Journal of Environmental Radioactivity 138 (2014) 227e237

Contents lists available at ScienceDirect

Journal of Environmental Radioactivity journal homepage: www.elsevier.com/locate/jenvrad

Hierarchical modeling of indoor radon concentration: how much do geology and building factors matter? Riccardo Borgoni a, *, Davide De Francesco b, Daniela De Bartolo c, Nikos Tzavidis d a

Department of Economia, Metodi Quantitativi e Strategie d'Impresa, University of Milano-Bicocca, Building U7, Piazza dell'Ateneo Nuovo 1, 20126 Milano, Italy b Research Department of Infection & Population Health, UCL, London, UK c Agenzia Regionale per la Protezione dell'Ambiente della Lombardia, Milano, Italy d Southampton Statistical Sciences Research Institute and Department of Social Statistics and Demography, University of Southampton, UK

a r t i c l e i n f o

a b s t r a c t

Article history: Received 19 January 2014 Received in revised form 14 August 2014 Accepted 31 August 2014 Available online

Radon is a natural gas known to be the main contributor to natural background radiation exposure and only second to smoking as major leading cause of lung cancer. The main concern is in indoor environments where the gas tends to accumulate and can reach high concentrations. The primary contributor of this gas into the building is from the soil although architectonic characteristics, such as building materials, can largely affect concentration values. Understanding the factors affecting the concentration in dwellings and workplaces is important both in prevention, when the construction of a new building is being planned, and in mitigation when the amount of Radon detected inside a building is too high. In this paper we investigate how several factors, such as geologic typologies of the soil and a range of building characteristics, impact on indoor concentration focusing, in particular, on how concentration changes as a function of the floor level. Adopting a mixed effects model to account for the hierarchical nature of the data, we also quantify the extent to which such measurable factors manage to explain the variability of indoor radon concentration. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Indoor radon concentration Hierarchical mixed models Building factors Floor effect

1. Introduction Although evidence since the early 16th century indicates that elevated radon exposure was probably responsible for excess prevalence of lung cancer mortality among miners in some Central European mines (Jacobi, 1993), the attention started focusing on radon exposure in the 1970s with the discovery that, in many countries, some houses had extremely high indoor radon concentrations. Radon occurs both indoors and outdoors, however indoor levels are considerably higher simply because the radon enters into a much smaller atmosphere and tends to accumulate therein. Hence, the primary interest of this paper is in indoor radon concentration (IRC hereafter). Radon has been established to be a Group 1 and Group A human carcinogen, according to the classification used by the International Agency for Research on Cancer (IARC) and by the US Environmental Protection Agency (EPA), respectively. Other studies have aimed at investigating the radon-related increase in incidence rate of cancer

* Corresponding author. E-mail address: [email protected] (R. Borgoni). http://dx.doi.org/10.1016/j.jenvrad.2014.08.022 0265-931X/© 2014 Elsevier Ltd. All rights reserved.

types other than lung cancer (Rericha et al., 2006; Smith et al., 2007) and multiple sclerosis (Bølviken et al., 2003). Monitoring surveys have been promoted in many countries all over the world, The aim of these studies is to assess the exposure of people to this radioactive gas and to identify areas more prone to high IRC. These surveys often provide tools for investigating relevant factors affecting IRC. Understanding these factors is important both in the construction of new buildings (prevention) and in dealing with existing buildings (mitigation or remediation). Elevated IRC levels often originate from the radon content in the underlying rocks and soils and are detected in those dwellings close to the ground, e.g. in detached houses or at the ground floor. Radon may enter in dwellings by diffusion or pressure driven flow if suitable pathways between the soil and living spaces are present. Hence, other soil characteristics such as porosity and permeability can impact on indoor accumulation. Building materials are generally the second main source of indoor concentration. Radon exhalation from building materials depends not only on the radium content, but also on factors such as the fraction of radon produced which is released from the material, the porosity of the material, the surface preparation and the finishing of the walls. Building materials containing by-product

228

R. Borgoni et al. / Journal of Environmental Radioactivity 138 (2014) 227e237

gypsum and concrete containing alum shale may have high radium concentration as well as volcanic tuffs and pozzolana, where radium and thorium content can reach hundreds of Bq/kg. The relationship between radon measurements and housing factors as well as geologic indicators of high radon potential has been widely documented in many countries, see amongst others, Buchli and Burkart (1989), Gates and Gundersen (1992), Gunby
vesque et al. et al. (1993), Tell et al. (1994), Price et al. (1996), Le (1997), Zhu et al. (1998), Apte et al. (1999), Sundal et al. (2004), Shi et al. (2006), Smith and Field (2007), Kemski et al. (2009), Hunter et al. (2009) Ielsch et al. (2010), Borgoni (2011), Borgoni et al. (2011), Cinelli et al. (2011), Fronka (2011), Sahoo et al. €uner et al. (2013). (2011), Ashok et al. (2012), Bra By adopting a mixed model approach, in this paper we show to what degree these factors are relevant for explaining the variability of radon concentration in indoor environments and how much of the pollutant variability remains unexplained. Although it might be expected that IRC tends to decline by moving towards the higher floors of a building, this relationship has been systematically documented only occasionally. An interesting exception is the paper by Antignani et al. (2009) which, however, adopted a methodological approach different from what we propose in this paper. We investigate this particular aspect in some detail showing not only the shape of the relationship between the IRC and the floor level, but also how the variability of this phenomenon which remains unexplained by measured variables, changes from one floor to another. The paper is organized as follows. In the next section we introduce the data. The hierarchical mixed model for modeling IRC is described in detail in Section 3. Section 4 presents the main results of the study. Finally, Section 5 provides some concluding remarks. 2. Data: the Lombardy campaign of 2009e2010 Since radon exposure in dwellings and workplaces is one of the major causes of lung cancer, in 2002 the Italian Ministry of Health approved a National Radon Plan for reducing the risk of lung cancer in the country. This plan consists of a set of coordinated actions aiming at protecting the general population and workers from the health risk due to exposure to radon. Responding to this initiative, the Lombardy Region in Italy planned a first survey in 2003 and a second one which took place in 2009/2010. The more recent survey, which was designed in agreement with regional and national health and environmental organizations, is considered in this paper. For sampling purposes Lombardy municipalities have been stratified into five groups defined by quintiles of the IRC distribution resulted from a survey carried out in 2003/2004 and 22 municipalities were randomly selected within the strata. Within each municipality selected, from 5 to 15 families were randomly drawn from the municipality archive. In order to evaluate how IRC changes as a function of floor, if the building in which the selected family lived consisted of more than one floor, a detector was placed also in the other floors up to the third floor and, also, to the top floor. Hence, for almost each building, more than one observation was obtained corresponding to the rooms at different levels of the same building. The selected buildings should be housing units. Holiday houses, shops, stores, garages and entrance halls to public facilities were not eligible for inclusion in the sample. Similarly, rooms exceeding 300 m3 volume were excluded. Inside a building, detectors were not supposed to be placed in storerooms or similar rooms without ventilation or rarely used. Rooms such as kitchen and bathrooms were also excluded because of the humidity that could affect the measurements. In houses dosimeters were

