Rejoinder to the discussions of "Spatial+: A novel approach to spatial confounding".

In this rejoinder, we set out some of the main points that we took from the discussions of our paper "Spatial+: A novel approach to spatial confounding." The comments provided by the discussants include excellent questions and suggestions for extensions and improvements to spatial+. The discussions also highlight the growing interest in understanding spatial confounding, underpinned by the many recent contributions to the literature on this topic.

Spatial+: A novel approach to spatial confounding.

In spatial regression models, collinearity between covariates and spatial effects can lead to significant bias in effect estimates. This problem, known as spatial confounding, is encountered modeling forestry data to assess the effect of temperature on tree health. Reliable inference is difficult as results depend on whether or not spatial effects are included in the model. We propose a novel approach, spatial+, for dealing with spatial confounding when the covariate of interest is spatially dependent but not fully determined by spatial location. Using a thin plate spline model formulation we see that, in this case, the bias in covariate effect estimates is a direct result of spatial smoothing. Spatial+ reduces the sensitivity of the estimates to smoothing by replacing the covariates by their residuals after spatial dependence has been regressed away. Through asymptotic analysis we show that spatial+ avoids the bias problems of the spatial model. This is also demonstrated in a simulation study. Spatial+ is straightforward to implement using existing software and, as the response variable is the same as that of the spatial model, standard model selection criteria can be used for comparisons. A major advantage of the method is also that it extends to models with non-Gaussian response distributions. Finally, while our results are derived in a thin plate spline setting, the spatial+ methodology transfers easily to other spatial model formulations.

Longitudinal evaluation of the impact of standardised packaging and minimum excise tax on tobacco sales and industry revenue in the UK.

BACKGROUND: Standardised packaging for factory-made cigarettes (FM) and roll-your-own tobacco (RYO), and a minimum excise tax (MET) were fully implemented in the UK in May 2017 following a 12-month transition period. This paper is the first to examine effects on tobacco sales volumes and company revenues. METHODS: Analysis of UK commercial supermarket and convenience store electronic point of sale data on tobacco sales. FM and RYO products' data (May 2015-April 2018) yielded 107 572 monthly observations. Expected values from additive mixed modelling were used to calculate trends in: (1) volumes of tobacco sold overall, by cigarette type (FM and RYO) and by seven market segments; and (2) company net revenues. A 10-month period (June 2015-March 2016) before the transition to standardised packs was compared with a 10-month period after the introduction of the MET and full implementation of standardised packs (June 2017-March 2018). RESULTS: Postimplementation, the average monthly decline in stick sales was 6.4 million (95% CI 0.1 million to 12.7 million) sticks faster than prelegislation, almost doubling the speed of decline. Sales of cheap FM brands, previously increasing, plateaued after implementation. Company monthly net revenues declined from a stable £231 million (95% CI £228 million to £234 million), prelegislation, to £198 million (95% CI £191 million to £206 million) in April 2018. CONCLUSIONS: The concurrent introduction of standardised packaging and MET in the UK was associated with significant decline in sales and in tobacco industry revenues, and the end of the previous growth in cheap cigarette brands that appeal to young and price conscious smokers.

A shared frailty model for multivariate longitudinal data on adverse event of radiation therapy.

Oral mucositis is an inflammatory adverse event when treating head and neck cancer patients with radiation therapy (RT). The severity of its occurrence is believed to mainly depend on its site and the distribution of a cumulative radiation dose in the mouth area. The motivating study investigating differences in radiosensitivities (mucositis progression) at distinct sites where the severity of mucositis is assessed regularly at eight distinct sites on an ordinal scale results in multivariate longitudinal data and thus poses certain challenges. To deal with the multivariate longitudinal data in this particular setting, we take a time-to-event approach focusing on the first occurrence of severe mucositis at the distinct sites using the fact that the site-specific cumulative radiation dose thought to be the main driver of oral mucositis develops over time. Thereby, we may address multivariate longitudinal processes in a simpler and more compact fashion. In this article, to find out differences in mucositis progression at eight distinct sites we propose a shared frailty model for multivariate parallel processes within individuals. The shared frailty model directly incorporating 'process indicators' as covariates turns out to adequately explain the differences in the parallel processes (here, mucositis progressions at distinct sites) while taking individual effects into account. The parallel result with the one from the previous analysis based on the same data but conducted with an alternative statistical methodology shows adequacy of the proposed approach.

