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Article

Macroscopic State-Level Analysis of Pavement Roughness Using Time–Space Econometric Modeling Methods

by
Mehmet Fettahoglu
1,
Sheikh Shahriar Ahmed
2,
Irina Benedyk
1 and
Panagiotis Ch. Anastasopoulos
1,3,*
1
Department of Civil, Structural and Environmental Engineering, University at Buffalo—The State University of New York, Buffalo, NY 14260, USA
2
Steer Group, Brooklyn, NY 11201, USA
3
Stephen Still Institute for Sustainable Transportation and Logistics, University at Buffalo—The State University of New York, Buffalo, NY 14260, USA
*
Author to whom correspondence should be addressed.
Sustainability 2024, 16(20), 9071; https://doi.org/10.3390/su16209071
Submission received: 27 August 2024 / Revised: 9 October 2024 / Accepted: 15 October 2024 / Published: 19 October 2024
(This article belongs to the Section Sustainable Transportation)
Browse Figures
Figure 1
<p>Methodological Framework Adopted to Conduct Spatial Regression Analysis of State-level Pavement Roughness Indicators.</p> "> Figure 2
<p>U.S. Climatic Weather Zones (Adapted from Smith et al., 1993) [<a href="#B50-sustainability-16-09071" class="html-bibr">50</a>].</p> "> Figure 3
<p>State-Level Surface Geology Distribution (Bletcher, 1943) [<a href="#B51-sustainability-16-09071" class="html-bibr">51</a>].</p> "> Figure 4
<p>Predominant Soil Type Frequency Histogram.</p> "> Figure 5
<p>Climate Type Frequency Histogram.</p> "> Figure 6
<p>Moran’s <span class="html-italic">I</span> Scatterplot for (<b>a</b>) Rural Collector Roads, (<b>b</b>) Rural Interstate Roads, (<b>c</b>) Rural Minor Arterial Roads, (<b>d</b>) Rural Principal Arterial Roads, (<b>e</b>) Urban Collector Roads, (<b>f</b>) Urban Interstate Roads, (<b>g</b>) Urban Minor Arterial Roads, (<b>h</b>) Urban Other Expressway Roads, (<b>i</b>) Urban Principal Arterial Roads.</p> ">
Versions Notes

Abstract

:
This paper used pavement condition data collected by the Federal Highway Administration (FHWA) between 2001 and 2006 aggregated by U.S. states to identify macroscopic factors affecting pavement roughness in time and space. To account for prior pavement conditions and preservation expenditure over time, time autocorrelation parameters were introduced in a spatial modeling scheme that accounted for spatial autocorrelation and heterogeneity. The proposed framework accommodates data aggregation in network-level pavement deterioration models. Because pavement roughness across different roadway classes is anticipated to be affected by different explanatory parameters, separate time–space models are estimated for nine roadway classes (rural interstate roads, rural collectors, urban minor arterials, urban principal arterials, and other freeways). The best model specifications revealed that different time–space models were appropriate for pavement performance modeling across the different roadway classes. Factors that were found to affect state-level pavement roughness in time and space included preservation expenditure, predominant soil type, and predominant climatic conditions. The results have the potential to assist governmental agencies in planning effectively for pavement preservation programs at a macroscopic level.

1. Introduction

Transportation agencies in the United States (US) face significant challenges in maintaining and preserving their roadways at a desirable level, notwithstanding the billions of dollars that are annually allocated for this purpose. For example, the allocated budget for the Federal Highway Administration for the 2017 fiscal year was USD 43 billion, with the majority of this budget being dedicated to pavement maintenance and preservation [1]. Despite the substantial amount of allocated funds, the American Society of Civil Engineers (ASCE) reported that the U.S. road infrastructure had a condition letter grade of “D”, and that the system was being underfunded [2]. In light of this, transportation agencies continuously seek strategies to appropriately allocate limited budgets. This is important for both individual pavement sections at the project level, and for the entire pavement infrastructure at the network level [3].
In order to develop effective strategies for highway infrastructure funding, a comprehensive understanding of pavement conditions over time and space is necessary. To that end, a substantial amount of research has been conducted to explain factors affecting pavement conditions. In previous research efforts, space-related independent variables have been found to play a significant role in pavement conditions. However, the effects of spatial trends, such as spatial autocorrelation and heterogeneity, cannot be captured adequately through the use of independent variables alone [4]. To that end, the intent of this paper was to develop a framework that identified macroscopic factors affecting pavement conditions in a spatial modeling scheme, aggregated at a U.S. state level. To achieve this goal, pavement condition data obtained from the FHWA (2006) [3] across a six-year study period (2001–2006) were modeled by using spatial econometric modeling techniques. Along with spatial autocorrelation, time autocorrelation parameters were introduced in the models to account for the prior pavement condition. Other factors that were shown to significantly affect pavement conditions were surface geology, climatic conditions, and pavement preservation expenditures. The analysis was carried out separately for nine different roadway classes with the motivation that the factors affecting pavement performance were expected to vary between distinct roadway classes [5].

2. Literature Review

Lytton (1987) suggested that pavement condition models can be broadly classified into two categories: deterministic and probabilistic models [6]. To further categorize pavement condition models, Haas and Hudson (2015) indicated that there are four types of deterministic pavement condition models: (1) purely mechanistic, (2) mechanistic–empirical, (3) regression-based, and (4) subjective [7]. A purely mechanistic model has not yet been developed in the literature, but the closest approach to purely mechanistic models are mechanistic–empirical models, in which structural or functional deterioration parameters are modeled. Regression-based models generally aim to explain the deterioration level of performance measures as a function of various independent variables. Finally, subjective approaches are used in agencies where historical pavement data are not available. They are experience-based and aim to estimate serviceability loss or other measures of deterioration against age [8].
With reference to mechanistic–empirical models, Pereira and Costa (2007) incorporated parameters such as reversible deflection, horizontal tensile strain, and active compression strain to develop models predicting pavement deformation [9]. Mechanistic–empirical models were developed to predict pavement performance measures such as cracking and the International Roughness Index (IRI) using variables representing pavement material characteristics and structural characteristics of the pavement as explanatory factors [10].
With respect to regression analysis, there is prior research that focused on developing models to predict functional or structural deterioration measures (e.g., International Roughness Index (IRI), Pavement Condition Rating (PCR), Pavement Serviceability Index (PSI), rutting depth, etc.) by utilizing various univariate and system of equations regression techniques (linear, nonlinear, Seemingly Unrelated Regression Equations (SUREs), time series, Three-Stage Least Squares (3SLS), survival models, etc.) as a function of various independent variables such as axle loads, pavement properties, subgrade conditions, weather and environmental considerations, traffic loads, and so on [5,11,12,13,14,15,16,17,18,19,20,21,22,23]. In addition to classical statistical regression-based models, various machine learning (ML)-based models (e.g., artificial neural network (ANN), gradient boosting, and so on) were also developed in the pavement performance literature. ANN-based models have been developed to predict pavement cracking [24], pavement roughness [25,26], and pavement serviceability [27]. Flintsch and Chen (2004) used ANN and fuzzy logic systems to estimate pavement performance models [28].
As an alternative to deterministic modeling approaches, which provide a single value as the output, probabilistic models fit a probability distribution to the desired pavement performance measure. The methodological approaches that fall under this category are Markov chains and Bayesian statistics. Examples of pavement condition/performance modeling studies that employed Markov chains are those of Butt et al. (1987), Li et al. (1996), and Yang et al. (2005) [29,30,31]; meanwhile, Bayesian techniques were employed by Smith et al. (1979) and Hong and Prozzi (2006) [32,33].

