Variation in the Seasonal Diagnosis of Acute Lymphoblastic Leukemia: Evidence from Singapore, the United States, and Sweden

Fei Gao1, Per Nordin2, Ingela Krantz2, Kee-Seng Chia3 and David Machin1,4

1 Division of Clinical Trials and Epidemiological Sciences, National Cancer Centre, Singapore
2 Skaraborgsinstitutet, Skövde, Sweden
3 Centre for Molecular Epidemiology, National University of Singapore, Singapore
4 Medical Statistics Unit, School of Health and Related Research, University of Sheffield, Sheffield, United Kingdom

Correspondence to Fei Gao, Division of Clinical Trials and Epidemiological Sciences, National Cancer Centre, 11 Hospital Drive, Singapore 169610 (e-mail: ctegfe{at}nccs.com.sg).

Received for publication December 29, 2004. Accepted for publication May 13, 2005.


    ABSTRACT
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 References
 
This study investigated, by summing data over successive years, the evidence for the seasonal diagnosis of acute lymphoblastic leukemia. To do so, the authors estimated the dates of peak diagnosis over a range of geographic locations including Singapore (1968–1999), Hawaii and mainland United States (1973–1999), and western Sweden (1977–1994) at latitudes of 1.16°N to 58.24°N. In contrast to other studies, the authors used case-by-case information on dates, gender, and age rather than grouped data for analysis. The seasonal pattern was estimated by fitting a von Mises distribution to the data from each location. No seasonal pattern was found in Singapore, which is close to the equator and does not have marked climatic changes. Likewise, seasonality was not demonstrated in Hawaii or mainland United States despite a 26.18° range of latitudes. In contrast, a significant peak (early January) was observed for western Sweden that appeared strongest for males (December 22, 95% confidence interval: November 16, January 16) and those less than age 20 years (January 14, 95% confidence interval: December 8, March 27). Thus, despite a wide geographic range of localities, there is little evidence of any seasonality in the diagnosis of acute lymphoblastic leukemia in most populations studied and no strong evidence of any influence of climate (as expressed by latitude).

leukemia, lymphoblastic, acute; seasons


Abbreviations: ALL, acute lymphoblastic leukemia; SEER, Surveillance, Epidemiology, and End Results


    INTRODUCTION
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 References
 
Many infectious and some chronic diseases have characteristic seasonal onsets (1Go), and, in certain instances, these seasonal rhythms can be related to etiologic precipitating factors (2Go). Thus, it has been speculated that acute lymphoblastic leukemia (ALL) in children is virus related (3Go) and may be the consequence of an abnormal response to common infections occurring in early life (4Go). If infections are involved, then some seasonal pattern of onset might be expected to be associated with them (5Go). In general, demonstration of seasonal variation in the onset of leukemia may provide insight into potential risk factors (6Go).

The first published studies on the seasonality of leukemia appear to be from Belgium (7Go, 8Go), which reported a November–February peak in acute leukemia. Since then, numerous reports (we identified more than 30 in a systematic literature search) from 15 countries have specifically investigated the seasonality of ALL.

While many of these studies have identified an obvious seasonal pattern, others have not found a significant seasonal variation. For example, for England and Wales combined, a summer peak was noted in two age groups (0–19 and 20–44 years) in studies from the early 1960s (9Go, 10Go), while one study some 30 years later showed no such pattern (11Go).

Inconsistency between results may itself be informative, perhaps reflecting various levels of between-population heterogeneity and different patterns of seasonality induced by possible causative agents. It is also possible that seasonality may be more pronounced within subtypes of leukemia or, for example, a particular locality. There was little evidence of seasonality in a national data set from Great Britain (England, Scotland, and Wales), but seasonality (summer: May–October) was evident in one regional data set from the West Midlands, England (12Go). The authors concluded that "further work on seasonality needs more sophisticated analysis, controlling for broad geographical heterogeneity" (12Go, p. 678). However, to our knowledge, no formal synthesis of published reports has been attempted to date. As a consequence, the etiology of leukemia in this respect, in both children and adults, still remains essentially undetermined.

