Original Research ArticleLikelihood function for estimating parameters in multistate disease process with Laplace-transformation-based transition probabilities
Introduction
While a variety of stochastic processes have been applied to biomedical field such as cancer progression from early pre-clinical detectable phase (PCDP) to late clinical phase (CP) [1], [2], [3], [4], [5], [6], [7], intractable mathematical forms are often implicated in modeling a complex multistate disease process. For example, while a three-state disease progression model [8] is extended to the five-state model [9] a complicated and lengthy mathematical formula would be expected.
To tackle the thorny issue of complex calculation [10], the Laplace transformation that has been widely used in engineering is therefore proposed. The advantages of Laplace transform not only make use of the exponential decay kernel to reduce the problem of improper integral but also transform the convolution form into multiplication form.
There are several applications to medical field with Laplace transformation. Weiss and Zelen [11] applied a semi-Markov model to modeling the first passage times or the probabilities for curative treatment based on a renewal equation with convolutional forms following the Laplace transformation. A similar application has been seen in other studies [12], [13]. Such a method is also extended to generalized exponential function to approximate the underlying Weibull or Gamma distributions [14]. Webster et al. demonstrated the usefulness of taking Laplace transformation in multi-stage cancer model in order to avoid multi-fold convolution forms embedded in probability density function. Following the one-to-one property, the explicit forms can be calculated by taking inverse Laplace transformation [15].
While the methodology used in these studies are useful and theoretically sound, one of thorny issues is pertaining to the estimation of parameters after the Laplace transformation of multistate transition probabilities. The conventional likelihood function often requires the detailed time-stamped history data that are very demanding for data collection. It is therefore interesting to build up the likelihood function coupled with the transition probabilities after Laplace transformation rather than the original likelihood. The likelihood based on Laplace transformation may only require aggregate data of counts for various states with the principle of data reduction. Such statistical thoughts have been never attempted before and are worthy of being investigated.
Section snippets
Methodology
The combination of matrix computation and Laplace transformation may shed light on the solution to such a problem by providing a concise formula for multi-fold convolution between the states of disease progression and avoid the requirement of time-stamped data based on the principle of data reduction.
We tempted to develop the methods on the derivation the transition kernel for the process using empirical data. This novel method was first demonstrated by using the three-state Markov model and
Three-state model for breast cancer with simulated data
Using the method proposed by Wu in 2013, we simulated 1,000,000 subjects considering the scenario of breast cancer screening with biennial inter-screening interval [19] to evaluate the proposed method on the derivation of estimated results on with the Laplace transformed method. A three-state progressive model of breast cancer including the states of free-of-disease, PCDP, and CP were considered. The observed frequencies of subjects with each type of transition mode given prevalent and
Discussion
Based on the method of Laplace-transformation, we developed a series of likelihood functions for the states of cancers detected in screening program. Transition kernels used in multistate Markov models for complex multi-compartment progression of cancer was turned into a series of simple partial fraction forms with the multi-fold convolution of Laplace transformation. The proposed Laplace-transformed likelihood functions were applied to cancers characterized by transition probabilities and
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Funding
This study was supported by the Ministry of Science and Technology, Taiwan (grant number MOST 108-2118-M-002 002-MY3, and MOST 108-2118-M-038-001-MY3). The funding source had no role in study design, data collection, analysis, or interpretation, report writing, or the decision to submit this paper for publication.
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Laplace Transform (PMS-6)
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