3. Methodological Approach

As the literature review suggested, regression techniques have been extensively used to develop models for pavement performance prediction, both at aggregate and disaggregate levels. These models unveil important insights in the understanding of pavement deterioration over time. However, a limitation of regression-based pavement performance modeling studies is their inability to account for (or unveil) possible spatial trends, namely spatial dependence, or heterogeneity, and spatial autocorrelation. Prior research has shown that when such spatial trends are not properly accounted for, the models may encounter misspecification issues, which results in a reduction in the accuracy of models and validity of inferences [4,34,35]. To address any spatial dependence that may exist in macroscopic spatial and temporal factors affecting pavement performance, spatial econometric modeling was used in this study.
In the following subsections, various aspects of spatial data modeling are explained, including spatial trends, spatial weights, exploratory spatial data analysis, and spatial regression modeling; the instrumental variable approach that enabled accounting for previous conditions is described; and the steps of the methodological framework that was developed in this study are presented.

3.1. Methodological Framework to Determine Pavement Deterioration

Step 1: Collect the International Roughness Index (IRI) data at the U.S. state level from accessible sources per year between 2001 and 2006 (sources provided in the data section). Some of the available data were in the form of intervals (e.g., IRI less than 65 in/mi, between 65 and 94 in/mi, etc.), whereas some were available as a single IRI value, which made it necessary to implement an aggregation process.
Step 2: Aggregate the IRI data available in intervals and point values (this aggregation is further elaborated on in the data section, under the dependent variable subsection) to come up with one IRI per U.S. state per year for each year between 2001 and 2006.
Step 3: Further aggregate the yearly IRI data to three-year periods to comprise the IRI in time period t (referred to as the IRIt, hereinafter), obtained by averaging the state IRI in the years 2004–2005–2006, and the IRI in time period t−1 (referred to as the IRIt−1, hereinafter), by averaging the state IRI in the years 2001–2002–2003. The three-year averaging aggregation step was implemented to account for possible drastic changes in the IRI values in the years where maintenance and rehabilitation activities were performed.
Step 4: Regress the IRIt−1 variable against its exogenous regressors (i.e., weather type, soil type, pavement expenditure information) to use the resulting estimated variable as an independent variable in the IRIt models (i.e., instrumental variable approach). As explained previously, using the IRIt−1 directly without this step would create endogeneity issues.
Step 5: Upload the IRI, expenditure, climate, and subgrade data to GeoDA software [35] to perform the analyses explained in the following subsections, i.e., spatial weight matrix creation, exploratory spatial data analysis, ordinary least squares regression, and spatial regression for each roadway class.
The methodological framework is summarized as a flowchart in Figure 1.

3.2. Spatial Trends: Autocorrelation and Heterogeneity

Used interchangeably with spatial dependence, spatial autocorrelation occurs when values of a variable in one location are correlated with the values of the same variable in other locations which are defined as neighbors [4,34]. This refers to the presence of co-variation of individual properties within a spatial system, resulting in systematic spatial patterns or observable clusters, consequently violating assumptions of independence of the errors and the exogeneity of regressors, leading to inconsistent and biased parametric estimates [36]. When the spatial autocorrelation is not uniform across all observations, this trend is called spatial heterogeneity, which causes heteroskedasticity issues [35].

3.3. Spatial Weights

One of the crucial operational tasks in spatial econometrics is the structural expression of spatial dependence and its incorporation in a model. The notion of spatial dependence implies the need to determine which units in a spatial system are dependent on each other. Formally, this is achieved through a spatial weight matrix, denoted as “ W ” [4,34]. The elements of a spatial weight matrix, W i j , express the neighboring relationships between the observations i and j . To avoid scale effects and to normalize the external influence on each spatial unit, the weight matrix is standardized, implying that all elements are scaled so that the sum of each row equals to one [36].
There are two ways to construct a weight matrix and define neighborhood relationships amongst spatial units (in this paper, U.S. states were defined as the spatial units). The first approach is contiguity-based, where two observations are defined as neighbors if they share a common border. The second approach is distance-based, where the interaction between spatial units i and j depends on the distance d i j between the centroids of spatial units (states). After carrying out an extensive analysis of the data using various types of weight matrices, the best results were obtained with the distance-based k -nearest neighbor matrix. The k -nearest neighbor matrix was constructed by choosing a number, k , which defined the number of neighbors of a spatial unit. For instance, when k was chosen as 4, each state would have its four closest states defined as its “neighbors”. Elements of weight matrices have binary values: 1 if two states are considered neighbors and 0 otherwise. In this paper, the k value (number of neighboring states) was chosen differently for different roadway classes to obtain the best results in terms of statistical fit.

3.4. Exploratory Spatial Data Analysis

Anselin (1999) defined exploratory spatial data analysis (ESDA) as a collection of techniques to help visualize spatial trends in the spatial structure of data [37]. ESDA is a subset of exploratory data analysis (EDA) methods, where EDA generally searches for any patterns and trends in data while ESDA concentrates on geographical patterns in data and helps identify spatial dependence and heterogeneity.
In this study, Moran’s I statistic was used to identify spatial dependence [38]. The statistic was calculated as the following:
I = n S 0 × ∑ i ∑ j W ( x i − x ¯ ) ( x j − x ¯ ) ∑ i ( x i − x ¯ ) 2
where x i and x j are the attributes of interest for spatial units i and j; x ¯ is the mean value of the attribute across n number of spatial units; W is the spatial weight matrix; and S 0 = ∑ i ∑ j w i j is the sum of the weights determined in the specified weight matrix W . The expected value of the I statistic was calculated as
E I = − 1 / ( n − 1 )
If the I value was found to be larger than E ( I ) , positive spatial autocorrelation was assumed, whereas values of I lower than the E ( I ) pointed to negative spatial autocorrelation [39].

