.731552 2.149954 3.755172 2.511676 3.302706 2.362742 1.555794 2.97958 3.967696 4.248084 2.797564 3.68112 4.15706 3.41016 3.180648 3.663516 2.692096 4.129868 2.846132 3.950988 2.876696 2.005356 3.555852 3.967696

TRD

9200195 9000087 2201421 9100182 8700008 2301694 2101387 8900021 9300201 2401756 2401903 2201450 2201203 9500490 9900779 9100176 9000090 9300199 3001561 8600003 3301696 9300199 8900033 3501931 2301761 9500583 3301355 2313030 9500588 3301391

Intensity

EXP

ADFI

DRM

Patient number

MOD MOD L MOD MOD VL VL MOD VL VL VL

Table 9 – Case study of patients diagnosed.

MOD L L MOD VL VL L L VL INT L

DIAG

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L VL VL VL MOD INT INT VL MOD MOD MOD INT MOD L VL MOD INT VL VL MOD MOD VL L L L MOD L MOD MOD INT VL INT INT

BKH MSH HDC

L VL VL L MOD VL MOD INT MOD MOD L L VL MOD INT L VINT L VL MOD VINT MOD 1 2 3 4 5 6 7 8 9 10 11

SHV EXT FVR No

R1: If fever = low and headache = very low and joint pain = intense and nausea = moderate and stomach discomfort = low and coughing = low and loss of appetite = very

Table 10 – The fuzzy rule base.

The next step in the fuzzification process is the development of fuzzy rules. The fuzzy rules for this research were developed with the assistance of six medical doctors (domain experts) with a good level of experience and expertise in malaria diagnosis. One hundred and twelve fuzzy rules were extracted from the 30 data sets. A few of those rules are presented in Table 10 as the fuzzy rule base. The rules for malaria disease have 22 symptoms (Fig. 2) as premise parameters. In cases where there are resemble rules (rules with the same criteria but different variables), the rule with the least firing strength is pruned leaving those with high firing strengths. Some of the rules are as follows:

MOD MOD MOD L VL VL MOD INT VL L VINT

(12)

VL L VL INT L INT MOD L L INT INT

if 4 < x ≤ 5

1,

MOD L L L VL MOD VL MOD INT INT MODL

x

⎩5

JTP

Very intense (x) =

⎧ ⎨ 0,

CHL

1,

17

18

c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 0 3 ( 2 0 1 1 ) 10–27

Table 11 – Fuzzy inference for a patient. No

FVR EXT SHV HDC MSH BKH JTP CHL NAU DRM TRD EXP DZN VMT DRH DRT SMC ABT CUG LOA YOE SUW

9200195

1.0

0.6

0.4

0.75

0.4

0.6

0.75 0.6

0.4

0.6

low and sweating = low and shivering = moderate and muscle ache = moderate and backache = low and delirium = moderate and tiredness = moderate and excessive sleeping = low and chills = low and dizziness = low and vomiting = low and diarrhea = moderate and abnormal breathing = intense and yellowish eyes = low and state of unwell = moderate and dehydration = moderate then malaria is moderate. R29: If fever = intense and headache = moderate and joint pain = moderate and nausea = moderate and stomach discomfort = low and coughing = very low and loss of appetite = moderate and sweating = intense and shivering = very intense and muscle ache = low and backache = moderate and delirium = intense and tiredness = intense and excessive sleeping = moderate and chills = intense and dizziness = low and vomiting = very low and diarrhea = moderate and abnormal breathing = very low and yellowish eyes = very intense and state of unwell = intense and dehydration = moderate then malaria is intense. R42: If fever = intense and headache = moderate and joint pain = moderate and nausea = moderate and stomach discomfort = low and coughing = very low and loss of appetite = very low and sweating = low and shivering = intense and muscle ache = low and backache = moderate and delirium = moderate and tiredness = very low and excessive sleeping = intense and chills = very low and dizziness = very low and vomiting = moderate and diarrhea = very low and abnormal breathing = very intense and yellowish eyes = low and state of unwell = moderate and dehydration = intense then malaria is low.

4.2.

