Research Report

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INFERENCE PRINCIPLES FOR MULTIVARIATE SURVEILLANCE
(2011-03-29) Frisén, Marianne; Statistical Research Unit, Department of Economics, GU
Multivariate surveillance is of interest in industrial production as it enables the monitoring of several components. Recently there has been an increased interest also in other areas such as detection of bioterrorism, spatial surveillance and transaction strategies in finance. Multivariate counterparts to the univariate Shewhart, EWMA and CUSUM methods have earlier been proposed. A review of general approaches to multivariate surveillance is given with respect to how suggested methods relate to general statistical inference principles. Multivariate on-line surveillance problems can be complex. The sufficiency principle can be of great use to find simplifications without loss of information. We will use this to clarify the structure of some problems. This will be of help to find relevant metrics for evaluations of multivariate surveillance and to find optimal methods. The sufficiency principle will be used to determine efficient methods to combine data from sources with different time lag. Surveillance of spatial data is one example. Illustrations will be given of surveillance of outbreaks of influenza.
STATISTICAL RANK METHODS FOR ORDINAL CATEGORICAL DATA
(University of Gothenburg, 1991-03-01) Holm, Sture; Svensson, Elisabeth
The aim of this paper is to present a new rank. method for analysing ordinal scale problems, and to give some of its basic properties. The method is suitable for the assessment of validity and reliability of health measurement instruments. We will be able to separate systematic and random differences between judges or scales and also. in a suitable way. measure the size of these two types of differences. Some methods for estimating systematic differences between raters will be given. The model is illustrated in a worked example.
ASPECTS OF MODELLING NONLINEAR TIME SERIES
(University of Gothenburg, 1992-01-01) Teräsvirta, Timo; Tjøstheim, Dag; Granger, Clive W J
It is common practice for economic theories to postulate non-linear relationships between economic variables, production functions being an example. If a theory suggests a specific functional form, econometricians can propose estimation techniques for the parameters, and asymptotic results, about normality and consistency, under given conditions are known for these estimates, see e.g. Judge et. a1. (1985) and White (1984) and Gallant (1987, chapter 7). However, in many cases the theory does not provide a single specification or specifications are incomplete and may not capture the major features of the actual data, such as trends, seasonality or the dynamics. When this occurs, econometricians can try to propose mort: general specifications and tests of them. There are clearly an immense number of possible parametric nonlinear models and there are also many nonparametric techniques for approximating them. Given the limited amount of data that is usually available in economics it would not be appropriate to consider many alternative models or to use many techniques. Because of the wide possibilities the methods and models available to analyze non-linearities are usually very flexible so that they can provide good approximations to many different generating mechanisms. A consequence is that with fairly small samples the methods arc inclined to over-fit, so that if the true mechanism is linear, say, with residual variance the fitted model may appear to find nonlinearity and the estimated residual variance is less than . The estimated model will then be inclined to forecast badly in the post-sample period. It is therefore necessary to have a specific research strategy for modelling non-linear relationships between time series. In this chapter the modelling process concentrates on a particular situation, where there is a single dependent variable Yt to be explained and:.!..t is a vector of exogenous variables.
Exact Semiparametric Inference About the Within-Subject Variability in 2 x 2 Crossover Trails.
(University of Gothenburg, 1992-03-01) Guilbaud, Olivier
The comparison of primary interest in a 2 x 2 crossover trial typically concerns the effect of the treatments, say A and B, on the mean response level. This article deals with another important aspect, namely the within-subject response variability under A and B. Differences in drug formulation and/or administration may lead to considerable differences in withinsubject variability, whatever is the difference in terms of mean level; and consideration of both these aspects may therefore be of considerable importance for the judgement of the treatments. It is shown that, although there are no within-subject treatment replications, it is possible to make various exact inferences about the AlB ratio of within-subject variances and about the (A - B)difference in mean level, simultaneously and marginally. These inferences are semiparametric in that no distributional assumption is made about the betweensubject variability, whereas a normality assumption is used for the within-subject variability. The inferences include tests, confidence regions, and a multiple test procedure. A power approximation is also given. The results are illustrated numerically.
