| Seasonal variation of suicides and homicides in Finland: With special attention to statistical techniques used in seasonality studies. | ||
|---|---|---|
| Prev | Chapter 2. Review of the literature | Next |
At least three study types can be distinguished in seasonality research. The first are the studies in which a researcher is interested in the seasonal distribution of a time series itself, such as the monthly or weekly pattern of suicides. The purpose of a study is usually to describe the main properties of the data, for example values for the minimum and maximum incidence of events, and/or to present a mathematical model fitted to the data. The statistical methods are mainly descriptive techniques or standard statistical methods, such as graphical presentations or a chi-square test (Siegel & Castellan 1988).
In the second type of study, the main object is to clarify whether there are differences in the seasonal patterns between subgroups of a population. For example, the seasonal patterns of events are compared between genders. The statistical methods consist mainly of techniques, which compare the seasonal distributions of events between two or more categories of a group variable, like for example, the chi-square test or one-way analysis of variance test (Armitage & Berry 1987, Siegel & Castellan 1988).
Thirdly, there are studies in which seasonal patterns of two or more time series are correlated with each other, for example the seasonal pattern of deaths due to suicides is related to the seasonal patterns of climatic variables. In that case the statistical methods are techniques, which correlate with or seek for a similar or lagged rhythmicity in two or more time series data, for example Pearson’s correlation or bivariate spectral analysis (Armitage & Berry 1987, Chatfield 1996).
The type of a study, thus, determines partly what kind of a statistical technique can be used for seasonal analyses of the data. In addition, there are features of the data, which influence the choice of an appropriate statistical method, such as the size of a sample and the type of a time clustering (e.g. whether the data is aggregated to monthly, weekly or seasonal totals of events). Researchers should always be aware of the basic assumptions required by a statistical technique, which they intend to use in their study. Otherwise, due to the inappropriate statistical method, spurious results or false negative findings are possible.
In the following chapters the most basic common statistical techniques used in seasonality studies in epidemiological research are briefly presented. Because the presentation does not include mathematical expressions of these techniques, references to the appropriate medical and statistical literature are given in the text.
“Anyone who tries to analyse a time series without plotting it first is asking for trouble” (Chatfield 1996). A graph serves two main purposes in a study. Firstly, it reveals easily the important features of a time series, such as trend, seasonal variation, outliers, and discontinuities. It also shows the possible errors and missing information. Thus, a graph presents the statistical information of a data in an aggregated and evocative way (Armitage & Berry 1987). Secondly, it serves as an aid to a statistical analysis. A graph shows the structure of a data, and suggests hypotheses (mathematical model), which may be further investigated (Marrero 1983, Haus et al. 1980, Chatfield 1996).
When presenting a seasonal pattern of a data-set, a simple graph may show, for example, the monthly frequencies of events in a form of simple histograms or a line diagram (Armitage & Berry 1987). A more complex graph is a periodogram obtained from a spectral analysis (Bloomfield 1976, SPSS 1994, Chatfield 1996). Plotting a graph needs careful considerations, so that a graph would not become misinterpreted. Researchers should pay attention, for example, to the choice of scales, the way that the points are plotted (e.g. as a continuous line or as separate dots), a clear title for the axes, and presentation of measurement units (Armitage & Berry 1987, Chatfield 1996).
Chi-square techniques include a variety of methods designed for different pursuits of a study. A common situation is that a researcher may want to test whether there is a statistically significant difference in the observed distribution of a variable between two or more categories of a grouping variable (Siegel & Castellan 1988). Thus, the purpose of this kind of analysis is mainly that of a significance test (Armitage & Berry 1987). Lester (1971) used the chi square test when comparing the monthly numbers of suicides between genders, methods (active, passive) and age groups (15-54 years, over 55 years). He did not find any statistically significant difference between seasonal distributions in any of these subgroups.
