Nepal Journal of Science and Technology 6 (2005) 47-52
A Probability Model to Describe the Distribution
of Age at Menarche in Nepal
T.R.Aryal
Central Department of Stc~tistics
Tribhuvan University, Kirtipur; Karhrntlndu
E-mail: traryal@ rediffmail.com.np & dais @ wlink.com.np
Abstract
The main aim of this paper is to propose a probability model to describe the distnbution of females accordmg to age
at menarche. The suitability of the proposed model has been tested with the real data set. Expected waiting time at
blrth for attaining menarche has also been computed by using life table technique. The data have been taken from
a sample survey in Palpa and Rupandehi districts. The proposed logistic distribution was found well fit for describing
the dstribution of age at menarche in Nepal. It was found that the estimated mean age at menarche was 14.6years.
The expected waiting time at bxth for getting menarche was found to be 14.7years, whch is more or less consistent
result. The findmgs may give useful information for policy implications to design the policies and programmes
especially in fertility and reproductive health related issues of a country.
Key words: fertility, life-table, menarche
Introduction
Demographc models can be construct'ed either by
aefining the process in mathematical forms or by using
certain empirical observations in numerical forms.
Deterministic and stochastic models have been
developed and used since long back but its application
to the problems of demography is of the recent origin
(Aryal2003). A limited use of model for studying the
demographic behaviour might perhaps be due to lack
of controlled experiments (Aryal2002).Demographers
and other social scientists are now being engaged in
formulating models/ techniques in demography and
to provide concise description of extensive data sets.
Thereproductive lifespan of a woman b e p s with
the onset of menstruation i.e. menarche and is
continued till the cessation of menstruation i.e.
menopause. Average age at menarche vary
substantially between women across different countries
and across different ethnic groups (Aryal2004, Thomas
et al. 2001). Age at menarche plays an important role
in deciding the level of fertility and women health in a
society. Age at menarche has received relatively little
attention in the existing demographc literature, whch
may perhaps be due to the lack of reliable data as well
as the lack of interest among the ~esearchers.
A number of studies have reported that the average
age at menarche ranges from about 15 to 19 years
among rural, backward tribes and developing countries
whereas it ranges from about 12 to 14 years among
well-off urban and developed populations (Aryal2002,
Becker 1993, Eveleth & Tanner 1976). ~ g ate
menarche affects fertility by lengthening the span of
reproductivelife, adolescent sub-fecundty, and foetal
wastage (Aryal2005, kley et al. 1993). Women with
early menarche have had shorter birth interval and
hgher lifetime fertility (Aryal2004). After menarche,
there is wide variation in the age at entry into sexual
union, the use of contraception, abortion, etc. (Davis
& Blake 1956).
No single attempt has been made to describe the
dstribution of age at menarche through mathematical
as well as probability model in Nepal or elsewhere,
whch may be due to the lack of reliable data or lack
of the interest among the demographers. The aim of
th~spaper is to propose and/or graduate the probability
model to describe the dutribution of age at menarche
and hence obtain an average age at menarche. An
attempt has also been made to compute waiting time
at birth for attaining menarche through life table
technique.
Methodology
The data for h s study have been taken from a sample
survey entitled "Demographc Survey on Fertility and
Mobility (DSFM) in Rural Nepal: A study of the Palpa
and Rupandehi Distncts" conducted between January
and June 2000. The details of the survey are found in
Aryal(2002).A total of 811households were surveyed.
In the survey a total of 1566 fenmles' ages of 12-55
T.R. AryaUNepal Journal of Science and Technology 6 (2005) 47-52
years were obtained. A sample of 1019 ever-married
women under aged 55 years was interviewed. Besides,
information on socio-economic, demographic and
cultural variables, the data on age at menarche have
also been collected fromever-married females and their
daughters. Each and every eligible woman was asked
whether she had a menarche. If the answer was yes,
the woman was asked about her age at menarche. A
total of 1258 females (includmg 239 daughters) had
attained rnenarche at the survey date. There were 308
girls who have not attained menarche at the time of
interview.Age at menarche was measured in completed
years at the survey date and is grouped into three birth
cohorts of females viz. early blrth cohort (below 1970),
intermdate birth cohort (1970-79) and the recent birth
cohort (1980 and above). The NFHS survey data of
Nepal have the limitation that there is no any
information about the age at menarche.
