An empirical study of a hybrid imbalanced-class DT-RST classification procedure to elucidate therapeutic effects in uremia patients

Abstract

The high prevalence and incidence of severe renal diseases exhaust constrained medical resources for the treatment of uremia patients. In addition, the problem of imbalanced-class data distributions induces negative effects on classifier learning algorithms. Hemodialysis is the most common treatment for uremia diseases due to the limited supply of donated organs available for transplantation. This study focused on assessing the adequacy of hemodialysis. The lack of available information represents the primary obstacle limiting the evaluation of adequacy, namely: (1) the imbalanced-class problem in a given dataset, (2) obeying mathematical distributions for a given dataset, (3) a lack of effective methods for identifying determinant attributes, and (4) developing effective decision rules to explain a given dataset. To address these issues for determining the therapeutic effects of hemodialysis in uremia patients, this study proposes a hybrid imbalanced-class decision tree-rough set model to integrate the knowledge of expert physicians, a feature selection method, imbalanced sampling techniques, a rough set classifier, and a rule filter. The method was assessed by examining the medical records of uremia patients from a medical center in Taiwan. The proposed method yields better performance compared to previously reported methods according to the evaluation criteria.

Appendices

Appendix 1: Definitions of rough set theory

Definition 1

Let $B \subseteq A$ and $X \subseteq U$ denote an information system (or an attribute–value system) that uses an information table to represent a basic knowledge framework in which the columns contain “attributes” and the rows contain “objects.” U is the defined universe, a non-empty set of finite objects, and A is a non-empty finite set of attributes $\varvec{C} \cup \varvec{D}$, such that $a:\varvec{U} \to V_{a}$, where $V_{a}$ is the set of values that attribute a may take and where C and D are finite sets of condition and decision attributes, respectively, for objects x, y such that for each condition attribute a, a(x) = a(y), and d(x) = d(y). Furthermore, B is a reduced set of attributes, and X is a subset of objects in the approximation space.

Definition 2

Let $B \subseteq A$ to indicate an associated equivalence relation denoted IND(B), which is termed a B-indiscernibility relation, and the equivalence classes of the B-indiscernibility relation are denoted [x] _B , as follows:

$$\varvec{IND}(\varvec{B}) = \left\{ {x,y \in U^{2} :a(x) = a(y),\quad \forall a \in B} \right\},$$

(1)

$$\left[ {x_{i} } \right]_{B} = \left\{ {x_{j} \in U^{2} :(x_{i} ,y_{j} ) \in \varvec{IND}(\varvec{B})} \right\},\,x_{i} \in U.$$

(2)

The equivalence class of IND(B) is termed the elementary set in B because it represents the smallest discernible objects.

Definition 3

The set X is approximated using the information table contained in B by constructing the B-lower and B-upper approximation sets as follows:

$$\underline{B(X)} = \bigcup\limits_{x \in U} {\{ B(x):B(x) \subseteq X\} } ,\quad ({\text{lower}}\,{\text{approximation}})$$

(3)

$$\overline{B(X)} = \bigcup\limits_{x \in U} {\{ B(x):B(x) \cap X \ne \emptyset \} } ,\quad ({\text{upper}}\,{\text{approximation}})$$

(4)

where x denotes an object in the universe U. The B-lower approximation is the union of all of the equivalence classes in [x] _B that the target set contains. However, the B-upper approximation is the union of all of the equivalence classes in [x] _B that have non-empty intersections with the target set. Given the lower and upper approximations of a set $X \subseteq U$, the universe is divided into boundary, positive, and negative regions, as follows:

$$BND(X) = \overline{B(X)} - \underline{B(X)} ,\quad ({\text{boundary}}\,{\text{region}})$$

(5)

$$POS(X) = \underline{B(X)} ,\quad ({\text{positive}}\,{\text{region}})$$

(6)

$$NEG(X) = U - \overline{B(X)} .\quad ({\text{negative}}\,{\text{region}})$$

(7)

The elements in $\underline{B(X)}$ (lower approximation) can be classified as positive members of target set X using the knowledge in B, but the elements in $\overline{B(X)}$ (upper approximation) can be classified only as possible members of target set X based on the knowledge in B. The boundary region set, BND(X), contains objects that cannot be clearly classified as members of X using the knowledge in B. That is, they can neither be ruled in nor be ruled out as members of the target set X. The target set X is termed “rough” (or “roughly definable”) with respect to the knowledge in B provided that the boundary region is non-empty. The positive region, POS(X), contains all of the objects of U that can be classified into classes using the knowledge in B. The negative region, NEG(X), contains the set of objects that can be definitely ruled out as members of the target set.