On the other hand, when a researcher is interested in certain forms of departure from the null hypothesis, he/she can use a chi-square goodness-of-fit test, which assesses a goodness-of-fit of an observed sample distribution in relation to a hypothesised distribution (Horn 1977, Siegel & Castellan 1988, Agresti 1990). The null hypothesis, for example, would state that the observed monthly distribution of events follows a uniform distribution over a year. If the counts, such as monthly frequencies of events, are aggregated over several years, the 12 monthly counts are independent Poisson observations with an equal intensity in each month, and it can be shown that the counts can be considered to derive from a multinomial distribution. The proportions of 1/12 should be adjusted according to the calendar effect and the effect of leap years (Cleveland & Devlin 1980, Walter 1994, Torrey et al. 1997). For example, a significant seasonal pattern in both suicides and homicides was revealed when Lester (1979) used large samples of suicides and homicides, and a chi-square goodness-of-fit test.
The main value of the chi-square goodness-of-fit test is not that of rejecting the null hypothesis, but rather that of a routine screening test making sure, that enough data is gathered to discern a departure from the null hypothesis. Confidence intervals are found to be a feasible way to locate the categories, like months, which actually show departures from the null hypothesis (Wonnacott & Wonnacott 1990).
The use of chi-square methods in seasonality analyses has some limitations, though. Firstly, the chi-square test does not take into account the ordered structure of the data, and it does not distinguish between irregular fluctuation and a smooth cyclical pattern (James 1976, Walter 1977a, Takei et al. 1992). Test is also sensitive to a variety of types of departures from the assumed distributional form (Shensky & Shur 1982, Sarmukaddam & Rao 1987, Freedman 1979). The chi-square methods, however, can be thought as “portmanteau techniques” able to assist in many different situations (Armitage & Berry 1987). Their popularity as one of the tests for seasonality may be accounted for by the simple mathematical theory behind them, which make them easy to calculate and understand. Sometimes chi-square tests have been utilised together with more sophisticated methods in order to “facilitate comparisons with previous studies” (Clarke et al. 1998b).
A correlation analysis is used either for significance testing or assessing the degree of a relationship. Thus, it does not only test whether two quantitative variables are associated but also estimate the extent to which two variables are related with each other (Sokal & Rohlf 1981). A correlation coefficient, however, does not indicate that the relationship is one of cause and effect (Bowie & Prothero 1981, Armitage & Berry 1987).
Two most common techniques for correlations are the Pearson’s product-moment correlation and the Spearman’s rank order correlation coefficient (Armitage & Berry 1987, Siegel & Castellan 1988). In seasonality studies, the Pearson’s and Spearman’s correlation coefficients have been used to assess the relationships between two time series data, e.g. monthly totals of suicides are correlated with the monthly mean values of ambient temperature. However, since these analyses are intended to look for a linear relationship between two variables, their use in seasonal data including possible cyclic trends (i.e. non-linear trends) might be questionable. For example, Zung and Green (1974) studied the relationship between monthly numbers of suicides and climatic variables with the Pearson’s correlation, but no statistically significant correlations were observed. Preti (1997) reported that the monthly distribution of suicides correlated positively (Spearman’s correlation) with the mean monthly values of maximum and minimum temperature and indicators of exposure to the sun, and negatively with mean monthly values of humidity grade and indicators of rainfall.
The purpose of a regression analysis is to describe and/or predict variation in a dependent variable with variations in one or more predictors (called also independent or explanatory variables). There has been much confusion between a correlation and a regression analysis. While a correlation analysis establishes and estimates the strength of the relationship between two variables, a regression analysis describes a functional relationship and/or predicts one in terms of the other (Sokal & Rohlf 1981, Cambell & Machin 1993).