The fuzzy inference

After the diagnosis intensity has been determined the fuzzy inference process is performed to determine the degree of influence on the fuzzy parameter. The fuzzy inference employed in this research is the Root Sum Square (RSS). RSS is given by the formula r2 where r represents the value of a firing rule in the rule base. This method combines the effects of all applicable fuzzy rules, scales the function at the respective magnitudes, and computes the ‘fuzzy’ centroid of the composite area [50]. The functions represented in Eqs. (8)–(12) were used to evaluate the raw data sets obtained from the patients. The results of this evaluation were then employed on the rule base to obtain values for the membership functions before defuzzifying to obtain their crisp output. An example of the computation is demonstrated with patient number 9200195 and presented in Table 11, while the rule base evaluation is presented in Table 12. Using RSS for each of the linguistic variables; low, moderate, intense and very intense we have: Low = 0.22 Moderate = 0.93 Intense = 1.22 Very intense = 1.03

0.75 0.6

0.4

0.1

0.4

0.6

0.6

0.4

0.4

0.75 0.4

0.6

Table 12 – Transcript of evaluation of the rule base on patient no. 9200195. Rule no. (x)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56

Non-zero minimum value 0.4 0.1 0.1 0.1 0.1 0.1 0.6 0.6 0.1 0.4 0.6 0.1 0.1 0.1 0.4 0.1 0.4 0.6 0.1 0.1 0.1 0.4 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.6 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4 0.4 0.1 0.1 0.1

Rule diagnosis for rule x Mod Mod Mod Mod Mod Int Int Int L Int Vint Mod L Int Int Mod Vint Vint Mod Mod Mod Mod L Int Int Mod L Mod Int Int Int Mod Int Vint Mod Mod Int Mod Int Mod L Mod Vint Int Mod Int Int Int Vint Mod Mod Int Int Int Int Mod

Table 13 – Results summary. S/N

Patient number

Medical experts’ diagnosis

9200195 9000087 2201421 9100182 8700008 2301694 2101387 8900021 9300201 2401756 2401903 2201450 2201203 9500490 9900779 9100176 9000090 9300199 3001561 8600003 3301696 9300199 8900033 3501931 2301761 9500583 3301355 2313030 9500588 3301391

High Moderate High High Moderate Moderate High Moderate High Moderate Moderate Moderate Low Moderate Very high High Moderate Moderate High High Moderate High Moderate Very high High High Moderate Low High High

4 3 4 4 3 3 4 3 4 3 3 3 2 3 5 4 3 3 4 4 3 4 3 5 4 4 3 2 4 4

ADFI

Malaria intensity

3.807656 2.084208 3.112266 3.72112 2.707014 2.905654 2.731552 2.149954 3.755172 2.511676 3.302706 2.362742 1.555794 2.97958 3.967696 4.248084 2.797564 3.68112 4.15706 3.41016 3.180648 3.663516 2.692096 4.129868 2.846132 3.950988 2.876696 2.005356 3.555852 3.967696

High Moderate High High Moderate Moderate Moderate Moderate High Moderate High Moderate Low Moderate High Very high Moderate High Very high High High High Moderate Very high Moderate High Moderate Moderate High High

Fuzzy results Numeric scale 4 3 4 4 3 3 3 3 4 3 4 3 2 3 4 5 3 4 5 4 4 4 3 5 3 4 3 3 4 4

% Possibility 59 42 48 56 44 48 54 41 62 49 54 33 27 44 68 62 48 59 66 65 55 62 48 72 53 61 49 39 63 61

Malaria intensity High Moderate Moderate High Moderate Moderate Moderate Moderate High Moderate Moderate Low Low Moderate High High Moderate High High High Moderate High Moderate Very high Moderate High Moderate Low High High

Matching diagnosis Numeric scale 4 3 3 4 3 3 3 3 4 3 3 2 2 3 4 4 3 4 4 4 3 4 3 5 3 4 3 2 4 4

DAF DAF DA DAF DAF DAF AF Dev 1 DAF DAF DAF DF DA DAF DAF AF Dev 1 DF DAF AF Dev 1 DF DAF DF DAF DAF DAF AF Dev 1 DAF DAF DF DAF DAF

c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 0 3 ( 2 0 1 1 ) 10–27

Numeric scale 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

AHP results

Key: DAF: all three diagnosis match (doctor, AHP, fuzzy); DF: only the doctor’s and fuzzy diagnosis matched; DA: only the doctor’s and AHP diagnosis matched; AF Dev 1: only the AHP and fuzzy diagnosis matched with a deviation of 1.