supposed to be placed in bedrooms. The dataset consists of measurements taken in 721 rooms located in 380 different buildings. Long term measurements, i.e. collected over two consecutive 6month periods, were carried out using CR-39 trace detectors. The detectors, contained in closed plastic canisters, were positioned in situ between September 2009 and November 2010 and changed after 6 months in order to avoid potential saturation. The two semester measures were recorded and the annual average is considered in the present paper. Some summary statistics and the histogram of frequencies are reported in Fig. 1. The arithmetic mean of the annual average concentration was equal to 163 Bq/m3. This is a much higher value than the national average of 111 Bq/m3 (Bochicchio et al., 2005, 2013) and confirmed the relatively high average concentration of this region also found in other studies (Borgoni et al., 2011; Bochicchio et al., 2005). Looking at the histogram, a strong asymmetry of the IRC distribution can be noted. In addition to IRC, for each measurement unit other information (other than the floor) about the building and the rooms were collected by means of a questionnaire administered to dwellers. Measurements recorded by dosimeters and building information were combined into a single dataset. In particular we focused on factors expecting to affect IRC as
vesque et al., 1997; found in previous studies (Price et al., 1996; Le Br€ auner et al., 2013), such as the material from which the walls are made (stone vs. other materials such as lateritious and hollow brick), the daily average hours with open windows, the year of construction or last renovation, the type of building (detached vs. non-detached) and the type of connection with the soil (in direct contact with the ground vs. with basement or crawl space). When entering the variables in the model we prefer to code them as binary or indicator variables for a number of reasons. Some of these variables have a natural binary form such as detached vs. non-detached or in direct contact with the ground vs. not in contact with the ground. The year of construction or last renovation and the daily average hours with open windows have been dichotomized using a cutpoint that best discriminates the IRC between categories (i.e. 1980 and 6 h, respectively). The rationale of this choice comes from the not infrequent finding that dichotomous variables are better at detecting the relationship between variables compared to continuous variables (Farrington and Loeber, 2000). A finer segmentation of explanatory variables can create issues with data sparseness. Furthermore, concerning the daily average time with open windows one has to consider that this variable was collected before the dosimeter was positioned. We believe that people may have a clear perception as to whether they will open a window for a long or for a sort time but a less clear idea about the actual time of the window opening. As radon mainly comes from the soil, we expected to find higher concentrations in lower floors. Table 1 summarizes the IRC distribution conditional on the floor level. As we move from the basement to upper floors, the mean and the standard deviation of IRC tend to decrease. Table 1 also shows that other factors are associated with IRC. Another important feature is the soil composition on which a building is built. The concentration of uranium and radium varies depending on the lithology of a rock. There are also other factors affecting the leak and release of radon which are strongly related to the soil composition like porosity, permeability and presence of faults or fractures. All these factors can facilitate or obstruct the outflow of radon. Geological composition of the Lombardy Region is extremely varied, presenting many lithological and soil typologies. In order to derive this information the data were linked to geologic and a soil maps constructed by Borgoni et al. (2011) using GIS. For analytical purposes we considered 11 geological classes. Fig. 2 (a)

R. Borgoni et al. / Journal of Environmental Radioactivity 138 (2014) 227e237

229

Fig. 1. Summary statistics and histogram of frequencies of indoor radon concentration (Bq/m3).

shows the frequency of the 721 rooms sampled in each of these categories. About one fifth of the rooms is located in the Deposits class, one fifth is located in the Triassic Dolomites class while, overall only the 5% was built on metamorphic rocks. These percentages roughly reflect the proportion of the geological composition of the region. The IRC distribution differs remarkably from one class to another. Fig. 2 (b) shows a boxplot for each of the geological classes where IRC values are averaged over buildings. The higher average concentrations occur for the “Triassic Dolomites”, “Metamorphic” and “Alluvial of Mountain” classes whereas “Moraines” and “Deposits” are characterized by lower concentration. 3. Statistical methodology Multilevel analysis (Goldstein, 2011; Snijders and Bosker, 1999) is a methodology for the analysis of data with complex patterns of variability. Hierarchical modeling is conveniently carried out by resorting to mixed-effects models (McCulloch and Searle, 2001) i.e.

Table 1 Summary statistics of radon concentration by floor and other room (top) and building (bottom) characteristics. N (%) Floor Basement 18 (2%) Ground 215 (30%) Mezzanine/1
317 (44%) 2
129 (18%)
3 þ 42 (6%) Wall material (X1) 0 ¼ Other material 569 (79%) 1 ¼ Stone 148 (20%) Missing value 4 (1%) Opened windows (X2) 0 ¼ 6 h or more 360 (50%) 1 ¼ Less than 6 h 350 (48%) Missing value 11 (2%) Type of building (Z1) 0 ¼ Non-detached 179 (47%) 1 ¼ Detached 199 (52%) Missing value 2 (1%) Type of soil connection (Z2) 0 ¼ Basement/crawl space 145 (38%) 1 ¼ Contact with ground 223 (59%) Missing value 12 (3%) Year construction/last renovation (Z3) 0 ¼ Before 1980 146 (38%) 1 ¼ After 1980 228 (60%) Missing value 6 (2%)

