Simple estimators of the intensity of seasonal occurrence
- M Alan Brookhart^{1}Email author and
- Kenneth J Rothman^{2}
https://doi.org/10.1186/1471-2288-8-67
© Brookhart and Rothman; licensee BioMed Central Ltd. 2008
Received: 21 December 2007
Accepted: 22 October 2008
Published: 22 October 2008
Abstract
Background
Edwards's method is a widely used approach for fitting a sine curve to a time-series of monthly frequencies. From this fitted curve, estimates of the seasonal intensity of occurrence (i.e., peak-to-low ratio of the fitted curve) can be generated.
Methods
We discuss various approaches to the estimation of seasonal intensity assuming Edwards's periodic model, including maximum likelihood estimation (MLE), least squares, weighted least squares, and a new closed-form estimator based on a second-order moment statistic and non-transformed data. Through an extensive Monte Carlo simulation study, we compare the finite sample performance characteristics of the estimators discussed in this paper. Finally, all estimators and confidence interval procedures discussed are compared in a re-analysis of data on the seasonality of monocytic leukemia.
Results
We find that Edwards's estimator is substantially biased, particularly for small numbers of events and very large or small amounts of seasonality. For the common setting of rare events and moderate seasonality, the new estimator proposed in this paper yields less finite sample bias and better mean squared error than either the MLE or weighted least squares. For large studies and strong seasonality, MLE or weighted least squares appears to be the optimal analytic method among those considered.
Conclusion
Edwards's estimator of the seasonal relative risk can exhibit substantial finite sample bias. The alternative estimators considered in this paper should be preferred.
Background
In a classic paper, Edwards [1] describes a geometrically motivated, moment-based method to fit a sine curve to a time series of square-root transformed monthly frequencies. From this basic framework, he derived both a test of the null hypothesis of no seasonality and an estimator of the intensity of seasonal occurrence (i.e., the peak-to-low ratio of the fitted sine curve). Owing to its intuitive appeal and computational simplicity, Edwards's and related methods have been widely used in epidemiology in studies of seasonality, e.g., [2–7].
Although there has been considerable discussion of the hypothesis testing procedure described by Edwards and a variety of alternative tests have been proposed [8–12], there has been relatively little discussion of the properties of Edwards's estimator of the intensity of seasonal occurrence. St. Leger discusses some computational difficulties involved with maximum likelihood estimation of the parameters in Edwards's model [13]. Nam compared the performance of the MLE with a moment-based "locally reasonable" estimator, similar to Edwards's estimator, and concluded that the MLE was preferable when the seasonal trend was strong [14].
In this paper, we review various approaches to the estimation of the intensity of seasonal occurrence, including Edwards's methods, least squares, weighted least squares, and the MLE. We then propose a new closed-form moment estimator of the peak-to-low ratio based on non-transformed data and a second-order moment statistic. Through an extensive Monte-Carlo simulation study, we compare the finite sample performance of the estimators discussed in this paper across a variety of data generating distributions, including some that involve overdispersion and autocorrelation of the outcome and thus depart from the assumed model. All estimators and confidence interval procedures discussed in this paper are applied in a reanalysis of data on the seasonal incidence of monocytic leukemia.
Methods
Data and Probability Model
Edwards's approach is used to study the seasonality of rare events that arise from an underlying non-homogeneous Poisson process with a rate given by the periodic function
λ(t) = μ{1 + αcos(2πt + θ)},
where μ is the total number of expected events in the year, t is the time in years, θ is the phase angle, and α is the hemi-amplitude of the periodic process.
We consider the situation in which the year is divided into k equally-sized intervals and aggregate data are available on the frequency of events occurring in each interval across T years. We denote the observed frequencies with N _{ i }, i = 1, ..., k.
where i is the interval (e.g., quarter, month, week), and φ + 0.5 is the time of peak incidence. The parameter n is the total expected number of events across all years, i.e., n = μT.
Edwards's Method
In the subsequent sections, we borrow this geometric framework todevelop alternative estimators of α and R.