19

20

c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 0 3 ( 2 0 1 1 ) 10–27

Table 14 – System performance summary. AHP Mean Variance Observations Df F P(F ≤ f) one-tail F critical one-tail Percentage matching diagnosis (AHP/doctor) Percentage matching diagnosis (fuzzy/doctor) Percentage matching diagnosis (AHP/fuzzy) Pearson correlation [AHP/doctor] (overall diagnosis) Pearson correlation [Fuzzy/doctor] (overall diagnosis) Pearson correlation [AHP/fuzzy] (overall diagnosis) Pearson correlation [AHP/fuzzy] (false diagnosis)

3.6 0.52413793 30 29 1.0155902 0.48352982 1.86081143

Defuzzification: The defuzzification interface is a mapping from a space of fuzzy actions defined over an output universe of discourse into a space of non-fuzzy actions. This is due to the fact that the output from the inference engine is usually a fuzzy set, while for most real life applications, crisp values are often required. The three common defuzzification techniques are: max-criterion, center-of-gravity and the mean of maxima. Though the max-criterion is the simplest to implement because it produces the point at which the possibility distribution of the action reaches a maximum value [51], the CoG method is adopted in this study because it is more accurate in representing fuzzy sets of any shape [52]. The center of gravity (CoG) is an averaging technique. The difference is that the (point) masses are replaced by the membership values. This method is called the center of area defuzzification method in the case of one-dimensional fuzzy sets. In a discrete form, COG (crisp output) is defined by: Nq 

COG(BO ) =

BO (yq )yq

q=1 Nq 

Fuzzy

BO (yq )

q=1

where Nq is the number of quantization used to discretize membership function. B o (y) of the fuzzy output Bo . Defuzzifying (for patient 9200195) using the method of centre of gravity we obtain: 1.977 = 0.59. 3.38 This represents 59% possibility of intensive (high) malaria diagnosis for patient number 9200195. The other patients’ data were similarly computed and the results for the 30 patients are presented in Table 13. If two or more patients have the same possibility with the same outputs, then it is likely the patients have the same symptoms with the same fuzzy values. In treating such patients, the doctor or the system is reminded of such a previous case. The system can however filter some cognitive feelings in the

3.366667 0.516092 30 29

Doctor 3.5 0.534483 30 29 1.019737 0.479195 1.860811

AHP

Doctor

3.6 3.5 0.524138 0.534483 30 30 29 29 1.035635 0.462766 1.860811

Fuzzy 3.366667 0.516092 30 29

67% 80% 76.67% 0.716647 0.820695 0.822124 0.785714

decision support engine to see if the patients have similar physiological characteristics before prescribing the same family of drugs. Low eigenvalue contributes to low eigenvector which finally contributes to the low weighted value and the Aggregate Diagnostic Factor Index (ADFI) for a particular patient. It is this parameter that is used to determine the degree of intensity of malaria in a patient. In effect, if the possibility of diagnosis is not obvious, AHP still goes on to determine the ADFI and from there obtain the intensity of malaria, if low (mild) therapy can be prescribed so that the patient can start on time to prevent a more serious attack.

5.

