Mathematical Problems in Engineering

Volume 2019, Article ID 9797584, 11 pages

https://doi.org/10.1155/2019/9797584

## Prediction of Bridge Component Ratings Using Ordinal Logistic Regression Model

^{1}Upper Great Plain Transportation Institute, North Dakota State University, Fargo, ND 58108, USA^{2}Department of Civil and Environmental Engineering, Rutgers, The State University of New Jersey, Piscataway, NJ 08854, USA

Correspondence should be addressed to Hao Wang; ude.sregtur.eos@162wh

Received 2 January 2019; Revised 3 March 2019; Accepted 8 April 2019; Published 16 April 2019

Academic Editor: Jian Li

Copyright © 2019 Pan Lu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Prediction of bridge component condition is fundamental for well-informed decisions regarding the maintenance, repair, and rehabilitation (MRR) of highway bridges. The National Bridge Inventory (NBI) condition rating is a major source of bridge condition data in the United States. In this study, a type of generalized linear model (GLM), the ordinal logistic statistical model, is presented and compared with the traditional regression model. The proposed model is evaluated in terms of reliability (the ability of a model to accurately predict bridge component ratings or the agreement between predictions and actual observations) and model fitness. Five criteria were used for evaluation and comparison: prediction error, bias, accuracy, out-of-range forecasts, Akaike’s Information Criteria (AIC), and log likelihood (LL). In this study, an external validation procedure was developed to quantitatively compare the forecasting power of the models for highway bridge component deterioration. The GLM method described in this study allows modeling ordinal and categorical dependent variable and shows slightly but significantly better model fitness and prediction performance than traditional regression model.

#### 1. Introduction

The highway bridge system is generally considered an essential part of the US transportation infrastructure. The efficient use of public funds for repairing and maintaining bridges requires an effective bridge asset management framework. Transportation management agencies worldwide have begun to adopt bridge management systems (BMS) to determine the optimum future bridge maintenance, repair, and rehabilitation (MRR) strategy at the lowest possible life-cycle cost based on forecasted bridge conditions [1–4]. Use of various forecasting models has played a critical role in predicting future bridge conditions for decision makers.

In the United States, highway bridge ratings typically consist of three major components: deck, superstructure, and substructure. The current method for monitoring them relies heavily on visual inspections which only take into account the observed physical health of the bridge. During visual inspections, a condition rating of the three major components is given on an integer scale of 0 to 9 with 8 equal interim levels. On this scale, 0 is failure and 9 is near-perfect condition [5]. Bridge components deteriorate as a result of operating conditions and external environmental loads [6]. Because of the importance of these components for normal operation and safety, prediction models for component conditions are routinely developed to assess the condition of bridges for a given future time span.

A review of the literature reveals that many researchers have used various models to predict the future condition rating of bridges ([2–4, 7–15]; Lu and Zheng 2017). These prediction models include straight-line extrapolation, linear regression, Markovian, nonlinear regression, logistic regression models, artificial neural networks, Bayesian network, simulation, and some data mining based algorithms [7].

Applications of deterioration models such as Markov-chains and simulation are gaining popularity in forecasting bridge condition ratings; however they are limited by their inability to provide specific information on the deterioration of an individual infrastructure element [16]. In addition, the assumption in the Markovian method that probabilistic deterioration in a given period is independent of history can be unrealistic for bridges [9]. Artificial intelligence such as neural networks is advantageous because of their ability to model nonlinearities automatically. Neural networks can handle binary categorical inputs by using 0/1 inputs, but it would be difficult to handle multiple categories that are ordinal in nature. Moreover, neural networks are more of a “black box” method that produces results that are difficult to interpret [16]. Multiple linear regression’s simplicity and explanatory relationship explains its popularity in literature, but this method may not be appropriate to model bridge condition because it does not take into consideration the ordinal discrete dependent variable. Use of multiple linear regression in that case will result in a violation of the normality assumption [9, 16]. Logistical methods such as the probit model allow the capture of the latent nature of infrastructure performance and incremental discrete dependent variables but do not adequately account for the multilevel discrete and ordinal nature of bridge ratings [9].

Multinomial regression is a variant of nonlinear regression that is capable of handling discrete dependent variables with multiple levels. However, bridge condition ratings are commonly represented as variables that are both discrete and ordinal in nature. In multinomial logistic regression, values of the dependent variable do not indicate any order or ranking. Ordinal logistic regression is an extension of multinomial regression that is believed to be theoretically appropriate and practically feasible for modeling bridge component rating changes. Those logistic models have been widely adopted in modeling discrete choices in motor vehicle crash severity and, to a lesser degree, in pipeline deterioration and wastewater utility deterioration [16]. However, use of the method to model bridge component or element rating changes has rarely been found in previous studies [17]. Madanat, Mishalani, and Ibrahim [17] presented an ordered probit method for the estimation of infrastructure deterioration models and associated transition probabilities from condition variables.

