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Let us assume I am trying to develop a predictive model that will give an indication of the progression of the percentage crack area on bridge decks. Engineering knowledge indicates that the crack area will be zero up to a point (crack initiation time) after which the crack area will grow (typically an S-curve) is assumed. I do not have any longitudinal data. I only have cross-sectional data representing the bridge condition at present. Below is a simplified example of the type of data (randomly created example) I have:

Hypothetical Example Data

Notice that there are lots of zeros in the crack area column. Also, I have simplified the regressors to only two – in reality there may be more information such as traffic loading, rainfall etc. My data sets may contain anything from several hundred to several thousand observations.

My Question: What approaches, if any, are viable to build a predictive model based on a single cross sectional set as shown above. I realize the limitations of cross-sectional data in this context, but intuitively I feel there is useful information in the data I have.

I expect that the model will be of the form:

enter image description here

Notice that for making predictions, I WILL know what the crack area at present is. I want to know either what the crack area will be one year from now, or the probability of the crack area exceeding certain thresholds, conditional on the values for the predictors.

Approaches I have considered:

  1. Combining multiple Survivor Analysis models with endpoints such as Crack Area = 0%, Crack Area = 3%, Crack Area = 10% etc. The problem with this approach is that I will only have left and right censored data available (no uncensored data where I actually observe when the endpoint is reached).
  2. Bayesian/Markov approaches in which I predict the probability of the crack area being Y (or less/more than Y) in year (i+1) given that the crack are in year Y is M, and the predictors have values a, b, c etc.

I would appreciate any ideas or suggestions on how this type of data can be approached to provide some sort of evidence based predictive model.

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  • $\begingroup$ Does your knowledge of the subject matter provide a general parametric form for the S-shaped development of crack area over time, following initiation of the crack? $\endgroup$
    – EdM
    Commented May 6, 2022 at 21:23
  • $\begingroup$ @EdM - thanks for taking an interest in this. There is no generally accepted form for the S-shaped equation. Often, three linear segments are simply used. I am not so much interested in fitting a curve as I am in knowing methods by which I can use this data as evidence to building any sort of model. My feeling is that the danger of confusing longitudinal and cross-sectional data means one has to treat carefully. But I feel there has to be SOME value in the above type of information? Any advice will be appreciated. $\endgroup$
    – Fritz45
    Commented May 6, 2022 at 23:44
  • $\begingroup$ For the other covariates (traffic loads, rainfall, etc.) do you have longitudinal data or are you similarly limited to current values? $\endgroup$
    – EdM
    Commented May 7, 2022 at 11:36
  • $\begingroup$ @EdM - for some of them (e.g. traffic loads), I may have longitudinal data, or I can infer it from the type of parameter (e.g. traffic grows linearly with a given/known growth rate). Other covariates may be stationary (e.g. rainfall, ignoring long term effects). Ideally, I am interested in how to approach "evidence extraction" under the assumption that I have ONLY the above set available. But if it is a game changer, I can assume that I have longitudinal data for some (definitely NOT all) covariates. Thanks again for your interest in this question. $\endgroup$
    – Fritz45
    Commented May 7, 2022 at 22:26

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