Results and discussion

The study evaluated the diagnosis of 30 patients using the AHP and the fuzzy methodology separately. The essence of the study was to ascertain the degree to which each method represents the true diagnosis of the patient. Table 13 presents the summary of the diagnosis from each method, as compared with the diagnosis of medical experts, while the system performance results are displayed in Table 14. The intensity of malaria was rated as: very low (1), low (2), moderate (or mod) (3), high (or int) (4), and very high (or very int) (5). Table 14 shows that the AHP had 67% exact diagnosis with the medical expert, while the fuzzy system had 80% exact diagnosis. The correlation of diagnosis between the medical experts and the two methods indicates that the fuzzy system had a better correlation (0.821) than the AHP method (0.717). On the overall, 76.67% of the fuzzy and AHP diagnosis matched exactly. The correlation of the fuzzy/AHP overall diagnosis was very high (0.822), and the correlation of diagnosis that did not match the experts’ diagnosis (‘False diagnosis’) was also high (0.7857). Table 13 indicates that all the ‘false’ diagnosis of AHP and fuzzy, which matched each other, had a deviation of one linguistic value (AF Dev 1) from the doctors’ diagnosis. Though medical diagnosis requires exactness because of its sensitivity, it is believed that a more robust method of extracting expert’s experiential knowledge in the model development process could produce results that would almost perfectly match the medical experts’ diagnoses. This study is built on the hypothesis that there is a significant difference between the AHP-powered malaria diagnosis

21

c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 0 3 ( 2 0 1 1 ) 10–27

Table 15 – Comparative analysis of peer diagnosis performance between fuzzy and AHP. AHP

Fig. 10 – Comparison of AHP and fuzzy diagnosis with medical doctors’ diagnosis.

system, the fuzzy-powered malaria diagnosis, and the medical experts’ diagnosis. While the descriptively show that not all cases have a matching diagnosis, and the fuzzy system seemed to perform better than the AHP in terms of exact matches with the medical experts’ diagnoses, the Analysis of Variance test (Table 14) indicates that the F values in all three comparisons (AHP/fuzzy, doctor/AHP, doctor/fuzzy) are less than the critical values. Thus, the alternative hypothesis is upheld, indicating that the differences in diagnoses are not statistically significant. Fig. 10 shows the graphical comparison of the AHP/fuzzy/doctor diagnoses. The graph shows a few AHP outliers, which could also be corrected by increasing the number of second level criteria used in the AHP modeling. The number of top level criteria used in the decision model plays a significant role in the diagnosis decision model, and the higher the number of attributes (seven or more), the better the model would be [52]. One advantage fuzzy diagnoses has over other methodologies such as AHP is that, it resembles human decision making with its ability to work from approximate reasoning and ultimately find a precise solution. In order to standardize the diagnoses for both fuzzy and AHP methodology, a comparative analysis of peer diagnosis performance is carried out. This analysis has helped to improve the fuzzy diagnoses especially where patients have low fuzzy membership degree and nearly the same diagnostic value in the AHP methodology. The results of these computations are presented in the Table 15. The standard value Si of a patient is defined by the equation: Si = Mi + ((di − a) ∗ t)

Si = standard diagnostic value for each patient. Mi = minimum diagnostic value between AHP methodology and fuzzy methodology for each patient. di = raw diagnostic value for AHP methodology or fuzzy methodology for each patient. a = mean diagnostic value for AHP methodology or fuzzy methodology of the 30 patients. t = standard deviation diagnostic value AHP methodology or fuzzy methodology of the 30 patients.

Fuzzy

Mean Standard deviation

0.525983 0.117823

0.530667 0.106478

Diagnostic values Patient number 9200195 9000087 2201421 9100182 8700008 2301694 2101387 8900021 9300201 2401756 2401903 2201450 2201203 9500490 9900779 9100176 9900090 9300199 3001561 8600003 3301696 9300199 8900033 3501931 2301761 9500583 3301355 2313030 9500588 3301391

0.599991 0.341884 0.476752 0.568854 0.432373 0.474342 0.448576 0.345666 0.615457 0.420912 0.544363 0.313186 0.227568 0.436425 0.683599 0.639607 0.454835 0.599453 0.679071 0.593843 0.541106 0.626336 0.445123 0.699674 0.498054 0.626114 0.468698 0.291937 0.599387 0.626613

0.596318 0.349505 0.474605 0.563123 0.430346 0.47439 0.457729 0.351824 0.615538 0.427657 0.540994 0.308634 0.231267 0.430346 0.682887 0.629512 0.456939 0.596318 0.673771 0.599396 0.541571 0.62527 0.448251 0.701526 0.500927 0.618447 0.470406 0.301628 0.602227 0.618447

6.

