Modelling seasonally varying data: A case study for Sudden Infant Death Syndrome (SIDS)

Many time series are measured monthly, either as averages or totals, and such data often exhibit seasonal variability – the values of the series are consistently larger for some months of the year than for others. A typical series of this type is the number of deaths each month attributed to SIDS (Sudden Infant Death Syndrome). Seasonality can be modelled in a number of ways. This paper describes and discusses various methods for modelling seasonality in SIDS data, though much of the discussion is relevant to other seasonally varying data. There are two main approaches, either fitting a circular probability distribution to the data, or using regression-based techniques to model the mean seasonal behaviour. Both are discussed in this paper.     

A seasonal mixed-POT model to estimate high flood quantiles from different event types and seasons

Flood events can be caused by several different meteorological circumstances. For example, heavy rain events often lead to short flood events with high peaks, whereas snowmelt normally results in events of very long duration with a high volume. Both event types have to be considered in the design of flood protection systems. Unfortunately, all these different event types are often included in annual maximum series (AMS) leading to inhomogeneous samples. Moreover, certain event types are underrepresented in the AMS. This is especially unsatisfactory if the most extreme events result from such an event type. Therefore, monthly maximum data are used to enlarge the information spectrum on the different event types. Of course, not all events can be included in the flood statistics because not every monthly maximum can be declared as a flood. To take this into account, a mixture Peak-over-threshold model is applied, with thresholds specifying flood events of several types that occur in a season of the year. This model is then extended to cover the seasonal type of the data. The applicability is shown in a German case study, where the impact of the single event types in different parts of a year is evaluated.     

ARIMA modelling of birth data

In this article the fitting of ARIMA models to time series relating to the births at Edendale Hospital in Natal, South Africa, over a 16-year period is discussed. The model (011)X(011)12 provides andexcellent fit to the monthly totals of mothers delivered but serious discrepancies between estimates of the moving average parameters obtained by the method of unconditional least squares using various statistical computer packages were observed. These can be ascribed to the fact that certain of the packages use the method of backcasting to calculate the unconditional sum-of-squares function in the estimation procedure and this approach breaks down for values of the moving average parameters close to the invertibility boundary. A simulation study to compare the different methods of estimation for the model (011) X (011)12 and to assess the seriousness of discrepancies in the ULS estimates is described. ARIMA models which provided the best fit to the series of monthly totals of caesarean sections performed, breechs births and instrumental deliveries are non-seasonal and reflect a dependence of present on past observations. In contrast the series involving stillbirths and neonatal deaths are white noise indicating that perinatal deaths are random events.     

Log-linear models for correlated marginal totals of a contingency table

The marginal totals of a contingency table can be rearranged to form a new table. If at least twoof these totals include the same cell of the original table, the new table cannot be treated as anordinary contingency table. An iterative method is proposed to calculate maximum likelihood estimators for the expected cell frequencies of the original table under the assumption that some marginal totals (or more generally, some linear functions) of these expected frequencies satisfy a log-linear model.In some cases, a table of correlated marginal totals is treated as if it was an ordinary contingency table. The effects of ignoring the special structure of the marginal table on thedistributionof the goodness-of-fit test statistics are discussed and illustrated, with special reference to stationary Markov chains.     

A Bayesian benchmarking of the Scott–Smith model for small areas

When the finite population ‘totals’ are estimated for individual areas, they do not necessarily add up to the known ‘total’ for all areas. Benchmarking (BM) is a technique used to ensure that the totals for all areas match the grand total, which can be obtained from an independent source. BM is desirable to practitioners of survey sampling. BM shifts the small-area estimators to accommodate the constraint. In doing so, it can provide increased precision to the small-area estimators of the finite population means or totals. The Scott–Smith model is used to benchmark the finite population means of small areas. This is a one-way random effects model for a superpopulation, and it is computationally convenient to use a Bayesian approach. We illustrate our method by estimating body mass index using data in the third National Health and Nutrition Examination Survey. Several properties of the benchmarked small-area estimators are obtained using a simulation study.     

Row similarity in binary-valued matrices

An algorithm is developed for calculating the probability distribution of the number of matches between two specified rows of a matrix of zeroes and ones. Cases covered include row totals fixed, column totals fixed, and column totals and the two specified rows' totals fixed. The results are applied to presence-absence data on six species of ground finches on 23 Galàpagos islands and two constructed examples.     