3.5. Spatial Regression Modeling

After visually and statistically analyzing the data to examine the presence of spatial trends, spatial econometric models were estimated for cases when the test statistic (Moran’s I) turned out statistically significant. The spatial models estimated in this paper were the following: spatial lag, spatial error, and spatial lag and error models.
Using the notation from Anselin (2013) [4], the spatial lag model is expressed as
y = ρ W y + X β + u
where y is an (n x 1) vector consisting of dependent variable values; ρ is the spatial autoregressive coefficient; W y is a spatially lagged dependent variable; X is an n x k matrix of observations; β  is a k x 1 vector of regression coefficients; and u is an n x 1 vector of disturbances (errors). The spatial error model was formulated as the following [35]:
y = X β + u ,   u = λ W u + ε
where λ is the spatial autoregressive parameter; W is the weight matrix; and ε is an idiosyncratic error vector. The spatial lag and error model was formulated, with all terms as previously defined, as the following [35]:
y = ρ W y + X β + u ,   u = λ W u + ε
To investigate whether regression residuals were spatially correlated, a Moran’s I test for regression residuals was performed [38] as the following:
I = e ′ W e / S 0 e ′ e / n
where S 0 is the sum of the weights and e is a vector of regression residuals [35]. Inference on Moran’s I was based on an asymptotic standard normal approximation and therefore a z-test was performed.
To assess whether standard ordinary least squares (OLS) models needed to be re-estimated as spatial models to account for spatial dependence, Lagrange Multiplier (LM) tests derived by Anselin (2013) [4] were performed, where L M ρ and L M λ denote the formulation of the LM test for parameters ρ and λ , respectively:, as the following
L M ρ = 1 / n J ρ . β × ( u ′ W y / s 2 ) 2
L M λ = 1 / T × ( u ′ W u / s 2 ) 2
where s 2 is the maximum likelihood variance u ′ u / n ; T is a scalar computed as the trace of quadratic expression of the weight matrix with T = t r ( W ′ W + W 2 ) ; J ρ . β is a part of the estimated information matrix with J ρ . β = W X b ′ M W X b + T s 2 / n s 2 ; b is the vector of estimated parameters in OLS; M is the projection matrix I − X X ′ X − 1 X ′ ; and all other terms are as previously defined. The null hypotheses in the L M tests for respective parameters were ρ = 0 and λ = 0 , as the alternative hypotheses were ρ   ≠   0 and λ   ≠   0 . Both LM tests followed a χ 2 distribution.
For higher-order spatial autocorrelation, the tests determined whether the inclusion of both   ρ and λ was statistically justified. The test statistic is shown as L M ρ λ , where the null hypothesis was that both parameters were equal to 0. The L M ρ λ test statistic, as shown in Anselin (2013) [4], was χ 2 -distributed with two degrees of freedom, and it was computed as the following:
L M ρ λ = d λ 2 T + d λ − d ρ 2 D − T ~ X 2 2
with D = W X β ^ ′ [ I − X X ′ X − 1 X ′ ] ( W X β ^ ) / σ M L 2 + T .
The L M ρ and L M λ tests, as shown by Anselin et al. (1996) [40], were also sensitive to the presence of the other parameter; i.e., the L M ρ test may have truly been suggesting spatial error correlation although testing for the lag coefficient, and vice versa. To address this issue, robust forms of both statistics were also computed. In other words, the potential influence of the alternative coefficient was corrected by subtracting a value. The robust test statistics of each form are given as the following:
L M ρ * = ( d ρ − d λ ) 2 ( D − T ) ~ X 2 ⁡ ( 1 )  
L M λ * = ( d λ − T D − 1 d ρ ) 2 [ T ( 1 − T D ) ] ~ X 2 ⁡ ( 1 )
In the case that the spatial lag model was determined as the best fitted model, an additional diagnostic test was performed to determine if there was any remaining spatial autocorrelation that was not completely addressed with the spatial lag model. This test was a statistic developed by Anselin and Kelejian (1997) [41], noted as A K L and formulated as the following:
A K L = n ∗ I 2 ϕ 2
where the numerator is the multiplication of Moran’s I statistic computed from regression residuals to the sample size and the denominator is an estimate of the variance in the Moran’s I statistic. This statistic was χ 2 -distributed with one degree of freedom.
Along with regression diagnostics to examine spatial parametric significance, aspatial regression diagnostics were performed to investigate the presence of multicollinearity, error normality, and heteroskedasticity in the estimated models. To examine multicollinearity, a multicollinearity condition number was calculated, which indicated strong multicollinearity when the value exceeded 30 or 50 [35].
For error normality, the test value was based on the p-value of the Jarque–Bera (JB) statistic [42]. This statistic was distributed as an χ 2 variate, and it used the third and fourth moments of regression residuals m 3 and m 4 , defined as b 1 and b 2 , respectively, where b 1 = m 3 / σ 3 and b 2 = m 4 − 3 , formulated as
J B = n 6 ( b 1 ) 2 + n 24   ( b 2 ) 2
To test the presence of heteroskedasticity (non-constant error variance), the Breusch–Pagan (BP) statistic [43] was calculated as the following:
B P = 1 2   [ f ′ Z ( Z ′ Z ) − 1 Z ′ f ]
where f is the squared residual divided by the consistent estimate for error variance, i.e., f i = e i 2 / σ 2 M , and Z is an n x (P + 1) matrix consisting of a constant vector and P number of auxiliary variables z p , where z p is defined as the squares of original regression variables xi, i.e., z p = ( x i ) 2 . This statistic had a χ 2 distribution with P + 1 degrees of freedom.
The BP test assumed normal errors, which may or may not have been true even in the presence of heteroskedasticity [35]. A robust form of the BP statistic was suggested by Koenker and Bassett (1982) [44]. The KB test statistic was χ 2 -distributed with P + 1 degrees of freedom. If the test result was statistically significant, spatial heterogeneity was assumed. With a new error square variance ( u i ) 2 formulation, this statistic was computed as the following:
V a r ∧ [ u i 2 ] = ∑ i [ e i 2 − σ M L 2 ]
K B = ( 1 / V a r ∧ u i 2 ) [ f ′ Z ( Z ′ Z ) − 1 Z ′ f ]
Another aspect of spatial modeling is that for models with a spatial lag parameter, the estimated coefficients of non-spatial independent variables are interpreted differently. The marginal effect of an independent variable, which calculates the change in the dependent variable when the independent variable in consideration is increased by one unit, was computed as follows in a spatial lag model [4]:
d y i d x i , h = β h  
β h * = β h / ( 1 − ρ )
where β h is the marginal effect in the standard OLS model, and β h * is the marginal effect in a spatial lag or spatial lag with an error model.

3.6. Time-Lagged Instrumental Variable

To address endogeneity issues that arose from using the pavement condition value of previous years in modeling the current year condition, an instrumental variable approach was employed. The pavement conditions of previous years were regressed against their exogenous regressors, and the resulting estimated variable was used in the regression model for current pavement conditions. Although not seen as an entirely strict solution of the problem, this approach has proven to be effective in addressing endogeneity and yields consistent parameter estimates [45].