One study, reporting some 20 years ago, examined in particular the influence of latitude on presentation of ALL for US subjects aged 0–19 years by using monthly data from the Surveillance, Epidemiology, and End Results (SEER) Program and from an independent survey of a 57-county study area in the eastern half of Nebraska (13Go). The authors examined the seasonal variation in ALL by using periodic regression of monthly rates; they noted up to three peaks (April, August, and December) in registries north of 40°N and the same, but in different months (February, July, and October), for the southern counterparts. They commented that "the observed peaks in monthly ALL risk coincide with seasonal elevations in the rates of allergenic and infectious diseases, elements of which are capable of promoting lymphocytic proliferation and transformation" (13Go, p. 915).

The SEER Program records, and makes available to researchers, individualized data on each case from 11 registries in the United States over a range of latitudes from 21.18°N to 41.36°N. We have the same detail for cases of ALL from part of western Sweden (latitude 58.24°N) and the whole of Singapore (latitude 1.16°N). In these two registries, the actual day is recorded and not just the month, as with SEER.

The purposes of our study were to search for a seasonal component in the presentation of ALL in the registry data from these three countries, to determine the influence of latitude (if any), and to investigate the role of subject age and gender.


    MATERIALS AND METHODS
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 References
 
For all registries, we restricted our analysis to cases of ALL (International Classification of Diseases, Eighth Revision, code 204.00 or International Classification of Diseases, Ninth Revision, code 204.0) (14Go, 15Go). For the US locations, the state capitals were used to define latitude and longitude.

Data
Singapore (latitude 1.16°N, longitude 103.51°E).
We used Singapore Cancer Registry data that included all cases of cancer occurring in citizens and permanent residents of the island (population 3.5 million) over the period 1968–1999. Disease classification follows the International Classification of Diseases, Ninth Revision. In addition, age, gender, ethnic group, date of birth, place of birth, and date of diagnosis are available on an individual case basis. A total of 941 ALL cases were registered for that time period. Two cases were excluded from our analysis of age (for one, only the year was recorded; for the other, the diagnosis was recorded a few days earlier than the birth).

United States (latitude 21.18°N–47.36°N, longitude 72.41°W–157.52°W).
The SEER Program of the National Cancer Institute records data from 11 cancer registries over a wide geographic area in the United States, and we used the data for the time period 1973–1999. The SEER locations included the island of Hawaii and the following mainland areas: metropolitan Atlanta, Georgia; Connecticut; metropolitan Detroit, Michigan; Iowa; New Mexico; the San Francisco-Oakland Standard Metropolitan Statistical Area, California; Seattle, Washington; Utah; and Los Angeles and San Jose-Monterey, California. Seattle contributed data for the years 1974–1999, metropolitan Atlanta for 1975–1999, and Los Angeles and San Jose-Monterey for 1992–1999 only.

SEER collects patient-specific information on tumor site, histology, gender, age, ethnicity, and date (month and year only) of diagnosis from all residents diagnosed with cancer in collaborating states or localities. Data for 9,158 ALL patients were collected, but 38 cases were excluded from our seasonality analysis because their month of diagnosis was not available.

Western Sweden (latitude 58.24°N, longitude 13.50°E).
Individual dates of each arrival at the hospital for any disease were available for part of the West Götaland Region in western Sweden from 1977 to 1994 (the population of the catchment area was 270,000 in 1995), and we used these data. Disease classification follows a Swedish version of the International Classification of Diseases, Eighth Revision before 1987 and a Swedish version of the International Classification of Diseases, Ninth Revision thereafter. For our purposes, the date of admission to the hospital was considered the date of diagnosis. In addition, age and gender are available on an individual-case basis.

