The Use of Transfer Function Models, Intervention Analysis and Related Time Series Methods in Epidemiology

In epidemiology, data often arise in the form of time series e.g. notifications of diseases, entries to a hospital, mortality rates etc. are frequently collected at weekly or monthly intervals, Usual statistical methods assume that the observed data are realizations of independent random variables. However, if data which arise in a time sequence have to be analysed, it is possible that consecutive observations are dependent. In environmental epidemiology, where series such as daily concentrations of pollutants were collected and analysed, it became clear that stochastic dependence of consecutive measurements may be important. A high concentration of a pollutants today e.g. has a certain inertia i.e. a tendency to be high tomorrow as well. Since the early 1970s, time series methods, in particular ARIMA models (autoregressive integrated moving average models) which have the ability to cope with stochastic dependence of consecutive data, have become well established in such fields as industry and economics. Recently, time series methods are of increasing interest in epidemiology. Since these methods are not generally familiar to epidemiologists this article presents their basic concepts in a condensed form. This may encourage readers to consider the methods described and enable them to avoid pitfalls inherent in time series data. In particular, the following topics are discussed: Assessment of relations between time series (transfer function models). Assessment of changes of time series (intervention analysis), forecasting and some related time series method

The Use of Measures of Influence in Epidemiology

Helfenstein U (Biostatistical Centre for the Medical Department, University of Zurich, Plattenstrasse 54, CH-8032, Zurich, Switzerland) and Minder C. International Journal of Epidemiology 1990, 19: 197-204. In epidemiological studies the units of observation often consist of political entities such as countries, each of which has its own specific inner structure. When a multiple regression is performed it is therefore of particular interest to ana-lyse not only the overall behaviour of the dataset, but in addition, to investigate how each individual country contributes to, and deviates from, this overall behaviour. By means of the example ‘relation between infant mortality and structural data of countries' several ways are discussed of how each individual country can influence the regression model. Firstly the potential influonco which each country might exhibit due to the explanatory variables alone is analysed. Then the actual influence of each country is analysed by taking the explanatory variables and the target variable into account simultaneously. This is done by means of statistical measures not generally familiar to epidemiologists, which have been developed in recent years (leverage values, Cook's distances). These measures also point to deviations of countries from the model, and suggest directions in which to search for explanation. Finally the influence of the ‘size' of the countries is investigate

The use of measures of influence in epidemiology

In epidemiological studies the units of observation often consist of political entities such as countries, each of which has its own specific inner structure. When a multiple regression is performed it is therefore of particular interest to analyse not only the overall behaviour of the dataset, but in addition, to investigate how each individual country contributes to, and deviates from, this overall behaviour. By means of the example 'relation between infant mortality and structural data of countries' several ways are discussed of how each individual country can influence the regression model. Firstly the potential influence which each country might exhibit due to the explanatory variables alone is analysed. Then the actual influence of each country is analysed by taking the explanatory variables and the target variable into account simultaneously. This is done by means of statistical measures not generally familiar to epidemiologists, which have been developed in recent years (leverage values, Cook's distances). These measures also point to deviations of countries from the model, and suggest directions in which to search for explanation. Finally the influence of the 'size' of the countries is investigated

When did a reduced speed limit show an effect? Exploratory identification of an intervention time

In a statistical analysis of accident data before and after a speed limit reduction, the time of the countermeasure is, of course, well known. Our understanding of the accident process may, however, be increased if we assume in a thought experiment that this time is unknown. We ask if the data themselves can tell us something about such a possible time. By means of time series of traffic accidents in Zurich before and after a speed limit reduction, different exploratory methods are presented to identify the "unknown" time of this measure. For most of the investigated series, the most likely time was found to lie in the three months before the true introduction. A possible explanation of this result may be that the media already informed the public before the countermeasure was actually introduced. This finding leads to an improved parsimonious intervention model which distinguishes between a possible "preintervention effect" and the usual "intervention effect.

Box-Jenkins modelling in medical research

Notifications of diseases, entries in a hospital, injuries due to accidents, etc., are frequently collected in fixed equally spaced intervals. Such observations are likely to be dependent. In environmental medicine, where series such as daily concentrations of pollutants are collected and analysed, it is evident that dependence of consecutive measurements may be important. A high concentration of a pollutant today has a certain 'inertia', i.e. a tendency to be high tomorrow as well. Dependence of consecutive observations may be equally important when data such as blood glucose are recorded within a single patient. ARIMA models (autoregressive integrated moving average models, Box-Jenkins models), which allow the stochastic dependence of consecutive data to be modelled, have become well established in such fields as economics. This article reviews basic concepts of Box-Jenkins modelling. The methods are illustrated by applications. In particular, the following topics are presented: the ARIMA model, transfer function models (assessment of relations between time series) and intervention analysis (assessment of changes of time series)