4. Data

4.1. Dependent Variable

The dependent variable in this paper was the “Average State International Roughness Index (IRI)” in a U.S. state in time period t, where the state-level IRI data were collected from various sources [3,5,23,46,47,48,49]. Due to inconsistencies in the data, the following spatial units were removed from the analysis: Alaska, Delaware, Rhode Island, Hawaii, and District of Columbia. As such, the analysis was performed using data from the remaining 46 contiguous U.S. states.
To elaborate on “Average State IRI”, the FHWA database providing the IRI data [3] had the state-level IRI data available per year between 2001 and 2006; however, rather than providing one IRI value per state, the IRI was split into four performance categories in units of inches/miles as “less than 65, between 65 and 94, between 95 and 170, more than 170”. The threshold numbers defined by the FHWA refer to “excellent”, “good”, “acceptable”, and “unacceptable” ride quality for IRI values of <65, 65–94, 95–170, and >170, respectively. The numbers of sample pavement segment miles in each category were provided per state, per roadway functional category, and per year. To calculate one average IRI value from this data format, an aggregation method was employed by multiplying the number of miles in each category with the mean value of the category thresholds and dividing this sum with the total number of miles in each category. However, the first and the last categories did not have upper and lower thresholds, hindering the calculation of their corresponding mean category values. Although the values of 65 and 170 could have been used in the calculation, in real-life conditions, much lower and higher IRIs are encountered, calling for the need to determine lower and upper thresholds.
To have a general idea for the determination of these two threshold values, a literature review was conducted. Through aggregating the minimum and maximum IRI values in the data, the final lower and upper IRI thresholds were determined as 39.33 and 230.24 in/mi, respectively. These values redefined the pavement performance category values as “39.33–65, 65–94, 95–170, and 170–230.24”, allowing for the calculation of a mean value for each category. The average values of the categories were multiplied by the number of miles in each respective category in the calculation year, and this sum-product was divided into the total number of miles, giving the average state IRI per year in inches/mile. As a result, one average state IRI was determined for each state in the study per year, yielding a total of six IRI values: IRI in 2001, 2002, 2003, 2004, 2005, and 2006. Equation (19) below shows the procedure, with the IRIt as IRI in year t; 49.66, 77, 132.5, and 200.12 as the mean category values; and xij as the numbers of miles in year t in the pavement performance category j, respectively.
I R I t = 49.66 × x i 1 + 77 × x i 2 + 132.5 ×   x i 3 + 200.12 × x i 4   ∑ j = 1 4 x i j
As the last step of aggregation, the state IRIs in years 2004, 2005, and 2006 were averaged to constitute the “Average State IRI in time period t”, and years 2001, 2002, and 2003 were averaged to define the “Average State IRI in time period t−1”, as shown in Equations (20) and (21). Of these aggregated values, the IRIt was determined as the dependent variable. The IRIt−1 was used as an independent variable by incorporating an instrumental variable approach, meaning the variable was regressed against exogenous independent variables and the resulting estimated IRIt−1 vector was used as the time-lagged instrumental variable.
I R I t = I R I 2004 + I R I 2005 + I R I 2006 3
I R I t − 1 = I R I 2001 + I R I 2002 + I R I 2003 3

4.2. Independent Variables

The independent variables included in the analysis were the state climate type, subgrade type, and state pavement expenditures. To start with the effects of the climate on pavement performance, freeze–thaw interactions caused pavement deformations, hence they were included in the deterioration model. To account for the impact of weather, the United States were divided into four different climatic regions: wet–freeze, wet–nonfreeze, dry–freeze, and dry–nonfreeze [50]. Each state was grouped under its corresponding dominant climatic zone, as shown in Figure 2 [50]. The categorization of these climatic zones was based on the minimum, maximum, and mean daily temperatures; daily precipitation; and daily snowfall each zone received.
Concerning the subgrade; natural surface soil plays an important role on pavement performance. Based on the size of soil particles, soil was classified under four major categories: clay, silt, sand, and gravel. In this ordering of soil, its granularity and competence increased, whereas its plasticity decreased as a subgrade material. For increased accuracy in determining the soil type, subcategories were determined for each soil type category. Regarding soil formation in the United States, eight surface geological zones were identified in the country as presented in Figure 3 [51]: glacial soils, residual soils, filled valleys and outwash, coastal plain soils, alluvium, lacustrine soils, loess, and clay/organic soils. The soil type of a surface plays an important role in pavement subgrade soil. Similar to the climatic zone, each state was assigned with its most dominant and second most dominant soil types. However, local variability was seen within a state in terms of the subgrade type.
Different soil types exhibit different levels of structural strength. Glacial, loess, and coastal soils have relatively higher structural strength and provide a strong subgrade for pavement, whereas residual, lacustrine, clay organic, alluvium, and filled valley soils do not exhibit stable structural strength and behave as weak pavement subgrade material [5].
As far as the impact of pavement treatment expenditure, higher values of this variable were expected to increase pavement performance. To represent expenditure, two different variables were used as independent variables: capital outlay expenditure and preservation expenditure. The FHWA (2010) made the distinction between the two expenditure types by defining capital outlay as “expenditures that are associated with highway improvements such as land acquisition and other right-of-way costs; preliminary and construction engineering; new construction; reconstruction, resurfacing, rehabilitation, and restoration; installation of guardrails, fencing, signals” and defining preservation as the “maintenance of highways, highway and traffic services, administration, highway law enforcement, highway behavioral safety, and interest on debt” [52].

4.3. Summary Statistics

Figure 4 and Figure 5 show the frequency distribution histograms for predominant soil types and climate types. The soil type histogram shows that residual soils were the most commonly observed soil type. Residual soils are known for performing poorly as a subgrade material, specifically in Midwest regions [5]. The second most frequent soil type was glacial, known for performing well as a subgrade material. The climate type frequency histogram showed that the most commonly observed climate type definition was the wet–freeze areas, which should have been expected to decrease pavement performance. Table 1 shows the summary statistics for the average state IRI [3,5,23,46,47,48,49,52], instrumented average state IRI, capital outlay expenditure [3], and preservation expenditure [3] values separately for different roadway classes.

5. Results and Discussion

5.1. Exploratory Spatial Data Analysis (ESDA) Results

Exploratory spatial data analysis (ESDA) was performed by calculating Moran’s I value [38]. For each roadway type, Table 2 shows the Moran’s I values and Figure 6a through 6i illustrate Moran’s I scatterplots. For Moran’s I value calculation, the K-nearest neighbor weights were used. After testing the statistical fit of various neighboring structures with different k values, the best results were obtained with k = 6 for rural roads and k = 4 for urban roads.
Based on the Moran’s I test statistic results and the z-values shown in Table 2, aside from rural principal arterials and urban other expressways, all roadway types showed statistically significant results, indicating spatial clustering of a similar pavement condition for these roadways. To investigate local spatial dependence and/or cluster centers, Moran scatterplots [53] were developed, as shown in Figure 6a through 6i for each roadway type. The number of data points in four quadrants of the scatterplot was interpreted to examine the spatial clustering analysis. The first and third quadrants referred to high value–high value and low value–low value IRI clustering, respectively, and the second and fourth quadrants indicated that spatial units with high IRI values were neighbors with spatial unites with low IRI values, indicating a clustering pattern of dissimilar values. The scatterplots further supported the spatial clustering interpretation of Moran’s I statistic results for seven out of nine roadway classes.