Min Median Arithmetic Max mean

SD

38 14 13 16 20

185 114 67 70 84

348 233 118 134 159

1936 1754 1862 1764 778

456 307 199 226 190

13 31

68 137

136 271

1936 222 1754 335

16 15

68 93

129 199

1497 202 1936 295

15 13

65 87

132 188

1622 214 1936 279

21 13

82 77

136 186

1434 180 1936 296

13 15

80 77

129 187

1109 145 1936 303

statistical regression models which incorporate both fixed effects (that are constant across groups), and random effects (that randomly vary across groups). By associating common random effects to observations in the same group, mixed-effects models flexibly represent the covariance structure induced by the grouping of the data. Multilevel models have been employed on some occasions in the past for modeling IRC (Price et al., 1996; Apte et al., 1999; Gelman, 2006; Bochicchio et al., 2013; Almasri et al., 2009). These papers mainly focused on two-level models aiming at explaining the spatial variability of IRC and mapping IRC values across space to identify radon prone areas. The goal of this section is to construct a three-level model that, taking into account the hierarchical structure of the data described above, expresses how IRC depends on, or is explained by, a set of secondary variables which may refer to each level of the hierarchy and also provide an estimate of the amount of variability which remains unexplained. In particular the considered model allows one to assess the impact of the floor where a room is located on IRC level and variability. 3.1. The hierarchical nature of the data Given the sampling design, IRC measurements taken in different rooms are naturally clustered within buildings. Measurements taken at different floors inside the same building can be expected to be more alike than measurements taken in rooms belonging to different buildings, hence data can be hardly considered uncorrelated. Data with such a structure are termed hierarchical, nested, or multilevel. According to this terminology, rooms may be the level-1 units in a 2-level structure where the level-2 units are the buildings. A third level of hierarchy can be also introduced given that measurements taken in rooms which share the same typology of soil or rock can be expected to be more alike than those which are located on different geological classes. Fig. 3 sketches the hierarchical structure graphically. The existence of this data hierarchy is neither accidental nor ignorable and to ignore this risks overlooking the importance of group effects, and may also render invalid many of the traditional statistical analysis techniques. It is therefore worthwhile to deepen the similarities between units in the same building or lithology and the differences between different building and geological classes. The degree of resemblance between concentrations measured at different floors in the same building can be expressed by the intraclass correlation coefficient. The term “class” is conventionally used to refer to the units in the outer levels of the classification under consideration (i.e. buildings with respect to rooms and lithologies with respect to buildings). For example, the within-

230

R. Borgoni et al. / Journal of Environmental Radioactivity 138 (2014) 227e237

Fig. 2. (a) sample frequency and IRC distribution (b) by lithology.

building variance is the variance within the buildings about their means, while the between-building variance is the variance between the buildings' means. The total variance then can be decomposed as the sum of these two components. The intraclass correlation coefficient is defined as the ratio between the betweenbuildings variance and the total variance, i.e. the proportion of variance accounted by the building (2-level in the hierarchical structure). This value measures the correlation between values of two randomly drawn rooms in the same, randomly drawn, building (Goldstein et al., 2002). When more than two levels exist, as in the data at hand, several kinds of complex correlation structures may occur. In addition to the intra-buildings correlation, an intralithology correlation coefficient can be calculated, i.e. the proportion between the between-lithology variance and the total variance, and represents the correlation between two rooms in the same geological class. The existence of a non-negligible intra-unit correlation and the consequent violation of the assumption of independence suggest that standard OLS multiple regression can lead to incorrect inferences due to biased estimates of the standard errors for the regression coefficients.

In summary, the 3-level hierarchical structure induces three sources of variability and the total variability of IRC (s2) can be decomposed as follows:

s2 ¼ s2v þ s2u þ s2e where s2v is the between-lithologies variability, s2u is the betweenbuildings within-lithologies variability and s2e represents the variability associated to rooms within buildings and lithologies. A detailed discussion of the multilevel model we use is given in the rest of this section and in Section 4. Here we just notice that in our sample the lithology explains 10.2% (¼ s2v =s2 ) of the total variability while 72.9% ð¼ 100ðs2v þ s2u Þ=s2 ) of the total variability is due to the building and lithology levels together. In other words the levelthree intraclass correlation, which expresses the likeness of concentration in rooms in the same geological class, is estimated to be about 0.1, while the intraclass correlation expressing the likeness of IRC in rooms of the same building and, consequently, in the same geological class, is estimated to be about 0.7. A further aspect that clearly emerges is that the building level contributes more to variability than lithology level (given that the

Fig. 3. Graphical scheme of the hierarchical structure of the data.

R. Borgoni et al. / Journal of Environmental Radioactivity 138 (2014) 227e237

between-building variability is considerably larger than betweenlithology variability) and that measurements randomly taken in the same building are much more similar than measurement taken in the same geological class. In any case, the similarity between units belonging to the same grouping level seems considerable and cannot be neglected. Taking into account the manner in which units are grouped within buildings and lithologies when modeling IRC data, has several advantages. Firstly, it enables to obtain statistically efficient estimates of regression coefficients and adjusts the estimated standard errors, confidence intervals and significance tests for the lack of independence. Inference under the multilevel model will be, in general, more conservative (i.e. leading to more trustworthy results) than under the traditional regression model which ignores correlation within groups. Finally, by allowing the use of covariates measured at either the room (for example frequency of window opening) or the building (year of construction) level, it enables to explore the extent to which differences in the IRC are accounted for by measurable factors acting at those levels. This can be done by using a multilevel modeling approach as it is further discussed in the rest of the paper. 3.2. The mixed-effects model Hereafter the following notation, already sketched in Fig. 3, is being adopted. Yijk is the dependent variable measured in room i (i ¼ 1, …, njk) belonging to building j (j ¼ 1, …, Nk) in lithology k (k ¼ 1, …, K). In the case study presented in the next section the logarithmic transformation of IRC is considered since it is largely accepted that IRC follows approximately a log-normal distribution as it was firstly observed by Nero et al. (1986) and subsequently found in many empirical studies (see Gunby et al., 1993, for a more theoretical discussion and Murphy and Organo, 2008, for a comparison of alternative parametric models). Furthermore, model diagnostics (not reported here but available upon request) showed that the usual normal assumption for random effects adopted in multilevel modeling was not appropriate for the data at hand, whereas working on the log-scale produced much more satisfactory results. xijk represents the value of an explanatory variable for room i nested within building j nested in lithology k. Covariates measured at higher level of the hierarchy are denoted by dropping the index associated with the lower levels, for instance zjk denotes an explanatory variable at the building level, whereas wk is an explanatory which varies across lithologies. The explanatory variables are those presented in Table 1, where corresponding encoding is also shown. Since one of the main goals of this paper is to investigate the floor effect, we started by considering a random coefficient model which allows the floor impact to change from floor to floor. A specific notation is also introduced for this variable despite the fact that this is a level-1 variable. Since there are no reasons to consider the relationship between IRC and the floor to be linear, the floor effect is modeled using indicator variables. More specifically, the floor effect is modeled by a set of four indicator variables, namely FB (basement), F1* (first floor), F2* (second floor) and F3* (third floor or above) which take value 1 if the room is located in the respective floor and 0 otherwise. The ground floor is adopted as the reference category. All other explanatory variables are indicator variables that take value 1 in the presence of the corresponding characteristic. The full model can be formally defined using the following equation.

