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Our aim is to introduce a Bayesian intensity model for the Lexis diagram, which represents individual follow-up ... prior according to Arjas and Gasbarra (A-G) for.
UNIVERSITY OF HELSINKI FACULTY OF MEDICINE

BAYESIAN INTENSITY MODEL FOR LEXIS DIAGRAM (1) University of Helsinki, Department of Public Health, Helsinki Finland 00114 (2) National Institute for Health and Welfare, Helsinki Finland 00271

Anna But, M.Sc. (1) Jari Haukka, Ph.D. (1) Tommi Härkänen, Ph.D. (2)

INTRODUCTION In survival analysis, a unifying idea is to see the occurrence of events as a dynamic process in time, and to model it conditioning on past. The effect of event history on future risk can be modeled by assuming an intensity process characterized by a non-negative hazard function. In many applications, there is more than one time scale at which time to event can be represented, and exploring the intensity on two time scales can provide an essential information on failure mechanism. Our aim is to introduce a Bayesian intensity model for the Lexis diagram, which represents individual follow-up times on two time scales. We focus on right-censored survival data with either a single event or censoring time recorded on two time scales.

MODEL

RESULTS

Our approach to model the event history data is based on the point process on the Lexis diagram. We proceed from Lexis diagram to its isomorphic representation allowing a simple discretization of the process with respect to one of the two time scales (Figure 1). Discretization is accomplished through the partition of the observational plane into strips Ak, k=1,…,K. For each strip, we assign a hazard function λk, which is modeled by piecewise constant function gk.

We generated 500 data sets to test the method. The hazard rate underlying the generated data was defined to incorporate the areas of different hazard levels (Figure 3).

PRIOR

Piecewise constant functions are parameterized by jump points ϵk,j j=0,1,…, and corresponding constant hazard levels hk,j between them [1]. The prior for the jump points is specified as a time-homogeneous Poisson process with parameter μ. The initial hazard level of first strip is assumed to have the Gamma (∙|α0, β0) and a random walk Gamma prior is assigned for subsequent hazard levels hk,j ~Gamma (∙|α, βk,j), where βk,j is defined as

Figure 3. The intensity with three rate areas (left), according to which the data were simulated (right).

Figure 1. Lexis diagram (upper plot), and its isomorphic representation. V1 and V2 are the time scales.

For subsequent strips, the weighted mean of neighboring hazard levels ℎ𝜃−ℎ is given by

The 95 % pointwise coverage probabilities were much better using the Lexis prior and the Bayesian intensity model (BIM) than by using the prior according to Arjas and Gasbarra (A-G) for intensity function on real line or a Poisson regression model with splines and frequentist inference (Table 1).

Table 1. Number (percentage) of points at which coverage probability is within [0.94, 0.96].

𝑗,𝑘

Method

Number (%)

BIM, Lexis: α=0.1, φ=0.5

581 (39%)

BIM, Lexis: α=0.1, φ=1

546 (37 %)

BIM, Lexis: α=0.1, φ=2

431 (29%)

BIM, A-G: α=0.1 Figure 2. The prior distribution of hazard levels hk,j.

The prior distribution is specified by five hyperparameters μ, α0, β0 , α, and φ. Number of jumps is controlled by the intensity μ. In addition, the hazard levels hk,j depend on the weighted mean of preceding hazard levels in within- (j-1) and between-strip (k-1) directions (Figure 2). For small values of smoothing parameter α>0, the distribution of hazard level hk,j is likely to be flat corresponding to little prior information. Smoothing direction (weight) parameter is φ>0. If φ=0 then gk are independent a priori. By the definition, conditional prior expectation for hazard levels equals the weighted mean of neighboring hazard levels, and tightness of hk,j is regulated by 𝛼.

INFERENCE

Inference under such non-parametric Bayesian model requires a numerical integration. We apply the reversible jump Metropolis-Hastings algorithm to sample from the posterior distribution and to allow changes in the dimension of the parameter space. Bayesian analyses were performed using the BITE software [2].

Poisson regression

94 (6%) 388 (26%)

CONCLUSIONS

The Lexis prior appeared to provide much better fit to the survival data according to the coverage probabilities. The approach introduced here can be directly extended to repeating events and cases with more than two dimensions. The research has received funding from the European Community’s Seventh Framework Programme (FP-7) under grant agreement number 282526, the CARING project.

References

[1] Arjas E., Gasbarra D. (1994) Nonparametric Bayesian inference from right censored survival data, using the Gibbs sampler. Statistica Sinica 4, 505-524 [2] Härkänen T. (2003). BITE: Bayesian intensity estimator. Computational statistics 18, 565-583