Suicide seasonality : Evidence of 11-year cyclic oscillations in Brazilian suicide rates

Several researchers have studied suicide seasonality through different statistical approaches. In the present research, we performed a detailed statistical study of Brazilian suicide data registered from 1980 to 2010 using a new approach known as Threshold Bias Model (TBM). Eleven-year cyclic oscillations were observed in suicide rates that, at first sight, appear as being negatively correlated with the cycles of solar activity. Such oscillations are more noticeable for males although they also have been observed in the female rates.


INTRODUCTION
Suicide seasonality is a multifactorial phenomenon that is often associated with the increased winter-summer variation in the number of sunlight hours that occurs in temperate and polar zones of the globe 1,2 .Consequently, its occurrence has been little studied in the tropics.
In Brazil, a previous study conducted between 1996 and 2004 in the city of São Paulo (located on the Tropic of Capricorn) did not revealed significant correlation between suicides and the number of sunlight hours 3 .However, beyond the hypothetical influence of the daily duration of sunlight on suicide rates, several authors have reported that long-term seasonal patterns in such rates may be correlated with solar and geomagnetic activities [4][5][6][7] .
Since seasonal oscillations in suicide rates may be small in comparison with data fluctuations due to death undercounting and/or cause of death misclassification, researchers are using several statistical methods in order to quantify this phenomenon 8,9 .Considering this, we performed a preliminary study of suicide seasonality in Brazil employing the recently proposed Threshold Bias Model (TBM) 10 -a model that provides useful comparison parameters to help us identify seasonality patterns.

METHOD
The present research considers male and female suicide rates structured by age that were recorded in Brazil from 1980 to 2010.Such data are openly available from the website of DATASUS 11 .In the same way, the average sunspot numbers considered here (relating to solar cycles 21, 22 and 23) can be freely obtained from the website of the Solar Influences Data Analysis Center (SIDC) 12 .
All analyses were made using the software Microcal Origin version 5.0 13 .In this software, suicide rates were edited in proper worksheets and plotted as functions of age.Then, we employed the Levenberg-Marquardt method to fit Equation 1 (TBM distribution) to these datasets considering β=11 as discussed in Folly 10 .The meanings of all TBM parameters were explained in this reference. (1) exp After all the distributions were fitted, we used the values of the parameters b, V 0 and C ap to calculate the middle-age suicide rates SR (τ 1/2 ) for each year from 1980 to 2010 using the expression SR (τ 1/2 ) = (b-V 0 ).C ap .Such distributions were numerically integrated (using Origin's integration tool) from the critical age t c to infinity in order to determine the areas under the curves.The values of t c can be calculated using Equation 2.
The areas under the TBM distributions and the calculated values of SR (τ 1/2 ) were plotted together with sunspot numbers in a same time scale to facilitate comparisons.In order to quantify the period and amplitude of seasonal oscillations of these data, they were fitted by the function f (T) defined by the Equation 3, in which T is the time in years.The year 1980 was considered as the zero on the time scale.
The Equation 3 defines f (T) as being the sum of a third order polynomial (necessary to fit the secular trend) and a sinusoidal term in which the parameters P 5 , P 6 and P 7 are the amplitude, period in years and initial phase respectively.

RESULTS
The TBM distributions were fitted to yearly suicide rates by age from 1980 to 2010 (Figure 1).The obtained parameter values with standard errors, chi-squares, degrees of freedom and p-values are shown in the Table 1.Despite the distributions fitted better the female data, even for males, the differences between fitting curves and statistical data were not significant (all p>0.05), with the worst fit observed for the year 1987 (p=0.975).
We organized the curves for the male datasets in a 3D graph in order to depict their annual evolution over the studied period (Figure 2).By performing numerical integrations, we determined the areas under the fitted distributions for males and females.Such areas are shown in the Figure 3a with the sunspot numbers for solar cycles 21, 22 and 23 on the same time scale.In the same way, the calculated middleage suicide rates SR (τ 1/2 ) are shown together with sunspot data in Figure 3B.
As previously explained, the values of SR (τ 1/2 ) and the areas under the TBM distributions were fitted by the    B).

DISCUSSION
Albeit some researchers have reported positive correlations between suicide rates and sunspot numbers 4 , our preliminary results are in agreement with the fi ndings of several other authors who have reported negative correlations between these phenomena [5][6][7] .
Since the middle-age suicide rates SR (τ 1/2 ) and the areas under the TBM distributions are relatively immune to data fl uctuations over the studied age groups, the results of present research can be considered as relatively robust.
Despite the observed oscillations in suicide rates be more prominent for males, they were also noticeable for females.For males, these oscillations are observed over all age groups, but become more intense with increasing ages (Figure 2).Such a progression with age was not clearly observed in the female rates.In Brazil, due to occupational factors, men are more likely to be exposed to sunlight than women 14 .Th is fact may hypothetically explain the greater amplitude of oscillations observed in the male suicide rates although this hypothesis has not been tested in the present study.Th e reader should also be aware that the present results (obtained from nationwide population data) might not be valid at regional level.Interpretations of such results as being valid for small cohorts can lead to ecological fallacy 15 .
Th e neurobiological mechanism linking suicide with solar activity remains unknown.Recent studies have positively correlated suicide occurrence with low levels of melatonin 16 in populations exposed to increased number of sunlight hours.As such correlation was not investigated in the present work, further researches are required in order to clarify this point.

Figure 1 .
Figure 1.Annual suicide rates by age and gender in Brazil from 1980 to 2010.The continuous curves shown in red are the Threshold Bias Model (TBM) distributions fitted to each statistical dataset

Figure 2 .
Figure 2. 3D representation of male Threshold Bias Model distributions plotted as functions of age and year.The continuous curve shown in red represents the annual evolution of middle-age death rates by suicide 1980 1985 1990 1995 2000 2005 2010 y e a r

Figure 3 .
Figure 3. Sunspot numbers during solar cycles 21, 22 and 23 plotted together with (A) the areas under Threshold Bias Model (TBM) distributions and (B) the middle-age death rates by suicide.The continuous curves shown in red and blue were obtained by fitting the function defined by the Equation 3 to these data for males and females respectively