The choice of an appropriate regression analysis depends, for example, on the statistical distribution of a dependent variable (e.g. a normal or binomial distribution), the type of independent variables (continuous, dichotomous etc.) and what kind of association is assumed to exist between dependent and independent variables (linear, non-linear etc). An ordinary regression analysis is usually concerned with the linear relationship between the mean value of one variable and the value of another variable, i.e. whether a change in a dependent variable will lead directly to a change in another dependent variable. A multiple regression analysis gives a regression model in which the dependent variable (or outcome) is expressed as a combination of the several independent variables (explanatory variables, predictors, covariates) (Armitage & Berry 1987, Tabachnick & Fidell 1989, Altman 1991, Munro 1997). A logistic regression analysis can be utilised when a dependent variable is a dichotomous variable (Kleinbaum 1994, Fleiss et al. 1986).
Occasionally, a regression analysis with dummy variables has been utilised in seasonality studies. For example, a regression with 11 dummy variables (one month is used as a reference month) representing the 12 months of the year has been used to test whether the events of some phenomenon is different from some months than others (Tennenbaum & Fink 1994). After adjusting for the number of days in each month Tennenbaum and Fink (1994) found that July and August were months with significantly more homicides than January. By means of regression analysis, which included dummy variables for each of the six national public holidays, Phillips and Wills (1987) noted that suicides did not increase around the holiday after adjusting for the effect of extraneous variables, such as days of the week.
Anderson & Anderson (1984) utilised an ordinary linear regression analysis in their study examining temperature-aggression relationships. A dependent variable was the daily number of criminal assaults (homicide, rape, battery, armed robbery), aggressive crimes (murder, rape) and non-aggressive crimes (robbery, arson), and the day of a week, temperature and temperature squared were used as the predictor variables. The results showed that both criminal assaults and aggressive crimes increased linearly in frequency as temperature increased, and that they were significantly associated with the day of the week. The non-aggressive crimes were unrelated to all predictors. Souetre et al. (1990) revealed with the regression analyses that the significant factors affecting the regional distribution of suicides were ambient temperature and sunlight duration. Wasserman and Stack (1994) noticed that people are not significantly more likely to commit a suicide near the time of their birthday than they are at other time of the year after controlling for seasonal effects (month of death), gender, marital status, ethnicity and educational level in the multivariate regression analysis.
The main idea of analysis of variance techniques is to describe the characteristics of a variable under study and to compare the means of a dependent variable between categories of a grouping variable. There are many assumptions required before the analysis of variance methods can be used, all of which should be carefully taken into account (Armitage & Berry 1987, Munro 1997). However, when used in seasonality analysis, these methods only reveal whether time can be considered as a statistically significant source of variation (Haus et al. 1980).
Student’s t-test is used to analyse the difference between two independent group means, for example to compare the mean values of suicides between winter and summer. The one-way analysis of variance allows comparisons between two or more group means, for example to compare mean value of homicides between 12 calendar months. A two-way analysis of variance compares the mean differences simultaneously between two grouping variables, for example monthly mean values of suicides by gender. One- or two-way analyses of variance technique include the possibility to make multiple group comparisons (Sokal & Rohlf 1981, Armitage & Berry 1987).
By means of ANOVA, for example, Maes et al. (1993a) showed that a significantly higher number of violent suicides occurred in spring and summer than in other seasons. Meares et al. (1981) found with the help of the two-way analysis of variance that daily mean suicide incidence differed significantly from the equal monthly incidence and that there was a significant between the years effect also.
The non-parametric techniques make few, if any assumptions about the distribution of the dependent variable in the population. They are usually used as alternative statistical techniques in situations, when the statistical assumptions required by the parametric methods are not fulfilled (Siegel & Castellan 1988).
The Kolmogorov-Smirnov one-sample test is used to assess the degree of agreement between the distribution of observed and expected values determined according to some specified distribution, for example a uniform distribution. It is suitable to assess the goodness-of-fit for variables, which are measured on at least in ordinal scale. When sample sizes are small, the Kolmogorov-Smirnov test has been found to be more powerful than its alternative, the chi-square goodness-of-fit test (Siegel & Castellan 1988).
