Cox and Random Walk Statistical Models for Dynamics of Intractable Ordinal Data: An Example of Fecal Hemoglobin Concentration
Date Issued
2015
Date
2015
Author(s)
Peng, Szu-Min
Abstract
Background As fecal hemoglobin concentration (f-Hb) is a good predictor for colorectal cancer (CRC) incidence and mortality, the dynamics of f-Hb is therefore of great interest in the face of large population-based screening data on periodical examination of f-Hb. Modeling the evolution of f-Hb is intractable as it is an ordinal property and often involves with correlated, censoring, truncating, and dynamic movement with absorbing barriers in the province of the random walk model. Aims This thesis was first to assess the values of f-Hb across three groups (normal, adenoma, and CRC), estimate the effective median f-Hb concentration (f-Hb50) and its threshold when the adenoma and CRC were detected. The second aim was to apply the random walk model to quantify the dynamic change of f-Hb considering the upper limit because of occurrence of adenoma and CRC. Methods Conventional survival analysis was employed to test the difference in the mean (or median) value of f-Hb across three groups. The Cox proportional hazards (PH) regression model, making allowance for correlated property, was applied to estimating the hazard ratio (HR) of reaching the ranking of f-Hb across three groups after controlling for relevant covariates. The non-parametric method was used to estimate effective median value of f-Hb (f-Hb50) and the threshold value of f-Hb to hit colorectal adenoma and CRC. To consider the dynamic (stochastic) property, a random walk model with asymptotic distribution and multi-nominal distribution was further developed to elucidate the evolution (repeated measurement) of f-Hb data to estimate the forward probability (p) and backward probability (q) by three types of diseases status. These parameters were also exploited for calculating the gambler’s ruin probabilities of hitting adenoma and CRC. Results The result of ANOVA shows that the differences in the mean value of f-Hb across three groups were statistically significant. The result of Cox PH regression after adjusting for other covariates (gender, age, family history and brand), compared to the normal group, the HR of the CRC group was 0.181 (0.178, 0.184) and the adenoma group was 0.204 (0.202, 0.205), which suggest that screenee who had higher f-Hb may have higher probability to be diagnosed with disease. The estimated results on the random walk logistic regression model is that the drift rate (p-q) was the highest in the CRC patients followed by adenoma, and the lowest in subjects free of colorectal neoplasia. With the random walk logistics regression model merely considering forward (p) and backward probability, the calculation probabilities gave 0.733 and 0.267 for patents diagnosed as CRC, 0.575 and 0.425 of p and q for patients diagnosed as adenoma, and 0.358 and 0.642 of p and q for the normal subjects. Compared with the normal group, the odds ratio of moving forward was 4.923 for CRC and 2.426 for adenoma. If we set 400 μg/g for CRC, 300 μg/g for adenoma and 20 μg/g for normal as the absorbing barrier the gambler’s ruin probability of reaching the barrier was 0.867, which was higher than 0.455 of adenoma whereas the ruin probability for the normal subject was very low. If the initial value (x) was set 1 it takes, on average, 740 steps for CRC, 893 steps for adenoma, and 7.05 steps for normal to reach absorbing barrier. Conclusions The thesis has applied the Cox PH regression model and developed a random walk regression model to accommodate the ordinal data with long tail distribution at extremely high value, undetectable circumstance at extremely low value, and missing values and also in relation to multi-state outcome. These proposed models have been applied to nationwide population-based screening for CRC with FIT to estimate the hazard ratio for CRC and adenoma as opposed to the normal subjects, also to estimate the f-Hb50 and threshold of developing CRC and adenoma, and get a better understanding of how f-Hb moves forward and backward with time, providing what is the chance of having gambler’s ruin (reaching to the barriers of f-Hb) and how many steps are expected to be taken before ruining. These findings provide a new insight into policy-making for colorectal cancer screening and also the surveillance of early-detected colorectal cancer.
Subjects
Random walks
gambler’s ruin
colorectal cancer
screening
fecal hemoglobin
FIT
SDGs
Type
thesis
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