Moment-based Estimation of a Using Non-transformed Data
Let D _{ y }= $\frac{1}{k}{\displaystyle {\sum}_{i=1}^{k}{N}_{i}}$ sin(θ _{ i }) be the vertical component and D _{ x }= $\frac{1}{k}{\displaystyle {\sum}_{i=1}^{k}{N}_{i}}$ cos(θ _{ i }) be the horizontal component of the distance from origin to the center of gravity of the k masses. Let N = ΣN _{ i }. From the exact expressions for E[D _{ x }|N] and E[D _{ y }|N] (see Additional file 1), a first-order approximation for E[D|N] is given by:
This estimator is the same as Nam's locally reasonable estimator [14]. It can also be derived from least-squares estimation of the parameters of the periodic model:
N _{ i }= β _{0} + β _{1} sin(θ _{ i }) + β _{2} cos(θ _{ i }) + ε _{ i },
This relation also suggests a two-step weighted least-squares estimator of R. In the first step, least squares is used to estimate the parameters in (3) and then predicted values of each ${\widehat{N}}_{i}$ are generated. In the second step, the parameters of (3) are estimated using weighted least squares with weights given by w _{ i }= 1/${\widehat{N}}_{i}$. The optimality of these weights assumes that the variance of N _{ i }is equal to the expected value of N _{ i }. This procedure could be iterated until the estimates and weights converge.
This modification insures that the quantity inside the square root is always greater than or equal to zero. For small values of D ^{2}, ${\widehat{\alpha}}_{D2}\approx {\widehat{\alpha}}_{D}$. As D ^{2} increases, f converges to 1 and $\widehat{\alpha}$ _{ D2 }corresponds to the estimator using the exact expression for E[D ^{2}|N].
Ratio estimators such as $\widehat{R}$ are known to be biased upwards, particularly with sparse data. Later we discuss a bias-correction term for this estimate of R.
Confidence Intervals for R
Constructing confidence intervals for R is problematic because the null value lies on the boundary of the points of support for R. Frangakis and Varadhan recently proposed an approach for computing exact confidence limits for the seasonal relative risk derived from simulation and maximum likelihood estimation of parameters in a circular normal probability model.[19] Their approach can be adapted to estimate confidence intervals for any of the moment estimators proposed in this paper.
The approach involves finding the roots of the function h(R) = |$\widehat{R}$ - R| - q(R; α), where q(R; α) is the 1 - α quantile of |$\widehat{R}$ - R|. Note that q(R; α) depends on a particular estimator, although we do not make this explicit in the notation. The lower confidence limit is either zero or the value of the smaller root, whichever is larger. The upper confidence limit is the value of the larger root. Since q cannot be expressed in closed form, it is estimated via simulation. For a given value of R, data are simulated from the probability model and |$\widehat{R}$ - R| is computed for each simulated data set. In the simulation, the parameter φ can be held fixed at its estimated value. The value of q(R; α) is then estimated by taking the empirical 1 - α quantile of the simulated values of q. The roots of h can be found by using an iterative algorithm.
For the estimators considered in this paper, it is possible that the function h will have only one root. This situation occurs when the number of events is small and/or the seasonality is strong enough so that no upper bound can be placed on the strength of seasonality (the fitted trough of the sine curve is close to zero). When only a single root is found, we set the upper confidence limit to infinity.
For all estimators, $\text{V}\widehat{\text{A}}\text{R}(\widehat{\alpha})\approx 2/N$.
Simulation Study
We compared the various estimators discussed in this paper in a comprehensive Monte Carlo simulation study. Initially, we set k = 12 (corresponding to monthly observations) with n = 150, n = 500, and n = 2500. For each setting of k and n, we simulated data for values of R ranging from 1.05 to 3.05 in increments of 0.25.
For each simulated data set, we evaluate the following five estimators of R:
1. $\widehat{R}$ _{ E }: an estimate of $\widehat{R}$ using Edwards's estimator of α,
2. $\widehat{R}$ _{ LS }: an estimate of R using least squares,
3. $\widehat{R}$ _{ D2}: an estimate of R using $\widehat{\alpha}$ _{ D2},
4. $\widehat{R}$ _{ WLS }: an estimate of R using weighted least squares,
5. $\widehat{R}$ _{ MLE }: an estimate of R using the maximum likelihood estimate of α.
