The study was designed as a cross-sectional multilevel study. In addition to individual variables, contextual variables were taken into account to explain an outcome assessed at the individual level. The individual variables were obtained from the Brazilian Oral Health Survey - SBBrasil 2010 , and the contextual variables were collected at the municipal level from official public databases.
SBBrasil 2010 was a national epidemiological survey on oral health funded by the Ministry of Health. For representation of the complete Brazilian population, individuals aged 5 and 12 years and those in age groups 15-19, 35-44, and 65-74 years from 177 Brazilian municipalities were evaluated. Sampling was carried out at different domains of the state capitals, federal district, and municipalities within defined geopolitical macro-regions (North, Northeast, Central West, Southeast, and South), using probabilistic sampling in multiple stages with a design effect (DEFF) of 2. The primary sampling units were: (a) municipality, for the interior of the regions, and (b) census tract for the state capitals. Interviews and oral examinations were conducted in the subjects' homes. Oral examinations were performed under natural light, by trained and calibrated examiners (Kappa ≥ 0.65), using a handheld computer to record the data. The diagnostic criteria of Oral Health Surveys: Basic methods (4th edition) from World Health Organization (WHO) were used . In addition to assessment of the individual’s oral condition, interview was conducted with each household and comprised questions related to the socioeconomic profile of the family, use of dental services, self-reported oral morbidity, and self-perception of oral health. Details of the methodology used in SBBrasil 2010 have been described in a previous study . In the present study, data of 9,779 individuals in SBBrasil 2010 between the ages of 35 and 44 years were used, which is the standard age group for evaluation of oral health conditions in adults .
Contextual variables were collected from official public databases for each of 177 participating municipalities of SBBrasil 2010: Demographic census of 2010 by the Brazilian Institute of Geography and Statistics (IBGE) ; Atlas Brazil of the United Nations Development Program (UNDP) ; National Survey of Basic Sanitation of IBGE ; and the Department of Informatics of the Unified Health System (DATASUS) . In the databases of Atlas Brazil  and National Survey of Basic Sanitation , data of IBGE 2010 demographic census were acquired between August 1, 2010 and October 30, 2010 from 316,574 census tracts with predefined territorial boundaries .
In this study, the dependent variable was total number of missing teeth defined as any natural tooth missing due to extraction, for any reason corresponding to codes 4 and 5 of the DMFT index (decayed, missing and filled teeth) . This was assessed according to its discrete numerical nature whose values are whole numbers (counts).
The conceptual model for inequities in oral health of Watt & Sheiham (2012)  was building based on Conceptual Framework for Action on the Social Determinants of Health (CSDH) . In our study, that model was used for the grouping of contextual and individual independent variables. This theoretical model takes into account the social determinants of inequalities in oral health, in contrast to preventive approaches, that focus on the behavioral changes of the individual. According to this conceptual model, the variables that influence the oral health can be grouped into structural determinants (socioeconomic & political context) and intermediary determinants (socioeconomic position, behavioral & biological factors, and health services) (Figure 1).
In the socioeconomic & political context, all contextual variables were included: geographical location of the municipality (capital; interior) , Municipal Human Development Index (MHDI) (very high; high; medium/low) , Gini coefficient , percentage of the population with access to garbage collection , percentage of the population with access to a bathroom and piped water , coverage of oral health teams , number of dentists per 1000 inhabitants , and public water fluoridation (yes; no) . MHDI reflects composite information on income, education level, and longevity in each municipality, and the scores are on a scale from 0 to 1, where higher values reflect a higher level of human development. Gini coefficient measures inequality in income distribution on a scale from 0 (absolute equality) to 1 (absolute inequality) . The percentage of the population with access to garbage collection refers to the proportion of the population of each municipality with access to public garbage collection services . The percentage of the population with access to a bathroom and piped water refers to the proportion of households in the municipality with simultaneous access to water supply (running water) by the distribution network, and bathroom or toilet facilities exclusively for residents . The coverage of oral health teams refers to the proportion of the population in the municipalities that receive primary care of oral health teams . All contextual variables were analyzed as quantitative data expressed as numbers, except variables of the geographic location of the municipality, MHDI, and public water fluoridation.
