There is no evidence that MMRPs are effective for prevalent clinical pain conditions. The majority of the evidence remains ambiguous and susceptible to biases due to the small sample size of participants and the limited number of studies included.

Estimation and prediction in generalized linear mixed models are often hampered by intractable high dimensional integrals. This paper provides a framework to solve this intractability, using asymptotic expansions when the number of random effects is large. To that end, we first derive a modified Laplace approximation when the number of random effects is increasing at a lower rate than the sample size. Secondly, we propose an approximate likelihood method based on the asymptotic expansion of the log-likelihood using the modified Laplace approximation which is maximized using a quasi Newton algorithm. Finally, we define the second order plug-in predictive density based on a similar expansion to the plug-in predictive density and show that it is a normal density. Our simulations show that in comparison to other approximations, our method has better performance. Our methods are readily applied to non-Gaussian spatial data and as an example, the analysis of the rhizoctonia root rot data is presented.

Spatial generalized linear mixed models (SGLMMs) are popular models for spatial data with a non-Gaussian response. Binomial SGLMMs with logit or probit link functions are often used to model spatially dependent binomial random variables. It is known that for independent binomial data, the robit regression model provides a more robust (against extreme observations) alternative to the more popular logistic and probit models. In this article, we introduce a Bayesian spatial robit model for spatially dependent binomial data. Since constructing a meaningful prior on the link function parameter as well as the spatial correlation parameters in SGLMMs is difficult, we propose an empirical Bayes (EB) approach for the estimation of these parameters as well as for the prediction of the random effects. The EB methodology is implemented by efficient importance sampling methods based on Markov chain Monte Carlo (MCMC) algorithms. Our simulation study shows that the robit model is robust against model misspecification, and our EB method results in estimates with less bias than full Bayesian (FB) analysis. The methodology is applied to a Celastrus Orbiculatus data, and a Rhizoctonia root data. For the former, which is known to contain outlying observations, the robit model is shown to do better for predicting the spatial distribution of an invasive species. For the latter, our approach is doing as well as the classical models for predicting the disease severity for a root disease, as the probit link is shown to be appropriate. Though this article is written for Binomial SGLMMs for brevity, the EB methodology is more general and can be applied to other types of SGLMMs. In the accompanying R package geoBayes, implementations for other SGLMMs such as Poisson and Gamma SGLMMs are provided.

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ObjectiveTo study trends of infant mortality rate (IMR) and neonatal mortality rate in Greece during the period 2004–2016 and explore the role of sociodemographic factors in the years of crisis.DesignNationwide individual data for live births and infant (0–11 months) deaths provided by the Hellenic Statistical Authority were examined using Poisson, joinpoint regression and interrupted time series (ITS) analyses.SettingGreece.ParticipantsAll infant deaths (n=4862) over the 13-year period, of which 87.2% were born to Greek mothers, and respective live births.Main outcome measuresEvolution of IMR (0–364 days), early (<7 days) neonatal mortality rate (ENMR), late (7–27 days) neonatal mortality rate (LNMR) and post neonatal (28–364 days) mortality rate (PNMR) trends, by maternal nationality, place of residence and Human Development Index (HDI).ResultsBy Poisson regression, overall, during the study period, among infants of Greek mothers, IMR and PNMR declined significantly (−0.9%; 95% CI −1.7% to −0.1% and −1.6%; −3.0% to −0.2% annually, respectively), although differentially by place of residence (IMRurban: −2.1%; −2.9% to −1.3%, IMRrural: +10.6%; 7.6% to 13.6%). By contrast, among infants of non-Greek mothers, the low starting IMR/ENMR/LNMR/PNMR increased significantly (max ENMR:+12.5%; 8.6% to 16.5%) leading to a non-significant time–trend pattern overall in Greece. The inverse associations of HDI with IMR, ENMR and PNMR were restricted to Greek mothers’ infants. Joinpoint regression analyses among Greek mothers’ infants indicated non-significant increasing trends of IMR and ENMR following the crisis (+9.3%, 2012–2016, p=0.07 and +10.2%, 2011–2016, p=0.06, respectively). By contrast, the high (+17.1%; 8.1% to 26.9%, p=0.002) IMR increases among non-Greek infants were restricted to 2004–2011 and equalised to those of Greek mothers’ infants thereafter. ITS analyses in preset years (2008, 2010, 2012) identified significantly increasing trends in IMR, LNMR and PNMR after 2012, and in ENMR after 2010, among Greek mothers’ infants.ConclusionsHDI and rural residence were significantly associated with IMR. The strongly decreasing IMR trends among Greek-mothers’ infants were stagnated after a lag time of ~4 years of crisis approximating the previously sharply increasing trends among non-Greeks.