Conclusion

Medical diagnosis is a complex function that requires the combinatorial analysis of decision variables, most of which are qualitative in nature. It is possible to group these variables and arrange them in a hierarchical structure for the purpose of analysis. There is a high level of uncertainty management medical diagnosis. This is because human reasoning and decision making is fuzzy, involving a high degree of vagueness in evidence, concept utilization and mental model formulation [54]. The introduction of the analytic hierarchy process (AHP) by [38] enhanced understanding of the hierarchical structuring of decision variables and popularized the methodology of pairwise comparison of such variables to determine their relative importance in the decision matrix. However, the application of the traditional AHP to evaluation of alternatives has the following limitations [55]: (1) AHP uses crisp values for scoring purposes. The derivation of weights attached to symptom is carried out by medical experts, whose perceptions and feelings about a given weight are vague; (2) the diagnosis of a given patient is subjective and relative by nature. It would be inappropriate to assign crisp value to subjective judgment, especially when the data is imprecise or fuzzy. FL provides a simple way to arrive at a definite conclusion based upon vague,

22

c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 0 3 ( 2 0 1 1 ) 10–27

ambiguous, imprecise, noisy, or missing input information. FL does not require precise inputs, is inherently robust, and can process any reasonable number of inputs and numerous outputs can be generated, though system complexity increases rapidly with more inputs and outputs. Simple, plain-language rules are used to describe the desired system response in terms of linguistic variables rather than mathematical formulas, which makes FL work closely to the way human reasoning does. Besides dealing with uncertainty, fuzzy logic models commonsense reasoning, which is difficult in conventional systems that model mainly exact reasoning [56]. AHP is known to be very good in processing multidimensional and hierarchically structured data which calls for the relative importance of criteria groupings. This feature is not found in fuzzy logic methodology, which handles vague and imprecision of data and moreover it resembles human decision making with its ability to work from approximate reasoning and ultimately find a precise solution. This study presents one of the efforts aimed at comparing the use of AHP and fuzzy logic in medical diagnosis, with the aim of determining the component that is more effective in analysis, synthesis and evaluation of medical symptoms and diseases. The more effective technology would be more suitable as the entrant technology of the inference engine in a hybrid diagnosis system. The results show that there is no statistical difference between the AHP and fuzzy logic in terms of effectiveness of diagnosis of malaria. However, a close inspection of the system performance summary (Table 14) shows that the fuzzy logic is slightly better than the AHP in obtaining matching diagnoses with the medical experts’ diagnosis. Since the 1980s a number of works have been done in the use of fuzzy or AHP or a combination of both in medical decision making. However, not much research has been conducted on the comparison of both methods. A literature search shows only one of such studies, which was conducted by [42] on the comparison of AHP and fuzzy methodologies in hospital equipment acquisition decision. The results of their study are similar to those obtained in our study – that the fuzzy method produces a non-significant superiority over the AHP method. While this study has utilized malaria as a case, it is important to note that this may not present the level of generalization necessary to conclude the slight relative superiority of fuzzy logic in the inference process, especially when there is no statistical significance shown by the output variations. It is postulated that several experimental trials utilizing varying

diseases, large number of cases, and a large number of medical experts may make the difference in results to be very infinitesimal. We therefore conclude that the fuzzy logic shows a better but non-statistically significant performance over the AHP methodology, and is better utilized in the building of the entrant module in a hybrid system involving both methodologies.

Appendix A. Appendix A AHP Pseudocode.

Appendix B. Appendix B

JTP

CHL

NSA

DRM

TRD

EXP

DZN

VMT

DRH

DRT

SMC

ABT

CUG

LOA

YOE

SUW

9200195 9000087 2201421 9100182 8700008 2301694 2101387 8900021 9300201 2401756 2401903 2201450 2201203 9500490 9900779 9100176 9000090 9300199 3001561 8600003 3301696 9300199 8900033 3501931 2301761 9500583 3301355 2313030 9500588 3301391