SEPARATION OF SYSTEMATIC AND RANDOM ERRORS IN ORDINAL RATING SCALES
(University of Gothenburg, 1992-04-01) Svensson, Elisabeth; Holk, Sture
The aim of this paper is to introduce a new rank method which enables us to separate the inconsistency of repeated measurements into random and systematic differences and to quantify this lack of consistency in a few measures. The key of the separation approach is to make a particular type of ranking of the repeated judgements in the same experimental unit. It means that cases which have the same classification from one rater will be internally ranked according to the classifications from the other. This enables us to extract the random variation. The variance of the rank differences between the judgements is a suitable measure of the random interrater variability. The systematic differences are described by empirical measures of relative position and of relative concentration. These measures are normed into the interval [-1,1]. Our method has been applied to several medical rating scales both for construction and analysis. We use one of the data sets as an illustration.
COMPARISON BETWEEN TWO METHODS OF SURVEILLANCE: EXPONENTIALLY WEIGHTED MOVING AVERAGE VS CUSUM
(University of Gothenburg, 1993-01-01) Frisén, Marianne; Åkermo, Göran
When control charts are used in practice it is necessary to know the characteristics of the charts in order to know which action is appropriate at an alarm. The probability of a false alarm, the probability of successful detection and the predictive value are three measures (besides the usual ARL) used for comparing the performance of two methods often used in surveillance systems. One is the "Exponentially weighted moving average" method, EWMA, (with several variants) and the other one is the CUSUM method (Vmask). Illustrations are presented to explain the observed differences. It is demonstrated that a high probability of alarm in the beginning (although it gives good ARL properties) might cause difficulties since a low predicted value makes action redundant at early alarms.
EXACT PROPERTIES OF McNEMAR'S TEST IN SMALL SAMPLES
(University of Gothenburg, 1993-02-01) Jonsson, Robert
The exact distribution of McNemar's test statistic is used to determine critical pOints for two-sided tests of equality of marginal proportions in the correlated 2x2 table. The result is a conservative unconditional test which reduces to the conditional binomial test as a special case. Exact critical points are given for the significance levels 0.05, 0.01 and 0.001 with the sample sizes n=6(1)50. A computer program for tail probabilities makes the calculation of power easy. It is concluded that McNemar's test is never inferior to the conditional binomial test and that much can be gained by using the McNemar test if the main purpose is to detect differences between the marginal proportions in small samples. A further conclusion is that the chi-square approximation of McNemar's test statistic may be inadequate when n<=50. Especially the 5% critical points are constantly too small.
RESAMPLING PROCEDURES IN LINEAR MODELS
(University of Gothenburg, 1993-03-01) Gellerstedt, Martin
We will study here different resampling procedures for creating confidence sets in linear models. A special technique called abstract resampling makes it possible to use the true residuals and the true model for resampling. This may seem to be peculiar since the true residuals contains unknown parameters and thus are non observable; but for each specified parameter value the residuals are observable and can be used for resampling. Furthermore simulating the null distribution of some appropriate statistic gives the possibility to test the accuracy of a hypothetic parameter value. Finally a confidence set can be created by finding the parameter values which can not be rejected. Bootstrapping the true residuals will be called abstract bootstrapping. We will show that the abstract bootstrap method is closely related to a permutation method. A balanced abstract bootstrap method will also be presented, a method which treats the grand mean in linear models and can be applied in ordinary bootstrapping as well. The resampling methods; bootstrap, abstract bootstrap and the permutation method are all closely related. Which method to use is discussed from a practical point of view.
Statistical surveillance of business cycles
(University of Gothenburg, 1994-01-01) Frisén, Marianne
Methods for timely detection of turning-points in business cycles are discussed from a statistical point of view. The theory on optimal surveillance is used to characterize different approaches advocated in literature. This theory is also used to derive a new method for nonparametric detection of turning-points. It utilizes the characteristics of monotonic and unimodal regression. Estimation of parameters m a more or less stable model is thus avoided. Different new ways to evaluate methods are used and discussed. The principles are illustrated by data from Sweden and the USA.
CHARACTERIZATION OF METHODS FOR SURVEILLANCE BY OPTIMALITY
(University of Gothenburg, 1994-02-01) Frisén, Marianne
Different criteria of optimality are discussed. The shortcomings of some earlier criteria of optimality are demonstrated by their implications. The correspondences between some criteria of optimality and some methods are examined. The situations under which some commonly used methods have a certain optimality are thus illuminated. Linear approximations of the LR (likelihood ratio) method, which satisfies several criteria of optimality, are presented. These linear approximations are used for comparisons with other linear methods, especially the EWMA (exponentially weighted moving average) method. These comparisons specify the situations for which the linear methods can be regarded as approximations of the LR method.
