Clinical results

The following points summarise the effect of the risk factors on the hazard of each type of

outcome.

• Age of patient. Young patients have higher chance of local, regional and metastasis recurrence than old patients, whereas they have lower hazard of ”died from breast

cancer” than old patients.

• Stage of the disease. Patients in stage2 and stage3 have significantly higher hazard of local and regional than patient in stage1. The hazard of metastasis of stage1 is

significantly lower than the other three stages. Only patients in stage4 have significantly

higher hazard of ”died from breast cancer” than stage1.

• Surgery type. Patients without surgery or with incision biopsy surgery have higher hazard of all recurrence than patients with radical mastectomy and axillary clearance

except metastasis no difference. Patients with excision biopsy surgery have higher

axillary clearance and lower hazard for metastasis no difference for ”died from breast

cancer”.

• Histology. There is more chance for patients with ductal than patients with Other histology for local recurrence and higher hazard of regional than all other three types

of histology. The hazard of metastasis for patients with ductal is the same for lobular

but higher than the other two types. The hazard of metastasis for patients with ductal

is higher than dcis (ductal carcinoma in situ) but the same for the other two types.

• Cohort. Patients with primary surgery before 1990 have the same change of local recurrence as those after 1990 but they have lower hazard of regional recurrence and

metastasis. However, the hazard of ”died from breast cancer” is higher for patients

with primary surgery before 1990 than after 1990.

• Chemotherapy. Patients with chemotherapy have higher hazard of metastasis than patients without. There is no significant effect of chemotherapy on the hazard of local,

regional and ”died from breast cancer”.

• Menopausal status. Patients pre-menopausal have lower hazard of metastasis than patients post-menopausal. There is no significant effect of menopausal status on the

hazard of local, regional and ”died from breast cancer”.

• Radiotherapy. Patients with radiotherapy have lower hazard of local recurrence than patients without. There is no significant effect of radiotherapy on the hazard of regional,

metastasis and ”died from breast cancer”.

• Side of the body affected. There is no significant effect of the side of body with cancer (right or left) on the hazard of all types of outcome (competing risks).

4.13

Summary

In this chapter, different types of frailty models are discussed, shared frailty, correlated frailty

and multivariate frailty. Most of the existent models are correlated frailty models rather

than multivariate model. One of the limitations of the correlated frailty model is that they

have restricted correlation coefficients between frailties; e.g. Correlated Gamma and Inverse

Gaussian frailty discussed in sections 4.6 and 4.7. Whilst the multivariate frailty models

have unrestricted correlations, but they have the limitation that the marginal likelihood

function does not have a closed form and numerical integration is needed to get the maximum

likelihood estimator; e.g. Log-Normal frailty discussed in section 4.9. The interpretation of

the frailty is not straight forward in multivariate frailty models. A competing risks model

with multivariate frailty is introduced and employed in the analysis of the time to the first

recurrence of breast cancer. The analysis was carried out by a Cholesky decomposition

of multivariate Log-Normal frailty and by a non-parametric multivariate frailty. The non-

parametric frailty model is much less time consuming in fitting the data with the smallest

standard errors of parameters estimates. The simulations showed that only a few numbers of

mass points are needed to fit the data using non-parametric frailty compared to multivariate

Log-normal where at least eight points are needed to get acceptable results. In the analysis

of competing risks models, including frailty is important to take into account the potential

relation between different failure types. Ignoring this fact and employing the commonly used

estimation procedures underestimate the parameters of interest and could lead to inaccurate

inference about relevant risk factors. Another way to overcome the problem of mis-specifying

the frailty distribution is breaking the frailty distribution in sub-distribution the so-called

finite mixtures. in the next chapter, several simulation studies of mixture of different frailly

Frailty and finite mixture

5.1

Introduction

In previous chapters, it was shown that the choice of the frailty distribution is crucial for

making valid inferences. Fitting the model using a non-parametric frailty is one way to

overcome this problem. In this chapter, a different prospective is used to solve the problem.

The model is fitted by breaking the frailty distribution into a finite number of components,

the so-called finite mixture. Mixture models are usually used to model data that come from a

heterogeneous population. Using a mixture of frailty distributions increases the flexibility of

modelling the unobserved heterogeneity, especially if the frailty distribution is not unimodal.

The finite mixture models of a parametric frailty distribution can be viewed as a semi-

parametric models, as they can be written in terms of J components of a specific distribution.

In general, a random variable T with a probability density function f (t) can be decomposed

into a sum of J class probability density functions. Let fj(t) denote the jth class probability

density function. The finite mixture model with J -component has the following general form

f (t|θ1, · · · , θJ; π1, · · · , πJ −1) = J

X

j=1

where πj represents the probability that the realisation t is coming from the jth component.

Furthermore, these probabilities must satisfy the following constraints

0 < πj < 1 and J

X

j=1

πj = 1.

The mean and the variance of the finite mixture are

µ = E[T ] = J X j=1 πjµj σ2 = V [T ] = J X j=1 πj(µ2j + σ2j) − µ2 (5.1.1)

For more information see (Everitt and Hand, 1981) and (Frhwirth-Schnatter, 2006).