In the socioeconomic position, individual variables were included as follows: declared skin color (white; yellow/black/brown/indigenous), education level (years of study), and family income in USD (> 2,557; 853-2,556; 285-852; ≤ 284); and the minimum wage at the time of data collection was USD 290.0.
In relation to behavioral & biological factors, individual variables were included as follows: sex (female; male), age (years), self-perception of the need for treatment (yes; no), and pain in the teeth and gums in the last 6 months (no; yes). Also at this level, considering health services, the following individual variables were included: previous use of dental service (yes; no), time since last consultation (≤ 1 year; > 1 year; no previous use of dental service), reason for consultation (review/prevention; oral health problems; no previous use of dental service), type of dental service (public; not public; no previous use of dental service), and satisfaction with dental services (satisfied; dissatisfied; no previous use of dental service).
To explore the dependent variable, a map was drawn with the average number of lost teeth for each one of the five Brazilian geopolitical macroregions, state capitals, and federal district. For each Brazilian macroregion, besides mean teeth lost, a confidence interval of 95% (95% CI) was estimated corrected by Deff. Geographic information system (GIS)-based Quantum GIS (QGIS) software (General Public License; GNU), which is freely available online, was used for manipulation of spatial data and construction of a map.
The data relating the individual and contextual variables was initially organized in the statistical software Predictive Analytics Software (SPSS/PASW®) version 18.0 for Windows®. The descriptive analysis of the contextual variables used only the municipal data. The descriptive analysis of the individual variables was conducted according to the need of correction for the effect of sample design, because they are from samples by conglomerates. For such, the Complex Samples module was used, considering the weights resulting from the sampling process adopted. Measures of central tendency and variability were estimated for the numerical independent variables and simple (n) and relative (%) frequencies for categorical independent variables. The association between the total number of lost teeth and the individual characteristics was verified by the non-parametric tests: Spearman correlation (ƿ) for numerical independent variables; Mann-Whitney test for dichotomous independent variables; and Kruskall-Wallis test for the polytomous independent variables.
The data was exported to the STATA® software, version 14.0, and the Multilevel Hierarchical Negative Binomial Regression (stepwise backward method) model was used with use of contextual and individual data. The Negative Binomial Regression model is appropriate when the dependent variable is quantitative and with non-negative, integer values (counting data) and when there is overdispersion in the data (the variance of the dependent variable is greater than the mean) . Before starting the modeling, the adequacy of the dependent variable for this regression modality was verified and confirmed. For estimation of adjustment between outcome (total number of teeth lost) and the independent variables from first (contextual) and second (individual) levels of analysis, the fixed effect model was used . Initially, an empty model was used with only a random intercept and the dependent variable, without the others variables. Subsequently, a hierarchical block design was used as proposed by the adopted theoretical model  (Figure 1). Model 1 included only the contextual variables (socioeconomic & political context). All eight contextual variables adopted in our study were included in this first model. Adjustment was made in Model 1 and only the contextual variables that were significantly associated with the outcome (p ≤ 0.05) were maintained. From the second model, the individual variables were taken into account. Model 2 kept the contextual variables adjusted in model 1, and added the socioeconomic position. This model was also adjusted (p ≤ 0.05). Model 3 comprised the variables adjusted in models 1 and 2 and added behavioral & biological factors and health services. This final model was adjusted again (p ≤ 0.05). The menbreg, irr function was used to obtain the Mean Ratio (MR) and its 95% CI.
The analysis of the fit of the models was performed through Deviance, obtained through the Log Likelihood multiplied by (-2), where it is expected that there will be significant differences between the models (difference greater than 3.84) . In addition, multicollinearity was tested by verifying the correlations of independent variables, with no values above 0.7 being identified.
SBBrasil 2010 was conducted according to the ethical principles of the Resolution of the National Health Council (CNS; number 196/96), related to research on human beings; it was approved by the Research Ethics Committee of the Ministry of Health and registered at the National Research Ethics Committee of Brazil (CONEP) (CNS approval number: 15.498/2009). In all participants, information on the research was provided, and signed informed consent forms were obtained .