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AGING PERSPECTIVES IN SOME NONDEGRADATION
STOCHASTIC PROCESSES
Sen, Pranab K.
University of North Carolina
https://doi.org/10.5109/12586
出版情報：Bulletin of informatics and cybernetics. 37, pp.1-11, 2005-12. 統計科学研究会
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権利関係：
AGING PERSPECTIVES IN SOME NONDEGRADATION
STOCHASTIC PROCESSES
by
Pranab K.
Sen
Reprinted from the Bulletin of Informatics and Cybernetics
Research Association of Statistical Sciences,Vol.37
- r¾ ¾- r¾-
FUKUOKA, JAPAN
2005
Bulletin of Informatics and Cybernetics, Vol. 37, 2005
AGING PERSPECTIVES IN SOME
NONDEGRADATION STOCHASTIC PROCESSES
By
Pranab K.
Sen
∗
Abstract
In a stochastic environment, a degradation process, inspite of showing a mono-
tone trend, may contain stochastic variations which may camouflage the statistical
picture to a certain extent. There are, however, some other processes which may
not exhibit a degradation phenomenon. For some of these nondegradation stochas-
tic processes, associated aging perspectives are appraised, without being confined
to a semiparametric fashion, and their application in health related quality of life
assessment are considered.
Key Words and Phrases: DMRL, degradation and failure, HRQoL, mean residual life, recurrent
events, semiparametrics, stopped counting process.
1. Introduction
An observed degradation process Y = {Y
t
, t ∈ R
+
} is a nonnegative stochas-
tic process characterized by a nonincreasing trend of the associated intensity process.
Bagdonaviˇcius and Nikulin (2002) have considered some parametric as well as semipara-
metric degradation models for aging. Typically, it is assumed that
Y
t
= g(t, Z)U
t
, t ∈ Z
+
, (1)
where Z = (Z
1
, . . . , Z
r
)
0
is a (possibly) stochastic vector, representing auxiliary (ex-
planatory) or concomitant variables, g(t, .), in a parametric setup, is a given monotone
decreasing and continuously differentiable function of t, and
U
t
= exp{σt
−1/2
W (t)}, t ∈ R
+
, (2)
where W = {W (t), t ∈ R
+
} assumed to be a standard Wiener process (independent of
Z), and σ a positive (unknown) scale parameter. It is also desired, in some cases, to allow
the covariates to be possibly time-dependent, so that in (1.1), Z needs to be replaced by
an auxiliary stochastic process Z
t
, t ≥ 0. In a semiparametric formulation, less stringent
assumptions have been made on the form of g(.), mostly along the lines of the classical
Cox (1972) proportional hazard model (PHM). If g(., .) is not monotone, the process Y
is not characterized as a degradation process; in real applications, a degradation model
may only be adopted under such a characteristic feature. If Y
t
is observable, under the
above setup, we have
log Y
t
= log g(t, Z) + σt
−1/2
W (t), (3)
∗
University of North Carolina, Chapel Hill, NC 27599-7420 pksen@bios.unc.edu
2 P. Sen
so that conventional stochastic partial differential equations (SPDE) can be incorpo-
rated to prescribe statistical resolutions. However, in a class of problems arising in the
context of health related quality of life (HRQoL) assessments, the survival time is not
the variable Y
t
in (1.1), and a somewhat different approach than the SPDE is needed.
The degradation phenomenan also needs to be appraised in those contexts.
There are some notable instances in survival analysis, as well as reliability theory,
where we may have a stochastic process of the type (1.1) though the degradation phe-
nomenon may not b e apparent or even tenable. In the next section, we motivate such
a nondegradation process with a noteworthy hereditary disease, Thalassemia minor (or
Cooley’s anemia; also known as Mediterranean anemia), and appraise the plausibility
of a stochastic degradation model, paying due emphasis on HRQoL perspectives. In
such a case, the survival time and the QoL state during the survival both need to taken
into account in formulating the stochastic flow of the events of interest. Motivated by
this example, in Section 3, we proceed to formulate a class of nondegradation processes,
and in Section 4, we appraise their aging perspectives with an eye on the quality of life
adjusted (QAL) mean residual life (MRL) analysis. Some of these statistical analyses
are presented in Section 5. Some general statistical results and broad summaritative
remarks are made in the concluding section.
2. Thalassemia Minor
Anemia is a condition in which the number of red blood cells p er cu mm, the
amount of hemoglobin in 100ml of blood, and the volume of packed red blood cells per
100 ml of blood are less than normal. Clinically, anemia generally pertains to the oxygen-
transporting material in a designated volume of blood, in contrast to total quantities
as in oligocythemia, oligochromemia and oligemia. Anemia is frequently manifested by
paller skin and mucous membrane, shortness of breath, palpitation of the heart, soft
systolic murmers, lethargy and fatigability. Among the varieties of anemia, we may
mention (i) hypochromic anemia and (ii) thalassemia, both being marked by deficient
hemoglobin and usually microcytic blood cells; microcyte relates to small red blood cell
present especially in some anemia. Splenomegaly, i.e., the enlargement of the spleen, is
also observed in some case.