5.2. Spatial Regression Modeling

The results of the regression modeling for the nine different roadway classes are discussed below. Table 3 and Table 4 show the model estimation results for rural and urban roads, respectively. Table 5 presents the regression diagnostic test results, and Table 6 summarizes the goodness-of-fit statistics of the estimated models. As seen in Table 5, the test results indicated that the models did not suffer from multicollinearity, error non-normality, or heteroskedasticity issues. Also, the statistical insignificance of the Koenker–Bassett tests showed that spatial heterogeneity was not present. The Anselin–Kelejian test results showed that all of the spatial lag models successfully captured spatial autocorrelation, when present.
Due to the large number of separate models, the interpretation of the results was made based on the impact of each independent variable in models that were statistically significant. To start with the time-lagged instrumental variable, it was found to be significant in models for all roadway classes. This result showed that the previous pavement condition was an important factor in estimating future pavement deterioration.
The spatial lag and error parameters were found to be significant in models for various roadway classes. For rural collectors (RC) and urban principal (UP) arterials, spatial lag parameter was significant, resulting in a better fit over the OLS model in the spatial lag model. For rural interstates (RI), the spatial error parameter was significant, outperforming the OLS model compared to the spatial error model. For urban collectors (UC), both parameters were significant, resulting in the spatial lag and error model outperforming the OLS model. In these spatial models, the significant spatial parameter(s) captured the spatial autocorrelation amongst observations in the dependent variable, which violated the OLS assumption of independent observations. For the remaining roadway classes (rural minor arterial, rural principal arterial, urban interstate, urban minor arterial, and other urban expressways), the spatial models did not provide statistically significant improvements in fit when compared against the OLS models due to the statistical insignificance of the spatial parameters.
Variables related to pavement preservation expenditure were found to be significant in the various rural and urban roadway models. For rural minor (RM) arterials and urban minor (UM) arterials, the states with a preservation value below the mean expenditure value of all states in time period t demonstrated higher IRI values. A similar result was observed for rural collectors (RC) and rural interstates (RI) with the difference that the time period for this specific variable was t−1. For urban collectors (UC), preservation expenditure in time period t was a significant variable with a negative coefficient. For urban principal (UP) arterials, inversed preservation expenditure in time period t−1 was a significant variable with a positive coefficient. The statistical significance and coefficients of all of the expenditure variables in the models demonstrated that treatment expenditure was inversely correlated with pavement deterioration. In other words, an increase in expenditure resulted in lower IRI values, i.e., a better pavement condition.
The variables related to climate type were also significant in the various models. For rural interstates (RI), rural principal (RP) arterials, and other urban (UO) expressways, states classified by the wet–freeze or dry–freeze weather classifications were found to have higher IRI values as compared to the states that were not classified as such. For urban interstates (UI), wet–freeze states demonstrated the same trend. This result is consistent with the negative impact of freeze–thaw transitions on pavement conditions.
Regarding soil type, states with residual or clay organic subgrade soil were found to have higher IRI values compared to other subgrade types in the rural collector (RC) model. Considering the weak structural strength of these soil types, the model demonstrated a negative impact on pavement performance, corresponding to empirical observations.

6. Summary and Conclusions

In this work, pavement performance data of the U.S. states were collected from the Federal Highway Administration [3] to identify factors affecting the pavement roughness—assessed with the International Roughness Index (IRI), measured in inches per mile—at a macroscopic level (state level), while accounting for spatial and temporal heterogeneity and dependency. A spatial econometric modeling framework was leveraged to estimate separate models for nine different roadway functional classes. The spatial aggregation was made at the state level, i.e., all states had one average IRI measurement per nine functional classes. The explanatory factors used to model pavement roughness were the climatic conditions, subgrade soil type, and pavement treatment expenditures, along with spatial and temporal autocorrelation parameters.
The contributions of this study are threefold as the following:
  • Identifying the presence of spatial autocorrelation of a pavement condition and accounting for it at network-level pavement deterioration modeling by using spatial regression modeling;
  • Developing a pavement deterioration modeling framework (with the steps explained in the methodology section) that could be used by policy makers and related agencies;
  • Conducting a national-level experiment using U.S. state-level pavement data and discussing insights revealed by the model estimation results.
The first contribution is important, as pavement data will always have a spatial element to them and the deterioration models available in the literature often ignore the spatial trends, yielding biased parametric estimates, resulting in problematic inferences from the models. It was shown in this study that analyzing pavement data in a spatial modeling scheme could significantly improve the statistical fit of the models. The second contribution is the development of a new framework by aggregating IRI data on time and space, and it can be useful for agencies while conducting pavement deterioration studies, which is one of the core elements of any pavement management system and provides valuable input in budget planning scenarios.
The key findings from this study can be summarized as the following:
  • The results of the national level experiment using U.S. state-level pavement data were in line with past studies that identified similar influential factors (e.g., weather, soil, previous conditions) affecting pavement performance;
  • Moreover, the spatial test results showed that the pavement condition in one state could possibly be correlated with the pavement condition in neighboring states, due to either the spatial correlation of factors affecting the pavement condition in neighboring states (it should be noted that this possibility was explored, but the results were statistically insignificant) or unobserved characteristics commonly shared among the neighboring states;
  • Factors affecting pavement performance in a state, such as the traffic volume, climate, subgrade soil type, and most importantly, pavement maintenance funding of a state, possibly play a role in the pavement performance of a “neighboring” state.
Furthermore, the spatial pavement IRI prediction approach presented in this work has the potential to inform efficient and sustainable pavement management and rehabilitation programs.
Although the spatial econometric modeling in this study demonstrated the potential to significantly improve pavement performance modeling performance, room for improvement exists. For example, the inclusion of additional pavement condition measures (e.g., rutting depth, pavement friction, etc.) and the simultaneous modeling of these measures have shown to improve the accuracy of pavement performance modeling in previous studies. In the context of a spatial econometric analysis, however, this was computationally cumbersome, and fell outside the scope of this study. In addition, the unobserved heterogeneity addressed in this paper was time- and space-specific. It is likely, however, that other layers of unobserved heterogeneity existed inertly in the data. Future studies could further explore such time-/space-centric econometric frameworks, while accounting for additional layers of unobserved heterogeneity through the use of random parameters or latent class modeling. Finally, using spatial Durbin modeling [35] and the inclusion of spatially lagged independent variables with the proposed framework (thus, exploring whether an independent variable in a location directly affects the dependent variable in a neighboring location), has the potential to further improve the model’s performance and provide additional macroscopic insights, such as quantifying the impact of rehabilitation expenditures, climatic conditions, subgrade soil type, or traffic counts in one state on the pavement condition of neighboring states.