Yijk ¼ b0jk þ b1jk FBijk þ b2jk F1*ijk þ b3jk F2*ijk þ b4jk F3*ijk þ þb5 X1ijk þ b6 X2ijk þ b7 Z1jk þ b8 Z2jk þ b9 Z3jk þ eijk

(1)

231

In order to model the building variability and account for the correlation induced by the hierarchical nature of the data, the intercept is allowed to vary across buildings and thus a random intercept, b0jk, was introduced. In the same way, assuming that building heterogeneity might also impact on the relation existing between floor and log-IRC, random coefficients for the floor variables, bhjk (h ¼ 1,…, 4), were included to allow for this effect by assuming.

b0jk ¼ b00k þ u0jk bhjk ¼ b0h þ uhjk ; with h ¼ 1; 2; 3; 4 where the u terms represent random components and the bs, on the right-hand-side of the equation, represent unknown fixed effects. So, for example, each building in the k-th lithology is characterized by its own specific intercept given by the average intercept (common to all buildings in lithology k) and a buildingspecific random variation u0jk. Finally, to allow for a third level of hierarchy, the intercept was also assumed to vary randomly across geological classes assuming

b00k ¼ b000 þ v0k ; where b000 is the average-population intercept (common to all lithologies) and n0k is the lithology-specific random variation for the generic k-th lithology. This specification introduces three levels of variability identified by the three types of residuals. Level 1 residuals, eijk, are assumed to be mutually independent and normally distributed with zero mean and variance equal to s2e . They represent the unexplained variability of log-IRC. A more complex structure is specified for the within building variability. Indeed buildings are characterized by five random coefficients, from u0jk to u4jk, that encompass the dependences between rooms in the same building and the floor where rooms are located. In other words, these random coefficients allow the difference between the IRC at each floor compared to that at the ground floor to vary across buildings. The vector (u0jk, …,u4jk) is assumed to be uncorrelated with eijk and distributed according to a multivariate normal law with mean 0 and a constant acrossbuildings covariance matrix i.e. varðuhjk Þ ¼ s2uh h ¼ 0, …, 4 and covðuhjk ; uh0 jk Þ ¼ shh0 . Therefore the covariance matrix of the random coefficients for a randomly selected building, denoted by U, has the following structure.

0 s2 u0 B B B U¼B B @

s01 2

su1

s02

s03

s12

s13

s2u2

s23 s2u3

s04 1 s14 C C C s24 C: C s A 34

s2u4

This correlation structure is assumed the same within all possible level-2 units. At the third level, i.e. lithology, only the intercept has a random coefficient, n0k, which represents the deviation of the k-th geological class intercept from the average intercept b000. Residuals n0k are assumed to be drawn from a normal distribution with zero mean, constant variance s2v0 and each other uncorrelated. Furthermore the n0ks are assumed uncorrelated with level-1 and level-2 residuals. s2v0 represents the within-lithologies variability and the corresponding random effect introduces a correlation between concentrations measured within the same lithology as it is discussed later on in this section.

232

R. Borgoni et al. / Journal of Environmental Radioactivity 138 (2014) 227e237

Hence, the specified multilevel formulation accounts for a 3level dependence structure and allows one to consider the natural correlation which may be expected between rooms within the same building, and between buildings within the same lithology unit as it is explained in detail subsequently. We note that the complexity of this model is considerable since it implies a quite large number of parameters: the fixed part has ten main effects, denoted by the bs, whereas the random part consists of 17 variance and covariance parameters (the 15 elements of matrix U, plus s2e and s2v0 ). For estimating the parameters of the multilevel models we used the Restricted Maximum Likelihood (REML) method that is recommended when, as in the present case, the number of clusters is small, or when the cluster sizes are small (Harville, 1977). An iteratively reweighted least squares algorithm is adopted to derive numerically the estimates as described in Bates (2012b). Hypothesis testing on the statistical significance of a single fixed parameter, b, of the model is carried out using a t-test. The test statistic is b b=SEð b bÞ where b b is the estimate of b and SEð b bÞ the associated standard error. This t statistic, under the null hypothesis, has, approximately, a t-distribution with an appropriate number of degrees of freedom which depends on the number of units and the number of variables that are pertinent to the hierarchical level to which the coefficient refers. 3.3. Correlation structure In a three-level hierarchical model, the values for the outcome variable are correlated within the same level-2 and level-3 unit. In addition, using random coefficients for the floor in model (1) implies that this correlation varies according to the floors in which rooms are located. Hence the marginal covariance matrix of the response variable is not diagonal and has a more complex structure. The variance of log-IRC taken in room i randomly selected at the ground floor i.e. for which FBijk ¼ F1*ijk ¼ F2*ijk ¼ F3*ijk ¼ 0, given the value x and z of the other covariates implied by model (1) is.

    var Yijk FBijk ¼ 0; F1*ijk ¼ 0; F2*ijk ¼ 0; F3*ijk ¼ 0; x; z ¼ s2v0 þ s2u0 1 þ 3s2e

(2)

whereas for another room located at the floor h (h ¼ B,1*,2*,3*) one gets

    var Yijk Fhijk ¼ 1; Fh0 ijk ¼ 0; x; z 0

¼ s2y0 þ s2u0 þ s2uh þ 2s0h þ s2e with h sh:

(3)

Analogously the correlation between measures taken in two rooms of the same building varies with the floor. For two different rooms i and i' in the same building and lithology, one at the ground floor and the other at the floor h, the model-implied covariance of log-IRC is.

  cov Yijk ; Yi0 jk ¼ s2y0 þ s2u0 þ s0h

(4)

while, for two rooms at floors h and h' different from the ground floor the model-implied covariance is equal to

  cov Yijk ; Yi0 jk ¼ s2y0 þ s2u0 þ s0h þ s0h0 þ shh0 :

(5)

Similarly the random level-3 intercept introduces a correlation between units in different buildings located in the same geological class (k). In this case the covariance is

  cov Yijk ; Yi0 j0 k ¼ s2y0 :

(6)

Note that this covariance is invariant with respect to the floor at which rooms are located. Taking into account that measurements taken in different lithologies are assumed to be uncorrelated, the correlation structure implied by the above model can be summarized by a block-diagonal matrix with respect to two pairs of rooms, one in lithology k and the other in lithology k'

 V¼

Vk 0

 0 V 0 : k

Vk and Vk' denote the correlation matrix of a randomly selected pair of measurements taken within lithology k and k' respectively and 0 denotes a null matrix. Since the model assumes that the dependence structure is invariant across level-3 units then Vk ¼ Vk'. This matrix can be decompose in a block form as:

