Raka Goyal, Ph.D., P.E.

Introduction

The survival-based models on Federal Highway Administration (FHWA) InfoBridge^TM (FHWA, 2020) are probabilistic bridge deterioration models based on a methodology that combines survival analysis and Markov chain theory. The modeling framework was first developed and implemented on bridge deck, superstructure, and substructure components of the North Carolina State bridge inventory (Goyal, 2015; Cavalline et al., 2015). In the current research, the model’s implementation has been substantially expanded and improved for the more complex application to the entire nationwide bridge inventory.

Survival analysis is a statistical approach that analyzes the time until death or failure—in the case of bridge components, which is the time spent in a condition rating until it deteriorates to a lower condition rating. Survival analysis allows determination of the survival probability of the component in that condition rating at any point of time. The main advantage of survival analysis is that it can account for incompletely recorded durations commonly found in duration-based data, such as condition-rating observations truncated at the beginning and end of the recording period. Instead of discarding these observations, a mechanism called “censoring” is used to include these observations in the analysis, which permits a more realistic estimate of duration compared to other statistical approaches.

In this study, the Cox proportional hazards model (PHM) (Cox, 1972) has been used for the analysis of condition-rating durations. Cox PHM is a semiparametric approach that does not make any assumptions about the shape of the distribution and can be used to analyze unimodal hazard functions associated with some infrastructure components. For the survival-based models developed in this study, the transition probabilities of the Markov chain are calculated from the PHM survival functions at each condition rating, carrying the advantage of survival analysis probabilistically over the entire lifecycle. Additionally, multivariable effects at each condition rating are quantified in terms of PHM hazard ratios and used to modify the Markov chain transition probabilities. In this way, the effects of time dependence and exogenous factors on deterioration, as analyzed through survival analysis, are incorporated in the Markov chain to develop a probabilistic lifecycle deterioration model.

Proportional-Hazards Bridge Deterioration Model

The survival function (S(t)) associated with a bridge-component condition rating is the cumulative survival rate of bridge components in the condition rating. The instantaneous risk of transitioning to a lower rating at time (t), conditional to survival until that time, is the hazard rate or the hazard function. In the Cox PHM, the hazard rate (h(t,z)) is defined as the product of a time-dependent nonparametric-baseline hazard function (h₀(t)) and a time-independent exponential function, representing the effects of design variables (z) through regression coefficients (β):

  (1)

In this function, the time-independent exponential function represents the effects of covariates, or explanatory factors, on the hazard rate. The baseline hazard rate is associated with the baseline variables, which are assigned a value of zero.

Proportional-hazards lifecycle deterioration modeling

The proportional-hazards bridge deterioration model (Goyal, 2015) can be represented in terms of structure-specific transition probability matrices (P_i) associated with each year (i) of the prediction period,

     (2)

The matrix elements P_ij represent the probability of the bridge component transitioning from condition rating i to condition rating j, assuming there is no improvement in condition rating and that the component deteriorates by no more than a single rating in one year. In each row, P_kk, on the diagonal is the baseline stay-the-same transition probability for condition rating k. If S_k (t,z) is the baseline survival function of a bridge deck associated with condition rating k for a bridge described by the vector of covariates z, the baseline transition probability (Goyal, 2015; Goyal et al., 2020) of staying at the same condition rating over the next annual reporting cycle at any time t is given by

            (3)

Since the sum of probabilities in each row should be equal to one, the baseline transition probability of deteriorating to a lower rating is 1 ‒ P_kk. The baseline transition probabilities are uniquely modified for each bridge or category of bridges using structure-specific hazard ratios HR_k, calculated by multiplying the PHM hazard ratios for the factors associated with a bridge that are identified as significantly influencing deterioration at condition rating k. A stationary implementation of the proportional-hazards deterioration model is used in this study, in which stationary or constant transition probabilities (P_kk) are obtained by averaging the yearly transition probabilities across the duration of the PHM-baseline survival functions for each condition rating k. The resulting stationary-transition probability matrix (P) is used in a homogeneous Markov chain to predict the future condition-state vector Z_n of a bridge component after n years if its present condition-state vector Z₀ is known using

               (4)

The condition-state vectors comprise of the probabilities of the bridge component being at all possible condition ratings at any given time. The future state vector Z_n can be multiplied by the column vector (R) of condition ratings to produce the expected condition rating, E, of the bridge component after n years.