A total of 81 cases were diagnosed with ALL, and we reviewed their diagnoses noted at each visit. As a consequence, two cases were excluded because, on most occasions, their diagnosis was recorded as lymphosarcoma or reticulosarcoma but only once as ALL. Corrections to the date of presentation regarding ALL were made for seven cases.

Statistical analysis
Data from each calendar year were first standardized to 365 days and were then converted to an angle between 0° and 360°. We illustrate these data graphically in a rose diagram format (figure 1), where each petal is of a "standard" month or 360/12 = 30°. The segments are ordered from January to December (clockwise) starting from due north. In these diagrams, the square root of monthly totals is used to preserve equal areas for each unit of frequency, as in a conventional histogram.



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FIGURE 1. Rose diagram of the seasonal distribution of acute lymphoblastic leukemia cases in Singapore (1968–1999), Hawaii (1973–1999), mainland United States (1973–1999) (metropolitan Atlanta, Georgia; Los Angeles, California; New Mexico; San Jose-Monterey, California; the San Francisco-Oakland Standard Metropolitan Statistical Area, California; Utah; Iowa; Connecticut; metropolitan Detroit, Michigan; and Seattle, Washington), and western Sweden (1977–1994). Nov, November; Feb, February; Aug, August.

 
To examine the data for the presence of distinct annual peaks, histograms were also constructed (figure 2) in which the summated year representing the 12 months was repeated (once) to emphasize the circular nature of the calendar. The associated von Mises distribution was added. This distribution assumes a single peak for seasonality within a year and is described by the parameters µ and {kappa}, from which the peak magnitude, R, can be derived. All peak dates, µ, are presented in the format (month, day) of a 365-day year. Although this format may suggest spurious precision, it is used for clarity and to facilitate comparisons. The inverse of {kappa}, which indicates the amount of variation about the peak date, is also displayed in all tables in this paper. Small values of R (small {kappa}) would usually indicate the absence of seasonality, whereas those close to the maximum of 1 (large {kappa}) would indicate a very sharp peak in (here) the number of cases. In this study, all {kappa} were small, so that R {approx} {kappa}/2; therefore, only {kappa} is shown in the tables. Departures from a uniform distribution of cases over the year were tested by the Mardia {chi}2 statistic with 2 degrees of freedom (16Go). Details of the von Mises distribution and the Mardia test statistic are given in the Appendix.



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FIGURE 2. Repeated histogram of corrected monthly numbers of acute lymphoblastic leukemia cases, with the corresponding fitted von Mises distributions, in Singapore (1968–1999), Hawaii (1973–1999), and western Sweden (1977–1994). µ, peak date; {kappa}, estimated concentration parameter of the von Mises distribution; J, January; F, February; M, March; A, April; M, May; J, June; J, July; A, August; S, September; O, October; N, November; D, December.

 
The bootstrap technique (17Go) was used to obtain a 95 percent confidence interval for the peak date by using 2,000 bootstrap samples of the same size as the number of patients under consideration. Because the bootstrap method of calculation for confidence intervals requires individual dates, and SEER does not provide precise dates of diagnosis, for calculation purposes (95 percent confidence interval only) we randomly assigned to each case a day from their corresponding month of diagnosis. In this "simulation," the monthly total, fg, for each year was assumed to be distributed uniformly over the bin size of 30°, and a random sample of size fg was drawn from that bin. All of the peaks and the corresponding confidence intervals are displayed in a "forest" plot. We comment on the format of a confidence interval in this context when describing figures 3–5.



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FIGURE 3. Peak dates (95% confidence intervals) of acute lymphoblastic leukemia in Singapore (1968–1999), Hawaii (1973–1999), mainland United States (1973–1999), and western Sweden (1977–1994), ordered by latitude (°N). Jan, January; Mar, March; Jul, July; Sep, September; Nov, November. •, peak date of diagnosis; –, peak date with {kappa} < 0.1; {Delta}, lower limit of the 95% confidence interval; {nabla}, upper limit of the 95% confidence interval. The size of "•" is proportional to {kappa}, except that those peak dates with {kappa} < 0.1 are labeled "–," and the length of the vertical lines, one for each location, represents the 95% confidence interval for the peak date.

