Analysing Data of Childhood Acute Lymphoid Leukaemia by
Seasonal Time Series Methods
(University of Debrecen, Hungary
Abstract: We examined the periodicity of the childhood leukaemia
in Hungary using seasonal decomposition time series. Between 1988 and 2000
the number of annually diagnosed leukaemia (incidence) was analysed. The
time series of the number of patients between the age of 0 and 18 years
and the data series divided at the median were analysed. From the time
series the seasonal components, trends, the seasonally adjusted series,
the moving averages and the data series of the random components were examined.
The seasonal components of the dates of diagnosis revealed that a higher
percent of the peaks fell within the winter months than in the other seasons.
This proves the seasonal occurrence of the childhood leukaemia in Hungary.
These data seem to highlight the role of the environmental effects (viral
infections, epidemics, etc.) on the onset of the disease.
Keywords: time series analysis, seasonal decomposition time series
method, acute childhood lymphoid leukaemia
The method of seasonal time series analysis can be used in various fields
of the medicine. With such time series one can detect the periodic trend
of the occurrence of a certain disease [Barnett-Dobson,
2004], [Cani et al., 2001], [Fleming
et al. 2000], [Gallerani et al., 2004], [Oberg
et al., 2000], [Pereira, 2004], [Saynajakangas
et al., 2001]. Among other diseases, the seasonal periodicity of the
childhood lymphoid leukaemia was also analysed using statistical methods
[Cohen, 1987], [Harris-Harrel, 1987],
[Meltzer et al., 1996], [Sorenson-Pedersen,
2001], [Vienna-Polan, 1976]. The importance of
this theme was indicated that in the last few years in many international
publications were discussed with the seasonal variations of childhood leukemia/lymphoma
and the acute lymphoblastic leukemia [Karimi-Yarmohammadi,
2005], [Gao et al., 2005].
The pathogenesis of the childhood lymphoid leukaemia is still uncertain,
but certain environmental effects may provoke the manifestation of latent
genes during viral infections, epidemics or pregnancy.
The date of the diagnosis of patients were statistically analysed to
determine the role, which the accumulating viral infections and other environmental
effects may play during the conception and fetal period on the manifestation
of the disease. Because the available data is rather limited and controversial,
it seemed logical to make an in-depth analysis of the date of diagnosis
of the acute lymphoid leukaemia in Hungarian children.
The databank of the Hungarian Paediatric Oncology Workgroup contains
the data of all the patients with lymphoid leukaemia diagnosed between
1988 and 2000. In this time interval a total of 814 children were registered
(of which 467 were boys). The patients were 0-18 years old, with a mean
age of 6,4 years and a median of 5,4 years. The components of the time
series can be identified and isolated using mentioned statistical program
The periodicity of acute childhood lymphoid leukaemia in Hungary was
examined using seasonal decomposition time series method.
2.1 Seasonal Time Series
Time series consist of three components: the trend, the periodicity
and the random effects. The trend is a long-term movement representing
the main direction of changes. The periodicity marks cyclic fluctuations
within the time series. The irregularity of the peaks and drops form a
more-or-less constant pattern around the trend line. Due this stability
the length and the amplitude of the seasonal changes is constant or changes
very slowly. If the periodic fluctuation pattern is stable, it is called
a constant periodic fluctuation. When the pattern changes slowly and regularly
over the time, we speak of a changing periodicity. The third component
of the time series is the random error causing irregular, unpredictable,
non-systematic fluctuations in the data independent from the trend line.
An important part of the time series analysis is the identification
and isolation of the time series components. One might ask how these components
come together and how can we define the connection between the time series
and its components with a mathematical formula. The relationship between
the components of a time series can be described either with an additive
or a multiplicative model.
Let yi,j (i=1, ... , n; j=1, ... , m) marks the observed
value of the time series. The index i stand for the time interval (i.e.
a year), the j stands for a particular period in the time interval
(i.e. a month of the year). By breaking down the time series based on the
time intervals and the periods we get a matrix-like table. In the rows
of the matrix are the values from the various periods of the same time
interval; while in the columns are the values from the same periods over
various time intervals.
y1,1; y1,2; ... ; y1,m;
y2,1; y2,2; ... ; y2,m;
y3;1; y3,2; ... ; y3,m;
yn,1; yn,2; ... ; yn,m .