On trends and constants in periodic autoregressions

Periodic autoregressions are characterised by autoregressive structures that vary with the season. If a time series is periodically integrated, one needs a seasonally varying differencing filter to remove the stochastic trend. When the periodic regression model contains constants and trends with unrestricted parameters, the data can show diverging seasonal deterministic trends. In this paper we derive explicit expressions for parameter restrictions that result in common deterministic trends under periodic trend stationarity and periodic integration.     

On trends and constants in periodic autoregressions

Periodic autoregressions are characterised by autoregressive structures that vary with the season. If a time series is periodically integrated, one needs a seasonally varying differencing filter to remove the stochastic trend. When the periodic regression model contains constants and trends with unrestricted parameters, the data can show diverging seasonal deterministic trends. In this paper we derive explicit expressions for parameter restrictions that result in common deterministic trends under periodic trend stationarity and periodic integration.     

Analysis of adverse events in titration studies

The modeling of the dose–adverse event relationship in clinical studies of drugs that must be titrated is complicated due to confounding dose escalation and exposure time effects. We analyze the dose–adverse events over time of hypotension-related adverse events (dizziness, hypotension, postural hypotension, syncope, vertigo) in placebo-controlled benign prostatic hyperplasia studies of terazosin using two different methods. The first method uses a Cox regression model with time-dependent covariates to evaluate the time-to-first event data. The second method uses a logistic regression model with parameters estimated using generalized estimating equations to analyze multiple events. Doses were assigned to both placebo and terazosin patients according to the titration scheme in each study. Three combined titration-to-efficacy response studies (231 placebo, 230 terazosin patients) had a significant difference in the incidence of hypotension-related adverse events between placebo (7.4%) and terazosin (21.7%); however, they did not exhibit a significant difference between treatments in the rate of adverse events by dose. Applying these methods to a larger, longer duration titration to response study (1031 placebo, 1053 terazosin totals) produced similar results.     

Inference for domains under imputation for missing survey data

The authors study the estimation of domain totals and means under survey‐weighted regression imputation for missing items. They use two different approaches to inference: (i) design‐based with uniform response within classes; (ii) model‐assisted with ignorable response and an imputation model. They show that the imputed domain estimators are biased under (i) but approximately unbiased under (ii). They obtain a bias‐adjusted estimator that is approximately unbiased under (i) or (ii). They also derive linearization variance estimators. They report the results of a simulation study on the bias ratio and efficiency of alternative estimators, including a complete case estimator that requires the knowledge of response indicators.     

Spatial modeling of extreme rainfall in northeast Thailand

It is well recognized that the generalized extreme value (GEV) distribution is widely used for any extreme events. This notion is based on the study of discrete choice behavior; however, there is a limit for predicting the distribution at ungauged sites. Hence, there have been studies on spatial dependence within extreme events in continuous space using recorded observations. We model the annual maximum daily rainfall data consisting of 25 locations for the period from 1982 to 2013. The spatial GEV model that is established under observations is assumed to be mutually independent because there is no spatial dependency between the stations. Furthermore, we divide the region into two regions for a better model fit and identify the best model for each region. We show that the regional spatial GEV model reflects the spatial pattern well compared with the spatial GEV model over the entire region as the local GEV distribution. The advantage of spatial extreme modeling is that more robust return levels and some indices of extreme rainfall can be obtained for observed stations as well as for locations without observed data. Thus, the model helps to determine the effects and assessment of vulnerability due to heavy rainfall in northeast Thailand.     

A calibrated imputation method for secondary data analysis of survey data

In practical survey sampling, missing data are unavoidable due to nonresponse, rejected observations by editing, disclosure control, or outlier suppression. We propose a calibrated imputation approach so that valid point and variance estimates of the population (or domain) totals can be computed by the secondary users using simple complete‐sample formulae. This is especially helpful for variance estimation, which generally require additional information and tools that are unavailable to the secondary users. Our approach is natural for continuous variables, where the estimation may be either based on reweighting or imputation, including possibly their outlier‐robust extensions. We also propose a multivariate procedure to accommodate the estimation of the covariance matrix between estimated population totals, which facilitates variance estimation of the ratios or differences among the estimated totals. We illustrate the proposed approach using simulation data in supplementary materials that are available online.     