Additionally, the quality of approximation of the classification X by B describes the percentage of objects that can be correctly classified as $\gamma_{B} (X)$ [67]. If $\gamma_{B} (X) = 1$, the decision table is consistent; otherwise, it is inconsistent. The accuracy of the rough set representation of the target set X can be represented as $\alpha_{B} (X)$, as given in Eq. (8), and the quality of approximation of the classification X by B is given by Eq. (9) [67].

$$\alpha_{B} (X) = {{\left| {\overline{B(X)} } \right|} \mathord{\left/ {\vphantom {{\left| {\overline{B(X)} } \right|} {\left| {\underline{B(X)} } \right|}}} \right. \kern-0pt} {\left| {\underline{B(X)} } \right|}},\quad ({\text{classification}}\,{\text{accuracy}})$$

(8)

$$\gamma_{B} (X) = {{\sum {{\text{card}}\left( {\underline{{B(X_{i} )}} } \right)} } \mathord{\left/ {\vphantom {{\sum {{\text{card}}\left( {\underline{{B(X_{i} )}} } \right)} } {{\text{card}}(U)}}} \right. \kern-0pt} {{\text{card}}(U)}}.\quad ({\text{classification}}\,{\text{quality}})$$

(9)

Definition 4

The condition attributes that do not provide any additional information for the objects in U can be removed and reduced in a decision process of an information system. There is a subset of attributes that can, by itself, fully characterize the knowledge in the database, and such an attribute set is called a reduct. Given a condition attribute C and a decision attribute D, the reduct is defined as any $R \subseteq C$, such that $\gamma \left( {R,D} \right) = \gamma \left( {C,D} \right)$, and the reducted set is defined as the power set $R(C)$. The reduct set is the minimal non-redundant subset of attributes from the original dataset for IND(B) = IND(A) that all provide the same classification quality of the universe of discourse. That is, the reduct set is the subset of all possible reducts of the equivalence relation denoted by C and D, and the minimal reduct is the reduct of least cardinality for the equivalence relation.

Definition 5

The set of attributes that are common to all reducts is termed the core. The core is composed of the most relevant attributes in the original dataset and cannot be removed from an information system without causing collapse of the equivalence-class structure. There is no indispensable attribute if the core is empty. Let RED(A) denote all reducts of A. The intersection of RED(A) is labeled a core of A, shown as Eq. (10).

$$\varvec{CORE}(\varvec{A}) = \cap \varvec{RED}(\varvec{A})\quad ({\text{core}}\,\,{\text{concept}})$$

(10)

Appendix 2: Evaluation standard for imbalanced-class data

Traditionally, although the most common evaluation measurement is the overall accuracy rate, this criterion alone cannot assess classification performance, as accuracy is not a reliable metric for the real performance of a classifier and will yield misleading results if the dataset is unbalanced. Thus, other evaluation standards should be used when considering imbalanced-class data problems. In this study, it is absolutely necessary to adopt different criteria, such as accuracy (in %), precision, recall, F-measure, G-mean, true-positive (TP) rate, true-negative (TN) rate, false-positive (FP) rate, false-negative (FN) rate, receiver operation characteristic area (ROCA) [35], and the number of rules, to measure the performance of classifiers from various aspects. Furthermore, the mean, standard deviation, number of generated rules, and run time (in second) of each of the 14 methods were calculated for additional verification. The TP denotes the number of correct predictions that a sample is positive, FP represents the number of incorrect predictions that a sample is positive, FN is the number of incorrect predictions that a sample is negative, and TN is the number of correct predictions. Precision is the fraction of retrieved instances that are relevant, whereas recall is the fraction of relevant instances that are retrieved. High recall indicates that an algorithm returned most of the relevant results, whereas high precision means that an algorithm returned substantially more relevant results than irrelevant. The F-measure measures a combination of the precision and recall for given samples, and the G-mean measures the balanced performance of a learning algorithm between the TP-rate and the TN-rate. Particularly, for imbalanced-class data problems, the classification performance increases with the accuracy, TP-rate, TN-rate, precision, recall, F-measure, G-mean, and ROCA. However, as the FP-rate, FN-rate, and standard deviation decrease, the performance increases, and vice versa.

The related evaluation criteria are given as Eqs. (11–19).

$${\text{Accuracy}} = \frac{{{\text{TP}} + {\text{TN}}}}{{{\text{TP}} + {\text{FN}} + {\text{FP}} + {\text{TN}}}},$$

(11)

$${\text{Precision}} = \frac{\text{TP}}{{{\text{TP}} + {\text{FP}}}},$$

(12)

$${\text{Recall}} = \frac{\text{TP}}{{{\text{TP}} + FN}},$$

(13)

$${\text{F - measure}} = \frac{{2 * {\text{precision}} * {\text{recall}}}}{{{\text{precision}} + {\text{recall}}}},$$

(14)

$${\text{G - mean}} = \sqrt {\frac{\text{TP}}{{{\text{TP}} + {\text{FN}}}} \times \frac{\text{TN}}{{{\text{FP}} + {\text{TN}}}}} ,$$

(15)

$${\text{TP}}_{\text{rate}} = \frac{\text{TP}}{{{\text{TP}} + {\text{FN}}}},$$

(16)

$${\text{TN}}_{\text{rate}} = \frac{\text{TN}}{{{\text{FP}} + {\text{TN}}}},$$

(17)

$${\text{FP}}_{\text{rate}} = \frac{\text{FP}}{{{\text{FP}} + {\text{TN}}}},$$

(18)

$${\text{FN}}_{\text{rate}} = \frac{\text{FN}}{{{\text{TP}} + {\text{FN}}}}\cdot$$

(19)