11.2.1 Standard MIP Approach

Consider a classification problem in which there are two groups (G₁ and G₂). Here, G₁ is a set of observations in the first group above an estimated discriminant function and G₂ is a set of observations in the second group below that estimated discriminant function. The sum of the two groups contains n observations ( j = 1,…, n). Each observation is characterized by m independent factors (i = 1,…, m), denoted by z_ij. Note that there is no distinction between inputs and outputs in DEA‐DA, as found in the previous chapters on DEA. The group membership of each observation should be known before the DA computation. The observation (z_ij) may consist of a data set that contains zero, positive and/or negative values.

The single‐stage approach, often referred to as a “standard MIP approach,” is originally formulated as follows:

(11.1)

where M is a given large number (e.g., 10000) and ε is a given small number (e.g., 1). Both should be prescribed before the computation of Model (11.1). The objective function minimizes the total number of incorrectly classified observations by counting a binary variable (μ_j). The binary variable (0 or 1) expresses correct or incorrect classification. A discriminant score for cutting‐off between two groups is expressed by cf (  j G₁) and cf‐ε (  j G₂), respectively. The small number (ε) is incorporated into Model (11.1) in order to avoid a case where an observation(s) exists on an estimated discriminant function. Thus, the small number is for perfect classification. Furthermore, a weight estimate regarding the i‐th factor is expressed by λ_i. All the observations (z_ij for all i and j) are connected by a weighted linear combination, or , that consists of a discriminant function. The weights on factors are restricted in such a manner that the sum of absolute values of λ_i (i = 1,..., m) is unity in Model (11.1). The treatment is because we need to deal with zero and/or negative values in an observed data set. It is easily thought that λ_i is a weight on each factor, not each observation (i.e., DMU), as found in a conventional use of DA. The weight simply expresses the importance of each factor in terms of estimating a discriminant function. The importance among weight estimates is specified by a percentage expression.

It is possible for us to formulate Model (11.1) by maximizing the number of correct classifications. Both incorrect minimization and correct maximization does not produce a major difference between them. Because of different MIP formulations and different computations (i.e., minimization or maximization), the two approaches may produce different but similar results in terms of their classification capabilities.

In examining Model (11.1), we understand that it is difficult for us to directly solve the model because =1 is incorporated in the formulation. Hence, Model (11.1) needs to be reformulated by separating each weight λ_i() for i = 1,…, m. Here, and indicate the positive and negative parts of the i‐th weight, respectively.

Following the research effort of Sueyoshi (2006), this chapter reformulates Model (11.1) as follows:

(11.2)

In Model (11.2), binary variables ( and for i = 1,…, m) are incorporated in the formulation. The first two groups of constraints in Model (11.2) are originated from those of Model (11.1). The third constraint expresses |λ_i| by for i = 1,…, m. The third one is often referred to as “normalization” in the DA community. The fourth group of constraints indicate the upper and lower bounds of and those of , respectively. The fifth group of constraints implies that the sum of binary variables ( and for all i) is less than or equal to unity. The restriction avoids a simultaneous occurrence of > 0 and > 0. The last constraints are incorporated into Model (11.2) in order to utilize all weights (so, being a full model for a discriminant function).

All observations, or Z_j = (z_1j, …, _zmj)^Tr for all j, are classified by the following rule:

1. If cf *, then the j‐th observation belongs to G₁, and
2. If ≤ , then the j‐th observation belongs to G_2.

Here, (i = 1,…, m) and cf * are obtained from the optimality of Model (11.2). The classification of a new sample, or Z_k = (z_1k, …, _zmk)^Tr for the specific k, is determined by the same rule by replacing the subscript j by k in the classification rule.

A methodological strength of the standard MIP approach is that we can utilize a single classification ( for all i = 1,…, m and cf *). A drawback of the approach is that the computation effort on Model (11.2) is more complex and more tedious (i.e., more time‐consuming) than that of the two‐stage MIP approach, discussed in Section 11.2.2. The computational concern seems inconsistent with our expectation. The rationale concerning why the computational efficiency occurs in the two‐stage MIP approach will be discussed in the next subsection.