4 2 3 2 2 3 2 2 2 2 2 3 3 2 3 5 3 3 3 3 3 3 3 3 3 3 3 3 3 3

3 3 3 2 2 4 3 2 2 3 4 4 3 3 5 3 3 3 5 3 5 3 3 3 3 5 5 3 3 5

1 1 3 3 2 2 2 1 3 2 2 1 1 3 3 1 1 3 3 3 3 3 1 3 3 3 1 1 3 3

3 2 2 3 3 2 2 2 3 2 4 1 2 3 3 3 3 3 3 3 3 3 3 3 3 5 1 3 3 3

4 2 3 3 2 2 4 3 4 2 2 1 1 3 3 5 3 3 3 3 3 5 3 5 3 3 1 1 3 3

3 2 2 3 3 4 2 3 3 2 3 2 2 3 5 3 3 3 3 3 5 3 3 3 3 3 3 3 3 5

2 2 1 3 3 2 3 2 3 3 4 2 2 3 3 3 3 1 3 3 3 3 3 3 3 5 3 3 3 3

1 1 2 2 1 1 2 1 1 2 1 2 3 1 5 1 1 3 3 1 1 3 1 1 3 1 3 3 1 5

2 2 2 3 1 2 2 2 2 3 2 1 2 1 3 3 3 3 3 1 3 3 3 3 3 3 1 3 1 3

3 3 2 3 2 2 2 3 3 4 2 3 2 3 3 3 3 3 3 3 3 3 3 3 5 3 3 3 3 3

1 2 2 3 3 2 1 1 2 3 2 1 3 2 3 1 3 3 3 3 3 1 1 3 3 3 1 3 3 3

2 1 2 3 2 3 2 2 1 1 2 1 3 2 3 3 1 3 3 3 3 3 3 1 1 3 1 3 3 3

2 1 1 1 2 1 1 1 2 1 1 3 2 1 3 3 1 1 1 3 1 1 1 3 1 1 3 3 1 3

4 3 3 4 3 4 2 4 3 4 5 3 2 3 5 5 3 3 5 3 5 3 5 3 5 5 3 3 3 5

2 3 2 1 1 2 1 3 2 3 4 2 1 1 3 3 3 3 1 1 3 1 3 3 3 5 3 1 1 3

3 3 2 3 3 2 2 3 4 4 5 3 1 2 5 3 3 3 3 3 3 3 3 5 5 5 3 1 3 5

5 2 4 5 4 3 4 2 5 3 4 2 1 4 5 5 3 5 5 5 3 5 3 5 3 5 3 1 5 5

3 2 3 2 1 1 2 1 3 2 2 3 2 2 3 3 3 3 3 1 1 3 1 3 3 3 3 3 3 3

2 3 2 3 2 1 2 1 2 2 3 2 1 2 3 3 3 3 3 3 1 3 1 3 3 3 3 1 3 3

4 2 3 4 2 5 2 3 4 1 3 3 2 3 3 5 3 3 5 3 5 3 3 5 1 3 3 3 3 3

2 1 2 3 2 2 2 1 3 2 2 2 1 3 3 3 1 3 3 3 3 3 1 3 3 3 3 1 3 3

3 1 2 2 3 2 2 1 4 3 3 2 2 2 3 3 1 3 3 3 3 3 1 5 3 3 3 3 3 3

c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 0 3 ( 2 0 1 1 ) 10–27

Rating of patients on the malaria diagnosis variables. Patient number FVR EXT SHV HDC MSH BKH

23

24

Appendix C. Appendix C

BKH

JTP

CHL

NSA

DRM

TRD

EXP

DZN

VMT

DRH

DRT

SMC

ABT

CUG

LOA

YOE

SUW

ADFI

0.053 0.026 0.053 0.079 0.053 0.053 0.053 0.026 0.079 0.053 0.053 0.053 0.026 0.079 0.079 0.079 0.026 0.079 0.079 0.079 0.079 0.079 0.026 0.079 0.079 0.079 0.079 0.026 0.079 0.079

0.079 0.026 0.053 0.053 0.079 0.053 0.053 0.026 0.106 0.079 0.079 0.053 0.053 0.053 0.079 0.079 0.026 0.079 0.079 0.079 0.079 0.079 0.026 0.132 0.079 0.079 0.079 0.079 0.079 0.079

