Entropy for Business Failure Prediction: An Improved Prediction Model for the Construction Industry

This paper examines empirically the effectiveness of entropy measures derived from information theory combined with discriminant analysis in the prediction of construction business failure. Such failure in modern complex supply chains is an extremely disruptive force, and its likelihood is a key factor in the prequalification appraisal of contractors. The work described, using financial data from the Taiwanese construction industry, extends the classical methods by applying Shannon’s information theory to improve their prediction ability and provides an alternative to newer artificial-intelligence-based approaches.


Introduction
Over the last 35 years, business failure prediction has become a major research domain especially with increased global business competition [1]. Business failure is an extremely disruptive force in the construction industry [2]. Kangari et al. [3] indicated that the construction industry in the USA has several unique characteristics that sharply distinguish it from other sectors of the economy. The bankruptcy rate within the American construction industry has been increasing in recent years and the USA has the highest percentage of construction company failures each year [4,5]. The construction industry is also a major industry in the UK and has the highest percentage of company failures each year [6,7]. Similarly, in Asian countries like Taiwan where there has been phenomenal growth in the last few decades, the construction sector also plays a major economic role.
Beaver [8] was one of the first researchers to study business failure prediction. He analysed financial ratios one by one to evaluate their predictive ability. He then developed their predictive abilities using cutoff scores to classify each company as either failed or nonfailed company. However, this classification technique uses one ratio at a time and conflicts arise when one ratio classifies the company as healthy whilst another detects distress. His work was followed by Altman's [9] model based on discriminant analysis and Ohlson's work [10] based on the use of logistic regression.
Like many other problems in science and engineering, popular machine learning techniques from the 1990s such as neural networks and genetic algorithms have also been applied to business problems such as bankruptcy or business distress detection [11,12] with some successes. When qualitative data and uncertainties abound, these techniques are very useful indeed. However, techniques such as artificial neural networks require large datasets for training purposes and large models are often less easy to interpret [13]. Recent research trends in this area have also employed hybrid methodologies combining both machine learning techniques with the traditional statistical approach with some successes [13]. In this paper, the research methodology employs quantitative financial data as applied in previous research, but augmented using Shannon's information theory to better predict business distress. As shown in previous works, particularly in the construction industry (i.e., the sector addressed in this paper), the financial ratios are very important characteristics when modelling and detecting 2 Advances in Decision Sciences business distress [14]. Thus, despite a move towards machine learning techniques, traditional approaches such as the use of financial ratios enhanced with other methods such as Shannon's information theory can be as good as the machine learning approach. This paper describes how.
Dimitras et al. [14] noted that most studies on business failure models used three types of firms in their sample: manufacturing firms, a combination of manufacturing and retailing firms or firms from several industrial sectors. Kangari et al. [3] and Dimitras et al. [14] stressed that models developed for the manufacturing industry are inappropriate for a construction company due to market segmentation and industry structure differences. The financial condition of a company is a very important factor in determining contractor selection in the construction industry. Birrell [15] reported that financial stability is one of the most important criteria for evaluating the performance of general contractors. A number of other researchers have also commented on the issue as follows.
(i) Russell and Jaselskis [16] compiled a sample of 344 construction professionals taken from professional organizations in the US. They noted that financial stability is the most important decision parameter for public contracts. Financial condition and experience are the most important composite decision factors (CDF) for private contracts.
(ii) Holt et al. [17] presented the findings of a survey of 53 major UK construction organisations. This revealed that financial stability was probably the most important factor of all those considered.
(iii) Bubshait and Al-Gobali [18] stated that contractor experience and financial stability are essential criteria for prequalification in Saudi Arabia.
(iv) The results obtained by Russell [19] for contractor prequalification in the US show that the first two criteria (experience and financial stability) have the same ranking as in Saudi Arabia.
(v) Ng and Skitmore [20] used a postal questionnaire survey of 192 client consultant organizations in the UK. They investigated the divergence of prequalification criteria (PQC) adopted by different types of organizations. The results revealed that the most important PQC is financial stability in both private and government contracts.
(vi) Wong et al. [21] noted that "lowest price" is now not necessarily the UK construction clients' principal selection criterion. They also revealed that both maximum resource and financial capacity are in the top 3 project-specific criteria for public construction.
(vii) Topcu [22] investigated the ability to complete construction projects on time in the Turkish public sector; 70% of the ability was assigned to "financial status. " (viii) Pongpeng and Liston [23] presented a study aimed at developing a common set of criteria with weights of relative importance to evaluate contractor ability for government and private sectors in the Thai construction industry. A result from the survey showed that financial ratio is one of five most important criteria in the private sector.
This literature emphasises that financial status is an important issue in the construction industry. Furthermore, Dimitras et al. [14] identified that many business failure prediction models are based on the financial characteristics of firms in the form of financial ratios. Most of these models evaluate the available financial characteristics (financial ratios) of the firms studied. Kangari [2] suggested that overall industry indictors must also be monitored and trends analysed to determine swings in overall industry failure probabilities to better help determine the exceptions among companies.