We consider various perturbations of these baseline parameters in sensitivity analyses. First, we set k = 52 (corresponding to weekly observations) with n = 1000, n = 5000 and n = 10000. We also simulated data under two different probability models that departed from the assumed model: 1) a negative binomial model with the mean given by Edwards's model (1), but in which the counts were overdispersed with variance given by VAR[N _{ i }] = 1.5E[N _{ i }]; and 2) a model that generated data with a marginal mean given by Edwards's model, but in which the counts were strongly autocorrelated and overdispersed. We created autocorrelation and overdispersion among the observations by simulating N _{1} using Edwards's model, and then generating each N _{ i }, i = 2, ..., k by simulating Q _{ i }from Edwards model and then letting N _{ i }= Q _{ i }+ 0.1{E[N _{ i-1}] - N _{ i-1}}.
Additionally, we use the simulation results to evaluate the adequacy of the ad hoc confidence interval procedure suggested in section 2.4. For each simulated data set, we compute a 95% confidence interval for $\widehat{R}$ _{ D2}, $\widehat{R}$ _{ WLS }, and $\widehat{R}$ _{ MLE }and record the relative frequency of estimated confidence intervals that contain the true parameter.
Computation
All simulations were performed in SAS V9.1 running on a Windows XP platform using software created by the authors. The maximum likelihood estimates were found using PROC NLMIXED in which the likelihood function (conditional on N) is maximized using a Newton-Raphson algorithm with a line search and boundary constraint (see Additional file 2 for example program). For the Monte Carlo simulation study, the true parameter value was used as the starting point for the maximization routine. The weighted least-squares estimates were obtained in a two-step procedure using PROC GENMOD.
Results
Estimated bias and MSE for each estimator from the baseline simuala-tion for n = 150, 500, and 2500 based on 1,000 simulated datasets
n = 150 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
BIAS × 10 | MSE × 10 | |||||||||
True R | $\widehat{R}$ _{ D2} | $\widehat{R}$ _{ LS } | $\widehat{R}$ _{ WLS } | $\widehat{R}$ _{ MLE } | $\widehat{R}$ _{ E } | $\widehat{R}$ _{ D2} | $\widehat{R}$ _{ LS } | $\widehat{R}$ _{ WLS } | $\widehat{R}$ _{ MLE } | $\widehat{R}$ _{ E } |
1.05 | 1.93 | 3.07 | 3.07 | 3.12 | 3.17 | 0.85 | 1.51 | 1.51 | 1.55 | 1.63 |
1.30 | 0.63 | 1.87 | 1.86 | 2.05 | 2.04 | 0.90 | 1.32 | 1.30 | 1.38 | 1.51 |
