br Mobile phone use was estimated from the national
Mobile phone use was estimated from the national number of cel-lular mobile phone subscriptions obtained and was obtained from the United Nations specialized agency for information and communication technologies (ITU, 2016).
data were obtained after a Data Access Agreement with ONS was put in place prior to release of these data to the researcher.
2.2. Statistical methodology
Bayesian structural time-series models in which the explanatory variables are functions of time and the parameters are time-varying were used. The methodology is described in detail elsewhere (Broderson et al., 2015; Scott and Varian, 2014) and was used in pre-vious analyses which informed the current analyses (De Vocht, 2016). In summary, structural time-series models are based on state-space models, which distinguish between a state equation that describes the transition of a set of latent variables from one time point to the next and an observation equation that specifies how a given system state relates to measurements (Broderson et al., 2015). Kalman filtering is used for time series decomposition and the errors of different state-component models are assumed to be independent. Bayesian spike-and-slab priors are placed on the regression coefficients to enable selection of pre-dictors, with the “spike” determining the probability of a non-zero coefficient based on independent Bernoulli distributions, and the “slab” a weakly informative Gaussian prior with a large variance (George, Mcculloch, 1997). The model uses Bayesian model averaging to com-bine results and includes a regression component which enables the construction of a synthetic time series (i.e. the counterfactual). The posterior predictive density is then a joint distribution over all coun-terfactual data points (Broderson et al., 2015), and through subtraction of the counterfactual from the measured time series at each point in time, a semiparametric Bayesian posterior distribution for the causal effect is obtained.
Prior distributions for the variance are set as Gamma distributions and posterior simulation is done using a Gibbs sampler and Kalman filter (Durbin and Koopman, 2002) to simulate from a Markov chain with a stationary distribution. Priors for σ, initial value for σ and its upper value were set to 20%, 20% and 150% of the standard deviations in the outcome prior to forecasting of the counterfactual (1985–2005; see below), and the prior and initial values for μ were set to the stan-dard deviation and first value of this Tunicamycin period. Prior probability of in-clusion of each covariate was set to 50% and the prior degrees of freedom to 20 (i.e. the number of years for modelling minus 1), and the prior expected explained variance and expected model size were set to 77% and 5 covariates, respectively, based on an initial trial run. All analyses were based on 100,000 Markov Chain Monte Carlo (MCMC) iterations to satisfy all diagnostic criteria (Heidelberger-Welch diag-nostic stationary test, Heidelberg-Welch halfwidth, and Durbin-Watson tests), Raftery and Lewis’ diagnostic (Dependence factor < 5), Geweke test < 1.96 , and visual inspection of ACF/PACF autocorrelation plots). Bayesian tail-area probabilities were calculated and interpreted as classical p-values (Vehtari and Ojanen, 2012).
A 10-year lag between the introduction of mobile phones (for which, similar to previous analyses, the year 1995 was used, when the penetration rate in the UK passed 10% (ITU, 2016)) and it was assumed that any effect on the incidence rates would be measurable; in other words, time series of the years 1985–2005 were used to create the counterfactual and differences between the 2006–2014 measured numbers of annual newly diagnosed cases and the counterfactuals was interpreted as the causal effect. Following this, for those outcomes where a causal effect was observed, inclusion of the national mobile phone penetration rates should, if an important putative factor, explain, at least in part, observed excesses.
Sensitivity analyses were conducted assuming 0, 5 and 15-year lags to test this hypothesis (Online supplementary materials).
Modelled causal effects for annual number of newly diagnosed cases of Glioblastoma multiforme (GBM) for different anatomic brain regions (1985–2014) assuming 10-year lag between exposure and measurable effect.
Glioblastoma (multiforme) ICD 9/10 Incident cases Cumulative Causal 95% Credible Bayesian tail-area
Results of the assessments of the time-series of annual newly diag-nosed cases of GBM in anatomical regions of the brain compared to their forecasted counterfactuals are shown in Table 1. The number of newly diagnosed cases of GBM was higher than expected in the Frontal (+35.8% [95% Bayesian Credible Interval (BCI) -7.7%, 76.7%]; Bayesian tail-area probability (p-value) 0.05) and Temporal (+37.6% [95%BCI -6.6%, 77.6%]; p-value 0.05) lobes, as well as in the Cere-bellum (+58.5% [-0.0%, +120.3%]; 0.03). Sensitivity analyses (Online supplementary material Table S1) indicate that these effects were not present for other modelled lag periods. These sensitivity analyses also indicated a deviance from the counterfactual for malig-nant neoplasm in the Cerebrum, but only when no lag was assumed (positive effect) and for 15-year modelled lag (negative effect).