Let di,j (i=1,2, ... , n; j=1,2, ... , m) mark the trend
of the time series, si,j (i=1,2, ... , n; j=1,2, ... , m),
the periodic fluctuation and εi,j (i=1,2, ... , n;
j=1,2, ... , m), the random error. Using these denotations the
additive seasonal model can be defined as
(i=1,2, ... , n; j=1,2, ... , m); the multiplicative model as
(i=1,2, ... , n; j=1,2, ... , m).
The trend of a time series can easily be computed with moving averages
or analytic trend calculation. Moving averaging generates the trend as
the dynamic average of the time series. Analytic trend calculation approximates
the long-term movement in the time series with a simple curve (linear,
parabolic or exponential curve) and estimates its parameters.
The indices of the periodic fluctuation are called seasonal differences
(in the additive model) or seasonal ratios (in the multiplicative model).
These indices represent the absolute difference from the average of the
time interval using the additive model or the percentile difference using
the multiplicative model. Seasonal adjustment is done by subtracting the
j seasonal difference from the j data value of each i season (additive
model) or by dividing the j data value of each i season by the j seasonal
ratio (multiplicative model). The seasonally adjusted data reflect only
the effect of the trend and the random error.
The components of the time series can be identified and isolated using
statistical program packages. The analysis of the seasonal periodicity
of the childhood leukaemia was done with the Seasonal Decomposition module
of the SPSS/PC+ statistical program package.
3.1 Analysing the Periodicity of Acute Childhood Lymphoid Leukaemia
The analysis of the periodicity of acute childhood lymphoid leukaemia
was performed on the basis of the date of the diagnosis (year and month)
of the disease (monthly incidence). We analysed three data series. The
first data series contained the number of all the patients diagnosed monthly,
the second contained the number of those patients younger than the value
of the median, the third contained the number those older than the value
of the median.
The seasonal components of all patients revealed 9 peaks. 6 of these
peaks fell within the winter months, 1 in the autumn period, 1 in the summer
months, 1 in the spring months (Figure 1.). The seasonal
components of the younger age group showed 7 peaks in the winter, 1 in
the spring and 1 in the summer months. The seasonal components of the older
age group showed 7 peaks in the winter, 1 in the spring, 1 in the autumn
and 4 in the summer months.
Figure 1: Seasonal components of all patients of acute lymphoid leukaemia
diagnosed monthly in the observed period. Peaks of seasonal components
show the seasonal occurrence of the childhood leukaemia mainly in winter
Figure 2 shows the number of all patients (incidence)
and the seasonally adjusted series.
Figure 2: The number of patients (incidence) and the seasonally
Figure 3 shows the moving average (window width
=12 months) of the number have diagnosed cases of childhood leukaemia for
three age groups.
Figure 3: Moving averages calculated for three groups (younger than
the value of the median, older than the value of the median, and all patients)
of patients with childhood leukaemia
Analysis of the seasonality of childhood lymphoid leukaemia in Hungary
was performed both on the total number of patients and on the data series
divided at the median. A certain periodicity was found in the dates of
the diagnosis in patients with leukaemia. Although there was some difference
in the patterns of the seasonal components peaks of the three time series,
the majority of the peaks fell within the winter months in all three time
series. This was more significant in the group of all the patients and
in the younger age group. The results of the analyses proved the seasonal
occurrence of the childhood lymphoid leukaemia in Hungary. These results
refer to that in winter months certain environmental effects, suspected
viral infections, may provoke the manifestation of disease.
Some studies reported similar seasonality [Vienna-Polan,
1976], while other studies denied any kind such periodicity [Sorenson-Pedersen,
2001]. Similarly our data in occurrence of childhood leukaemia (incidence)
an autumn-winter peak were established by Karimi and Yarmohammadi [Karimi-Yarmohammadi,
2003], but they underlined the problem is complex and need more definitive
studies. On the other hand Gao and collaborators were examined the seasonality
of acute lymphoblastic leukaemia in different geographic locations. No
seasonal pattern was found in Singapore, in Hawaii and in USA. In contrast,
a significant peak (early January) was observed for western Sweden. On
the basis of examinations they were concluded there is little evidence
of any seasonality in the diagnosis of acute lymploblastic leukaemia [Gao
et al., 2005].
Due to the controversial nature of the available international data,
further studies should be carried out.
Statistical data are courtesy of Prof. Pál Kajtár, head
of the Hungarian Paediatric Oncology Workgroup.
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