VISUAL EVALUATIONS OF STATISTICAL SURVEILLANCE
(University of Gothenburg, 1994-03-01) Frisén, Marianne; Cassel, Claes
A computer program which simultaneously gives visual information on important characteristics is presented. Surveillance, that is continual observation of a time series with the goal of timely detection of possible important changes in the underlying process, is used in quality control, economics, medicine and other fields. When surveillance is used in practice it is necessary to evaluate the method in order to know which action is appropriate at an alarm. The probability of a false alarm, the probability of successful detection and the predictive value are three measures (besides the usual ARL) which are illustrated by the self-instructive computer program.
A COMPARISON OF TWO DESIGNS FOR ESTIMATING A SECOND ORDER SURFACE WITH A KNOWN MAXIMUM
(University of Gothenburg, 1994-04-01) Ekman, Claes
Two level fractional factorial designs with a star are often used when working with lower polynomial models. In this paper an alternative design is discussed and compared with the fractional factorial design. We are working under the assumption that the true underlying model is of second order with a known maximum point.
COMPARING POWER AND MULTIPLE SIGNIFICANCE LEVEL FOR STEP UP AND STEP DOWN MULTIPLE TEST PROCEDURES FOR CORRELATED ESTIMATES
(University of Gothenburg, 1994-05-01) Palaszewski, Bo
We consider hypothesis testing problems arising in e.g. the context of comparing k treatments with a control. The case of equi-correlated estimates is mainly discussed, although also unequal correlated estimates (e.g. unequal sample sizes for the treatments, when compared to a control treatment) are mentioned briefly. So called step down test procedures are compared with step up test procedures, with respect to power functions. Comparisons of rejected null hypotheses are also performed. No practical differences in performances between step up and step down test procedures could be found for finite sample sizes.
CONSTANT PREDICTIVE VALUE OF AN ALARM
(University of Gothenburg, 1994-06-01) Åkermo, Göran
One main purpose of statistical surveillance is to detect a change in a process, often expressed as a shift from one level to another. When a sequence of decisions is made, measures, like the number of decisions that have to be taken before an alarm are of interest. In many situations a shift might occur any time after the surveillance was initiated. Prior knowledge of the probability of a change, the incidence, can become crucial when a method is selected and the parameter values of the method are set. The predictive value of an alarm is a measure of performance that takes this information into consideration and is an important tool for evaluating methods. Mostly an alarm is useful only if its predictive value is large. The predictive value of an alarm is the probability that a change has occurred given an alarm. In this paper it is demonstrated that the incidence in the first point has to be relatively high, or the alarm limits very wide, in order to achieve a predictive value greater than, say 0.5. The interpretation of an alarm is difficult to make if the predictive value of an alarm varies with time. For the ordinary Shewhart method and a selection of Moving Average Methods it is demonstrated how the predictive value increases with time if the incidence is constant. The incidence which would give the methods a constant predictive value are determined. The methods are thus demonstrated to give easily interpreted alarms only if the values of the incidence are strongly decreasing with time. Since in most applications a constant incidence is assumed a modification of the ordinary Shewhart method is suggested. With this modification it is possible to obtain a constant predictive value in the whole range of observations or in some interesting interval.
ON PERFORMANCE OF METHODS FOR STATISTICAL SURVEILLANCE
(University of Gothenburg, 1994-07-01) Åkermo, Göran
Statistical surveillance is used to detect a change in a process. It might for example be a change of the level of a characteristic of an economic time series or a change of heart rate in intensive care. An alarm is triggered when there is enough evidence of a change. When surveillance is used in practice it is necessary to know the characteristics of the method, in order to know which action that is appropriate at an alarm. The average run length, the probability of a false alarm, the probability of successful detection and the predictive value of an alarm are measures that are used when comparing the performance of different methods for statistical surveillance. In the first paper a detailed comparison between two important methods, the Exponentially Weighted Moving Average and the CUSUM, is made. Some consequences of using only the average run length as the measure of performance are demonstrated. Differences between the methods are discussed in regard to the measures mentioned above. The second paper is focused on the predictive value of an alarm, that is the relative frequency of motivated alarms among all alarms. The interpretation of an alarm is difficult to make if the predictive value of an alarm varies with time. Thus conditions for a constant predictive value of an alarm are studied. The Shewhart methods and some Moving Average methods are discussed and some general differences in performance are pointed out. Three different types of Exponentially Weighted Average are discussed and some differences established. It is further stated that if a Fast Initial Response feature is added to a method, this will in general lower the level of the predictive value of an alarm in the beginning of the surveillance. The increased probability of alarm in the beginning might thus be useless.