0.106 0.053 0.079 0.053 0.053 0.079 0.053 0.053 0.053 0.053 0.053 0.079 0.079 0.053 0.079 0.132 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.079

0.039 0.039 0.039 0.026 0.026 0.052 0.039 0.026 0.026 0.039 0.052 0.052 0.039 0.039 0.065 0.039 0.039 0.039 0.065 0.039 0.065 0.039 0.039 0.039 0.039 0.065 0.065 0.039 0.039 0.065

0.014 0.014 0.042 0.042 0.028 0.028 0.028 0.014 0.042 0.028 0.028 0.014 0.014 0.042 0.042 0.014 0.014 0.042 0.042 0.042 0.042 0.042 0.014 0.042 0.042 0.042 0.014 0.014 0.042 0.042

0.005 0.003 0.003 0.005 0.005 0.003 0.003 0.003 0.005 0.003 0.006 0.002 0.003 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.008 0.002 0.005 0.005 0.005

0.047 0.023 0.035 0.035 0.023 0.023 0.047 0.035 0.047 0.023 0.023 0.012 0.012 0.035 0.035 0.059 0.035 0.035 0.035 0.035 0.035 0.059 0.035 0.059 0.035 0.035 0.012 0.012 0.035 0.035

0.006 0.004 0.004 0.006 0.006 0.008 0.004 0.006 0.006 0.004 0.006 0.004 0.004 0.006 0.010 0.006 0.006 0.006 0.006 0.006 0.010 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.010

0.011 0.011 0.006 0.017 0.017 0.011 0.017 0.011 0.017 0.017 0.022 0.011 0.011 0.017 0.017 0.017 0.017 0.006 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.028 0.017 0.017 0.017 0.017

0.007 0.007 0.014 0.014 0.007 0.007 0.014 0.007 0.007 0.014 0.007 0.014 0.021 0.007 0.034 0.007 0.007 0.021 0.021 0.007 0.007 0.021 0.007 0.007 0.021 0.007 0.021 0.021 0.007 0.034

0.077 0.077 0.077 0.116 0.039 0.077 0.077 0.077 0.077 0.116 0.077 0.039 0.077 0.039 0.116 0.116 0.116 0.116 0.116 0.039 0.116 0.116 0.116 0.116 0.116 0.116 0.039 0.116 0.039 0.116

0.048 0.048 0.032 0.048 0.032 0.032 0.032 0.048 0.048 0.063 0.032 0.048 0.032 0.048 0.048 0.048 0.048 0.048 0.048 0.048 0.048 0.048 0.048 0.048 0.079 0.048 0.048 0.048 0.048 0.048

0.004 0.007 0.007 0.011 0.011 0.007 0.004 0.004 0.007 0.011 0.007 0.004 0.011 0.007 0.011 0.004 0.011 0.011 0.011 0.011 0.011 0.004 0.004 0.011 0.011 0.011 0.004 0.011 0.011 0.011

0.007 0.004 0.007 0.011 0.007 0.011 0.007 0.007 0.004 0.004 0.007 0.004 0.011 0.007 0.011 0.011 0.004 0.011 0.011 0.011 0.011 0.011 0.011 0.004 0.004 0.011 0.004 0.011 0.011 0.011

0.045 0.022 0.022 0.022 0.045 0.022 0.022 0.022 0.045 0.022 0.022 0.067 0.045 0.022 0.067 0.067 0.022 0.022 0.022 0.067 0.022 0.022 0.022 0.067 0.022 0.022 0.067 0.067 0.022 0.067

0.333 0.250 0.250 0.333 0.250 0.333 0.166 0.333 0.250 0.333 0.416 0.250 0.166 0.250 0.416 0.416 0.250 0.250 0.416 0.250 0.416 0.250 0.416 0.250 0.416 0.416 0.250 0.250 0.250 0.416

0.045 0.068 0.045 0.023 0.023 0.045 0.023 0.068 0.045 0.068 0.090 0.045 0.023 0.023 0.068 0.068 0.068 0.068 0.023 0.023 0.068 0.023 0.068 0.068 0.068 0.113 0.068 0.023 0.023 0.068