Research Objectives.
Hamer's [24, page 77] research findings showed that the decomposition information measure for financial statements has a power of discrimination with respect to failed and nonfailed firms but the predictive ability is less than that derived from financial ratios. The goal in this research is to appraise empirically the usefulness of information measures (derived from information theory) in the prediction of construction business failure. In this research, the model is based on a financial distress definition of failure, not a juridical (mostly bankruptcy) definition. It appears that researchers in the area have not considered using the information measures of financial ratios to analyse whether this could improve predictive ability. Previous research on this issue appears to be inadequate [25][26][27][28][29][30].
Our new method modifies discriminant analysis with information measures derived from financial ratios. The data set used to test the hypothesis was from Taiwanese construction companies. Using this data set helps compensate for the lack of studies in the time, location, and industry differences affecting business failure prediction. Therefore, the research presented in this paper attempts to bridge a gap in earlier studies. The objectives of this paper are (i) to find effective financial ratios as discriminant variables for predicting construction companies failures; (ii) to assess whether information measures derived from financial ratios with discriminant analysis can improve the prediction ability of business failure compared to just using information measures of financial ratios and financial ratios with discriminant analysis; (iii) to assess the value of a practical model that is able to predict the failure of companies in the Taiwanese construction industry. For example, the proposed model can assist the local authorities in the selection of an approved list of constructors for competitive tendering.

Failure Prediction Methods.
Researchers in the past decade have realized that failure does not happen suddenly. Usually, failure take years; therefore, it is necessary to develop an early warning model that can evaluate the strengths and weaknesses of the financial features of companies. A number  [8] 1968 Univariate analysis Altman [9] 1968 Multivariate analysis (discriminant analysis) Taffler and Tisshaw [32] 1977 Multivariate analysis (discriminant analysis)

Construction-specific models
Mason and Harris [6] 1979 Multivariate analysis (discriminant analysis) Abidali and Harris [33] 1995 Multivariate analysis (discriminant analysis) Russell and Jaselskis [16] 1 9 9 2 L o g i t a n a l y s i s Severson and Russell [34] 1 9 9 4 L o g i t a n a l y s i s Hybrid models Zavergen [35] 1985 Logit analysis and entropy Keasey and McGuinness [36] 1990 Logit analysis and entropy of failure prediction models have been developed, based on various techniques. Financial ratio analysis is a very common approach to diagnose the financial strengths and weaknesses in any company. However, these methods are often not employed early enough to predict business failure [37]. The majority of business failure prediction studies are based on the original research of Beaver [8] and Altman [9]. Beaver had made the greatest contribution to univariate analysis. Beaver's analysis involved the use of a single financial ratio in his failure prediction model. The approach had been criticised because just using a single financial ratio is not a sufficiently reliable way to predict failure. Altman [9] performed a multivariate analysis of failure using discriminant analysis. The main idea of this analysis is to combine the information from several financial ratios into a linear discriminant function. Then, a discriminant score is computed and an optimal cutoff point is determined. A number of other studies have followed this methodology to predict business failure in different industries. Although they are all based on the original method devised by Altman, they are all different since financial reporting standards vary according to local conditions, there are different politicaleconomic interrelationships, and there is not a great deal of similarity between different industries. Lev [38] investigated the use of decomposition measures (information measures) in the prediction of financial failure. Lev compared the decomposition measures of matched pairs of failed and nonfailed firms. Decomposition analysis is a measurement associated with information theory by Shannon [39]. The contribution from information theory is the use of an information measure (entropy) for the unevenness of a distribution of weights based on pragmatic considerations. Concepts from information theory normally belong to the area of communication engineering. This approach is generally neglected in the field of prediction methods. Lev [38] identified that there is some power of discrimination with respect to failed and nonfailed firms, up to as far as five years prior to failure. This result suggested that some information measures may be usefully incorporated into the models employed to predict financial failure.
It is generally believed that financial data constitutes the most significant and accessible element in monitoring the performance of a firm and in predicting the trend toward failure. Most business failure prediction models are based on financial data. Argenti [40, page 121] described the financial data values as "symptoms" of failure rather than "causes. " This means that financial figures can be considered an indicator in predicting the possibility of failure. There have been numerous studies using statistical techniques to develop a combination of financial ratios which would predict business failure. The most popular are the classical statistical techniques especially multivariate discriminant models and logistic models. Table 1 summarises the key approaches, and Table 2 identifies the ratios used in these approaches.