The Mann-Whitney U-test is a non-parametric alternative to Student’s t-test. Respectively, the Kruskal-Wallis test is an alternative for the parametric one-way analysis of variance test, if there are two or more independent groups to compare (Siegel & Castellan 1988). Barker et al. (1994), for example, found with the Kruskal-Wallis test that significant seasonal and monthly variations in mean daily frequency of suicide attempts were observed in women, but not in men. In addition, significant relationships (as assessed with the Mann-Whitney U-test) were found between female parasuicides and ‘hot’, ‘still’, ‘still/hot’ days as well as between male parasuicides and ‘windy’ days.
A time series data-set consists of a long series of observations, like those on daily temperature, for instance, made at successive points in time, usually at equally spaced intervals. Furthermore, it is allowed that neighbouring observations are correlated with each other over time. The latter assumption is different from most statistical methods requiring independence of observations.
The one purpose of a time series analysis is to describe the time series by a mathematical model, which provides an appropriate description of the systematic and random variation in a data-set. Furthermore, with the help of a time series analysis, a researcher may also want to predict future observations by taking into account recent changes in the series (Armitage & Berry 1987, Chatfield 1996). The time series analysis methods are appropriate for data, which include a large number of time intervals. Methods for time series analyses vary from simple graphical techniques (which are used to visualise or describe the temporal patterns in a time series data-set) to sophisticated modelling techniques, such as a spectral analysis or a Box-Jenkins analysis.
The time series analyses can be classified in two categories, namely those of the time and those of the frequency domain. These two approaches are in fact mathematically equivalent, in that one form of analysis can be derived from the other (Armitage & Berry 1987). The frequency domain approach (spectral approach) involves representing the time series data by a superposition of sinusoidal waves of different frequencies. Harmonic analysis and spectral analysis are the most popular time series methods used in the frequency domain. In the time domain approach (ARIMA or Box-Jenkins approach), the behaviour of a time series is described in terms of the way in which observations at different times are related statistically with each other. It consists of methods, which fit autoregressive and/or moving-average models to the data-set after differencing and/or seasonal adjustments (Box & Jenkins 1976, McCleary & Hay 1980, Armitage & Berry 1987, Chatfield 1996).
Harmonic analysis methods are widely utilised in chronobiological research for biological time series exhibiting various predictable variations (Haus et al. 1980). These methods, which are called “rhythmometric“ methods in chronobiology, describe all aspects of a rhythm in terms of an appropriate mathematical model fitted to an entire time series (Bloomfield 1976, Nelson et al. 1979).
Harmonic analyses (periodic regression) include various techniques, in which the time series is decomposed into a number of periodic components of sinusoidal form. The basic idea of this type of analysis is that a time series data-set can be described in terms of some parameters, such as a mesor, amplitude and an acrophase of a series of sinusoidal curves. Before using a harmonic analysis technique, any trend in the data needs to be removed (Armitage & Berry 1987). Näyhä (1983), for example, observed a significant seasonal pattern of suicides for males and females by using a harmonic analysis and after adjusting the monthly frequencies of suicides to months of equal length. A similar approach was also followed in other studies (Näyhä 1982, Miccolo et al. 1989, Miccolo et al. 1991, Ho et al. 1997).
The mathematical terms used in a harmonic analysis, and also in a spectral analysis, are not as familiar to researchers in psychiatry as they are for chronobiologists (Hakko et al. 1999). For example, when a time series data-set is described in terms of a fitted mathematical model like a sine wave, some terms need to be clarified and described carefully in an article. The term “mesor“ means the mean value of a rhythm defined by a mathematical mode, such as a sine curve, i.e. a point midway between the lowest and highest value of a sine function. The term “amplitude“ is the distance between a mesor and the highest point defined by a mathematical model. The term “acrophase“ indicates the timing of the highest rhythms defined by a mathematical model. A period is the duration of one complete cycle in a rhythmic function (Nelson et al. 1979, Haus et al. 1980).