0.031 0.031 0.020 0.031 0.031 0.020 0.020 0.031 0.041 0.041 0.051 0.031 0.010 0.020 0.051 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.051 0.051 0.051 0.031 0.010 0.031 0.051

3.80766 2.08421 3.11227 3.72112 2.70701 2.90565 2.73155 2.14995 3.75517 2.51168 3.30271 2.36274 1.55579 2.97958 3.96770 4.24808 2.79756 3.68112 4.15706 3.41016 3.18065 3.66352 2.69210 4.12987 2.84613 3.95099 2.87670 2.00536 3.55585 3.96770

9200195 9000087 2201421 9100182 8700008 2301694 2101387 8900021 9300201 2401756 2401903 2201450 2201203 9500490 9900779 9100176 9000090 9300199 3001561 8600003 3301696 9300199 8900033 3501931 2301761 9500583 3301355 2313030 9500588 3301391

1.849 0.740 1.480 1.849 1.480 1.110 1.480 0.740 1.849 1.110 1.480 0.740 0.370 1.480 1.849 1.849 1.110 1.849 1.849 1.849 1.110 1.849 1.110 1.849 1.110 1.849 1.110 0.370 1.849 1.849

0.285 0.190 0.285 0.190 0.095 0.095 0.190 0.095 0.285 0.190 0.190 0.285 0.190 0.190 0.285 0.285 0.285 0.285 0.285 0.095 0.095 0.285 0.095 0.285 0.285 0.285 0.285 0.285 0.285 0.285

0.082 0.123 0.082 0.123 0.082 0.041 0.082 0.041 0.082 0.082 0.123 0.082 0.041 0.082 0.123 0.123 0.123 0.123 0.123 0.123 0.041 0.123 0.041 0.123 0.123 0.123 0.123 0.041 0.123 0.123

0.635 0.317 0.476 0.635 0.317 0.794 0.317 0.476 0.635 0.159 0.476 0.476 0.317 0.476 0.476 0.794 0.476 0.476 0.794 0.476 0.794 0.476 0.476 0.794 0.159 0.476 0.476 0.476 0.476 0.476

c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 0 3 ( 2 0 1 1 ) 10–27

Patient evaluation using the ahp priority model. Patient number FVR EXT SHV HDC MSH

Appendix D. Appendix D

FVR EXT SHV HDC MSH BKH JTP CHL NSA DRM TRD EXP DZN VMT DRH DRT SMC ABT CUG LOA YOE SUW SUM

1

2

3

4

0.369886 0.095128 0.040986 0.158746 0.026418 0.026418 0.026418 0.013056 0.014064 0.001536 0.011712 0.002064 0.005616 0.00689 0.03874 0.01586 0.00351 0.003718 0.022282 0.083172 0.02262 0.010208 0.999048

0.739772 0.190256 0.081972 0.317492 0.052836 0.052836 0.052836 0.026112 0.028128 0.003072 0.023424 0.004128 0.011232 0.01378 0.07748 0.03172 0.00702 0.007436 0.044564 0.166344 0.04524 0.020416 1.998096

1.109658 0.285384 0.122958 0.476238 0.079254 0.079254 0.079254 0.039168 0.042192 0.004608 0.035136 0.006192 0.016848 0.02067 0.11622 0.04758 0.01053 0.011154 0.066846 0.249516 0.06786 0.030624 2.997144

1.479544 0.380512 0.163944 0.634984 0.105672 0.105672 0.105672 0.052224 0.056256 0.006144 0.046848 0.008256 0.022464 0.02756 0.15496 0.06344 0.01404 0.014872 0.089128 0.332688 0.09048 0.040832 3.996192

5 1.84943 0.47564 0.20493 0.79373 0.13209 0.13209 0.13209 0.06528 0.07032 0.00768 0.05856 0.01032 0.02808 0.03445 0.1937 0.0793 0.01755 0.01859 0.11141 0.41586 0.1131 0.05104 4.99524

c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 0 3 ( 2 0 1 1 ) 10–27

Determination of intensity scale.

25

26

c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 0 3 ( 2 0 1 1 ) 10–27

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