*3.2. Classification of Maturity Level*

The BATM assessment is inseparable from the level definition of the PBM and PBA maturities. The classification and definition of PBM and PBA maturity level should be clear.

The classification of the PBM maturity levels refers to that of the PMS-PMMM model, and its maturity levels, from 1 to 5, are the initial, managed, defined, quantitatively managed, and optimizing levels, respectively.

The classification of the PBA maturity levels refers to the information technology governance maturity model under the COBIT 4.1 framework. In COBIT 4.1, the maturity levels from 0 to 5 are the non-existent, initial, repeatable but intuitive, defined process, managed and measurable, and optimized levels, respectively [54], which match well with the maturity levels of PMS-PMMM.

The specific feature definitions of the PBM and PBA maturity level are shown in Table 1.


**Table 1.** Definition of maturity level.

#### *3.3. Definition of Model Structure*

PMBOK is an excellent reference for analyzing project management capabilities, in which an abundance of "best practice" information is outlined in the document [34]. Because the knowledge content of each PMBOK performance domain is abundant, each performance domain is categorized into several key desired outcomes [55]. In total, there are eight performance domains and 37 desired outcomes in the PMBOK 7th edition.

In the BATM model, the eight performance domains are used as the first level assessment indexes, and the 37 desired outcomes are used as the second level assessment indexes; this constitutes the assessment index system of BATM model. The second level assessment indexes are used to measure the PBM and PBA maturities from the two dimensions of the virtual and real worlds, and the maturities of the first level assessment indexes are summarized from the second level indexes. For example, under the delivery performance domain, the maturity level is measured using five desired outcomes. They include: (1) projects contribute to business objectives and advancement of strategy; (2) projects realize the outcomes they were initiated to deliver; (3) project benefits are realized in the time frame in which they were planned; (4) the project team has a clear understanding of requirements; (5) stakeholders accept and are satisfied with project deliverables. The specific assessment indexes are shown in Table 2.


**Table 2.** Assessment index system of BATM.

The BATM structure can be classified into three layers: project layer, performance domain layer, and desired outcome layer. Because the mode structure is based entirely on PMBOK 7th edition, we did not conduct an empirical analysis of it.

#### *3.4. Questionnaire*

The questionnaire consists of three parts. The first part introduces the purpose and requirements of the questionnaire, the second part explains the definition of PBM and PBA maturity levels, and the third part is the scoring table for experts to score. The left side of the scoring table lists the assessment indexes of the BATM model, including eight performance domains and their corresponding 37 desired outcomes. The right side is the selection area of PBM and PBA maturity levels, and the weights of desired outcomes can be set according to their performance domains.

In the questionnaire, a scale of 1 to 5 for PBM maturities and a scale of 0 to 5 for PBA maturities were adopted to measure the responses. Level 1 corresponds to score 1, level 2 corresponds to score 2, etc. Because the first, third, and fifth levels of the PBM and PBA maturities are the initial level, the defined level, and the optimized level, respectively, and their second and fourth levels are also similar, we can deem that their five maturity levels are relatively consistent. In addition, as there may be no PBA in some projects, the PBA maturity levels include the non-existent level. The format of the scoring table is shown in Table 3.


**Table 3.** Scoring table of BATM questionnaire.

#### *3.5. Calculation of BATM Level*

After experts return the scoring table of the BATM questionnaire, the questionnaire organizers first identify whether experts' responses are qualified (in the two-dimensional maturity options of a management process of the scoring table, multiple selections and no selection are regarded as unqualified), then deal with the qualified data to obtain the three-layer BATM. The courses of evaluating the BATM involve the maturity determination of desired outcome layers based on the two dimensions of PBA and PBM, and that of performance domain layers and the project layer according to certain weight ratios. In the following expressions, wikj is the weight of expert k in the desired outcome j of performance domain i, i.e. the weight in the scoring table; wik, decided by the organizers, is the weight of expert k in performance domain i (for simplicity, the weights among experts can be considered not to change with performance domains); wi, also decided by the organizers, is the weight of performance domain i; n is the number of desired outcomes in performance domain i; and l is the number of experts. When determining the expert weight wik, the questionnaire organizers need to consider the basic information, such as the experts' education background, their corresponding positions, and their working years in each position. However, in order to simplify the statistical workload, experts' scores can also be treated equally; that is, the default value 1 can be used as the experts' weights.

The BATM value of each desired outcome (*mij PBM*, *<sup>m</sup>ij PMA*) can be obtained by Formulas (1) and (2), which are equal to the weighted average of the desired outcome maturity values provided by the experts and the corresponding expert weights.

$$m\_{PBM}^{ij} = \frac{\sum\_{k=1}^{l} \left( m\_{PBM\_{ikj}} w\_{ik} \right)}{\sum\_{k=1}^{l} w\_{ik}} \tag{1}$$

$$m\_{PMA}^{ij} = \frac{\sum\_{k=1}^{l} \left( m\_{PMA\_{ikj}}, w\_{ik} \right)}{\sum\_{k=1}^{l} w\_{ik}} \tag{2}$$

The BATM value of each performance domain (*m<sup>i</sup> PBM*, *<sup>m</sup><sup>i</sup> PMA*) can be obtained using Formulas (3) and (4). They are equal to the weighted average of the experts' performance domain maturity values and corresponding expert weights, in which the experts' performance domain maturity values equal the weighted average of the desired outcome maturity values and corresponding desired outcome weights provided by the experts in their scoring tables.

$$w\_{PBM}^i = \frac{\sum\_{k=1}^l \left(\frac{\sum\_{j=1}^n \left(m\_{PBM\_{ikj}} w\_{ikj}\right)}{\sum\_{j=1}^n w\_{ikj}}, w\_{ik}\right)}{\sum\_{k=1}^l w\_{ik}}\tag{3}$$

$$\begin{aligned} \sum\_{k=1}^{l} \left( \frac{\sum\_{j=1}^{n} \left( m\_{PMA,ikj}, w\_{ikj} \right)}{\sum\_{j=1}^{n} w\_{ikj}}, w\_{ik} \right) \\\ m\_{PMA}^{i} = \frac{\sum\_{k=1}^{l} w\_{ik}}{\sum\_{k=1}^{l} w\_{ik}} \end{aligned} \tag{4}$$

The BATM value of the project layer (*mPBM*, *mPMA*) is equal to the weighted average of all performance domain maturity values and the corresponding performance domain weight. They can be obtained by Formulas (5) and (6), in which mPBM and mPBA represent the project layer's PBM and PBA maturities, respectively.

$$m\_{PBM} = \frac{\sum\_{i=1}^{8} \left( m\_{PBM}^i.w\_i \right)}{\sum\_{i=1}^{8} w\_i} \tag{5}$$

$$m\_{PMA} = \frac{\sum\_{i=1}^{8} \left( m\_{PMA}^i.w\_i \right)}{\sum\_{i=1}^{8} w\_i} \tag{6}$$