Although the standard MIP approach is more time‐consuming than the two‐stage MIP approach in solving various DA problems, the former has an analytical benefit that should be noted here. That is, it is possible for us to reformulate Model (11.2) by linear programing which minimizes the total amount of slacks for expressing distances between observations of independent factors and an estimated discriminant function. It is trivial that a linear programming formulation is more efficient than the proposed MIP formulation, or Model (11.2). This chapter, along with Chapter 25, does not describe the linear programming formulation, because the performance of DA is usually evaluated by a classification rate, as documented in this chapter. Of course, we clearly understand that if we discuss frontier analysis, a linear programming formulation is sufficient enough to develop an efficiency frontier for measuring a degree of OE. See Chapter 3 on frontier analysis.

11.2.2 Two‐stage MIP Approach

It is widely known that the main source of misclassification in DA is due to the existence of an overlap between two groups in a data set. This type of problem does not occur in DEA. If we expect to increase the number of observations classified correctly, then DA approaches must consider how to handle an existence of such an overlap. If there is no overlap, most DA approaches can produce a perfect, or almost perfect, classification. However, if there is an overlap in the observations, an additional computation process is usually necessary to deal with this overlap. Thus, it is expected at the initial stage of DA that there is a tradeoff between computational effort (time) and level of classification capability.

In this chapter, a two‐stage approach is proposed to handle such an overlap and thus increase the classification capability of DA. Surprisingly, this two‐stage approach may decrease the computational burden regarding DA, depending upon the size of the overlap. The proposed computation process of the two‐stage approach consists of classification and overlap identification, along with handling the overlap.

The first stage of the two‐stage approach, discussed by Sueyoshi (2006), is formulated as follows:

Stage 1: Classification and Overlap Identification (COI):

(11.3)

In Model (11.3), the unrestricted variable “ρ” indicates the absolute distance between the discriminant score and a discriminant function.

Let (=), cf* and ρ * be the optimal solution of the first stage (COI), or Model (11.3). On the optimality of Model (11.3), all observations on independent factors, or , for j = 1,..., n are classified by the following rule:

1. If then the j‐th observation belongs to G₁ (= C₁),
2. If , then the observation belongs to G₂ (= C₂) and
3. If , then the observation belongs to an overlap.

Based upon the classification rule, the original sample data set (G) is classified into the following four subsets ():

1. ,
2. ,
3. and
4. .

In the classification, C₁ and C₂ indicate a clearly above group and a clearly below group, respectively. Here, C₁ is a partial group of G₁ which are clearly above the estimated discriminant function in Stage 1 and C₂ is a partial group of G₂ which are clearly below the estimated discriminant function in Stage 1. The remaining two groups () belong to the overlap in which D₁ is a part of G₁ and D₂ is a part of G₂. A new sample (Z_k) is also classified by the above rules by replacing subscript j by subscript k.

Stage 2: Handling Overlap (HO):

The existence of an overlap is identified by at the first stage for overlap identification. In contrast, indicates no overlap between two groups. In the former case when the overlap is found between them, the second stage needs to reclassify all the observations, belonging to the overlap (), because the group membership of these observations is still unknown and not yet determined. Mathematically, the second (HO) stage is formulated as follows:

(11.4)

In Model (11.4), the binary variable (μ_j) counts the number of observations classified incorrectly. These observations to be examined at the second stage are reduced from to as a result of the first stage. If the number of observations belonging to is larger than m (the number of weights to be estimated), then = m is used in Model (11.4). However, if the number is less than m, then is used in Model (11.4) where the number “p” (< m) indicates a prescribed number in such a manner that p is less than or equal to the number of observations in . Thus, the degree of freedom needs to be considered for the selection of p. In reality, prior information is necessary for the selection of p.