1.55 | 0.23 | 1.59 | 1.57 | 1.65 | 1.94 | 1.51 | 1.90 | 1.84 | 1.83 | 2.32 |
1.80 | 0.16 | 1.63 | 1.60 | 1.62 | 2.28 | 2.35 | 2.82 | 2.66 | 2.63 | 3.86 |
2.05 | 0.18 | 1.75 | 1.68 | 1.72 | 2.84 | 3.31 | 3.94 | 3.65 | 3.71 | 6.29 |
2.30 | 0.37 | 2.06 | 1.99 | 1.99 | 3.85 | 4.68 | 5.62 | 5.04 | 5.01 | 10.06 |
2.55 | 0.60 | 2.44 | 2.33 | 2.34 | 5.18 | 6.30 | 7.67 | 6.72 | 6.85 | 17.16 |
2.80 | 0.83 | 2.84 | 2.66 | 2.69 | 7.35 | 8.53 | 10.51 | 8.94 | 9.22 | 46.54 |
3.05 | 1.10 | 3.30 | 3.03 | 3.06 | 10.27 | 11.34 | 14.20 | 11.67 | 12.29 | 193.5 |
n = 500 | ||||||||||
BIAS × 10 | MSE × 10 | |||||||||
True R | $\widehat{R}$ _{ D2} | $\widehat{R}$ _{ LS } | $\widehat{R}$ _{ WLS } | $\widehat{R}$ _{ MLE } | $\widehat{R}$ _{ E } | $\widehat{R}$ _{ D2} | $\widehat{R}$ _{ LS } | $\widehat{R}$ _{ WLS } | $\widehat{R}$ _{ MLE } | $\widehat{R}$ _{ E } |
1.05 | 0.75 | 1.29 | 1.29 | 1.33 | 1.30 | 0.16 | 0.28 | 0.28 | 0.30 | 0.29 |
1.30 | -0.07 | 0.49 | 0.49 | 0.52 | 0.53 | 0.26 | 0.28 | 0.28 | 0.28 | 0.29 |
1.55 | -0.13 | 0.40 | 0.40 | 0.41 | 0.52 | 0.43 | 0.44 | 0.43 | 0.42 | 0.48 |
1.80 | -0.10 | 0.41 | 0.41 | 0.41 | 0.66 | 0.62 | 0.63 | 0.61 | 0.61 | 0.73 |
2.05 | -0.07 | 0.45 | 0.44 | 0.44 | 0.93 | 0.85 | 0.88 | 0.85 | 0.84 | 1.12 |
2.30 | -0.05 | 0.49 | 0.47 | 0.47 | 1.29 | 1.14 | 1.19 | 1.12 | 1.12 | 1.66 |
2.55 | -0.02 | 0.55 | 0.52 | 0.52 | 1.80 | 1.51 | 1.58 | 1.47 | 1.47 | 2.48 |
2.80 | 0.03 | 0.63 | 0.59 | 0.60 | 2.47 | 1.97 | 2.08 | 1.90 | 1.90 | 3.70 |
3.05 | 0.08 | 0.72 | 0.67 | 0.67 | 3.31 | 2.52 | 2.67 | 2.40 | 2.40 | 5.45 |
n = 2500 | ||||||||||
BIAS × 10 | MSE × 10 | |||||||||
True R | $\widehat{R}$ _{ D2} | $\widehat{R}$ _{ LS } | $\widehat{R}$ _{ WLS } | $\widehat{R}$ _{ MLE } | $\widehat{R}$ _{ E } | $\widehat{R}$ _{ D2} | $\widehat{R}$ _{ LS } | $\widehat{R}$ _{ WLS } | $\widehat{R}$ _{ MLE } | $\widehat{R}$ _{ E } |
1.05 | 1.53 | 3.81 | 3.81 | 4.07 | 3.84 | 0.22 | 0.35 | 0.34 | 0.37 | 0.35 |
1.30 | -0.64 | 0.89 | 0.87 | 0.86 | 1.02 | 0.58 | 0.55 | 0.55 | 0.55 | 0.57 |
1.55 | -0.44 | 0.74 | 0.71 | 0.71 | 1.35 | 0.84 | 0.83 | 0.81 | 0.81 | 0.88 |
1.80 | -0.32 | 0.76 | 0.70 | 0.69 | 2.40 | 1.19 | 1.20 | 1.14 | 1.15 | 1.35 |
2.05 | -0.23 | 0.84 | 0.75 | 0.75 | 4.23 | 1.65 | 1.66 | 1.56 | 1.56 | 2.06 |
2.30 | -0.13 | 0.97 | 0.85 | 0.82 | 6.98 | 2.22 | 2.24 | 2.07 | 2.07 | 3.18 |
2.55 | 0.00 | 1.15 | 0.97 | 0.92 | 10.77 | 2.93 | 2.96 | 2.69 | 2.70 | 4.94 |
2.80 | 0.16 | 1.37 | 1.15 | 1.11 | 15.76 | 3.79 | 3.83 | 3.43 | 3.44 | 7.73 |
3.05 | 0.28 | 1.58 | 1.28 | 1.24 | 21.98 | 4.80 | 4.86 | 4.27 | 4.29 | 11.99 |