 


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FIGURE 4. Estimated peak dates (95% confidence intervals) of presentation of acute lymphoblastic leukemia in children and adults in Singapore (1968–1999), Hawaii (1973–1999), mainland United States (1973–1999), and western Sweden (1977–1994), ordered by latitude (°N). Jan, January; Mar, March; Jul, July; Sep, September; Nov, November. •, peak date of diagnosis; –, peak date with {kappa} < 0.1; {Delta}, lower limit of the 95% confidence interval; {nabla}, upper limit of the 95% confidence interval. The size of "•" is proportional to {kappa}, except that those peak dates with {kappa} < 0.1 are labeled "–," and the length of the vertical lines, one for each location, represents the 95% confidence interval for the peak date.

 


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FIGURE 5. Gender-specific peak dates (95% confidence intervals) for acute lymphoblastic leukemia cases in Singapore (1968–1999), Hawaii (1973–1999), mainland United States (1973–1999), and western Sweden (1977–1994), ordered by latitude (°N). Jan, January; Mar, March; Jul, July; Sep, September; Nov, November. •, peak date of diagnosis; –, peak date with {kappa} < 0.1; {Delta}, lower limit of the 95% confidence interval; {nabla}, upper limit of the 95% confidence interval. The size of "•" is proportional to {kappa}, except that those peak dates with {kappa} < 0.1 are labeled "–," and the length of the vertical lines, one for each location, represents the 95% confidence interval for the peak date.

 

    RESULTS
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 References
 
Table 1 summarizes the geographic details of the four locations (Singapore, Hawaii, mainland United States, and Sweden). The latitudes range from 1.16°N (Singapore) to 58.24°N (Sweden) and the longitudes from 157.52°W (Hawaii) to 103.51°E (Singapore). On a per-population basis, the numbers of cases of ALL were very similar across regions, although the variation in the annual number was considerable, and there was a suggestion of rising numbers in Singapore (data not shown). The average age at presentation was similar among the four location groups, and more males than females (approximately 1.5:1) were diagnosed.


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TABLE 1. Latitude and longitude of the four locations and information on number, age, and gender of acute lymphoblastic leukemia cases obtained from cancer registries in Singapore (1968–1999), Hawaii (1973–1999), mainland United States (1973–1999), and western Sweden (1977–1994)

 
The numbers of cases presenting at each location on a cumulated monthly basis over the years for which data were used are given in figure 1. Although no single peak seemed to occur in Singapore or in either Hawaii or mainland United States, for western Sweden a peak was suggested over the interval between 300° and 0°–45° or, equivalently, mid-November to the end of February.

To some extent, the rose diagram format obscures the shape of the distribution over the year; therefore, we plotted data for Singapore, Hawaii, and Sweden again in a (double-year) histogram format (figure 2). Superimposed on these plots are the corresponding von Mises distributions. It is clear from this figure that the von Mises distribution reasonably describes the seasonal pattern for these cases and suggests that the strength of the peak is increasing from Singapore to Sweden. However, a more detailed examination of the values of {kappa} in all of the intermediate-latitude registries (table 2) was far from suggestive of a general pattern. Furthermore, there was no clear evidence of any systematic change in the estimated date of peak diagnosis over the changing latitudes.


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TABLE 2. Circular analysis of acute lymphoblastic leukemia cases from cancer registries in Singapore (1968–1999), Hawaii (1973–1999), mainland United States (1973–1999), and western Sweden (1977–1994), ordered by latitude

 
The estimated peak diagnosis is given for each location in table 2. For the total of ALL cases, the peak at August 14 in Singapore was not strong (R = 0.040, 95 percent confidence interval: May 1, October 29). In addition, the peak at May 15 in the United States was not strong (R = 0.031, 95 percent confidence interval: April 19, June 8). In contrast, a statistically significant peak was apparent in early January (p = 0.02), with a moderate amplitude of R = 0.223 in western Sweden. However, the sample (N = 79) was small and the 95 percent confidence interval (November 17, February 10) rather wide.