Model
Usually the onset of menarche occurs at about the age
of 12 years and almost all females attain it up to the
age of about 20 years. The onset of menarche rarely
occurs outside h s interval. Onset of menarche first
increases with age, attains maximum in age range of
14-15 years, and thereafter declines abruptly. Talung
this cpncept into account and other empirical findmgs,
a probability distribution to describe the distribution
of females according to their age at menarche is
proposed as under:
Let X denote the occurrence of menarcheal age
of females. We assume that X follows a logistic
distribution. The probability density functionof logistic
distribution is given by:
a and flare the location and scale parameters of the
distribution.
The corresponding dstribution function is given as:
F(x) =
1
(x-a))]
[I + exp {- ---
... (2)
B
Rearranging expression (2) and taking logarithms it
beconles a linear t'onn ~LS:
Thus the parameters a and b can easily be
estirimtedby the usual method of least squares principle
from expression (3).
Applications and Discussion
The proposed model is applied to the data of rural
Nepal @SMF 2000). To examine the incidence of age
at menarche of females three birth cohorts viz: early
birth cohort (below 1970), intermediate birth cohort
(1970-79) and the recent birth cohort (1980 and above)
are taken. The expected proportion of females who got
menarche between the age (x, x+h) (say) is given by
1
1 + exp {-
. .. (5)
x-a
X
Further, the expected proportion Pe(x) have also
been adjusted as
X
Tables 1 & 2 show the estimated values of the
parameters, and observed and expected number of
females according to the age at rnenarche for all the
data sets. Values of chi-squares were found
insignificant, which indicate the validity of the
proposed model as a reasonable approximation to
describe the age pattern of menarche of females in
Nepal. Figures 1 to 4 also indicate the closeness
between the observed and expected distribution of
females according to their age at menarche. The
estimated values of a (location parameter) and b (scale
parameter) were found respectively 14.10327
and 1.00534 among females of early birth
cohorts, 13.83420 and 0.92056 among intermediate
birth cohorts and 13.33404 and 0.8 1250 among the
recent birth cohorts. Mean and variance of the
distribution are calculated as E(x)=a and V(x)=-
827l-2
3
respectively (Johnson & Kotz 1970) and the values
are presented in Tables 1 & 2.
The estimated mean age at menarche through
model was found to be 14.1, 13.8 and 13.4years among
an early, intermediate and recent birth cohorts
respectively. Age at menarche was found slightly
hlgher for older birth cohort (14.1 years) than the recent
birth cohort (13.4 years). Some adjustment is to be
needed in the estimated mean age at menarche through
model because the age at menarche was recorded in
completed years at the survey date. So the adjustment
factor would be c and it is reasonably equal to 0.5 years.
Hence the adjusted mean age at menarche would be
T.R. AryaVNepal Journal of Science and Technology 6 (2005) 47-52
Table 1. Observed and expected number of females according to their age at menarche
Birth cohort of females
Age at
menarche
X
Early (below 1970)
Observed.
Expected.
29.81
34
63.25
78
110
100.89
105.67
94
60
71.48
35.28
29
15
14.72
12
13
14
15
16
17
18
19
20+
13.47
Total
Paramem!:
/3
X
Intermediate (1970-1979
Observed
Expected
39
34.10
90
75.34
122
115.29
92
105.23
49
59.56
22
25.01
429
2
d.f.
Mean
s.d.(s)
7.90
2
429.00
428
428.00
14.10327
1.0053
9.31
5
13.83420
0.92056
7.87
4
14.1
1.82
13.8
1.67
I
Table 2. Observed and expected number of females according to their age at menarche
Blrth cohort of females
Age at
menarche
X
12
13
14
15
16
17
18
19+
Total
Mean
s.d.(s)
Recent (1980 and above)
Exp.