Relative mean squared error of $\widehat{R}$ _{ D2 }to $\widehat{R}$ _{ WLS }
Overdispersed | Autocorrelated | |||||
---|---|---|---|---|---|---|
True R | n = 150 | n = 500 | n = 2500 | n = 150 | n = 500 | n = 2500 |
1.05 | 0.64 | 0.66 | 0.69 | 0.69 | 0.61 | 0.66 |
1.30 | 0.71 | 0.88 | 1.03 | 1.03 | 0.93 | 1.05 |
1.55 | 0.82 | 0.96 | 1.01 | 1.01 | 1.00 | 1.04 |
1.80 | 0.91 | 0.96 | 1.00 | 1.00 | 1.01 | 1.04 |
2.05 | 0.92 | 0.97 | 1.01 | 1.01 | 1.01 | 1.06 |
2.30 | 0.94 | 0.97 | 1.02 | 1.02 | 1.02 | 1.07 |
2.55 | 0.99 | 0.98 | 1.02 | 1.02 | 1.03 | 1.09 |
2.80 | 1.01 | 0.99 | 1.03 | 1.03 | 1.04 | 1.11 |
3.05 | 1.07 | 1.00 | 1.04 | 1.04 | 1.07 | 1.13 |
Percentage of estimated ad hoc 95% confidence intervals that cover the true parameter
n = 150 | n = 500 | n = 2500 | |||||||
---|---|---|---|---|---|---|---|---|---|
True R | $\widehat{R}$ _{ D2} | $\widehat{R}$ _{ LS } | $\widehat{R}$ _{ MLE } | $\widehat{R}$ _{ D2} | $\widehat{R}$ _{ LS } | $\widehat{R}$ _{ MLE } | $\widehat{R}$ _{ D2} | $\widehat{R}$ _{ LS } | $\widehat{R}$ _{ MLE } |
1.05 | 95.1 | 91.6 | 92.6 | 96.1 | 92.4 | 92.9 | 97.8 | 95.7 | 95.4 |
1.30 | 98.3 | 96.2 | 97.0 | 97.8 | 97.2 | 97.4 | 93.6 | 96.0 | 95.9 |
1.55 | 98.9 | 97.5 | 98.2 | 95.1 | 97.3 | 97.7 | 94.2 | 95.4 | 95.5 |
1.80 | 97.1 | 97.1 | 98.4 | 95.7 | 96.1 | 96.9 | 94.3 | 94.8 | 95.2 |
2.05 | 96.3 | 97.5 | 98.4 | 95.8 | 95.7 | 96.8 | 94.8 | 94.8 | 95.1 |
2.30 | 95.8 | 96.9 | 98.2 | 95.9 | 95.9 | 96.9 | 94.8 | 94.8 | 95.2 |
2.55 | 95.8 | 96.7 | 98.1 | 95.7 | 95.9 | 97.1 | 95.3 | 94.4 | 95.5 |
2.80 | 95.8 | 96.2 | 98.0 | 96.4 | 96.1 | 96.9 | 95.5 | 94.5 | 95.9 |
3.05 | 96.2 | 96.8 | 98.1 | 96.5 | 96.1 | 97.1 | 95.4 | 94.7 | 96.3 |
As a side note, the algorithm that we used to find the MLE experienced convergence problems close to R = 1. For R = 1.05, the MLE failed to converge in roughly 20% of the simulated data sets. This problem diminished as R increased. For R = 1.5 the MLE was located for 95% of the simulated data sets. This is likely to be a result of near non-identifiability of φ when the seasonality is weak. More computationally-intensive approaches, such as a grid search, might alleviate this problem; however, in the context of a simulation study, we required an approach that could converge rapidly. For all results discussed below, we excluded simulated data sets for which the MLE was not found. We found that the simulation results for the non-missing estimators were largely unaffected by the inclusion/exclusion of the simulations for which the MLE was not located.
