Key Words censored quantile regression, degradation,

2 ROOFER Example
The data for this study comes from the U. S. Army Construction Engineering Research Laboratories (USACERL). They have acquired databases from several Army installations that contain inventory and inspection data as well as condition ratings on roofs. There are three di erent condition indices, each of them is measured on a scale from 0 to 100. They are calculated by deducting points for each combination of distress type and distress level. A more detailed description of the ROOFER project may be found in Bailey et. al. (1997). For the present paper, we choose the Membrane Condition Index (MCI). The ratings are given by individual inspectors. Each roof section is carefully inspected, and the membrane distresses are recorded on a standardized Roof Inspection Worksheet. For example, blister is a common distress type with three severity levels used: low, medium and high. Once the severity level is determined, the size of the blister or blisters is measured and the problem density is then computed as a percentage of the a ected area relative to the total area. Points are then deducted from the index based on a pre-designed chart. The same is done for other
distress types. There were two questions of interest here: modeling degradation of roofs as a function of age, and assessing the impact of several explanatory variables. In this paper, we will focus on the rst problem. We selected a relatively homogeneous subset of 270 roof sections that are less than 20 years old, with no edge-metal and no interior drains. Figure 1 gives the jittered scatter plot of MCI against age. We can identify three features that are common to such datasets: skewness, heteroscedasticity, and clustering at one end. Indeed, a sizeable proportion (about 30%) of the data is clustered at the upper limit of 100 in Figure 1. Figure 1: Jittered Scatter Plot of Roofer Data
Key Words: censored quantile regression, degradation, splines, Tobit regression. Abstract:
Percentile curves for natural growth or degradation over time are often of interest. Motivated by consulting in various infrastructure studies, we consider the problem of estimation and inference for conditional percentile models. In particular, we present a study of roof condition degradation with the goal of estimating predictive curves for condition as a function of age. In the database roof conditions are reported as bounded scores, and the method of analysis needs to preserve the bounds. We present methods for modeling the age dependent distribution of roof condition scores using raw data as well as grouped quantiles. We consider both direct and latent modeling to accommodate the bounded response. replicated values at the high end of the condition index. Second, the conditional percentile curves are not expected to be globally linear. In the earlier years, there are usualldition index values at its maximum, and thus, the percentile curves may start to decline only after an initial holding period. Any parametric form for the percentile curves has to be able to allow constancy at the beginning. In this paper, we discuss and compare three di erent approaches that model the conditional percentile curves for condition indices of such type. In particular, we present a cross-sectional study of roof condition with the goal of estimating predictive curves for condition as a function of age. The same methods may be used for a variety of infrastructure studies of similar nature.
100
100 is assigned to MCI whenever the assessor believes that the severity and extent of membrane condition distress are below some given threshold levels (say, not clearly visible). Under this view, we believe that perfect membrane conditions have a latent rating of over 100. This is a convenient means to avoid downward bias that arise from those boundary values when we t a global parametric model (e.g. linear regression). With this in mind, we rst turn to a semiparametric method of censored regression quantiles.
CONDITIONAL QUANTILE CURVES WITH AN APPLICATION TO INFRASTRUCTURE STUDIES
Hanga C. Galfalvy, Xuming He, Douglas G. Simpson, University of Illinois Xuming He, 725 S. Wright Street, Champaign, IL 61820 (he@)
1 Introduction
At the Illinois Statistics O ce (ISO) we have been involved with several infrastructure studies in recent years. They require estimation of percentile curves for degradation of certain condition indices over time. Condition indices for the subject under study are generally designed and provided by eld experts to measure the extent of degradation of the subject. Such indices may vary from 0 to 100 or be set in the scale of 0 to 5. It is often the case that a maximum value is assigned to subjects without noticeable tear and wear. As a result, a large number of subjects are given the maximum value, especially in the earlier years. This makes it di cult and indeed inappropriate to use the traditional regression analysis for the following two reasons. First, the traditional regression models go after the conditional mean curve of a condition index as a function of age. Although the averages are meaningful summaries, they do not usually provide su cient information for non-Gaussian distributions. The degradation data we deal with are clearly skewed to the left. The possibility of nding a transformation to normality is excluded by a large number of