SATURATED DESIGNS FOR SECOND ORDER MODELS
(University of Gothenburg, 1994-09-01) Ekman, Claes
Construction of saturated designs for different types of second order models are discussed. Also a comparison between two types of saturated designs for the full second order model is presented.
A NOTE ON ROTATABILITY
(University of Gothenburg, 1994-10-01) Ekman, Claes
The need of a measure of rotatability is discussed and exemplified through some examples. The examples also shows the difficulties with measuring rotatability. A graphical technique for exploring the variance function is discussed.
SURVEILLANCE OF RARE EVENTS. ON EVALUATION OF THE SETS METHOD
(University of Gothenburg, 1995-01-01) Arnkelsdóttir, Hrafnhildur
Continual surveillance aiming to detect an increased frequency of some rare event is of interest in several different situations in quality control, medicine, economics and other fields. Examples are continual surveillance of defect articles in a production process or surveillance of a business cycle. Surveillance of rare health events in general and especially surveillance of congenital malformations has been a field of unabating interest during the last decades. Since the Thalidomide episode in the early 60's, several registries of congenital malformations are in operation all over the world. The basic idea is that if a 'catastrophe' occurs an alarm should be signalled as soon as possible after the occurrence. A method developed for this situation is the Sets method that focuses on the intervals between events under surveillance, e.g. intervals between successive births of malformed babies. If a previously defined number of such intervals are 'short' an alarm is triggered. The traditional evaluation measure used when discussing the Sets method is the ARL (Average Run Length). Here, evaluation measures such as the probability of a false alarm, the probability of a successful detection and the predictive value of an alarm are derived and discussed for the Sets method. The information provided by these measures is important for the implementation and use of a system of surveillance in practice.
DETECTION OF GRADUAL CHANGES. STATISTICAL METHODS IN POST MARKETING SURVEILLANCE
(University of Gothenburg, 1995-02-01) Sveréus, Alecka
Surveillance can be viewed as continual observation in time where the goal is to detect a change in the underlying process as soon as possible after it has occurred. In many applications, such as post marketing surveillance, it is of special interest to detect gradual changes in the underlying process. When a drug has been marketed one needs a continuous surveillance of adverse drug reactions. This is now done by different statistical methods suggested by the Food and Drug Administration (FDA), among others. The ability to detect a linear increase by different methods of surveillance is analysed. This is compared to the case where a sudden change to a constant level occurs. Often, as in the FDA recommendations, repeated significance tests are made and this technique of surveillance is identical to the Shewhart method. Different significance tests correspond to different transformations of the observations to the variable to be used in the Shewhart test. This method is evaluated by different measures of goodness as the false alarm probability, the probability of successful detection and the predicted value, for different cases of the critical event. Evaluations of the transformations suggested by the FDA are done in the case of the critical event being a sudden shift to a constant level and for the case of a linear increase. Considerable differences are demonstrated. A method which is optimal to detect a linear increase is derived. It takes into consideration all the data up to the decision time. A linear approximation of this method is derived and the weights in this approximation are studied. A comparison with the linear approximation of the method which is optimal to detect a sudden shift, is made. For the cases studied the Shewhart method approximates the optimal method better when the critical event is a linear change than when it is a shift.
ON SECOND ORDER SURFACES ESTIMATION AND ROTABlLITY
(University of Gothenburg, 1995-03-01) Ekman, Claes
The design of an experiment is an important component when collecting data to gain a deeper understanding of a problem. It is from the data collected that inferential statements concerning some phenomenon have to be made; therefore, we wish to extract as much relevant infonnation as possible from the data collected. Depending on the nature of the problem, good designs may be very different. The special type of problem studied here is the estimation of second order response surfaces. This type of response sutfaces are often used to locally approximate the response in a neighborhood of its maximum. The fIrst of the three papers included in the present study provides a brief overview of one of the most co~on designs of handling this problem. This design is a fractional two-level factorial design augmented with a star. An alternative design, called the complemented simplex design, is developed and compared with the augmented fractional factorial design. It is shown that the simplex design (up tQ six dimensions) is at least as good as the fractional factorial design with respect to a defmed design criterion. The comparison is made within the class of rotatable designs. Unfortunately, it shows that the complemented simplex design cannot be made rotatable in more than six dimensions. The second paper shows how saturated designs can be constructed from the complemented simplex design. These designs are compared with improved Koshal designs (up to six dimensions). Neither design was found to be superior to the other in all dimensions. Also, which design is superior depends on the design criterion. The third paper illustrates the complexity of rotatability and the difficulties in measuring rotatability. A graphical method of presenting degree ofllack of rotatability is presented.

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