Example: Seasonality of Monocytic Leukemia
Estimated peak-to-low ratio and 95% CI for the seasonal incidence of monocytic leukemia in England and Wales (1974–98) using four different estimators and two confidence interval procedures
Ad hoc 95% CL | Method of F & V 95% CL | ||||
---|---|---|---|---|---|
Estimator | Point Estimate | Lower Limit | Upper Limit | Lower Limit | Upper Limit |
$\widehat{R}$ _{ E } | 1.20 | 1.07 | 1.35 | 1.07 | 1.37 |
$\widehat{R}$ _{ LS } | 1.20 | 1.06 | 1.34 | 1.07 | 1.36 |
$\widehat{R}$ _{ D2} | 1.18 | 1.05 | 1.32 | 1.07 | 1.33 |
$\widehat{R}$ _{ MLE } | 1.20 | 1.07 | 1.35 | * | * |
The different estimators do not lead to substantively different interpretations of the data. Nevertheless, consistent with the results of the simulation, the estimators $\widehat{R}$ _{ D2 }are smaller than R _{ LS }and Edwards estimator. Given the large number of events and the fact that the data exhibit only moderate seasonality, the simulation study suggests that Edwards estimator should be only moderately biased for these data. The confidence intervals computed by the ad hoc confidence interval procedure were nearly identical to those of Frangakis and Varadhan.
Discussion
In this paper we have proposed a new estimator of the peak-to-low ratio of a periodic process and compared it to several alternative estimators, including Edwards's estimator, the MLE, and weighted least squares. Studies employing Edwards's method often involve very rare events and moderate seasonality. For these studies, the estimator proposed in this paper appears to be optimal. It has less bias and a smaller MSE than any of the estimators considered, including the MLE and weighted least squares. Weighted least squares was preferable from a MSE perspective in the setting of frequent outcomes or strong seasonality. We speculate that the simple estimator proposed in this paper improves upon the estimator of Edwards and the other moment-based estimator because it is based on an exact rather than an approximate expression for the distance from the origin to the center of gravity. We further speculate that the bias and inefficiency in the MLE is due to the small event rates considered in this paper.
The ad hoc confidence interval procedure that we evaluated performed reasonably well for data generated from Edwards's probability model. If more precise confidence intervals are needed, the computationally-intensive approach proposed by Frangakis and Varadhan can be employed [19]. Users should be aware that both of the confidence intervals considered in this paper are model based. If the underlying model is wrong, for example, in the setting of strongly autocorrelated or overdispersed data, the true coverage probabilities may differ from the nominal 95%.
We found that estimators based on this correction factor tended to be somewhat over-corrected, possibly because they are based on an approximation of the variance of $\widehat{\alpha}$.
One important limitation of the estimators proposed in this paper is that they are based on the assumption of a single cyclical effect (harmonic) that can be well approximated by a sine curve. For more complex data, with multiple periodic components or a linear trend, alternative statistical methods should be used. For such data there exist more complex harmonic models [20, 12], spectral methods [21], and various periodic regression models. Also, we outline an approach to estimating seasonal intensity using a periodic generalized linear model that assumes a log link and a Poisson distributed outcome (see Additional file 3). This approach is based on a different model for the mean, i.e., that the log of the expected value of the counts is a sinusoidal function. However, it allows for the inclusion of covariates and extends naturally to variably-sized intervals through use of a Poisson offset.
Edwards's method has been widely used in epidemiology in studies of seasonality. In this paper we have shown that Edwards's estimator of the seasonal relative risk can be substantially biased. The estimator proposed in this paper represents a straightforward modification of Edwards's estimator. Like that of Edwards, it is a simple estimator that is available in closed form. For modest seasonality and small numbers of events, this estimator appears to have the best finite sample performance characteristics of those estimators considered.
For more frequent events or stronger seasonality, the weighted least-squares approach discussed in this paper is preferable and is easily implemented using standard statistical software.
Declarations
Acknowledgements
The authors are grateful for the helpful comments of Tim Lash and Claus Dethlefsen. M. Alan Brookhart is supported by a career development grant from the National Institute on Aging (AG-027400).
Authors’ Affiliations
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