Description
M2 Mathématiques,
: Hasting-Metropolis (and Gibbs) samplers
Exercise 1: Hasting-Metropolis within Gibbs – Stochastic Approximation EM We observe a group of N (independent) individuals. For the i-th individual, we have ki measurements yi,j E R. In studies on the progression of diseases, measurements yi,j can be measures of weight, volume of brain structures, protein concentration, tumoral score, etc. over time. We assume that each measurement yi,j are independent and obtained at time ti,j with ti,1 < … < ti�ki.
1.A – A population model for longitudinal data We wish to model an average progression as well as individual-specific progressions of the measurements from the observations (yi,j)iE1,NJj,jE1,k. To do that, we consider a hierarchical model defined as follows.
i. We assume that the average trajectory is the straight line which goes through the point p0 at time t0 with velocity v0
d(t) := p0 + v0(t — t0)
where
p0 N(p0,o2p0 ) ; t0 N(t0,o2 t0 ) ; v0 N ( v0, �2 v0 ) and op0, Ut0, av0 are fixed variance parameters. While we consider straight lines, we can also fix p0.
ii. For the i-th individual, we assume a trajectory of progression of the form
di(t) := d(ci(t — t0 — Ti) + t0 ).
The trajectory of the i-th individual corresponds to an affine reparametrization of the average trajectory. This affine reparametrization, given by t ‘-+ ci(t — t0 — Ti) + t0, allows to characterize changes in speed and delay in the progression of the i-th individual with respect to the average trajectory. Moreover, we assume that for all i-th individual measurements
I
i.i.d. yi,j = di(ti,j) + �i�j where �i�j � N(0, cr2)
ai = exp() where i.i.d. � N(0, o2)
Ti
i.i.d. � N(0,o2�)
�
The parameters of the model are 9 = (t0, v0, o, O,-, a-). For all i E [1, N, zi = (ci, Ti) are random variables called random effects and zpop = (t0, v0 ) are called fixed effects. The fixed effects are used to model the group progression whereas random effects model individual progressions. Likewise, we define Oi = (o, O,-, a) and 9pop = (t0, v0 ).