                                 (5)

This document describes briefly the implementation of the proportional-hazards deterioration model to develop deck deterioration models for the nationwide bridge inventory with more details to follow in a journal publication currently under preparation. Detailed background and theoretical development of the proportional-hazards deterioration model can be found elsewhere (Goyal, 2015; Goyal et al., 2017; Goyal et al., 2020).

Data Structuring

The nationwide survival-based models are based on the FHWA National Bridge Inventory (NBI) data spanning the years from 1983 to 2018 for highway bridges nationwide (FHWA, 2019). The NBI files store historical inspection records of all bridges in the United States with a span greater than 20 ft. with more than 100 data items per bridge per year of recorded service (FHWA, 1995). The first step in organizing the data for modeling was to identify, query, and extract the NBI fields relevant to deterioration modeling from the yearly inspection records of individual bridges, and assemble a continuous record of bridge condition-related data from 1983 to 2018.

Bridges with concrete decks constitute an overwhelming percentage (>80 percent) of the NBI and are a focus area of the Long-Term Bridge Performance Program. For survival-based deterioration modeling, bridges were classified based on deck material, main-structure material, and design type. Further, since duration-based analysis is best served by long duration records, and in the interest of optimizing the use of computational resources, researchers decided that only bridges with continuous records of 25 years or more be used for model development. The bridge categories for survival-based models with the number of bridges in the historical database selected for model development are provided in Table 1.

Table 1. Bridge categories for survival-based deterioration models. © Raka Goyal (2019).

* Rebuilt and reconstructed bridges separated based on NBI items “Year Built” and “Year Reconstructed.”

Design variables for proportional-hazards survival analysis

Data was further preprocessed to extract all observations of the response variable, which is the observed continuous duration at each NBI condition rating analyzed. For each observed duration, corresponding censoring information was compiled in a separate vector of the same size as the response variable, but stored as a binary variable of either 0 or 1 depending on whether the observations were classified as completely observed or censored. In this study, all continuous observations that were truncated at the beginning year (1983) or end year (2018) of the NBI database were classified as right censored, presuming that the actual duration of the condition rating was longer than observed due to the limited time span of the data-recording period. Further, all observations where an increase in condition rating was observed, rather than deterioration, were considered as complete, or uncensored, under the assumption that observed improvements in condition rating reflected maintenance action performed because of an unrecorded transition to the lower rating.

For proportional-hazards analysis, the descriptive information on each structure, such as its functional classification, traffic characteristics, design parameters, and other details contained within the NBI historical records that could potentially influence the deterioration rates of specific bridge components were considered as potential explanatory factors. Some factors were derived from two or more NBI fields, for example, “age” refers to the age of the deck at the beginning of the observed condition-rating duration and is calculated based on the corresponding data year and the year when the bridge was last built or reconstructed. Additionally, average daily traffic per lane (ADTL) and average daily truck traffic per lane (ADTTL) were calculated by dividing the ADT and ADTT, respectively, by the number of through-traffic lanes on a bridge also recorded in the NBI (ADTT mentioned here refers to the number of trucks, which was first calculated from the percentage ADTT recorded in the NBI). ADTL and ADTTL are new variables that were introduced in the nationwide study to account for the width of the roadway in studying the impact of traffic on deck deterioration.

Each explanatory factor is organized into categories designated by one or more design variables to which bridges are classified based on either binary or reference-cell coding. The variables of ADTL, ADTTL, age, and maximum span length, which are continuously recorded, are divided into categories of approximately equal frequency of occurrence for each component based on weighted averages computed across the available bridge records. The categorical ranges for these variables differ for disparate bridge categories (Table 1) depending on the different statistical distributions of variables associated with the various bridge categories. The design variables included in the development of survival-based models for concrete decks nationwide are listed in Table 2. Category ranges developed for steel stringer bridges are provided for illustration. The subsequent steps involving multivariable proportional-hazards regression were performed individually on each of the distinct condition-rating specific sets of dependent and independent PHM-regression inputs, extracted in this way, for each category of bridges.

Table 2. Design variables included in proportional-hazards survival analysis of concrete decks nationwide.
© Raka Goyal (2019).

*Category values in parentheses are for concrete-deck steel stringer bridges.