Author Contributions

Conceptualization, M.F. and P.C.A.; methodology, M.F. and P.C.A.; validation, S.S.A., I.B., and P.C.A.; formal analysis, M.F.; investigation, M.F.; writing—original draft preparation, M.F., S.S.A., I.B., and P.C.A.; writing—review and editing, S.S.A., I.B., and P.C.A.; visualization, M.F.; supervision, P.C.A. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Conflicts of Interest

Author Sheikh Shahriar Ahmed was employed by the company Steer Group. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Figure 1. Methodological Framework Adopted to Conduct Spatial Regression Analysis of State-level Pavement Roughness Indicators.
Figure 1. Methodological Framework Adopted to Conduct Spatial Regression Analysis of State-level Pavement Roughness Indicators.
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Figure 2. U.S. Climatic Weather Zones (Adapted from Smith et al., 1993) [50].
Figure 2. U.S. Climatic Weather Zones (Adapted from Smith et al., 1993) [50].
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Figure 3. State-Level Surface Geology Distribution (Bletcher, 1943) [51].
Figure 3. State-Level Surface Geology Distribution (Bletcher, 1943) [51].
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Figure 4. Predominant Soil Type Frequency Histogram.
Figure 4. Predominant Soil Type Frequency Histogram.
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Figure 5. Climate Type Frequency Histogram.
Figure 5. Climate Type Frequency Histogram.
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Figure 6. Moran’s I Scatterplot for (a) Rural Collector Roads, (b) Rural Interstate Roads, (c) Rural Minor Arterial Roads, (d) Rural Principal Arterial Roads, (e) Urban Collector Roads, (f) Urban Interstate Roads, (g) Urban Minor Arterial Roads, (h) Urban Other Expressway Roads, (i) Urban Principal Arterial Roads.
Figure 6. Moran’s I Scatterplot for (a) Rural Collector Roads, (b) Rural Interstate Roads, (c) Rural Minor Arterial Roads, (d) Rural Principal Arterial Roads, (e) Urban Collector Roads, (f) Urban Interstate Roads, (g) Urban Minor Arterial Roads, (h) Urban Other Expressway Roads, (i) Urban Principal Arterial Roads.
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Table 1. Summary Statistics of the State IRIs, Expenditure Variables, and Selected Variables by Roadway Type.
Table 1. Summary Statistics of the State IRIs, Expenditure Variables, and Selected Variables by Roadway Type.
RoadwayVariable MeanSDMinMax
Rural CollectorState IRI in time period t119.76621.68649.665153.801
State IRI in time period t−1119.18623.96049.665166.552
Capital Outlay in time period t63.73679.4684.351489.524
Capital Outlay in time period t−161.55973.5116.186447.817
Preservation Expenditure in time period t49.61756.1215.835329.014
Preservation Expenditure in time period t−141.70746.7554.037303.073
Soil Type Indicator: 1 if Dominant Soil Type was Residual or Clay Organic, 0 otherwise0.3690.48801
Expenditure Indicator: 1 if Preservation Expenditure in time period t−1 was less than the mean value per state (USD 41.707M), 0 otherwise0.6730.47401
Rural InterstateState IRI in time period t84.01813.39355.022114.250
State IRI in time period t−186.00816.19155.725117.924
Capital Outlay in time period t87.04992.2338.480528.809
Capital Outlay in time period t−199.34596.56618.717428.623
Preservation Expenditure in time period t60.09650.2218.535210.321
Preservation Expenditure in time period t−173.15867.7758.223296.378
Weather Type Indicator: 1 if Weather Type was Wet–Freeze or Dry–Freeze, 0 otherwise0.6950.46501
Expenditure Indicator: 1 if Average Preservation Expenditure in time period t−1 was less than the mean value per state (USD 73.158M), 0 otherwise0.6300.48801
Rural Minor ArterialState IRI in time period t105.62515.07465.442130.165
State IRI in time period t−1106.57116.81765.592140.982
Capital Outlay in time period t85.92383.4518.407411.269
Capital Outlay in time period t−181.89569.6069.774371.144
Preservation Expenditure in time period t62.28646.3487.027232.274
Preservation Expenditure in time period t−153.99142.3525.893228.707
Expenditure Indicator: 1 if Average Preservation Expenditure in time period t was less than the mean value per state (USD 62.286M), 0 otherwise0.6080.49301
Rural Principal ArterialState IRI in time period t96.15613.77056.647124.620
State IRI in time period t−197.76216.46557.034133.000
Capital Outlay in time period t180.647149.49922.300673.110
Capital Outlay in time period t−1173.631120.19211.556560.289
Preservation Expenditure in time period t68.50856.3557.020252.285
Preservation Expenditure in time period t−157.01645.6766.225223.305
Weather Type Indicator: 1 if Weather Type was Wet–Freeze or Dry–Freeze, 0 otherwise0.6950.46501
Urban CollectorState IRI in time period t150.96831.16549.780194.485
State IRI in time period t−1148.06031.72450.346194.485
Capital Outlay in time period t16.74226.8780.069158.878
Capital Outlay in time period t−113.22318.7060.237106.596
Preservation Expenditure in time period t13.83211.6570.80859.027
Preservation Expenditure in time period t−121.28523.2410.14496.480
Urban InterstateState IRI in time period t97.72813.82261.939128.400
State IRI in time period t−1101.33817.45362.331137.411
Capital Outlay in time period t225.237300.3560.8811294.367
Capital Outlay in time period t−1208.293253.9960.4091139.460
Preservation Expenditure in time period t61.42888.4300.351491.137
Preservation Expenditure in time period t−150.97161.2010.037222.534
Weather Type Indicator: 1 if Weather Type was Wet–Freeze or Dry–Freeze, 0 otherwise0.4340.50101