 Vk ¼

A B

B A

 (7)

where matrix A is the 5  5, 5 being the number of floors, covariance matrix between two randomly taken rooms within the same building given by equations from (2)e(5), whereas matrix B is the 5  5 covariance matrix between two randomly taken rooms in two different randomly selected buildings within the same geological class. Dividing the entries of these matrices by the square root of the variances of the corresponding log-IRC one can get correlation measures which represent the model-based estimate of the withinbuilding and between-building within-lithology correlation coefficients. It might be observed that, whereas all covariances between measurements taken in different buildings in the same lithology are equal and given by equation (6), the corresponding correlation coefficients change according to the floor where rooms are located since this influences the standard deviations found in the denominator. 4. Hierachical modeling of IRC: results for the Lombardy Region data (Northern Italy) The hierarchical model described in the previous section is hereafter applied to the IRC data of the Lombardy region presented in Section 2. A minor number of records in the dataset have missing data namely, 45 rooms in 22 buildings. These records were omitted from the analysis reducing the sample to 676 rooms in 358 buildings. 4.1. Model estimation and diagnostics We started by estimating a random intercepts model without fixed effects denoted by M0, and move through the sequence of more structured models below:

M0 : Yijk ¼ b0jk þ eijk M1 : Yijk ¼ b0jk þ b1 FBijk þ b2 F1* ijk þ b3 F2* ijk þ b4 F3* ijk þ eijk M2 : Yijk ¼ b0jk þ b1 FBijk þ b2 F1* ijk þ b3 F2* ijk þ b4 F3* ijk þ b5 X1ijk þ b6 X2ijk þ eijk

R. Borgoni et al. / Journal of Environmental Radioactivity 138 (2014) 227e237

Table 3 Fixed effects estimate. “Ground” is used as the reference (baseline) category of the floor variable.

M3 : Yijk ¼ b0jk þ b1 FBijk þ b2 F1* ijk þ b3 F2* ijk þ b4 F3* ijk þ b5 X1ijk þ b6 X2ijk þ b7 Z1ijk þ b8 Z2ijk þ b9 Z3ijk þ eijk M4 : Yijk ¼ b0jk þ b1jk FBijk þ b2jk F1* ijk þ b3jk F2* ijk þ b4jk F3* ijk þ b5 X1ijk þ b6 X2ijk þ b7 Z1ijk þ b8 Z2ijk þ b9 Z3ijk þ eijk where b0jk ¼ b00k þ u0jk, b00k ¼ b000 þ v0k and bhjk ¼ b0h þ uhjk with h ¼ 1, 2, 3, 4 as explained above. Table 2 shows the results concerning the variance component estimates. Model M0 points out that the variability due to rooms (residual variability) and buildings is equal, about 40% of the total variation, whereas a minor component is due to the lithology (20%). In model M1 the floor variable was introduced (only as a fixed effect), whereas in model M2 and M3 other room and building specific variables were included. The results indicate that the main source of level 1 variability amongst the measurable factors is due to floor level. The variables introduced in M2 and M3 had no substantial impact upon the residual variability. The latter variables operated, mostly, at the building and lithology levels. Moving from M0 to M3 a reduction of 18% in the overall variation (i.e. the sum of the three components) is obtained suggesting that indoor radon accumulation is a quite complex phenomenon whose variability can be difficult to dissect and explain by observed characteristics. Finally, in model M4, the random floor coefficients have been introduced to allow the relationship between floor and log-IRC to vary across buildings. Table 2 shows the estimates of the variance components for the series of models from M0 to M4. Table 3 shows the estimates of the fixed coefficients of the final model, M4. The analysis has been performed using the R package “lme4” (Bates, 2012a). In order to evaluate the appropriateness of the model assumptions, residual diagnostics at different hierarchical levels were investigated. These diagnostics, summarized below, show an overall good performance of the model. Fig. 4 (a) plots the values fitted by the model versus the observed values showing a good performance of the model in terms of log-IRC prediction. Had the model specification been correct, one would expect residuals at different levels close to the normal distribution. Normal QeQ plot for level 1 residuals is depicted in Fig. 4 (b). Even though the distribution of the residuals appears to have slightly heavier tails than what it should be expected under normality and a few potential outlying observations, most of the sample quantiles are in agreement with the Gaussian quantiles.

Table 2 Variance component estimates. Level

Random component

M0

M1

M2

M3

M4

Lithology ðs2v0 Þ Building ðs2uh ; h ¼ 0; :::4Þ

Intercept Intercept Basement Mezzanine/1
2
3
þ Residual

0.165 0.343

0.178 0.335

0.146 0.300

0.139 0.294

0.322 53%

0.245 63.5%

0.244 64%

0.244 64.5%

0.122 0.591 0.715 0.466 0.407 0.568 0.082 85%

Room ðs2e Þ Explained Deviance (%)

233

Intercept Floor: (FB) (F1*) (F2*) (F3*) Walls material (X1) Window opening (X2) Type of building (Z1) Soil connection (Z2) Year construction/last renovation (Z3)

(b0) Basement (b1) Mezzanine/1
(b2) 2
(b3) 3
þ (b4) Stone (b5) Less than 6 h (b6) Detached (b7) Contact with ground (b8) After 1980 (b9)