A flexible multivariate random effects proportional odds model with application to adverse effects during radiation therapy.

Radiation therapy in patients with head and neck cancer has a toxic effect on mucosa, the soft tissue in and around the mouth. Hence mucositis is a serious common side effect and is a condition characterized by pain and inflammation of the surface of the mucosa. Although the mucosa recovers during breaks of and following the radiotherapy course, the recovery will depend on the type of tissue involved and on its location. We present a novel flexible multivariate random effects proportional odds model that takes account of the longitudinal course of oral mucositis at different mouth sites and of the radiation dosage (in terms of cumulative dose). The model is an extension of the proportional odds model that is used for ordinal response variables. Our model includes the ordinal multivariate response of the mucositis score by site, random intercepts for individuals, and a nonlinear function of cumulative radiation dose. The model allows to test whether sensitivity differs by mouth sites after having adjusted for site-specific cumulative radiation dose. The model also allows to check whether and how the (nonlinear) effect of site-specific dose differs by site. We fit the model to longitudinal patient data from a prospective observation and find that after adjusting for cumulative dose, upper, lower lips, and mouth floor are associated with the lowest mucositis scores and hard and soft palate are associated with the highest mucositis scores. This implies the possibility that tissues at different mouth sites differ in their sensitivity to the toxic effect of radiation.

Standardised packaging, minimum excise tax, and RYO focussed tax rise implications for UK tobacco pricing.

BACKGROUND: Standardised packaging for factory made (FM) and roll your own (RYO) tobacco was fully implemented in the UK in May 2017. Around the same time, several changes to the tax system were applied (a Minimum Excise Tax (MET) for FM products and tax increases weighted towards RYO products). The tobacco industry claims that standardised packaging will lower prices (a disincentive for quitting) by commoditising the product, yet had itself taken advantage of the previous tax regime to achieve large profits from premium brands while also keeping some products' prices relatively low. Here we evaluate the impact of standardised packaging, the MET and the RYO focussed tax changes on price and industry profitability. METHODS AND FINDINGS: Nielsen electronic point of sale (EPOS) data (May 2015 to April 2018) were used to calculate real (inflation adjusted) monthly price per stick overall, by cigarette type (FM and RYO) and by seven market segments. Trend estimation, using additive mixed models, assessed weighted average price (weighted by volume of sales) and tobacco industry net revenue changes. The beginning and end of the data series were compared in terms of: (a) average monthly price growth, (b) average monthly net revenue growth, and (c) undershifting and overshifting patterns after tax changes. FM and RYO real prices changed little over the 3-year period-overall prices rose by about 1p per stick. There was no evidence of commoditisation with prices of all FM segments (but not RYO) rising faster after the implementation of standardised packaging than immediately beforehand. The prices of the cheapest FM brands rose with the implementation of the MET. RYO price increases did not close the gap to FM pricing levels despite RYO focussed tax increases. Tax changes following the implementation of standardised packaging and the MET were more widely and quickly passed on to smokers in the form of higher prices than the tax change pre-implementation. The main limitations are first that because we do not know the exact mechanism by which Nielsen scales up sample data to provide UK estimates, we could only use data for a set three year period during which the same adjustments are made. Second, the tax and standardised packaging events were sometimes too close in time to separate their consequences statistically. Third, tobacco prices may also be affected by external factors such as changes in smokers' disposable income or availability of electronic nicotine delivery systems. CONCLUSIONS: There was no long-term lowering of tobacco prices after the implementation of standardised packaging as predicted by the industry. The introduction of the MET was successful in increasing the price of the cheapest FM cigarettes and narrowing the price gap between FM brands. The RYO tax increases were, however, insufficient to narrow the price gap between RYO and FM. Overall, undershifting became less extensive indicating that tobacco industry manipulation of the tax system which had previously kept cheap products available had declined. This suggests that standardised packaging and a MET will likely contribute to further declines in UK tobacco use.