The 13 estimated peak dates ranged over 10 months of the year from January 3 in western Sweden to October 23 in San Jose-Monterey, southern California (figure 3). Ten of the 11 peak dates in the United States were located on the arc between Singapore and Sweden, suggesting a weak latitude effect. Only the date for San Jose-Monterey, California, was found within the opposite arc. However, a plot of the estimated peaks, with corresponding 95 percent confidence intervals, by increasing latitude showed no clear, systematic pattern. Furthermore, all but one of the corresponding estimates of {kappa} were less than 0.12 (small), implying little evidence of seasonality in most registries except for western Sweden ({kappa} = 0.458). The size of the symbol "•" representing peak date of diagnosis is proportional to {kappa}, except that those peak dates with {kappa} < 0.1 are labeled "–," and the length of the vertical lines, one for each location, represents the 95 percent confidence interval for the peak date. Because the confidence intervals may "wrap around" December 31 and January 1, different symbols are used to indicate the lower ("{Delta}") and upper ("{nabla}") confidence limits, which indicate the direction to go along the arc from that point to find the peak.

For the places from Singapore to San Francisco, at a latitude of <40°N—the cutpoint used previously for comparison purposes (13Go)—the estimated peaks ranged from April to October (table 2, figure 3). For the locations at a latitude of ≥40°N, from Utah to western Sweden, the estimated peaks ranged from January to February (winter) except in Seattle, where the latitude is 47.36°N and the estimated peak was July 19 (summer).

In the broad age categories of 0–19 and ≥20 years, there appeared to be no trend across the latitudes for children or adults (table 3, figure 4). Although the values of {kappa} tended to increase once the subjects, within each registry, were divided into these two age groups, suggesting a difference between the younger and older cases in the corresponding peak dates of diagnosis, no systematic pattern emerged of one age group experiencing a peak earlier in the year than another.


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TABLE 3. Circular analysis of acute lymphoblastic leukemia cases by age in Singapore (1968–1999), Hawaii (1973–1999), mainland United States (1973–1999), and western Sweden (1977–1994), ordered by latitude

 
A more detailed examination using six age groups (0–2, 3–9, 10–19, 20–49, 50–69, and ≥70 years; details not shown) failed to identify any subgroup for which there was a clear seasonal peak. In addition, we found no evidence of any systematic trend across the ages in date of peak diagnosis.

There were no systematic patterns within or between genders across the registries (table 4, figure 5). Furthermore, no association between peak date of diagnosis and longitude was observed.


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TABLE 4. Circular analysis of acute lymphoblastic leukemia cases by gender from Singapore (1968–1999), Hawaii (1973–1999), mainland United States (1973–1999), and western Sweden (1977–1994), ordered by latitude

 

    DISCUSSION
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 References
 
Many studies have investigated the seasonal patterns in the presentation of ALL, but (to our knowledge) none has used individual case-by-case methods in their approach to analysis, instead grouping the data into monthly counts. Such an approach will be less statistically sensitive and may therefore fail to identify patterns that may be present. Our approach with individual data enables the von Mises distribution to be used to describe the variation over the calendar year, and estimates of the corresponding parameters allow the peak date to be identified, the strength of the peak to be determined (over a range from 0 to 1), and confidence intervals to be calculated. If there is little or no evidence of a peak, one is nevertheless estimated; however, the corresponding confidence interval will cover the majority of the year, reflecting the uncertainty associated with such an estimate.

For this study, we were able to obtain individualized data from western Sweden, the Singapore Cancer Registry, and the SEER database of 11 registries in the United States for a total of 10,599 cases over the years 1968–1999.