Moreover, the accuracy of the decision-making relies heavily on the outcomes of a reliable bridge condition forecasting model ([2, 15]; Lu and Zheng 2017). Many recent researches focus on improving model forecasting accuracy and shed some light on improving forecasting accuracy [2, 4, 12–15]. However, many of the researches ignore the forecasting reliability and provide incomplete picture regarding forecasting accuracy. Accuracy reported through previous research are often the statistical relative closeness measurement of model estimation to the actual condition [12, 18]; those measurements are critically important to demonstrate statistical soundness of the model’s forecasting power; however, they do not provide the full picture of the forecasting capability. For example, for discrete values such as bridge component ratings, estimation closeness along with exact estimation and estimation within certain rating difference will provide more complete pictures. Moreover, many previous researches ignore the forecasting reliability issue [18]. Their forecasting model can perform really well with certain data or dependent variables but work really poor with others. Thomas and Sobanjo [12] proposed a semi-Markov chain deterioration model, working really well with pourable joint seal element condition forecasting with relative closeness of 0.981 as 1 being perfect estimation, but working really poor with reinforced concrete abutment element condition forecasting with relative closeness of 0.154.

In this research, the authors will evaluate the model forecasting capability based on various measurements including relative closeness measurements and exact accurate. Moreover, this research will validate the model forecasting power with not only in-sample data but also external data validation for three bridge component ratings.

#### 2. Objective

In this study, an ordinal logistic regression method was developed to predict network-level bridge component ratings with North Dakota 2012 NBI data. A multiple linear regression model was also developed with the same data set as a reference for comparing model fitness and forecasting skill. The model is not perfectly suited for handling ordinal data as stated earlier; however it can be used for comparison since this type of model is popular within engineers and are straight forward to develop and use. Five criteria were used to evaluate and compare the two models: prediction error, bias, accuracy, out-of-range forecasts, Akaike’s Information Criteria (AIC), and log likelihood (LL). The developed model was validated with North Dakota 2013 and 2014 NBT data. The application of the model for predicting MAP-21 bridge performance indicator was conducted and discussed.

#### 3. Ordinal Logistic Regression

Ordinal logistic regression is used to model the relationship between an ordered multilevel dependent variable and independent variables. In the modeling, values of the dependent variable have a natural order or ranking. One example of ordinal variables is bridge component ratings (ranging from 0 to 9, with 0 being fail and 9 being near-perfect). When the response categories are ordered, in ordinal logistic regression model, the event being modeled not only is having an outcome in a particular category but also preserves information about response categories which are ordered. Ordinal logistic regression models, also known as proportional odds models, utilizing proportional odds, have the following general form [19] shown in where Y is response variable with k ordered categories; j= 1,2,…,k-1; is cumulative probability for j=1,2,…,k-1. Note , so it should not be modeled; are dependent observations which are statistically independent i=1,2,…,n; are p explanatory variables; correspond to the regression coefficients for the respective independent variables; are the cut-off points between categories.

Multinomial logit models do not consider proportional odds and ignore ordered response categories. For k possible outcomes, running k-1 independent binary logistic regression models in which one outcome, say k, are chosen as a reference and then the other k-1 outcomes are separately regressed against the reference outcome. The general form is followed by the following equation:

The restriction of ordinal regression originates from the proportional odds assumption even though ordinal regression takes care of ordinal relationship between levels of the dependent variable [16]. The proportional odds assumption is that is independent of j. In other words, the effects of independent variables, , are constant between different levels of the dependent variable. The proportional odds assumption can be tested by using a likelihood ratio score test to determine whether allowing the effects of independent variables to change will result in significant improvements in model fitness [16]. If the proportional odds assumption is not met, there are still several options, such as using the partial proportional odds model [20]. Our models meet the proportional odds assumption, possibly because of the large sample size and continuous latent response. The proportional odds cumulative-logit model acts well with its connection to the idea of a continuous latent response. Bridge condition is actually a categorized version of a latent continuous variable. The 9-point scale is a coarsened version of a continuous variable indicating degree of component condition. The continuous scale is dived into 9 regions by 9 cut-points: 0-9. If we have normal errors rather than logistic errors, or in other words when an error term is a random error from a logistic distribution with mean zero and constant variance, the coarsened version of a continuous variable will be related to the independent variables by a proportional odds cumulative-logit model. It worth mentioning that the 9-point scale of bridge component ratings is subjective and there is a great need to model the relationship between the inspection rating and the actual condition of the bridge components. However, in this research the main focus is to demonstrate the forecasting improvement of the proposed model with 9-point scale measuring components due to the data availability.

#### 4. National Bridge Inventory Database

The National Bridge Inventory (NBI) ASCII database is a unified database compiled by the Federal Highway Administration (FHWA) for all bridges and tunnels in the United States that have public roads passing above or below them [21]. The database provides the most comprehensive bridge information in the United States. Detailed information regarding NBI data can be found in the FHWA NBI reference report [22]. The data in the NBI is collected by state highway agencies and reported to FHWA annually/biennially.

As stipulated in the National Bridge Inspection Standards, bridges are inspected at least once every 24 months. During these inspections, the conditions of the three major bridge components (deck, superstructure, and substructure) are rated using a standard scale developed by Federal Highway Administration (Table 1). One can tell easily that bridge component ratings are ordinal discrete data from Table 1. North Dakota 2012 NBI data is selected for model formulation and North Dakota 2013 and 2014 NBI data are used for external model validation purpose.