Obs
45
46.09
117
100.27
125
125.33
69
81.35
31
32.99
Y}
14.97
2
401
401.00
13.3
1.47
Total
Exp.
Obs
118
285
357
255
140
61
27
9
6
.
1258
121.58
248.59
328.98
276.25
165.06
73.61
29.03
10.89
4.01
1258.00
13.7
1.79
T.R.AryaVNepal Journal of Science and Technology 6 (2005) 47-52
l4.6,14.3 and 13.9years among an early, intermdate
and good quality of nutrition.
and recent birth cohorts respectively.A low mean age
among the recent birth cohort
at
may perhaps be due to the effects of modernization
Menarche is a universal event and every fenlale
would eventually have experienced it. Under this
assumption, the expected waiting time for petting
42
13
14
15
16
17
18
19
20+
Age at menarche
.
Figure 1 Observed and expected distribution of age at
(birth cohort below 1970)
o- Obs
<
2I
13
14
15
16
17
menarche
- - - - EXP
18
19
I
20+
Age at menarche
Figure 2 Observed and expected distribution of age at menarche (birth
cohort 1970-1979)
J
T.R. AryalINepal Journal of Science and Technology 6 (2005) 37-52
menarche in a certain age or age group is obtained by
using life table technique. Thenumber of females who
have had menarche in a certain age or age group was
considered as the numher of death cases (Nombodiri
& Suchilidnin 1987).Accordingly, a life table has been
cons~ructedwith this information.
--
Obs
<I2
13
14
15
16
17
----~4
.
-
18
19
Age at menarche
Figure 3 Observed and expected of age at menarche (birth cohort
1980 and above)
Age at menarche
Figure 4 Observed and expected distribution of age at menarche (all
females)
20+
T.R. AryaVNepal Journal of Science and Technology 6 (2005) 47-52
Table 3. Expected waiting time for menarche according
to age
(in years)
Number of
Number of
females do not females have
have menarche menarche
0-12
12-14
14-16
16-18
18-20
21and+
na
168
95
36
9
0
Age group
0
419
601
196
42
2
Expected waiting
time (in yars)
14.66
2.65
1.63
1.18
1.13
0.50
Table 3 shows the expected waiting time for
getting menarche according to age of females. The
expected waiting time at brth was found to be 14.7
years, which is consistent with the median age at
menarche of 14.8 years (Aryal 2002). The expected
waiting time was found be 2.7, 1.6, 1.2 and 1.1 years
for females belonging to the age group of 12-14, 1416,16-18and 18-20respectively. The model provided
the consistent estimate of mean age at menarche 14.6
years (Aryal 2004). Hence both the techniques
proposed in this paper provided more or less consistent
estimate of average at menarche.
Conclusion
The proposed logstic distribution was found well fit
to the data of Nepal for describing the distribution of
age at menarche. Estimated mean age at menarche
through model was found to be 14.6, 14.3 and 13.9
years among an early, intermediate and recent birth
cohorts respectively. A low mean age at menarche
among the recent birth cohort females may perhaps be
due to the effects of modernization and good quality
of nutrition. The expected waiting time at birth was
found to be 14.7 years, whlch is a consistent result
with the meQan age at menarche (14.8 years). Hence
both the techniques provided more or less consistent
estimate of average at menarche in the study
population.
Author believes that the finding of h s paper
provides some interesting and valuable results, which
may inspire future research in the field of demography
especially, age at menarche, reproductive health and
hence fertility. Modelltechnique developed andlor
graduated in ths paper may be useful nlaterials for
researchers, planers and other social scientists working
in the related areas. The findings may also give
functional input for policy implications to design the
policies and programmes of a country.
Acknowledgements
Thanks go to an anonymous reviewer for useful
suggestions.
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