Observations belonging to the overlap (D₁D₂) are classified at the second (HO) stage by the following rule:

1. If , then the j‐th observation in the overlap belongs to G₁ and
2. If then the j‐th observation belongs to G₂.

A new sample (Z_k) is also classified by the above rule by replacing the subscript (j) by the one (k).

Figures 11.1 to 11.4 visually describe the classification processes of the two‐stage approach. For our visual convenience, Figure 11.1 has coordinates for z₁ and z₂, the two factors which depict observations belonging to two groups G₁ and G₂. Stage 1 is used to identify an occurrence of an overlap between two groups by two lines. Thus, Stage 1 classifies all observations into these three groups as follows: a part of G₁, a part of G₂ and an overlap group. In the classification, the clearly above group (C₁) is part of G₁ and the clearly below group (C₂) is part of G₂. See Figure 11.3 for a visual description of C₁ and C₂. The remaining group belongs to the overlap. Stage 1 cannot classify observations belonging to the overlap.

Figure 11.2 visually describes how Stage 1 determines the size of an overlap between two groups by using a distance measure (ρ). All the observations are classified into these three groups: C₁ (clearly above), C₂ (clearly below) and an overlap between them, as depicted in Figure 11.3. The distance measure (ρ) in Figure 11.2, which is unrestricted in the sign, has an important role in identifying the overlap size. That is, Stage 1 determines the lower bound of G₁ by subtracting ρ from an estimated discriminant function and it determines the upper bound of G₂ by adding ρ to an estimated discriminant function. The minimization of ρ in Model (11.3), which is visually specified in Figure 11.2, determines the size of the overlap located between the upper and lower bounds of the two groups.

Figure 11.3 depicts the classification of three groups. At the end of Stage 1, all the observations are separated into three groups: C₁ (clearly above), C₂ (clearly below) and an overlap that contains observations that are further separated into two subgroups (D₁ and D₂) at Stage 2.

Figure 11.4 visually describes the function of Stage 2 that separates observations belonging to the overlap into D₁ and D₂. D₁ is the group of observations classified above the discriminant function and D₂ is the group of observations classified below the discriminant function at the second stage. Thus, the group classification from G₁ and G₂ to C₁, C₂, D₁ and D₂ indicates the end of Stage 2.

Altman’s Z score: Following the work of Altman (1968, 1984), the performance of the j‐th DMU is measured by a score where is the i‐th weight estimate of a discriminant function. DEA‐DA cannot directly compute the Altman’s Z (AZ) score because the two‐stage approach produces two discriminant functions (so, double standards) for group classification. Hence, it is necessary for us to adjust the computation process to measure the AZ score.

To adjust the problem of double standards, this chapter returns to the proposed approach in which the first stage estimates a discriminant function and the second stage estimates a discriminant function . The AZ score for the j‐th DMU ( j = 1,…, n) is defined as follows:

(11.5)

This chapter refers to Equation (11.5) as “Altman’s Z score,” or simply “AZ score.” Exactly speaking, the equation is slightly different from the equation for the original AZ score proposed by Altman (1968). However, this chapter uses the term to honor his research work.

The proposed AZ score consists of weight estimates that are adjusted by the number of observations used in the first and second stages of DEA‐DA. The adjustment is due to the computational process of DEA‐DA in which the first stage (for overlap identification) estimates weights of a discriminant function by using all the observations. The total sample size is “n” for the first stage. Meanwhile, the second stage utilizes only observations in the overlap. The sample size used in the second stage is #(overlap). Thus, this chapter combines two groups of weight estimates measured by DEA‐DA into a single set of weight estimates in order to compute the AZ score. Using the AZ score, DEA‐DA can determine the financial performance of each DMU. See Chapter 25 on the selection of common multipliers.