Information Decomposition Analysis Using Entropy Measures for Business Failure Prediction
The main problem with both univariate and multivariate analysis models described in the previous section is that they are not dynamic in nature. Models including trend variables to improve the selected financial ratios without dynamic attributes are a step in the right direction [6,33]. This research explores the transformation of financial ratios to information decomposition measures using information theory. Information theory is primarily directed at defining and measuring the amount of information contained in a message. Information is defined in this context as a function of the two sets of probabilities: the one before the reception of the message and the other after it. Therefore, knowledge of the changes in the probabilities permits measurement of the amount of information contained in the message that induced these changes. Such transformation can adjust the data to be naturally dynamic. Then, accumulative dynamic information measures of financial ratios can be compared with static financial ratios model in terms of their failure prediction ability. Previous studies of the decomposition measures for failure prediction are based on elements of financial statements (e.g., assets, liabilities, and equity). The findings from previous research show that decomposition measures for financial statements have the power of discrimination with respect to failed and nonfailed firms, but the predictive ability is less than that for financial ratios. However, researchers in the area Table 2: A summary of results from previous studies based on the financial factors taken into account by the study. Beaver, 1968 [8] Altman, 1968 [9] Taffler and Tisshaw, 1977 [32] Mason and Harris, 1979 [6] Abidali and Harris, 1995 [33] Ratio: profitability Earnings before interest and tax/net capital employed (net assets + short-term loan)

• •
Earnings before interest and tax/net assets (total assets − current liabilities) • appear to have not considered using the information measures of financial ratios to analyse whether that could improve predictive ability. Moreover, both univariate analysis and the multivariate discriminant analysis model are not dynamic in nature. Thus, the focus of this new work was to explore combining discriminant analysis with the information measures of financial ratios in order to address omission in the previous work. Thus, in this work, if the chosen financial data is appropriate, then consideration of time-series contribution, location and industry differences is incorporated into the analysis. Lev [38] applied the decomposition method to a sample of failed and trading firms in order to test its predictive ability.
Advances in Decision Sciences 5 The assumption being that failed firms have a greater degree of instability in their financial behaviour and therefore the measures of change will be greater than those in the case of the trading ones. His empirical results confirm this. The average decomposition measure was 0.0423 nits in the case of the failed firms whereas the continuing firms had an average of 0.0075 nits. The effectiveness of this method as a predictive tool is further enhanced by dichotomous classification, which demonstrated that he achieved a lower misclassification rate than all other ratio tests by Beaver, except for the cash flow to total debt ratio. The authors have considered whether multivariate predictive ability using accumulative information measures of financial ratios is better than just using the financial ratios model. Access to Taiwanese construction industry data for the empirical study in the investigation is a potential opportunity to apply dynamics to failure prediction. There appears so far to have been relatively little research in this area [41,42]. Thus, the question is one that deserves empirical scrutiny. The investigation of predictive ability for business failure is undertaken through the following steps: (1) a multivariate model which uses financial ratios alone; (2) transformation of financial ratios to information measures; (3) a multivariate model which uses the information measures of financial ratios; (4) comparison of the multivariate predictive ability of financial ratios and the information measures of financial ratios.
The financial ratios selected are listed in Table 3.

Development of the Models
Three models are developed and tested in this research:  Each of these models was used to generate estimated coefficients of independent variables for a particular year. These estimated coefficients were then used to classify firms as failed or nonfailed for the estimation sample. In this research, the discriminant analysis performed uses the STAT module of the SAS statistical software package for this research. This includes the " "-test, the Shapiro-Wilks " " test, Wilks' Lambda test, linear discriminant function, and data crossvalidation.  Table 4).   [33] research, they used the sample of 11 failed and 20 nonfailed UK construction companies. Nonfailed companies were selected on a random basis and Table 5 shows the nonfailing companies selected.

Discriminator Selection.
A summary of the financial ratios for the whole sample of companies examined using statistics, including failed and nonfailed companies, is presented in Table 6. These financial ratios (discriminator) are validated for developing the discriminant models for Advances in Decision Sciences 7 this research. Of the 24 selected financial ratios (see Table 3), findings from the posttest show that eight ratios outperform other ratios on a statistical basis (see Table 6). These eight ratios could distinguish significantly between nonfailed firms and failed firms. Table 7 thus lists the selected eight financial ratios as discriminator variables for the research. The eight selected financial ratios were used in the construction of the prediction models. These ratios may also be used to evaluate a company's performance.