The most common modifications of a harmonic analysis are presented in the following sections, because these methods have frequently been used to study different types of seasonal distributions in epidemiological research.
Edwards’ test: Edwards’ model (1961) tests whether the distribution of events follows a simple harmonic curve having one peak and one trough within a single 12-month period. In Edwards’ method the data must consist of the frequencies of events grouped into appropriate time intervals, for example months of a year or days of a week. The assumptions required by the original method are that the length of the time intervals must be equal and that the events are independent. The data are presented in the form of the rim of a circle divided into sectors. For example, twelve sectors are defined when the data consists of monthly totals of events, and seven sectors when the data includes the totals by days of a week. If there were no seasonal variation in the data, the expected centre of the gravity of a time series of events would be in the centre of the circle. The position of the actual centre of gravity indicates the time period for peak incidence and its distance from the centre of the circle indicates the relative strength of the seasonal variation (Edwards 1961, MacMahon & Pugh 1970, Hewitt et al. 1971, Pocock 1974, Walter 1977a, Jones 1988). The efficiency of the method is dependent on the underlying model with a single peak and a single trough (MacMahon & Pugh 1970).
Although Edwards’ test is a popular method for seasonality in epidemiological research, it is only occasionally utilised in psychiatry. For example, Bazas et al. (1979) identified with the help of Edwards’ test a statistically significant seasonality of suicides in Greece with a June peak. Takei et al. (1992) revealed significant seasonality in hospital admissions for schizophrenia and affective disorders with a common peak in July for patients in England and Wales.
Edwards’ test suffers from the lack of power for small sample sizes (Hewitt et al. 1971, Roger 1977). It is sensitive to occasional extreme values in the data as well as cyclic variations of other forms than a simple harmonic curve (Hewitt et al. 1971, Wehrung & Hay 1970). Also, the assumption of the equally spaced time intervals may not be fulfilled in practice, e.g. due to the unequal lengths of the months. Edwards’ test does not take into account the size of the population at risk (Walter & Elwood 1975). Various modifications of this test have later been employed (Hewitt et al. 1971, Cave & Freedman 1975, Walter & Elwood 1975).
Hewitt’s test for seasonality: Hewitt et al. (1971) proposed a non-parametric test for seasonality as an alternative for the parametric Edwards’s test, because Edwards’ method has been noted to suffer from a lack of power in relation to small samples. In Hewitt’s method the monthly data of events was ranked from lowest to highest, and then all possible sequences of six consecutive months were examined. Hewitt’s test makes the assumption that the year is split into two equally wide intervals of 6 months each. Although no lower limits for sample size are stipulated, at least six of the 12 months must have non-zero frequencies of occurrences (Hewitt et al. 1971).
Rogerson’s (1996) generalised Hewitt’s test to situations in which a predetermined 3, 4 or 5-month period of raised frequencies of occurrences are hypothesised. For example, Walter (1977b) applied the non-parametric Hewitt’ test for admissions due to mania and reported a significant seasonal trend for both sexes with peak values occurring during May-October.
Hewitt’s test is criticised due to its assumption to divide a year into two 6-month periods (Rogerson 1996, Marrero 1983). Also, it does not allow estimating parameters of the simple harmonic curves even if a significant departure from the null hypothesis is indicated. When sample size is substantial large, Hewitt’s test has been found to possess a low power to detect a seasonal trend, unless it is fairly marked (Walter & Elwood 1975).
Pocock’s method: Pocock (1974) developed a method, which generalises Edwards’ approach and allows the alternative hypothesis of a seasonal pattern of arbitrary shape. In that test, the seasonal pattern is estimated by the application of a harmonic analysis to a complete time series of frequencies observed for a fixed population in equally spaced time intervals.