Singapore (latitude 1.16°N) is very close to the equator; therefore, if there was indeed an influence of season on the date of peak diagnosis of ALL, it should be weak in this country because there are no marked seasonal weather patterns. In contrast, western Sweden (latitude 58.24°N) has seasons that differ markedly from winter, with few daylight hours and temperatures below 0°C, to summer, with almost continual daylight and temperatures ranging from 25°C to 30°C. Thus, one might expect to identify a seasonal pattern for ALL if one exists, and western Sweden was indeed the only location of the 13 studied in which a statistically significant peak (January 3) was identified. However, the sample size was small (79 cases in total), so there is a danger of a spurious finding especially because there are regions covered by the US registries where the climate (but not day length) is similar to that of western Sweden, but no peaks were established there. For the US registries as a whole, and despite a wide range of latitudes extending from Hawaii to Seattle (a difference of 26.18°), there is little evidence to suggest a climatic effect.

Apart from the Swedish data just noted, in which the peak was confined to young males, there was little other evidence to suggest that, within specific age groups, strong peaks are present or that there is any systematic trend across ages regarding diagnosis, even when age was categorized into groups ranging from 0–2 years to the elderly. There was also little evidence of any gender effect in any location or any suggestion of a consistent effect across locations.

Our failure to find any such influences may have resulted from not studying registries in more regions of the world, weaknesses in the analytical approach, and (nonrecorded) etiologic factors playing a major role and obscuring any latitude, longitude, gender, and age effects. Thus, we are aware that we were not able to perform an "individual-case," systematic overview of the ALL studies (we identified more than 30) that have been conducted. Also of concern is the loss of statistical sensitivity because the SEER data provide only the month and not the day of diagnosis. Were the precise days available, then angular regression models (18Go) could be used to investigate the associations rather than the somewhat descriptive approach that we had to use. Further work is also required to confirm (or otherwise) the utility of the von Mises distribution as an adequate description of seasonality; once more, only precise dates would allow a thorough investigation.

For some diseases, seasonal variation in known or unknown precipitating factors will depend on climate and a range of population characteristics (19Go), and these in turn will induce seasonal patterns in the disease itself. However, apart from gender and age, unavoidably all of the possible precipitating factors are expressed merely through the latitude (and longitudes) of the individual registry locations. These clusters will clearly obscure the influence of, for example, case-specific socioeconomic status and some environmental factors (20Go, 21Go) that have been linked to the risk of developing ALL.

Our findings are consistent with a report concluding that there was no seasonality in the United States across all ages (22Go) but not with a later reanalysis of these same data (13Go). This reanalysis identified up to three peaks in each of the nine locations studied for subjects aged 0–19 years. Our view is that this result more likely reflects the statistical methodology applied to data that are essentially random (but grouped into 12 bins), so that a trigonometric model with many parameters will mirror the vagaries of such a toothy distribution and not reflect the smoothed (and more likely) pattern of a uniform distribution over the year. We could not substantiate the claimed summer peak in the diagnosis of ALL in children (23Go).

In contrast, for western Sweden, the significant peak in winter (early January) is close to the peak of onset reported for Capetown, South Africa (winter: June–August) (24Go) and the peak of symptoms occurring in Shiraz, Iran (winter: October) (5Go). However, this information differs from the summer peaks reported for onset in England and Wales (9Go, 10Go) and diagnosis in England (6Go). These contrasting results for ALL may be due to the different statistical approaches that have been used, the different age groups chosen for analysis and reporting, or the different date of "onset" of ALL considered, although the delay between clinical symptoms in children and diagnosis is not likely to be great (4Go).

The seasonality observed in western Sweden (if it could be firmly established) may, in any event, be related more to precipitating than to etiologic factors. For example, eventual cases might first present as a result of lowered immunity to infections; these may be common in winter or summer, depending on their type and their ALL being detected as a result. The overall health care system in Sweden is ranked highly (25Go) and provides relatively open access to care. Consequently, the peak of moderate magnitude in January may reflect post-Christmas and New Year festivities delaying self-referral and not the presence of an etiologic determinant.