Thalassemia or Thalassanemia: a group of inherited disorders of hemoglogin meta-
bolism in which there is a decrease in net synthesis of a particular globin chain without
change in the structure of that chain; several genetic types exist, and the corresponding
clinical picture may vary from barely detectable hematologic abnormality to severe and
fatal anemia. The Lepore thalassemia syndrome is due to production of abnormally
structured Lepore (a group of abnormal) hemoglobin which are clinically indistinguish-
able, but the non α-globin chains are structurally altered. β- Thalassemia relates to
heterozygous state. α-Thalassemia is due to one of two or more genes that depress
(partially to completely) synthesis of β-globin chains by the chromosome bearing the
abnormal gene. In a homozygous state, one may have a severe type with erythrob-
lastosis fetails and fatal death, only Hb Barts and Hb H present; a mild-type is not
clinically defined. In a heterozygous state, severe type, Thalassemia minor with 5 - 15
per cent of Hb Barts at birth and only traces of Hb Barts in adult; in mild-type, 1-2
per cent Hb Barts at birth, not detectable in adults. Thalassemia minor is thus the
heterozygous state of a Thalassemia gene or a hemoglobin Lepore gene, usually asymp-
tomatic, and mild hypochromic microcytosis; often slightly reduced hemoglobin level with
Aging Perspectives in Some Nondegradation Stochastic Processes 3
slightly increased erythrocyte count. Types of hemoglobin are variable and depend on the
gene involved. There may be a production of about 10 per cent of the Hb Lepore, Hb F
moderately increased, and Hb A
2
normal.
It is clear from the above description that the type of the disorder and degree of
severity can vary considerably, and as a result, the clinical picture may vary considerably;
we therefore need to focus on a specific case. In this study, we specifically keep the
Talassemia minor disorder (TMD) in mind, and proceed to assess its impact on HRQoL
as well as longivity (MRL), following a clinical detection of TMD. A particular measure
of the hemoglobin level is the primary variable, denoted by Y
t
, while the other recordable
characteristics are to be treated as covariates. Familial factors as well as other clinical
observations are also included in the set of covariates and explanatory variables, which
is denoted by Z
t
, t ∈ R
+
. Let us focus primarily on Y
t
, t ≥ 0 and note that there is
generally a normal hemoglobin level, N which, for people not afflicted with the disorder,
is a central value of the distribution, and for a person in the TMD group, Y
t
consistently
lies somewhat below this level. N is also subject to small interpersonal variation even
among the TMD-free people, and so also Y
t
among the TMD-identified people. There
is also a threshold level, denoted by L, such that as soon as Y
t
goes below L, there
is some clinical symptom which calls for a medical attention. Generally, following an
effective, brief treatment, Y
t
jumps to its preepisode level, and fluctuates around it until
the next episode when it dips again below L. This process continues and apart from
these possible episodes, the survival picture is not that much affected, alb eit the anemic
condition may be reflected in some living characteristics, in the manner described before.
There is also a level C, the comatic state level, and a lower level D, the death state
level. If Y
t
plunges below C, it needs a serious medical attention and a comatic state
may evolve. Further, such a treatment may not be very effective in the longrun, and as
Y
t
approaches the level D, the individuals survival is at stake. In this way, the survival
time X is defined to be the time until a person with TMD enters the absorbing state
D, and the number of episodes occurring prior to entering the absorption state, denoted
by M, though stochastic in nature, may cast valuable information on the severity of
the TMD. Further, in this setup, generally one does not bother to record Y
t
as long
as Y
t
> L, so that essentially, the observable random element are the epoch times
τ
j
, j ≥ 0 along with some little observations on the Y
t
in the clinical stage when it
dips below L. Also, the episodes are generally associated with high fever or some other
disease factors, and hence, that information being generally available, is an important
explanatory variable. From HRQoL perspectives, the events of interest are the inter-
episode times T
j
= τ
j
− τ
j−1
, j ≥ 1, along with the clinical information for the sub-
threshold state and X itself. Simply the survival time itself may not capture the whole
picture. Figure 1 pertains to this phenomenon.
3. Aging Perspectives
In characterizing aging aspects of a life distribution, generally, the hazard rate is
more commonly used instead of the survival function itself. The celebrated Cox (1972)
model allows incorporation of explanatory and concomitant variables in a semiparamet-
ric way, though there is a need to check the validity of such a model in a specific case, as
may be the present one. We denote the survival time by X, and all other explanatory