Under this model, the variation between the months is described as a sum of sinusoidal curves. The seasonal variation consists of those components with cycles, which repeat themselves an exact number of times per year. Or in other words, the seasonal variation is presented by the harmonics of period 1, 1/2, 1/3, 1/4, 1/5 and 1/6 of a year, and the significance of each of these and the percentage variation that each represents can be estimated (Pocock 1974, Barraclough & White 1978a,b). The sample variance could be divided into random, seasonal and non-seasonal components. In order to compare the relative importance of seasonal variation in two or more different time series, the adjusted ratio of seasonal and random components of variance can be calculated (Pocock 1974). For example, Barraclough and White (1978a) found with Pocock’s method that for suicides the harmonics with yearly period 1, 1/2, 1/4 and 1/6 were all significant, but for undetermined deaths only the harmonic with a 1/4 year period was significant.
Walter and Elwood test for seasonality: Walter and Elwood (1975) modified Edwards’ test in situations, which allow for an arbitrary pattern of variation in the population at risk and also for unequal lengths of time intervals. Different from Edwards’ test, the expected number of events (e.g. expected number of events in a month) is calculated to be proportional to the population at risk. From the test it is possible to estimate the amplitude of the seasonal variation and the time at which the maximum occurs in a postulated simple harmonic fluctuation.
The Walter and Elwood method has also been used in situations, in which the frequencies of an event of primary interest were compared directly to the frequency of another event, for example admissions with schizophrenia versus admissions with neurosis rather than all admissions (Clarke et al. 1998a). Because both events are assumed to being seasonally distributed, the adjustment for the comparison event is supposed to reveal the “true“ biological seasonal pattern in an event of main interest (Hare & Walter 1978).
By using the Walter and Elwood method for seasonality, Walter (1997b) found a peak incidence during August in admissions for mania after adjusting for total admissions. Hare and Walter (1978) showed a significant July peak in admissions for schizophrenia and an August peak for mania after adjusting for various comparative diagnoses. Recently, Clarke et al. (1998a) reported the July peak was seen in the admission patterns of schizophrenia, and the peak of admissions for bipolar disorder extended from June to August, when in both analyses the comparison event were the admissions with neurosis.
Other modifications for Edwards’ test: Cave and Freedman (1975) modified Edwards’ test (1961) for seasonality so that a sinusoidal curve with two peaks and two troughs was proposed, instead of Edwards’ hypothesis of a sinusoidal curve with only one peak and one trough during the year. St Leger (1976) advised, how Edwards’ method could be refined by using a maximum likelihood technique.
Roger (1977) presented a test statistic, which originated from the methods provided by Edwards (1961) and Walter & Elwood (1975), because according to Roger himself, these methods were “based on an intuitive approach to the seasonality problem“. Roger derived a test statistic suitable for small samples. By using Roger’s test statistic, for example, Shensky and Shur (1982) showed that both “low genetic risk” and “high genetic risk” groups of schizophrenic patients showed a significant seasonal variation in birth-rates, with peaks in October and April-May, respectively. Poikolainen (1982) found that alcohol-related hospital admissions followed a simple harmonic process with a peak incidence mainly during summer months.
Kolmogorov-Smirnov type statistic: The Kolmogorov-Smirnov type statistic is a non-parametric method that is used to test whether the distribution of an ordinal variable differs significantly between two samples (Clarke et al. 1998a). Freedman (1979) developed a variation of this test in a situation, when there is no particular reason to expect a specific parametric alternative, for example a sinusoidal curve, to the null hypothesis of a uniform seasonal pattern. The data can be exact dates of occurrences or, for example, grouped into months.
Clarke et al. (1998a) found that both schizophrenia and affective disorder had a significant seasonal variation in their monthly admissions as compared with admissions for neurosis, when Kolmogorov-Smirnov type statistic were used in the statistical analyses. In another study they found that the quarterly birth distribution of unipolar forms of affective disorder differed significantly from the general population (Clarke et al. 1998b).