Urban Minor ArterialState IRI in time period t137.60922.84959.789176.311
State IRI in time period t−1134.38525.70561.859180.558
Capital Outlay in time period t54.19875.9031.863434.645
Capital Outlay in time period t−146.37761.1030.337334.722
Preservation Expenditure in time period t28.94941.3770.230211.123
Preservation Expenditure in time period t−120.93128.8730.106168.245
Expenditure Indicator: 1 if Average Preservation Expenditure in time period t−1 was less than the median value per state (USD 8.372M), 0 otherwise0.5000.50501
Urban Other Express
way		State IRI in time period t107.00916.96869.361139.602
State IRI in time period t−1110.25519.35967.635148.944
Capital Outlay in time period t101.200199.5860.0001168.276
Capital Outlay in time period t−186.667177.2750.000959.994
Preservation Expenditure in time period t36.83950.5000.450275.150
Preservation Expenditure in time period t−131.97538.3490.514175.858
Urban Principal ArterialState IRI in time period t130.86519.23974.033163.373
State IRI in time period t−1132.34721.99478.639172.428
Capital Outlay in time period t134.334214.2840.5411282.708
Capital Outlay in time period t−1125.840187.7871.379979.940
Preservation Expenditure in time period t36.12446.9560.889209.800
Preservation Expenditure in time period t−124.37931.3490.127180.361
Inversed Average Preservation Expenditure in time period t−1 multiplied by 1 million64.240129.6181.020724.987
Note 1: “Time period t” refers to the averaged values of variable in years 2004, 2005, and 2006, and “time period t−1” refers to the averaged values of variable in years 2001, 2002, and 2003. Note 2: Capital Outlay and Preservation Expenditure values are given in USD million.
Table 2. Moran’s I values for the dependent variable IRIt.
Table 2. Moran’s I values for the dependent variable IRIt.
Roadway ClassMoran’s I (z-Value)Scatterplot
Rural Collector0.094 (1.724)Figure 6a
Rural Interstate0.211 (2.970)Figure 6b
Rural Minor Arterial0.104 (1.725)Figure 6c
Rural Principal Arterial0.064 (1.119)Figure 6d
Urban Collector0.197 (2.470)Figure 6e
Urban Interstate0.184 (2.319)Figure 6f
Urban Minor Arterial0.150 (2.012)Figure 6g
Urban Other Expressway0.067 (1.022)Figure 6h
Urban Principal Arterial0.338 (3.936)Figure 6i
Table 3. Model Estimation Results for Rural Roads.
Table 3. Model Estimation Results for Rural Roads.
OLSSpatial LagSpatial ErrorSpatial Lag and Error
VariableRCRIRMRPRCRIRMRPRCRIRMRPRCRIRMRP
Constant29.202 (2.149)21.005 (1.977)20.924 (1.639)28.883 (1.851)−34.583 (−1.111)−10.434 (−0.292)34.960 (0.535)−7.068 (−0.230)31.834 (2.465)18.704 (1.985)20.889 (1.687)30.438 (1.985)−92.802 (−1.265)−0.137 (−0.006)20.575 (0.811)−7.207 (−0.356)
Instrumented Time Lag Variable0.646 (5.396)0.593 (5.315)0.756 (6.481)0.629 (3.904)0.640 (5.678)0.590 (5.357)0.763 (6.516)0.508 (2.826)0.625 (5.462)0.617 (6.121)0.756 (6.690)0.612 (3.877)0.651 (5.867)0.601 (5.674)0.748 (6.146)0.485 (2.557)
Weather Type Indicator
1 if wet–freeze or dry–freeze, 0 otherwise–6.876 (2.103)–7.409 (1.925)–4.224 (0.976)–4.656 (1.097)–8.149 (3.446)–7.392 (1.922)–6.857 (2.171)–4.539 (1.464)
Soil Type Indicator
1 if residual or clay organic, 0 otherwise7.727 (1.699)–––7.887 (1.844)–––8.048 (1.825)–––9.179 (2.115)–––
Expenditure Indicator
1 if mean preservation expenditure for RM in time period t was less than USD 62.286M, 0 otherwise ––5.905 (1.797)–––6.141 (1.838)–––5.941 (1.864)–––5.525 (1.694)–
1 if mean preservation expenditure for RC in time period t−1 was less than USD 41.706M, 0 otherwise14.716 (3.115)–––13.240 (2.948)–––14.343 (3.035)–––13.124 (3.199)–––
1 if mean preservation expenditure for RI in time period t−1 was less than USD 73.158M, 0 otherwise –6.045 (1.947)–––5.669 (1.834)–––4.875 (1.847)–––3.164 (1.187)––
Spatial Lag Parameter (ρ)––––0.547 (2.246)0.400 (0.920)−0.141 (−0.219)0.512 (1.343)––––1.018 (1.627)0.266 (0.785)0.013 (0.048)0.540 (1.686)
Spatial Error Parameter (λ)––––––––0.165 (0.672)0.548 (1.758)0.019 (0.068)0.075 (0.344)−1.000 (−0.700)0.838
(−1.671)		−0.093 (−0.294)−0.595 (−1.211)
Note 1: t-statistics in parentheses. Note 2: RC = Rural Collector, RI = Rural Interstate, RM = Rural Minor Arterial, RP = Rural Principal Arterial.
Table 4. Model Estimation Results for Urban Roads.
Table 4. Model Estimation Results for Urban Roads.
OLSSpatial LagSpatial ErrorSpatial Lag and Error
VariableUCUIUMUOUPUCUIUMUOUPUCUIUMUOUPUCUIUMUOUP
Constant89.613 (3.192)18.769 (1.333)2.338 (0.174)51.001 (2.739)38.753 (1.871)−10.846
(−0.171)		−41.367
(−1.418)		−55.299
(−2.460)		17.559
(0.234)		−28.304
(−0.938)		89.669 (3.303)15.351 (1.135)5.440 (0.417)51.906 (2.810)58.089 (2.970)15.988 (0.397)−34.939
(−1.926)		−56.026
(−3.062) 		27.099 (0.299)−30.510
(−1.181)
Instrumented Time Lag Variable0.554 (2.896)0.748 (5.503)0.961 (10.028)0.453 (2.609)0.842 (4.441)0.624 (3.224)0.608 (4.132)0.949 (11.286)0.416 (2.254)0.570 (2.989)0.553 (2.986)0.783 (5.983)0.941 (10.146)0.448 (2.630)0.667 (3.793)0.508 (3.147)0.650 (4.273)0.942 (10.974)0.425 (2.313)0.620 (3.180)
Preservation Expenditure (M) in time period t−1.140 (3.415)––––−1.053 (3.132)––––−1.134 (−3.515)––––−1.136 (−3.681)––––
Inversed preservation expenditure in time period t−1––––0.034 (1.743)––––0.031 (1.878)––––0.032 (1.755)––––0.029 (1.832)
Expenditure Indicator
1 if median preservation expenditure for UM in time period t was less than USD 8.372M, 0 otherwise––10.539 (2.766)––––10.477 (3.137)––––10.176 (2.769)––––10.779 (3.275)––
Weather Type Indicator
1 if wet–freeze, 0 otherwise–5.750 (1.799)––––2.880 (0.850)––––5.629 (1.901)––––2.534 (1.225)–––