Small area estimates of labour force participation under a multinomial logit mixed model

Summary.  A new methodology is developed for estimating unemployment or employment characteristics in small areas, based on the assumption that the sample totals of unemployed and employed individuals follow a multinomial logit model with random area effects. The method is illustrated with UK labour force data aggregated by sex–age groups. For these data, the accuracy of direct estimates is poor in comparison with estimates that are derived from the multinomial logit model. Furthermore, two different estimators of the mean-squared errors are given: an analytical approximation obtained by Taylor linearization and an estimator based on bootstrapping. A simulation study for comparison of the two estimators shows the good performance of the bootstrap estimator.     

Fusing point and areal level space–time data with application to wet deposition

Summary.  Motivated by the problem of predicting chemical deposition in eastern USA at weekly, seasonal and annual scales, the paper develops a framework for joint modelling of point- and grid-referenced spatiotemporal data in this context. The hierarchical model proposed can provide accurate spatial interpolation and temporal aggregation by combining information from observed point-referenced monitoring data and gridded output from a numerical simulation model known as the 'community multi-scale air quality model'. The technique avoids the change-of-support problem which arises in other hierarchical models for data fusion settings to combine point- and grid-referenced data. The hierarchical space–time model is fitted to weekly wet sulphate and nitrate deposition data over eastern USA. The model is validated with set-aside data from a number of monitoring sites. Predictive Bayesian methods are developed and illustrated for inference on aggregated summaries such as quarterly and annual sulphate and nitrate deposition maps. The highest wet sulphate deposition occurs near major emissions sources such as fossil-fuelled power plants whereas lower values occur near background monitoring sites.     

Dynamic Bayesian forecasting of AFL match results using the Skellam distribution

The scoring and defensive abilities of Australian Rules Football teams change over time as a result of evolving player rosters, tactics and other management factors. We develop a dynamic model based on the Poisson difference (Skellam) distribution which simultaneously models the two different point scoring mechanisms in Australian Rules Football, the motivation for which comes from work on predicting outcomes in soccer matches. Our model is developed in a Bayesian framework and is fitted using the Stan modelling language. Model validation is performed on the 2015 Australian Football league (AFL) home and away season.     

Estimation for small areas in household surveys

In many socio-economic surveys the objective is estimation of total or proportion of persons with a particular attribute. Multi-stage area samples are drawn from geographic strata and population within areal units is used as an auxiliary variable in ratio estimation. For large administrative areas, the auxiliary variable totals are available as population projections based on the last census. However, for small areas population changes are significantly affected by non-demographic factors and hence projections with high enough reliability are not available for small areas. In such situations the efficiency of design-based estimators for small areas can be improved by a ratio adjustment based on the auxiliary variable total for a large area. An inequality on the efficiency of the ratio adjusted estimator is established and its bias and variance is investigated     

Calibrated random imputation for qualitative data

In official statistics, when a file of microdata must be delivered to external users, it is very difficult to propose them a file where missing values has been treated by multiple imputations. In order to overcome this difficulty, we propose a method of single imputation for qualitative data that respect numerous constraints. The imputation is balanced on totals previously estimated; editing rules can be respected; the imputation is random, but the totals are not affected by an imputation variance.     

A flexible parametric survival model for fitting time to event data in clinical trials

Time‐to‐event data are common in clinical trials to evaluate survival benefit of a new drug, biological product, or device. The commonly used parametric models including exponential, Weibull, Gompertz, log‐logistic, log‐normal, are simply not flexible enough to capture complex survival curves observed in clinical and medical research studies. On the other hand, the nonparametric Kaplan Meier (KM) method is very flexible and successful on catching the various shapes in the survival curves but lacks ability in predicting the future events such as the time for certain number of events and the number of events at certain time and predicting the risk of events (eg, death) over time beyond the span of the available data from clinical trials. It is obvious that neither the nonparametric KM method nor the current parametric distributions can fulfill the needs in fitting survival curves with the useful characteristics for predicting. In this paper, a full parametric distribution constructed as a mixture of three components of Weibull distribution is explored and recommended to fit the survival data, which is as flexible as KM for the observed data but have the nice features beyond the trial time, such as predicting future events, survival probability, and hazard function.