Spectral analysis is used to find various kinds of periodic behaviour in time series. The rationale behind a spectral analysis is that a time series, which is recorded at equal time intervals, can be decomposed into a sum of trigonometric periodic functions with different frequencies, amplitudes and phases. Frequencies range from zero to the highest frequency discernible in the data. Periodicities in the series can then be examined for each frequency (Chatfield 1996, SPSS 1944, Thrall & Engelman 1990).
Spectral analysis is a modification of Fourier analysis, which is basically concerned with approximating a function by a sum of sine and cosine terms, called the Fourier series representation (Bloomfield 1976, Chatfield 1996). The difference between the spectral and harmonic analysis is, that while a harmonic analysis examines periodic fluctuations with predetermined frequencies, a spectral analysis allows the whole frequency band to be analysed simultaneously (Armitage & Berry 1987).
The spectral analysis can also be used to study pairs of series (bivariate spectral analysis). The coherence between two series can be calculated and it represents the degree of linear association between the two series in different frequency bands (Bloomfield 1976, Chatfield 1996, Thrall & Engelman 1990). As with many time series analysis methods, the data usually need to be prepared prior to a spectral analysis. For example the missing values in a series must be replaced and seasonal means or a linear trend must be removed (Thrall & Engelman 1990).
Various forms of a spectral analysis have been applied in seasonality studies. Torrey et al. (1996) showed with bivariate spectral analysis that statistically significant coherence in birth patterns among diagnostic groups were found between bipolar disorder and major depression, paranoid schizophrenia, and schizoaffective disorder, as well as between schizoaffective disorder and ‘process’ schizophrenia. The same method was used by Linkowski et al. (1992), who found that the death probabilities due to violent suicides were positively correlated to the humidity grade and negatively to the sunlight duration in both sexes.
Under the label of “time series analysis”, two correlation techniques must firstly be mentioned. The first one is a cross-correlation, which shows the correlation between two series at the same time or with each series leading by one or more lags. It should be used only on series that fulfil the criteria of stationary (i.e. mean and variance of each series stay about the same over the length of the series). Another important correlation is an autocorrelation, which measures the correlation between observations of a series at different distances apart (Box & Jenkins 1976, SPSS 1994).
ARIMA (Box-Jenkins) models are widely used time series techniques in time domain. ARIMA means AutoRegressive Integrated Moving Average, after the three components (autoregression, integration, moving averages) of the general ARIMA model (SPSS 1994). Sometimes ARIMA models are related to regression models. According to McCleary and Hay (1980) “the only real difference between ARIMA and regression approaches to time series analysis is a practical one. Whereas regression models can be built on the basis of prior research and/or theory, ARIMA models must be built empirically from the data”. The latter condition determines quite strictly that relatively long time series are needed for ARIMA models. Otherwise, regression approaches informed by prior research and/or theory may be a more appropriate solution than ARIMA models (McCleary & Hay 1980).
ARIMA modelling is usually conducted in three stages. Firstly, a reasonable model based on patterns of autocorrelation is identified. Secondly, the parameters of the tentative model are estimated so that their values are statistically significant and consistent with assumptions of stationary. Thirdly, the adequacy of the model with its estimated parameter values is evaluated. This iterative identification/estimation/diagnosis procedure continues until an adequate and most parsimonious ARIMA model has been achieved for a given time series (Box & Jenkins 1976, McCleary & Hay 1980, Armitage & Berry 1987, Martinez-Schnell & Zaidi 1989, Tennenbaum & Fink 1994, Chatfield 1996).
For example, Tennenbaum and Fink (1994) applied ARIMA model in their data of monthly homicides occurred during 1976-89. A significant seasonality was found to homicides. After adjusting for the number of days per month, the best fitted ARIMA model was first-order autoregressive, i.e. the number of homicide in the current month was a linear function of the numbers of homicides in the previous month and the same month one year prior.