Table of Contents
1 Introduction 1
1.1 Example Data Sets 1
1.1.1 Cardiovascular Disease Deaths 1
1.1.2 Schizophrenia 2
1.1.3 Influenza 4
1.1.4 Exercise 5
1.1.5 Stillbirths 6
1.1.6 Footballers 7
1.2 Time Series Methods 8
1.2.1 Autocovariance and Autocorrelation 9
1.3 Fourier Series 14
1.3.1 Cosine and Sine Functions 14
1.3.2 Fourier Series 18
1.3.3 Periodogram 19
1.3.4 Cumulative Periodogram 23
1.4 Regression Methods 25
1.4.1 Scatter Plot 26
1.4.2 Linear Regression 27
1.4.3 Residual Checking 29
1.4.4 Influential Observations 33
1.4.5 Generalized Linear Model 35
1.4.6 Offsets 38
1.4.7 Akaike Information Criterion 39
1.4.8 Non-linear Regression Using Splines 40
1.5 Box Plots 42
1.6 Bayesian Statistics 44
1.6.1 Markov Chain Monte Carlo Estimation 45
1.6.2 Deviance Information Criterion 46
2 Introduction to Seasonality 49
2.1 What is a Season? 49
2.1.1 Seasonality and Health 50
2.2 Descriptive Seasonal Statistics and Plots 53
2.2.1 Adjusting Monthly Counts 53
2.2.2 Data Reduction 55
2.2.3 Circular Plot 61
2.2.4 Smooth Plot of Season 63
2.3 Modelling Monthly Data 65
2.3.1 Month as a Fixed Effect 66
2.3.2 Month as a Random Effect 69
2.3.3 Month as a Correlated Random Effect 69
3 Cosinor 75
3.1 Examples 76
3.1.1 Cardiovascular Disease Deaths 76
3.1.2 Exercise 78
3.1.3 Stillbirths 80
3.2 Tests of Seasonality 80
3.2.1 Chi-Squared Test of Seasonality 83
3.2.2 Sample Size Using the Cosinor Test 85
3.3 Sawtooth Season 86
3.3.1 Examples 87
4 Decomposing Time Series 93
4.1 Stationary Cosinor 96
4.1.1 Examples 97
4.2 Season, Trend, Loess 98
4.2.1 Examples 101
4.3 Non-stationary Cosinor 104
4.3.1 Parameter Estimation 106
4.3.2 Examples 109
4.4 Modelling the Amplitude and Phase 111
4.4.1 Parameter Estimation 114
4.4.2 Examples 116
4.5 Month as a Random Effect 118
4.5.1 Examples 119
4.6 Comparing the Decomposition Methods 121
4.7 Exposures 122
4.7.1 Comparing Trends with Trends and Seasons with Seasons 123
4.7.2 Exposure-Risk Relationships 124
3 Controlling for Season 129
5.1 Case-Crossover 129
5.1.1 Matching Using Day of the Week 132
5.1.2 Case-Crossover Examples 133
5.1.3 Changing Stratum Length 135
5.1.4 Matching Using a Continuous Confounder 135
5.1.5 Non-linear Associations 136
5.2 Generalized Additive Model 138
5.2.1 Definition of a GAM 138
5.3 A Spiked Seasonal Pattern 142
5.3.1 Modelling a Spiked Seasonal Pattern 143
5.4 Adjusting for Seasonal Independent Variables 146
5.4.1 Effect on Estimates of Long-term Risk 147
5.5 Biases Caused by Ignoring Season 149
6 Clustered Seasonal Data 351
6.1 Seasonal Heterogeneity 151
6.2 Longitudinal Models 153
6.2.1 Example 154
6.3 Spatial Models 155
6.3.1 Example 156
References 159
Index 163