11.2.3 Differences between Two MIP Approaches

Before extending DEA‐DA into the classification of multiple groups, the following five comments are important for understanding the two MIP versions:

1. First, the selection of M and ε influences the weight estimates ( and cf *). Different selections may often produce different weight estimates. That is a drawback of the MIP versions of DEA‐DA. Thus, DEA‐DA may suffer from a methodological bias.
2. Second, mathematical programming approaches for DA do not attain methodological practicality at the level of statistical and econometric approaches, because there is no user‐friendly computer software to solve various DA problems. Moreover, statistical tests are not available to us, because asymptotic theory is not yet developed to the level of these conventional (statistical and econometrics) approaches.
3. Third, the two‐stage MIP approach is effective only when many observations belong to an overlap. If the number of observations in the overlap is small, the performance of the standard MIP approach is similar to that of the two‐stage MIP approach in terms of classification performance.
4. Fourth, the standard MIP approach may be practical in persuading corporate leaders and policy makers, because it produces a single separation function. A problem of the standard MIP approach is that it is more time‐consuming than the two‐stage MIP approach. The rationale on why the latter approach is more efficient than the former in terms of their computational times is because of the analytical structure of the two‐stage approach. That is, the number of binary variables at the first stage is much lower than that of the second stage. The second stage is similar to the standard approach in their analytical structures. A difference between the two is that the second stage has binary variables used to count the number of incorrectly classified observations in . The number is less than that of observations in the standard approach under . Thus, the two‐stage approach is computationally more efficient than the standard approach. The former approach can handle much larger data sets than the latter approach.
5. Finally, although this chapter does not discuss it in detail, all DA research work must examine whether the sign of each weight is consistent with our prior information, such as previous experience and theoretical requirement. For example, when non‐default (G₁) and default (G₂) firms are separated by several financial factors (e.g., profit and return on equity; ROE), the sign of profit is expected to be positive. However, it may be negative due to some structural difficulty in a data set (e.g., multi‐collinearity). In that case, it is necessary for us to incorporate an additional side constraint(s), representing prior information, into the proposed DA formulations. See Sueyoshi and Kirihara (1998) for a detailed discussion on how to incorporate such prior information into DEA‐DA.

11.2.4 Differences between DEA and DEA‐DA

Table 11.1 compares DEA and DEA‐DA to summarize their differences. It is clearly identified that both DEA and DEA‐DA originate from GP, as discussed in Chapter 3. In their conventional uses, however, DEA is used for performance assessment and DEA‐DA is used for group classification.

11.4.1 First Example

This chapter uses two illustrative data sets to document the discriminant capabilities of the proposed DEA‐DA approaches. The first data set, listed in Table 11.2, contains 31 observations which are separated into two groups (G₁: the first 20 observations and G₂: the remaining second observations). Each observation is characterized by four factors.

The bottom of Table 11.2 summarizes two hit rates (apparent and leave one out: LOO) for six different DA methods. Here, “apparent” indicates the number of correctly classified observations in a training sample. In measuring the apparent of the first data set, all 31 observations become the training sample. In contrast, the LOO measurement drops each observation from the training sample and then applies a DA method to the remaining observations. LOO examines whether the omitted observation is correctly classified. Thus, the LOO measurement needs to be repeated until all the observations are examined.

Figure 11.6 depicts such a computation process to measure apparent and LOO. In the figure, the apparent measurement uses the whole data set as both a training sample set and a validation sample set. Meanwhile, the LOO measurement uses a single observation omitted from the whole data set as a test sample. The remaining data set is used as a training sample set.

As summarized in Table 11.2, the two‐stage MIP approach is the best performer in the apparent and LOO measurements (90.32 and 83.33%, respectively). Two econometric (logit and probit) approaches and Fisher’s linear DA perform insufficiently because the data set does not satisfy the underlying assumptions that are necessary for applying those DA methods. In contrast, a shortcoming of the MIP approaches is that they need computational times which are much longer than those of the other DA methods.

11.4.2 Second Example

The second data set, originally obtained from Sueyoshi (2001b, pp. 337–338), is related to Japanese banks that contains 100 observations. All the observations in the data set are listed by their corporate ranks. Therefore, G₁ contains 50 banks ranked from the first to the 50th. Meanwhile, G₂ contains the remaining 50 banks ranked from the 51st to the 100th. This data set has been extensively used by many studies on DA.