Analysing radon accumulation in the home by flexible M-quantile mixed effect regression

Abstract

Radon is a noble gas that occurs in nature as a decay product of uranium. Radon is the principal contributor to natural background radiation and is considered to be one of the major leading causes of lung cancer. The main concern revolves around indoor environments where radon accumulates and reaches high concentrations. In this paper, a semiparametric random-effect M-quantile model is introduced to model radon concentration inside a building, and a way to estimate the model within the framework of robust maximum likelihood is presented. Using data collected in a monitoring survey carried out in the Lombardy Region (Italy) in 2003–2004, we investigate the impact of a number of factors, such as geological typologies of the soil and building characteristics, on indoor concentration. The proposed methodology permits the identification of building typologies prone to a high concentration of the pollutant. It is shown how these effects are largely not constant across the entire distribution of indoor radon concentration, making the suggested approach preferable to ordinary regression techniques since high concentrations are usually of concern. Furthermore, we demonstrate how our model provides a natural way of identifying those areas more prone to high concentration, displaying them by thematic maps. Understanding how buildings’ characteristics affect indoor concentration is fundamental both for preventing the gas from accumulating in new buildings and for mitigating those situations where the amount of radon detected inside a building is too high and has to be reduced.

Appendices

Appendix A: Preliminary data analysis

Hereafter some preliminary data analyses is reported that motivates the need for a robust approach when modelling IRC data. To this aim an ordinary random effect model for the mean IRC that reflects the hierarchical structure of the data with buildings nested in the geological classes has been fitted using the function lmer of the R package lme4. Figure 9a shows the normal qq-plot of the individual residuals (i.e. residuals pertinent to the building level) whereas Fig. 9b displays the normal qq-plot of the residuals estimated from the model at the geological class level. These plots show that the normality assumptions of the ordinary mixed model are violated, which is also confirmed by the Shapiro-Wilk test (p values=0.0000078 for the geological class residuals and p value= 2.2e−16 for the building residuals). Figure 10a shows the histogram of the standardised building residuals obtained by the random effect regression model, whereas Fig. 10b displays the distribution of the standardised residuals by geological classes. The histogram appears very skewed and some classes have many large positive residuals (larger than 2). Thus, influential observations seem to be present in the data. This is also confirmed by Fig. 11 that displays the Cook’s Distance for the two sets of residuals.

Fig. 9

QQ-plot of building residuals (a) and of geological class residuals (b) estimated by the two-level random effect regression model for the mean IRC

Fig. 10

Histogram of standardised building residuals of the two-level random effect regression model for the mean IRC (a); boxplots of standardised building residuals by geological classes (b)

Fig. 11

Cook Distance of building residuals (a) and of geological class residuals (b) estimated by the two-level random effect regression model for the mean IRC

It is clear that the data may contain outliers and influential points that invalidate the Gaussian assumptions. In these circumstances, estimates of the model parameters are biased and inefficient and the robust approach suggested in this paper sounds more appropriate.

Appendix B: Additional results for modelling geocoded radon data

Appendix 1 provides a short comparison of the estimated parameters obtained from quantile and M-quantile regression models. The two approaches cannot be directly compared since they target different location parameters. However, both approaches try to model location parameters that are related to the same part of the conditional distribution of IRC. Table 5 reports the estimated regression coefficients for q = 0.5 for two approaches: (1) the proposed semiparametric M-quantile random effect regression model (semiMQRE), and (2) a semiparametric quantile regression model (semiQR). semiQR is based on an additive quantile regression model (Koenker et al. 1994) where the spatial structure is captured by bivariate splines but without accounting for the hierarchical structure in the data by a random component. The results indicate that the coefficients based on M-quantile regression models are in the same direction as the ones based on quantile regression. However, with quantile regression convergence problems of the algorithm sometimes occurred. On the other hand, estimation with the M-quantile approach was smoother but the interpretation of the estimated parameters is more difficult.

Table 5 Results—Semiparametric M-quantile and quantile regression models for q = 0.5: Point estimates with standard errors in parentheses

Finally, Fig. 12 presents the estimated effects obtained from M-quantile and quantile-mixed regression models by quantile for each explanatory variable that is considered in the model. In particular, the solid line represents the proposed semiparametric M-quantile random effect regression model and the dashed line stands for an additive quantile regression model (Geraci 2018) which includes a bivariate spline to capture the spatial structure as well as random effects to account for the hierarchy of the data (fitted by the R package aqmm). Note that we only plot the point estimates (without the point-wise 95% confidence intervals) in order to avoid an overload of Fig. 12. The results confirm that the results based on both models are in the same direction.

Fig. 12

Estimated coefficients of quantile regressions (dashed line) and M-quantile regressions (solid line): a intercept, b fault distance, c floor material, d wall material, e year from construction/last renovation, f single building, g not in contact with the ground, h air conditioning system