Despite investigation of seasonal patterns in the presentation of ALL, as indicated by the date of diagnosis, over a range of latitudes from 1.16°N to 58.24°N, there is no clear message with respect to their interpretation. Nevertheless, patterns may have emerged if the "date of first symptom" had been recorded and studied, because, for example, a significant seasonal variation for Hodgkin's disease has been reported in this date but not in the date of diagnosis (26Go). Thus, this and other studies (27Go, 28Go) suggest that date of first symptom versus date of diagnosis more closely reflects the event that precipitates the clinical onset of disease. However, the induction period of ALL is sufficiently short, so a similar seasonal pattern should be observed for date of first symptom and date of diagnosis (27Go). Although the interval is indeed short for the majority of pediatric ALLs, it has been claimed that ALL can be clinically silent for months or even years in some cases (4Go). Such late cases are likely to dilute seasonal patterns.

A similar lack of consistent findings has been reported for other forms of leukemia with respect to a peak in the date of diagnosis. For example, a significant summer peak was reported for ALL but not for acute myeloid leukemia in the United States (23Go), while a significant autumn peak (November) for ALL but (bimodal) peaks for acute myeloid leukemia in winter and spring were noted in Shiraz, Iran (5Go). In contrast, no clear evidence of seasonality for ALL, acute myeloid leukemia, and chronic myeloid leukemia was reported from England and Wales (11Go), but none of these studies have been analyzed on a case-by-case basis.

Given the small numbers of cases from western Sweden, no firm evidence from the United States, and the expected absence in Singapore, any suggestions for peak seasonality of diagnosis could all be ascribed to chance variations. Likewise, the corresponding degree of seasonality reported in other studies may be enhanced (or obscured) by local referral characteristics—even leading to a false indication of an underlying climatic component.


    APPENDIX
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 References
 
The von Mises Distribution
The von Mises distribution with a single peak at µ has the following probability density function:

where and {kappa} is positive.

Here, {theta} represents the angle, over a range of 0–360° (or 0–2{pi} radians), that can be equivalent to the date of diagnosis within a year. For example, if this date is February 28, 2004, then {theta} = 360 x (31 + 28)/366 = 58.03° or 1.0123 radians; if it is February 28, 2005, then {theta} = 360 x (31 + 28)/365 = 58.19° or 1.0151 radians. After converting each date to an angular day for the N subjects concerned, the peak µ is estimated as follows: First, calculate and note whether each is less than or greater than 0. Next, calculate µ0 = arctan (S/C), which gives a value in degrees (or radians) of between –90° and 90°. Finally, the estimated peak angle µ is equal to 1) µ0 itself, if S > 0 and C > 0; 2) µ0 + 180°, if C < 0 irrespective of the value of S; or 3) µ0 + 360°, if S < 0 and C > 0. Once µ is calculated in degrees, it can be converted to a date in a standard year of 365 days.

The second parameter of the von Mises distribution {kappa} is termed the concentration parameter and relates to the inverse of the variance of the distribution. The algebraic expression for the estimate of {kappa} from the data is complex. However, a very good approximation is given by the following:

where

R is the estimated magnitude of the peak at the date identified and takes values of 0–1. The larger the value of R, the stronger the peak identified at µ.

It can be shown that when {kappa} is large, suggesting a strong peak, the shape of the von Mises distribution is close to that of a Normal distribution, with the mean at µ and a standard deviation equal to In contrast, for small {kappa}, the von Mises distribution tends to the uniform distribution and is spread evenly over the whole 360°. This situation is clearly indicative of no peak in date of diagnosis being present.

A formal test of the null hypothesis, {kappa} = 0, is made by referring the value of M = 2NR2 to the chi-squared distribution with 2 degrees of freedom.


    ACKNOWLEDGMENTS
 
The authors gratefully acknowledge the Singapore Cancer Registry for allowing access to the data.

Conflict of interest: none declared.


    References
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 References
 

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