Significance of Individual Coefficients.
This study used discriminant analysis as a statistical methodology to determine which independent variables are most relevant in assessing failure risk. Applying the -test distinguishes whether the financial ratios are different for nonfailed firms and failed firms. For the selected eight financial ratios for constructing the discriminant model, the variables are significant since the value is below an -level (statistically significant) of 0.05. The test can judge the ability of the discriminator for selected financial ratios. After transforming the value to the value, the variable is significant when the value is below an -level (statistically significant) of 0.05. A more detailed understanding of the relationship can be gained from Table 8.
In previous research, the ratios shown in Table 9 are considered good discriminators between failed and nonfailed companies. However, in this research, the ratios considered good discriminators between failed and nonfailed companies have the maximum value and the value is less than 0.05. Operating profit: An indicator of the gross profit of a company or an industry after marketing and administrative cost is deducted from its gross profit

Financial structure
Total liability to total assets (debt ratio): This is the rate of loan capital. The bigger the rate, the heavier its debt will be. The rate would better be not larger than 0.5

Management efficiency
Operating expense to operating income Operating expense comprises marketing and administrative cost. This ratio tests the efficiency of expense management ↓ ↑ ↑ the higher the better, ↓ the lower the worse (debt-repaying and ability, earning ability); ↓ the lower the better, ↑ the higher the worse (financial structure and management efficiency). Our discriminators are not consistent with those of Beaver [8] and Abidali [44], as shown by the following: Return (1) Advances in Decision Sciences 9

Construction of the Linear Discriminant Model
The best discriminating variables are selected according to the maximum value and where the value is less than 0.05, using Wilks' Lambda measures, group discrimination according to the lowest Wilks' Lambda. The linear discriminant model produces a discriminant score ( -score) that overcomes these difficulties.  show that there is statistical significance in the three models. Discussions on these three models are also given in Sections 4.2, 4.3, and 4.4 respectively.

Prediction Model with Financial
where is discriminant score ( -score); EPS is earning per share; DR is total liability to total assets (debt ratio); ROE is return on equity (return on shareholders' equity); QR is acid test ratio (quick ratio); PBTP is profit before tax to paid-in capital (return on capital); ROA is return on assets (Return on total assets); OP is operating profit; OEOI is operating expense to operating income. The discriminant coefficient of the linear discriminant model and the associated -statistics are presented in Table 10. When the value is less than an -level of 0.05, this shows that the testing of the discriminator variable is significant. That is the discriminator variable could effectively distinguish the failed and nonfailed groups. Also presented in the table is the Wilks' Lambda, which provides an indication of the overall discriminant power for each model, and thestatistics test which shows its statistical significance. If the " " value is less than an -level of 0.05, this means that there is a significant discriminant power for the model.

Prediction Model with Information Measures.
Decomposition analysis is a measurement associated with information theory by Shannon [39]. Previous research paid most attention to the decomposition measure in the financial statement [25,27,29,38,45]. These researchers thought that financial ratios are a "measure of level." Financial ratios indicate the level of the ratio in a single accounting period. The decomposition measures discussed are "measures of variability." This research explored the transformation of financial ratios to information decomposition measures using information theory. The transformation can adjust the data to be naturally dynamic. Then, accumulative dynamic information measures of financial ratios are compared with static financial ratios model in terms of their failure prediction ability. This new work combines discriminant analysis with the information measures of ratio analysis.

Prediction Model with Static Information Measures of
Financial Ratios. The model was developed by selecting financial ratios as independent variables. The values of these independent variables are normalized for the same basis of measurement. The data are then transformed into the nits of information measures. The transformation equation is: where information unit: 1 bit = 0.693 nits and 1 nit = 1.443 bits. After data transformation, a discriminant function was developed using the discriminant analysis method. The where is discriminant score; EPS is earning per share; DR is total liability to total assets (debt rtio); ROE is return on equity (return on shareholders' equity); QR is acid test ratio (quick ratio); PBTP is profit before tax to paid-in capital (return on capital); ROA is return on assets (return on total EPS: earning per share; DR: total liability to total assets (debt ratio); ROE: return on equity (return on shareholders' equity) QR: acid test ratio (quick ratio); PBTP: profit before tax to paid-in capital (return on capital); ROA: return on assets (return on total assets) OP: operating profit; OEOI: operating expense to operating income.  assets); OP is operating profit; and OEOI is operating expense to operating income. The discriminant coefficient of the linear discriminant model and the associated -statistics are presented in Table 12. If the value is less than an -level of 0.05, this shows that the discriminator variable could effectively distinguish between the failed and nonfailed groups. Also presented in the table are the Wilks' Lambda which provides an indication of the overall discriminant power for each model and the -statistics test, which shows its statistical significance.