Proportional-Hazards Deterioration Model Development

The variables that were found to be statistically significant at each condition rating using PHM regression were further processed through a best subset-selection algorithm to optimize the size of the model without compromising its reliability. The survival functions developed using the best subset model incorporate the effect of the most significant explanatory variables on the deterioration rate over individual condition ratings. The condition-rating-dependent best subsets and associated hazard ratios obtained for the significant explanatory factors identified in the proportional-hazards deck model for steel stringer bridges nationwide are summarized in Table 3.

Table 3. Best subset factors and hazard ratios in proportional-hazards deck deterioration model for concrete deck steel stringer bridges nationwide. © Raka Goyal (2019).

An HR value of 1 signifies a lack of influence on the deterioration rate, and indicates that the factor was not included in the best subset for that rating. For example, in Table 3, deck type is included only in the best subset associated with condition rating 6. Absence of span type in Table 3 indicates that it was not included in the best subsets associated with any of the deck condition ratings throughout the lifecycle of concrete-deck steel stringer bridges. An HR value less than 1 indicates that the factor is associated with a reduced rate of deterioration, and a value greater than 1 indicates that the factor is associated with an increased rate of deterioration. As seen in the table, revealing the varying effects of the same factors at different condition ratings across the bridge-deck lifecycle is a unique aspect of the proportional hazards deterioration model.

The final step in the proportional-hazards deterioration-model development is the calculation of baseline-transition probabilities at each condition rating, and assembly of the Markov-chain transition-probability matrix, as shown in equation 2, for future condition forecasting. Sufficient historical condition-rating data was available in the nationwide NBI database to develop survival-function-based transition probabilities for all the condition ratings from 9 to 3. The stay-the-same transition probability for condition rating 2 was prescribed as 0.85 to prevent the deterioration models from converging abruptly to condition rating 2. Condition rating 1 is the lowest or absorbing state with a stay-the-same transition probability of 1. Figure 1 shows deck deterioration models illustrating the combined effects of the variables, deck type, age, ADTL, and maximum span length on deck-deterioration rates of concrete-deck steel stringer bridges, illustrating the depth of information revealed by the proportional-hazards deterioration models based only on the variables available within the NBI database.

Proportional-hazards deterioration-model plots for individual bridges in each bridge category (Table 1) are available on InfoBridge (FHWA, 2020). In addition to the mean-predicted condition rating, prediction curves associated with cumulative probabilities of 75 percent (lower bound), 50 percent (median), and 25 percent (upper bound) are also displayed as shown in Figure 2.

References

Cavalline, T. L., Whelan, M. J., Tempest, B. Q., Goyal, R., and Ramsey, J. D. (2015). Determination of Bridge Deterioration Models and Bridge User Costs for the NCDOT Bridge Management System. Final Report No. FHWA/NC/2014-07, North Carolina Department of Transportation, Raleigh, NC.

Cox, D. R. (1972). Regression Models and Life-Tables. Journal of the Royal Statistical Society. Series B (Methodological), 34(2), pp. 187–220.

FHWA. (1995). Recording and Coding Guide for the Structure Inventory and Appraisal of the Nation’s Bridges. Report FHWA-PD-96-001, Office of Engineering, Bridge Division, FHWA, Washington, DC.

FHWA. (2020). LTBP Infobridge Data, InfoBridge, obtained from: https://infobridge.fhwa.dot.gov/data, last accessed January 2020.

FHWA. (2019). Historical bridge inspection records with span greater than 20 ft., 100 data items/bridge/year, National Bridge Inventory, 1983 to 2018 datasets, obtained from: https://www.fhwa.dot.gov/bridge/nbi.cfm, last accessed March 18, 2020.

Goyal, R. (2015). Development of Survival-Based Framework for Bridge Deterioration Modeling with Large-Scale Application to the North Carolina Bridge Management System. PhD. Dissertation, University of North Carolina at Charlotte. Charlotte, North Carolina.

Goyal, R., Whelan, M., and Cavalline, T. L. (2017). Characterizing the Effect of External Factors on Deterioration Rates of Bridge Components Using Multivariate Proportional Hazards Regression. Structure and Infrastructure Engineering, 13(7): 894–905.

Goyal, R., Whelan, M., and Cavalline, T. L. (2020). Multivariable Proportional Hazards-Based Probabilistic Model for Bridge Deterioration Forecasting. Journal of Infrastructure Systems, 26(2): 04020007, DOI: 10.1061/(ASCE)IS.1943-555X.0000534.