Modelling a response as a function of high-frequency count data: The association between physical activity and fat mass.

Accelerometers are widely used in health sciences, ecology and other application areas. They quantify the intensity of physical activity as counts per epoch over a given period of time. Currently, health scientists use very lossy summaries of the accelerometer time series, some of which are based on coarse discretisation of activity levels, and make certain implicit assumptions, including linear or constant effects of physical activity. We propose the histogram as a functional summary for achieving a near lossless dimension reduction, comparability between individual time series and easy interpretability. Using the histogram as a functional summary avoids registration of accelerometer counts in time. In our novel method, a scalar response is regressed on additive multi-dimensional functional predictors, including the histogram of the high-frequency counts, and additive non-linear predictors for other continuous covariates. The method improves on the current state-of-the art, as it can deal with high-frequency time series of different lengths and missing values and yields a flexible way to model the physical activity effect with fewer assumptions. It also allows the commonly made modelling assumptions to be tested. We investigate the relationship between the response fat mass and physical activity measured by accelerometer, in data from the Avon Longitudinal Study of Parents and Children. Our method allows testing of whether the effect of physical activity varies over its intensity by gender, by time of day or by day of the week. We show that meaningful interpretation requires careful treatment of identifiability constraints in the light of the sum-to-one property of a histogram. We find that the (not necessarily causal) effect of physical activity on kg fat mass is not linear and not constant over the activity intensity.

Resources allocation in healthcare for cancer: a case study using generalised additive mixed models.

Our aim is to develop a method for helping resources re-allocation in healthcare linked to cancer, in order to replan the allocation of providers. Ageing of the population has a considerable impact on the use of health resources because aged people require more specialised medical care due notably to cancer. We propose a method useful to monitor changes of cancer incidence in space and time taking into account two age categories, according to healthcar general organisation. We use generalised additive mixed models with a Poisson response, according to the methodology presented in Wood, Generalised additive models: an introduction with R. Chapman and Hall/CRC, 2006. Besides one-dimensional smooth functions accounting for non-linear effects of covariates, the space-time interaction can be modelled using scale invariant smoothers. Incidence data collected by a general cancer registry between 1992 and 2007 in a specific area of France is studied. Our best model exhibits a strong increase of the incidence of cancer along time and an obvious spatial pattern for people more than 70 years with a higher incidence in the central band of the region. This is a strong argument for re-allocating resources for old people cancer care in this sub-region.

Modelling fat mass as a function of weekly physical activity profiles measured by actigraph accelerometers.

We show results on the Avon longitudinal study of parents and children (ALSPAC) using a new approach for modelling the relationship between health outcomes and physical activity assessed by accelerometers. The key feature of the model is that it uses the histogram of physical activity counts as a predictor function, rather than scalar summary measures such as average daily moderate to vigorous physical activity (MVPA). Three models are fitted: (1a) A regression of fat mass at age 12 (N = 4164) onto the histogram of accelerometer counts at age 12; (1b) A regression of fat mass at age 14 (N = 2403) onto the histogram of accelerometer counts at age 12 and (1c) a regression of fat mass at age 14 (N = 2413) onto the accelerometer counts at age 14. All three models significantly improve on models including MVPA instead of the histogram and improve the goodness of fit of models (2a), (2b) and (2c) from R(2) = 0.267, 0.248 and 0.230 to R(2) = 0.292, 0.263 and 0.258 for models (1a), (1b) and (1c) respectively. The proportion of time spent in sedentary and very light activity (corresponding to slow walking and similar activities) has a positive contribution towards fat mass and time spent in moderate to vigorous activity has a negative contribution towards fat mass.