Estimate

Std. error

t-value

p-value

4.54 0.25 0.47 0.50 0.43 0.46 0.11 0.12 0.07

0.15 0.18 0.06 0.07 0.10 0.08 0.05 0.07 0.07

29.62 1.39 8.27 7.69 4.34 5.58 2.21 1.62 0.92

0.000 0.164 0.000 0.000 0.000 0.000 0.028 0.106 0.358

0.09

0.07

1.27

0.204

Fig. 5 shows the distribution of first level residuals conditional on the buildings, graph (a), and lithologies, graph (b). The residuals appear to be centered around zero in all the considered classes and are roughly homoscedastic, although there are a minor number of situations showing a larger variability and a few potential outlying observations. Fig. 6 reports QeQ plots related to the level-2 random intercept and the random first floor coefficient. Overall both the intercept random effect and the first floor effect look close to normality, except for, the right and the left tail of the distribution respectively. The other floor random coefficients showed a similar behavior and are not reported here. 4.2. Interpretation of the model results The estimated intercept is equal to 4.54 which is an estimate of the overall expected log-IRC for the baseline room typology i.e. a room located at the ground floor of a non-detached house, with no-stone walls and frequent opening of windows (more than 6 h a day), built or renovated before 1980 and not in contact with the ground. The log-IRC estimate varies with the floor at which the room is located as displayed in Fig. 7 for the reference and a more radoneprone profile. Curves are parallel given the additive nature of the model. These results indicate that the ground and basement level do not statistically differ, while these two levels differ from the other three levels (1st, 2nd, 3rd floor) but these three levels do not statistically differ from each other. Hence, the model reinforces the idea that the relationship between IRC and the room floor is more like a level effect than linear or non-linear dependence. The estimated fixed parameters have a straightforward interpretation. Since the radon concentration has been modeled on the log scale, each parameter represents the increase or the decrease in the expected log concentration from the baseline typology. At the baseline the expected log IRC is estimated to be 4.54 (the intercept of the model). Moving from the baseline to a room where windows are open less than 6 h, leaving the other characteristics unchanged, increases the expected log concentration by 0.11. Since random effects have been specified, coefficients are allowed to vary randomly from one building to another, possibly, as a consequence of unobserved factors which act at the building level. For example, the variance of the intercept random effect at the building level is 0.591 (see Table 2) and summarizes the variation of the building-specific intercepts from the average intercept. This means that, assuming that the normality of random effects u0jk holds, the building-specific intercepts are centered at 4.54 and the expected IRC ranges between 3.03 and 6.05.

234

R. Borgoni et al. / Journal of Environmental Radioactivity 138 (2014) 227e237

Fig. 4. Scatter plot of observed versus fitted values (a); normal QeQ of residuals (b).

Fig. 5. Boxplot of residuals by building (a) and lithology (b).

On the original scale the fixed effects estimates represent a multiplicative increase or decrease compared to the baseline IRC value. However, if one wants to estimate the expected radon concentration instead, the relation between the normal and the

lognormal distribution has to be taken into account for deriving appropriate estimates. The following results from the normal distribution are used. If X is a normally distributed random variable with expected value m and variance s2, Y ¼ exp(X) is lognormally

Fig. 6. QeQ plots for the 2nd level random intercept (a) and 1st floor (b) effect.

R. Borgoni et al. / Journal of Environmental Radioactivity 138 (2014) 227e237

235

model is necessary when investigating the interplay of different sources of unobserved heterogeneity of IRC measures due to building and lithology factors. In any case, these potential associations amongst the predictors are not evident in our data: all the odds ratios between building-specific characteristics have been found to be statistically not significant. 4.3. Interpreting the lithology and building effects

Fig. 7. Model estimated average log-IRC (with 95% confidence interval) by floor for two different profiles of building.

distributed with expected value equal to exp(m þ s2/2). Since the regression model provides an estimate of m conditional on the values of the covariates on the log scale, the expected value of the radon concentration can be estimated by plugging in the estimated values in the previous formula. For example, in case of the baseline room the expected log concentration is 4.54 as mentioned above, and the variance is estimated equal to 0.795 ¼ 0.122 þ 0.591 þ 0.082 by plugging in equation (2) the estimates of the variance components obtained by model M4 (see Table 2). Hence the estimated expected value of Radon concentration for this room typology is exp(4.54 þ 0.795/2) z 139 Bq/m3. The expected IRC can be estimated in the same way for every desired room typology by selecting the corresponding estimates reported in Tables 2 and 3 For instance a room located at the basement has an average concentration of 192 z exp(0.248 þ 0.94/ 2)  exp(4.54). We obtaining the estimate of the variance by plugging in equation (3) the estimates obtained using model M4, that is b s 2y0 þ b s 2u0 þ b s 2uB þ 2b s 0B þ b s 2e ¼ 0.122 þ 0.715 þ 0.591  2  0.286 þ 0.082 ¼ 0.94. Note that 0.286 is the covariance between the ground and the basement random effect. Maximum likelihood estimates of the covariance matrix U of the random effects is routinely provided by many statistical software. Using these estimates, we found that the expected IRC in a room located in the basement is about 53 Bq/m3 higher than a room located at the ground floor. Using exactly the same calculation as above the expected IRC at the first floor is 78 Bq/m3 that is 61 Bq/m3 lower than the expected IRC at the baseline. In this case, the covariance between the ground and the first floor random effect has been estimated equal to 0.34. The influence of wall materials and the opening time were also found to be statistically significant (p < 0.05). Wall materials, in particular, seem to affect remarkably the value of log-IRC, rooms with stone walls having an expected value roughly 10% higher than the reference profile. Finally soil connection, year of construction or last renovation and whether or not the building is detached have been not found statistically significant (p > 0.05). The role of this set of variables in reducing the unobserved variability is indeed marginal as it can be seen from Table 2. As one of the reviewers of this paper noticed, one may remark here that the factors may be contingent. For example, one would not be surprised that stone buildings are generally older and are connected to the ground in many or most cases, so that a possible “stone wall” effect may be absorbed by the “connection to the ground” effect. Nonetheless, controlling for these potentially relevant factors via a multivariate

In model M4 the lithology only contributes marginally to the overall variability since the lithology level random intercept variance is considerably lower than random intercept variance at the building level. The lithology effect is shown in Fig. 8 where the socalled caterpillar plot is reported. The caterpillar plot displays the estimated random residuals of each lithology versus their rank order alongside a 95% confidence interval. This graph suggests that there are significant differences between lithological typologies with Deposit and Moraines having generally lower IRC and Triassic Dolomites, Alluvial of Mountain, Metamorphic and Detritus having higher IRC. The deviation of the intercept of these latter four geological classes from the average intercept seems to be statistically significant since their confidence interval is strictly positive. The predicted random effect of the Triassic Dolomites, the lithology characterized by the highest residual, is estimated as large as 0.4. This estimate, added to the overall intercept, gives an average logIRC for a room in the reference category and located in this lithology. An estimate of the expected IRC, conditional on this particular lithologic type, can then be obtained by adopting the log-normal correction mentioned above that gives a value as large as exp (4.54 þ 0.4 þ 0.795/2) ¼ 207 Bq/m3. In carbonate rocks like dolomite, radon emanation is generally higher in spite of the low bulk uranium concentration since residual clay coatings at fractures and solution cavities absorb radium and thus have a high radon emanation power from large internal surfaces. This may explain the high effect found in this lithology type. For the Deposits, the least radon prone lithology whose residual is estimated as large as 0.63, an estimate of the average IRC in a room of the reference category is 74 Bq/m3. Table 4 shows the estimated within-building correlation matrix between a pair of rooms randomly selected within the same building. These correlations result from the model estimate of matrix A introduced earlier. As it might have been expected it is

Fig. 8. Caterpillar plots comparing ranked lithology-level random effects along with 95% confidence intervals.

236

R. Borgoni et al. / Journal of Environmental Radioactivity 138 (2014) 227e237

Table 4 Estimated within-building correlation matrix.

