Mt e1 ln1t

Y^ieD(t) exp{ 1 [C21 + 2Ci1&i2Pi ln(l + t) + o^ ln2(l + t)]}

where

Ji1 biei- c2

4 Application to the PAQUID data

4.1 The estimated mean of the disablement process in men and women

The estimator of the parameter a2 is a2 = 0.57 for women and a2 = 0.69 for men. So the noise is large.

The mean disablement processes are represented in figure 1 for women and men. The score is always higher in women, and the magnitude of the difference increases after age 75.

Figure 1

Many studies have found a similar difference between older women and men, women being generally more disabled than men of the same age [SKCC92]-[REG02]. In particular, Verret [VER99] showed the influence of sex on mild and moderate disability in the PAQUID cohort. Using a five state Markov model with piecewise constant transition intensities on the same data, Regnault [REG02] confirmed that women were at higher risk at the beginning of the disablement process. The model of degradation shows that women are more disabled than men, but also that the difference increases with aging. The degradation process is faster in women.

Figure 1

Many studies have found a similar difference between older women and men, women being generally more disabled than men of the same age [SKCC92]-[REG02]. In particular, Verret [VER99] showed the influence of sex on mild and moderate disability in the PAQUID cohort. Using a five state Markov model with piecewise constant transition intensities on the same data, Regnault [REG02] confirmed that women were at higher risk at the beginning of the disablement process. The model of degradation shows that women are more disabled than men, but also that the difference increases with aging. The degradation process is faster in women.

4.2 The estimated mean of the disablement process in demented and non-demented subjects

The estimator of the parameter a2 is a2 = 0.79 for demented subjects and a2 = 0.51 for the non demented. The noise is large.

The mean degradation processes in demented and non-demented subjects are represented in figure 2.

The difference between these two groups of subjects is large. Subjects who will be diagnosed as demented at follow-up were more disabled at any age, even at entry in the cohort at age 65 before the diagnosis of dementia was made. This study confirms the strong impact of dementia on the disablement process. Future demented subjects were more disabled even before the clinical diagnosis of dementia was made and they had a higher speed of degradation. The model with time dependent covariates developed by Bag-donavicius and Nikulin [BN04] could be used to take the pre-clinical phase into account. Dementia is major cause of disablement in older persons [REG02], [DB-GG91]-[DGM91]. So the individual degradation curve of a non-demented individual is an important predictive factor of dementia in the future.

The five state Markov model with piecewise constant transition intensities used by Regnault [REG02] on the same data showed similar results : dementia was associated with progression from mild to moderate disability and then to severe disability.

Figure 2

Figure 2

4.3 The estimated mean of the disablement process in demented and non-demented men

The mean degradation processes in demented and non-demented men are represented in figure 3.

The difference between these two groups of men is large. Men who will be diagnosed as demented at follow-up were more disabled at any age, even at entry in the cohort at age 65 before the diagnosis of dementia was made. Future demented subjects were more disabled even before the clinical diagnosis of dementia was made and they had a higher speed of degradation.

Figure 3

Figure 3

4.4 The estimated mean of the disablement process in demented and non-demented women

The mean degradation processes in demented and non-demented men are represented in figure 4.

As in the case of men, the difference between demented and non-demented women is large. Future demented women were more disabled even before the clinical diagnosis of dementia was made and they had a higher speed of degradation.

Figure 4

Figure 4

4.5 The estimated mean of the disablement process in demented men and women

The mean degradation processes in demented men and women are represented in figure 5.

For the men and women who will be diagnosed as demented at follow-up the degradation process develops similarly. So the difference between the degradation of older women and men is observed only for non-demented individuals.

Figure 5

Figure 5

4.6 The estimated mean of the disablement process in non-demented men and women

The mean degradation processes in non-demented men and women are represented in figure 6.

For the men and women who will be diagnosed as demented at follow-up the degradation process develops differently. The disablement process develops quicker in women then in men.

Figure 6

4.7 The estimated mean of the disablement process in high and low educated subjects

The mean degradation processes in high (primary education present )and low educated (primary education absent ) subjects are represented in figure 7.

The disablement process develops quicker in low educated subjects. More about the influence of the level of education on the aging-degradation process one can see in Barberger-Gateau et al [B-GVP01], Dartigues et al [DB-GG91], etc.

Figure 7

5 Joint model for degradation-failure time data

Joint degradation and failure time data may be analized using the following model. We call a failure of an individual natural if the degradation process attains a critical level zq. Denote by T0 the moment of non-traumatic failure, i.e. the moment when the degradation attains some critical value zo:

We denote T the time to death. In this case the observed failure moment is

We shall consider the model when the hazard rate depends on degradation.

Following [BN04] let us consider the joint degradation model according to which the conditional survival of T given the degradation process has the form:

ST (t | A) = P {T > t | Z (s)), 0 < s < t} = exp j -J A0(s, a)A(g(s, A))d^, where A is the unknown intensity function, Ao(s,a) being from a parametric family of hazard functions.

Note that the function A is defined on the set of degradation values, not on the time scale.

The model states [BN04] that the conditional hazard rate AT (t | A) at the moment t given the degradation g(s, A), 0 < s < t, has the form

The term A(g(t,A)) shows the influence of degradation on the hazard rate, the term Ao(t, a) shows the-influence of time on the hazard rate not explained by degradation. If, for example,

Ao(t,a) = (1+ t)a, eat, then a = 0 corresponds to the case when the hazard rate at any moment t is a function of the degradation level at this moment.

Wulfsohn and Tsiatis [WT97] considered the so called joint model for survival and longitudinal data measured with error, given by

with bivariate normal distribution of of (Ai, A2). The difference: in our model the function A, characterizing the influence of degradation on the hazard rate, is non-parametric, in the Wulfsohn-Tsiatis model this function is parametric. On the other hand, the baseline hazard rate Ao (it is proportional to the hazard rate which should be observed if the degradation would be absent) is parametric in our model and non-parametric in Wulfsohn-Tsiatis model.

The analysis of the PAQUID data using the joint model see in Zdorova-Cheminade [Z-C03].

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