Prediction Model with Dynamic Information Measures of
Financial Ratios. The model was developed by considering the selected eight financial ratios as independent variables. These values of independent variables are normalized for the same basis of measurement. Then, the data are transformed to the nits of information measures. The transformation equation is The discriminant function was constructed by utilizing the 45 observations from the estimation sample. The linear discriminant function provided the following. The dynamic where is discriminant score; EPS is earning per share; DR is total liability to total assets (debt ratio); ROE is return on equity (return on shareholders' equity); QR is acid test ratio (quick ratio); PBTP is profit before tax to paid-in capital (return on capital); ROA is return on assets (return on total assets); OP is operating profit; OEOI is operating expense to operating income. The discriminant coefficient of the linear discriminant model and the associated -statistics are presented in Table 13. If the value is less than an -level of 0.05, this means that there is a significant discriminant power for the model.

Discussion
This study has been devoted to assessing the worth of a novel failure prediction model for the construction industry. The most important goal for the study was to develop an effective early warning model with a high predictive ability to forecast financial difficulty for construction companies, using a statistical methodology to test whether the results are significant and valid.
The financial ratios that occur most frequently in the literature may not be the most important for model construction because no theoretical or empirical justification exists to indicate that these popular ratios provide an adequate model. Prior research into the selection of the independent variables was not based on a justification or validation of empirical evidence. In this research, the selected variables (eight financial ratios) are found to be statistically significance using the -test.
This research found that "return on assets," "return on capital, " and "earning per share" are good discriminators between failed and nonfailed companies. They are not consistent with those of Beaver [8] and Abidali [44].
In Dimitras et al. 's [14] and Balcaen and Ooghe's [1] surveys, the discriminant analysis technique was still most frequently used in the development of failure prediction models. Discriminant analysis requires certain restrictive assumptions such as multivariate normality and equal covariance, which are often violated. These assumptions are not likely to significantly affect prediction models based on discriminant analysis. The drawbacks of discriminant analysis are the difficulty in interpreting time-series prediction test and the need for a prior probability of failure; however, it is not always easy to find any estimate for the prior probability of failure. Previous discriminant analysis models for business failure prediction, which are not dynamic in nature, have revealed certain weaknesses. Researchers thought of financial ratios as a "measure of level. " The information decomposition measures discussed are "measures of variability. " Information decomposition measures are dynamic whereas financial ratios are static. The transformation of financial ratios to information decomposition measures using information theory is based on the original model from early studies [38,45]. The transformation can adjust the data to be naturally dynamic. Therefore, the improved model (Model I9702) had to deal with the time-series contribution in an enhanced way. This model combined discriminant analysis with information measures of ratio analysis to improve the classification accuracy of the prediction model. This model (Model I9702) provides an approach to add dynamic consideration to failure prediction analysis.
The construction industry is a high-risk business. Therefore, it is important to minimize the risk cost by identifying potential failures at the earliest stage. It should be possible to respond quickly to the information which predicts financial failure in order to prevent future contract failure. Use of such methods could aid in contractor selection and help appraise risk better. Prediction models could also be designed to aid auditors, investors, and users in making their assessment of the likelihood of failure or nonfailure.

Further Work and Conclusion
Overall the approach adopted could serve as the basis for future related studies in different industries or countries. As an extension to this work, it may be possible to use the biostatistics method of survival analysis as an alternative approach to failure prediction for companies classified as at risk in the construction industry. Another possible extension is to build a conceptual framework of dynamic prediction, which applies agent technology developed as a dynamic prediction tool. To help in vetting construction companies on tender lists, it is possible to adopt the models indicated in the research or link these to other machine learning types of prediction methods. Other new methods could also include qualitative variables (such as management variables) that minimize the expected misclassification costs of using these methods.
In conclusion, this research modified discriminant analysis with entropy measures (information measures) derived from financial ratios. The use of discriminant analysis is usually based only on the dichotomous classification of failing and nonfailing groups. The drawbacks of discriminant analysis are the difficulty in interpreting the time-series prediction test and that a prior probability of failure is needed; however, it is not always easy to find any sensible estimate for the prior probability of failure. The estimate for the prior probability of failure is affected by the dynamic circumstances of the economy. This research establishes a way of incorporating this.