RA ¼

B G M/1
2
3
þ

B

G

M/1


2


3
þ

1

0.49 1

0.72 0.55 1

0.80 0.59 0.79 1

0.89 0.51 0.69 0.81 1

found that the measurements taken at different floors of the same building tend to correlate positively and quite strongly with each other, meaning that the higher (lower) the concentration is at a given floor, the higher (lower) the concentration is at any other floor of the same building. Table 5 shows the estimates of the correlation between a pairs of log-IRC measures taken in two rooms within two randomly selected buildings sharing the same lithological typology (they correspond to correlations given by the estimate of matrix B). It appears that these correlations are negligible. Values in matrix RA and RB represent model-based estimates of the intraclass correlation coefficients describing the similarity (or homogeneity) of observed responses within a given level-2 or level-3 unit. Hence the results reported in Tables 4 and 5 reinforce the idea that, in the interplay between lithology and building influence, the latter has a much higher impact on the indoor concentration variability. 5. Conclusion Mainly entering through cracks or holes in foundations and concrete floors as a consequence of pressure differential between buildings and the surrounding soil, radon gas concentrations can reach high indoor levels. However, gas accumulation also depends on building characteristics such as building materials, water supply and on air circulation. In this paper, the effect of various potential influential factors on IRC is evaluated using data collected from an indoor radon gas monitoring survey conducted by the Agency of Environmental Protection of Lombardy Region (Italy) in 2010. The survey was designed to evaluate how the indoor concentration changes as a function of the floor level by measuring IRC at different levels of the same building. Respecting other factors in addition, which are known to be relevant to IRC, this leads naturally to a hierarchical data structure where observations are nested within buildings. Since measures taken in the same geological type are expected to be more alike than measures taken in different geological classes a further level of hierarchy was introduced. Hierarchical modeling based on mixed effect models was adopted for analyzing the data as probably the most versatile tool to analyze such complex situations. A number of explanatory variables acting both at the room level and at the building level were included in the fixed part of the model. Amongst those variables and in accordance with the findings of other studies, window opening and wall materials were found to significantly impact on IRC. Quite surprisingly and

Table 5 Estimated between-building within-lithology correlation matrix.

RB ¼

B G M/1
2
3
þ

B

G

M/1


2


3
þ

0.13

0.14 0.15

0.17 0.18 0.21

0.17 0.18 0.21 0.22

0.15 0.16 0.19 0.19 0.17


vesque et al., 1997; contrarily to what found in previous studies (Le Borgoni et al., 2013) the type of soil connection and the type of building (attached vs detached) were not found to have a significant impact on radon concentrations. The floor was found to have a significant impact on indoor concentrations with the basement and ground floor being much more prone to radon than higher floors. This effect, however, was far from being linear and the decrease in radon concentration as the floor increases tended to level off from the first floor onwards. In addition, by considering random slopes we noted that the floor effect tended to be more variable at lower floors where concentrations were higher although the variability was found to increase again at the top floor possibly due to the scarcity of the sample measures taken at this level. Although fixed effects were found to be statistically significant, these effects managed to explain only a minor part of the variability of radon accumulation, roughly 12%, suggesting that radon accumulation in indoor environments is a quite complex phenomenon which can be hardly explained by measurable architectonic or other factors. Table 2 shows that the building and lithology components (model M0) explain roughly 53% of the overall deviance. Entering the two components in a stepwise manner in the model revealed that the lithology explains only 3% of the overall deviance. The intraclass correlation coefficient of lithology has been found to be small. All these results suggest a minor effect of lithology on IRC with respect to what has been found in other studies (for instance Hunter et al., 2009). This may also be due to the lack of accurateness of the lithology information. The geologic map adopted in this paper to classify lithology is at the scale 1:250,000; this may produce a somewhat fuzzy classification of the lithologic types that may weaken their effect. However, a direct comparison with the results obtained in other papers, where different approaches have been adopted (for instance a fixed effect for the lithology), is not straightforward. In addition and in order to shed some light on possible sources of indoor radon concentration the present paper contributes by identifying potential levers, such as building materials, which can be employed to efficiently protect the population, when at home, in more exposed areas. Finally, when considering buildings already present in the territory, our results help to identify what type of dwellings should be monitored more carefully and which parameters it would be best to intervene on for reducing IRC when necessary. Interestingly we found that, differently to what might be argued by common sense, critical situations can be encountered not only for those dwellings located at the ground floor or at the basements. For example, in a building located on a Triassic dolomites geological background, we expect to find an average IRC up to about 120 Bq/m3 even on the third floor of a building. References Almasri, A., Andersson, E.M., Barregard, L., 2009. A study of residential radon in Sweden using multi-level analysis. Health Phys. 96, 442e449. Antignani, S., Bochicchio, F., Ampollini, M., Venoso, G., Bruni, B., Innamorati, S., Malaguti, L., Stefano, A., 2009. Radon concentration variations between and within buildings of a research institute. Radiat. Meas. 44, 1040e1044. Apte, M.G., Price, P.N., Nero, A.V., Revzan, K.L., 1999. Predicting New Hampshire indoor radon concentrations from geologic information and other covariates. Environ. Geol. 37, 181e194. Ashok, G.V., Nagaiah, N., Shiva Prasad, N.G., 2012. Indoor radon concentration and its possible dependence on ventilation rate and flooring type. Radiat. Prot. Dosim. 148, 92e100. Bates, D., 2012a. Linear Mixed Model Implementation in Lme4. Available at: http:// ftp.uni-bayreuth.de/math/statlib/R/CRAN/doc/vignettes/lme4/Implementation. pdf (accessed 13.08.14.). Bates, D., 2012b. Computation Methods for Mixed Modes. Available at: http://cran.rproject.org/web/packages/lme4/vignettes/Theory.pdf (accessed 13.08.14.). Bochicchio, F., Campos Venuti, G., Piermattei, S., Nuccetelli, C., Risica, S., Tommasino, L., Torri, G., Magnoni, M., Agnesod, G., Sgorbati, G., Bonomi, M.,