1 if wet–freeze or dry–freeze, 0 otherwise–––10.847 (2.188)––––6.841 (0.688)––––10.307 (1.987)––––8.911 (0.765)–
Spatial Lag Parameter (ρ)–––––0.590 (1.763)0.770 (2.347)0.435 (3.012)0.374 (0.458)0.744 (2.759)–––––0.530 (2.026)0.665 (2.524)0.446 (3.436)0.263 (0.275)0.717 (2.893)
Spatial Error Parameter (λ)––––––––––−0.025 (−0.123)−0.055 (−0.196)0.435 (3.012)0.159 (0.668)0.485 (2.307)−0.666
(−2.167)		−0.668
(−1.555)		−0.317
(−0.742)		0.210 (0.278)−0.195
(−0.506)
Note 1: t-statistics in parentheses. Note 2: UC = Urban Collector, UI = Urban Interstate, UM = Urban Minor Arterial, UO = Urban Other Expressway, UP = Urban Principal Arterial.
Table 5. Regression Diagnostic Tests (P-Values in Brackets).
Table 5. Regression Diagnostic Tests (P-Values in Brackets).
Test StatisticRCRIRMRPUCUIUMUOUP
Moran’s I (error)1.146 [0.252]0.886 [0.376]0.532 [0.595]0.956 [0.339]0.183 [0.855]0.141 [0.888]1.850 [0.064]1.003 [0.316]2.184 [0.029]
L M ρ 2.503 [0.114]0.159 [0.690]0.000 [0.989]0.810 [0.368]0.652 [0.419]0.938 [0.333]9.001 [0.003]0.737 [0.391]7.291 [0.007]
Robust   L M ρ 3.752 [0.053]2.703 [0.904]0.018 [0.893]1.428 [0.232]3.289 [0.070]5.765 [0.016]7.096 [0.008]1.020 [0.312]6.191 [0.013]
L M λ 0.343 [0.558]3.650 [0.065]0.010 [0.920]0.119 [0.730]0.002 [0.967]0.091 [0.763]2.309 [0.129]0.280 [0.597]2.946 [0.086]
Robust   L M λ 1.591 [0.207]2.657 [0.103]0.028 [0.867]0.737 [0.391]2.639 [0.104]4.918 [0.027]0.404 [0.525]0.564 [0.453]1.847 [0.174]
L M ρ λ 4.095 [0.129]2.816 [0.245]0.028 [0.986]1.547 [0.461]3.291 [0.193]5.856 [0.054]9.405 [0.009]1.300 [0.522]9.138 [0.010]
Multicollinearity Number 16.81218.34718.30221.15616.46719.95115.93919.95118.410
Jarque–Bera (JB) statistic2.938 [0.230]2.742 [0.254]1.281 [0.527]1.803 [0.406]2.977 [0.226]0.571 [0.752]0.933 [0.627]0.734 [0.693]0.949 [0.622]
Breusch–Pagan (BP) statistic4.633 [0.201]1.449 [0.694]1.343 [0.511]2.106 [0.349]0.665 [0.717]3.344 [0.188]0.196 [0.907]0.065 [0.968]4.234 [0.120]
Koenker–Bassett (KB) statistic3.326 [0.344]1.193 [0.756]1.113 [0.573]1.683 [0.431]0.609 [0.737]3.833 [0.147]0.274 [0.872]0.064 [0.968]4.453 [0.108]
Anselin–Kelejian (AK) statistic1.311 [0.252]2.169 [0.148]0.001 [0.977]0.001 [0.977]3.638 [0.056]4.079 [0.043]0.670 [0.413]0.090 [0.764]0.917 [0.338]
Note 1: p-values are in brackets. Note 2: RC = Rural Collector, RI = Rural Interstate, RM = Rural Minor Arterial, RP = Rural Principal Arterial, UC = Urban Collector, UI = Urban Interstate, UM = Urban Minor Arterial, UO = Urban Other Expressway, UP = Urban Principal Arterial.
Table 6. Model Goodness-of-Fit Statistics and Forecasting Accuracy Measures.
Table 6. Model Goodness-of-Fit Statistics and Forecasting Accuracy Measures.
Roadway ClassificationModel TypeR2/Spatial Pseudo R2Adjusted R2/Spatial Pseudo Adjusted R2MADMAPERMSE
RCOLS0.6040.57610.2189.84513.642
S. Lag0.6410.6179.8779.45413.426
S. Error0.6040.60410.2959.95913.650
S. Lag and Error0.6450.60610.0189.56713.623
RIOLS0.4810.4447.7136.7219.643
S. Lag0.4840.4448.0756.9459.959
S. Error0.480–7.5056.5399.447
S. Lag and Error0.4810.4678.0407.0119.929
RMOLS0.5130.497.8117.87810.517
S. Lag0.5130.5127.8307.89310.521
S. Error0.513–7.8097.87410.517
S. Lag and Error0.5130.5127.8357.90010.522
RPOLS0.3410.349.0159.60211.436
S. Lag0.3420.3399.63510.17711.420
S. Error0.341–9.0059.59111.438
S. Lag and Error0.3430.3419.65510.19211.453
UCOLS0.3550.32520.93616.45725.220
S. Lag0.3900.33420.76816.14325.849
S. Error0.355–20.93216.46825.220
S. Lag and Error0.3990.34020.64315.89125.699
UIOLS0.4350.4098.1548.89510.386
S. Lag0.4180.5008.4879.00510.628
S. Error0.435–8.0968.82710.394
S. Lag and Error0.4320.5088.4158.98610.477
UMOLS0.7050.69210.0037.70312.400
S. Lag0.7580.7569.1817.00611.243
S. Error0.705–10.1337.95412.445
S. Lag and Error0.7580.7569.1716.99211.253
UOOLS0.2340.19811.98711.9614.849
S. Lag0.2610.24111.73711.65314.585
S. Error0.234–11.95211.92814.852
S. Lag and Error0.2570.24412.12512.05314.943
UPOLS0.3200.28912.74710.56815.861
S. Lag0.4590.36810.8538.80914.146
S. Error0.318–12.60310.59916.031
S. Lag and Error0.4600.37310.9538.86114.156
Note 1: RC = Rural Collector, RI = Rural Interstate, RM = Rural Minor Arterial, RP = Rural Principal Arterial, UC = Urban Collector, UI = Urban Interstate, UM = Urban Minor Arterial, UO = Urban Other Expressway, UP = Urban Principal Arterial. Note 2: MAD = Mean Absolute Deviation, MAPE = Mean Absolute Percentage Error, RMSE = Root Mean Squared Error. Note 3: The best performing model is bolded for each roadway class.
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Fettahoglu, M.; Ahmed, S.S.; Benedyk, I.; Anastasopoulos, P.C. Macroscopic State-Level Analysis of Pavement Roughness Using Time–Space Econometric Modeling Methods. Sustainability 2024, 16, 9071. https://doi.org/10.3390/su16209071

AMA Style

Fettahoglu M, Ahmed SS, Benedyk I, Anastasopoulos PC. Macroscopic State-Level Analysis of Pavement Roughness Using Time–Space Econometric Modeling Methods. Sustainability. 2024; 16(20):9071. https://doi.org/10.3390/su16209071

Chicago/Turabian Style

Fettahoglu, Mehmet, Sheikh Shahriar Ahmed, Irina Benedyk, and Panagiotis Ch. Anastasopoulos. 2024. "Macroscopic State-Level Analysis of Pavement Roughness Using Time–Space Econometric Modeling Methods" Sustainability 16, no. 20: 9071. https://doi.org/10.3390/su16209071

APA Style

Fettahoglu, M., Ahmed, S. S., Benedyk, I., & Anastasopoulos, P. C. (2024). Macroscopic State-Level Analysis of Pavement Roughness Using Time–Space Econometric Modeling Methods. Sustainability, 16(20), 9071. https://doi.org/10.3390/su16209071

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