R. Borgoni et al. / Journal of Environmental Radioactivity 138 (2014) 227e237 Minach, L., Trotti, F., Malisan, M.R., Maggiolo, S., Gaidolfi, L., Giannardi, C., Rongoni, A., Lombardi, M., Cherubini, G., D'Ostilio, S., Cristofaro, C., Pugliese, M., Martucci, V., Crispino, A., Cuzzocrea, P., Sansone Santamaria, A., Cappai, M., 2005. Annual average and seasonal variations of residential radon concentration for all the Italian regions. Radiat. Meas. 40, 686e694. 
Bochicchio, F., Zuni c, Z.S., Carpentieri, C., Antignani, S., Venoso, G., Carelli, V., Cordedda, C., Veselinovic, N., Tollefsen, T., Bossew, P., 2013. Radon in indoor air of primary schools: a systematic survey to evaluate factors affecting radon concentration levels and their variability. Indoor Air. http://dx.doi.org/10.1111/ ina.12073. Bølviken, B., Celius, E., Nilsen, R., Strand, T., 2003. Radon: a possible risk factor in multiple sclerosis. Neuroepidemiology 22, 87e94. Borgoni, R., 2011. Quantile regression approach to evaluate factors influencing residential indoor radon concentration. Environ. Model. Assess. 16 (3), 239e250. Borgoni, R., Tritto, V., Bigliotto, C., de Bartolo, D., 2011. A geostatistical approach to assess the spatial association between indoor radon concentration, geological features and building characteristics: the Lombardy case, Northern Italy. Int. J. Environ. Res. Public Health 8, 1420e1440. www.mdpi.com/journal/ijerph. Borgoni, R., Tritto, V., de Bartolo, D., 2013. Identifying radon-prone building typologies by marginal modelling. J. Appl. Stat. 40 (9), 2069e2086. €uner, E.V., Rasmussen, T.V., Gunnarsen, L., 2013. Variation in residential radon Bra levels in new Danish homes. Indoor Air 23, 311e317. Buchli, R., Burkart, W., 1989. Influence of subsoil geology and construction technique on indoor air 222Rn levels in 80 houses of the central Swiss Alps. Health Phys. 56, 423e429. Cinelli, G., Tondeur, F., Dehandschutter, B., 2011. Development of an indoor radon risk map of the Walloon region of Belgium, integrating geological information. Environ. Earth Sci. 62, 809e819. Farrington, D.P., Loeber, R., 2000. Some benefits of dichotomization in psychiatric and criminological research. Crim. Behav. Ment. Health 10 (2), 100e122. Fronka, A., 2011. Indoor and soil gas radon simultaneous measurements for the purpose of detail analysis of radon entry pathways into houses. Radiat. Prot. Dosim. 145, 117e122. Gates, A.E., Gundersen, L.C.S., 1992. Geologic Controls on Radon. Geological Society of America, Washington, DC, USA. Special Paper 271. Gelman, A., 2006. Multilevel (hierarchical) modeling: what it can and can't do. Technometrics 48, 432e435. Goldstein, H., Browne, W., Rasbash, J., 2002. Partitioning variation in multilevel models. Underst. Stat. Stat. Issues Psychol. Educ. Soc. Sci. 1 (4), 223e231. Goldstein, H., 2011. Multilevel Statistical Modelling, fourth ed. Wiley, New York. Gunby, J.A., Darby, S.C., Miles, J.C., Green, B.M., Cox, D.R., 1993. Factors affecting indoor radon concentrations in the United Kingdom. Health Phys. 64, 2e12. Harville, D.A., 1977. Maximum likelihood approaches to variance component estimation and to related problems. J. Am. Stat. Assoc. 72, 322e340.

237

Hunter, N., Muirhead, C.R., Miles, J.C.H., Appleton, J.D., 2009. Uncertainties in radon related to house-specific factors and proximity to geological boundaries in England. Radiat. Prot. Dosim. 136, 17e22. Ielsch, G., Cushing, M.E., Combes, Ph, Cuney, M., 2010. Mapping of the geogenic radon potential in France to improve radon risk management: methodology and first application to region Bourgogne. J. Environ. Radioact. 101, 813e820. Jacobi, W., 1993. The history of the radon problem in mines and homes. Ann. ICRP 23, 39e45. Kemski, J., Klingel, R., Siehl, A., Valdivia-Manchego, M., 2009. From radon hazard to risk prediction-based on geological maps, soil gas and indoor measurements in Germany. Environ. Geol. 56, 1269e1279.
vesque, B., Gauvin, D., McGregor, R.G., Martel, R., Gingras, S., Dontigny, A., Le
tourneau, E., 1997. Radon in residences: influences of Walker, W.B., Lajoie, P., Le geological and housing characteristics. Health Phys. 72, 907e914. McCulloch, C.E., Searle, S.R., 2001. Generalized, Linear and Mixed Models. Wiley, New York. Murphy, P., Organo, C., 2008. A comparative study of lognormal, gamma and beta modelling in radon mapping with recommendations regarding bias, sample sizes and the treatment of outliers. J. Radiol. Prot. 28, 293e302. Nero, A.V., Schwehr, M.B., Nazaroff, W.W., Revzan, K.L., 1986. Distribution of airborne radon-222 concentrations in US homes. Science 234, 992e997. Price, P.N., Nero, A.V., Gelman, A., 1996. Bayesian prediction of mean indoor radon concentrations for Minnesota counties. Health Phys. 71, 922e936. Rericha, V., Kulich, M., Rericha, R., Shore, D.L., Sandler, D.P., 2006. Incidence of leukaemia, lymphoma, and multiple myeloma in Czech uranium miners: a casecohort study. Environ. Health Perspect. 114, 818e822. Sahoo, B.K., Sapra, B.K., Gaware, J.J., Kanse, S.D., Mayya, Y.S., 2011. A model to predict radon exhalation from walls to indoor air based on the exhalation from building material samples. Sci. Total Environ. 409, 2635e2641. Shi, X., Hoftiezer, D.J., Duell, E.J., Onega, T.L., 2006. Spatial association between residential radon concentration and bedrock types in New Hampshire. Environ. Geol. 51, 65e71. Smith, B.J., Field, R.W., 2007. Effect of housing factor and surficial uranium on the spatial prediction of residential radon in Iowa. Environmetrics 18, 481e497. Smith, B., Zhang, L., Field, R., 2007. Iowa radon leukemia study: a hierarchical population risk model. Stat. Med. 10, 4619e4642. Snijders, T., Bosker, R.J., 1999. Multilevel Analysis. Sage, London. Sundal, A.V., Henriksen, H., Soldal, O., Strand, T., 2004. The influence of geological factors on indoor radon concentrations in Norway. Sci. Total Environ. 328, 41e53. €nsson, G., Daniel, E., 1994. Geochemistry and Tell, I., Bensryd, I., Rylander, L., Jo ground permeability as determinants of indoor radon concentrations in southernmost Sweden. Appl. Geochem. 9, 647e655. Zhu, H.C., Charlet, J.M., Tondeur, F., 1998. Geological controls to the indoor radon distribution in southern Belgium. Sci. Total Environ. 220, 195e214.