Data from the present analysis came from the 2010 National Oral Health Survey (SB Brazil 2010) conducted by the Brazilian Ministry of Health in Brazilian urban areas  between February and November 2010. The sample was obtained through the random selection of municipalities and census sectors, via multi-stage cluster sampling with probability of selection proportional to population size. Detailed information on the methods is found in other publications [40, 41]. Data for adults aged between 35 and 44 years were used in this study.
Individual interviews using a structured questionnaire was used to obtain demographic and socioeconomic characteristics. Oral health examinations were conducted in people’s homes by calibrated dentists (kappa>0.65) under natural light following the guidelines of the WHO manual for epidemiological studies . The DMFT (Decayed, Missing, and Filled Teeth) index was used to determine tooth status.
The outcome variable was the number of teeth (discrete quantitative variable) that was missing for any reason, determined by the sum of codes 4 and 5 of the DMFT index.
Education and income were used as measures of socioeconomic position at the individual level. Income was measured as total income received by all family members in the month preceding the survey (in seven categories from “R$250.00 or less” to “R$9500.00 or more”). For our analysis, the monthly household income was converted into multiples of the minimum wage, based on the current value at the time of survey (1 minimum wage = R$ 510.00, USD$303.57) and collapsed into four categories: up to 1, 1 to 2.9, 3 to 4.9, 5 or more times the minimum wage. This grouping was defined to distingue mechanisms in which income exerts its effects. Education was asked as the number of years of formal schooling and classified as less than four years (insufficient education), 4-7 (incomplete elementary education), 8-10 (completed elementary, but incomplete secondary education), and 11 or more (completed secondary, incomplete university education, or college graduate), according to the formal education system in Brazil.
We used the Municipal Human Development Index (HDI) as an indicator of municipal area socioeconomic status. The Brazilian HDI considers three dimensions: Longevity, Education (access to knowledge, based on average years studied of the population), and Income (living standards and purchasing power of the population according to the Municipal Gross Income per capita) . The HDI was obtained from the 2013 Brazil Atlas of Human Development, which allows a selection based on data extracted from the 2010 demographic. The values of HDI of sampled municipalities ranged from 0.481 to 0.847, and the groups are defined as very low (0-0.499), low (0.500 – 0.599), medium (0.600 – 0.699), high (0.700 – 0.799) and very high (0.800 – 1.00). According to this classification, the frequency of sampled municipalities was: low and very low (3.93%), medium (7.88), and high and very high (88.19). In this study municipalities were aggregated into low (<0.699) versus high (>0.70)
The covariables at individual level were age (adult: 35-39 and 40-45 years old), sex (female, male), skin color (white, black, yellow and brown/Ameridians), time since the last dental visit (< 12 months, between 1 and 2 years, > 3 years, did not visit). Skin color refers to the classification adopted in the demographic census performed in Brazil: whites, blacks, browns, yellows, and Amerindians. In this study, this variable was dichotomized: white versus blacks, browns, yellows, and Amerindians.
At the contextual level, we included the presence of fluoridated water supply (present or absent). The data regarding fluoridation was obtained on the National Basic Sanitation Survey performed by the Brazilian Institute of Geography and Statistics (IBGE) in 2008 . We also included the estimated coverage of the population by primary care oral health services, which corresponds to the mean monthly number of primary care oral health teams for every 3000 individuals to the total population of the municipality in the analyzed year. Higher oral health services coverage indicated higher potential access to basic dental services. The cut off for this variable was 40% that was the goal to be achieved in the biennium 2010/2011. Data about coverage were obtained from the website of the Department of Information Technology of the Unified Health System (DATASUS).
Comparison of income and education-based inequalities between municipalities with high and low HDI
Descriptive analysis was performed to obtain mean tooth loss for each municipality, and the results were shown separately according to the HDI level (high or low). The magnitude of relative and absolute educational and income-based inequalities in the tooth loss was calculated using the Relative Index of Inequality (RII), Slope Index of Inequality (SII) and Relative (RCI) and Absolute Concentration Index (ACI) for municipalities with high and low HDI. RII and SII are summary measures recommended when making comparisons across populations . These indices are regression-based and take the whole socioeconomic distribution into account, rather than only comparing the two most extreme groups. For municipalities with high and low HDI, the population in each education or income category was assigned a modified ridit-score based on the midpoint of the range in the cumulative distribution of the participants in the given category. We used generalized linear models (log-binomial regression), with a logarithmic link function to calculate RIIs (rate ratios) and with an identity link function to calculate SIIs (rate differences) . Both indices were estimated with 95% confidence intervals. The RII can be interpreted as the rate ratio, and the SII can be interpreted as the rate difference at the bottom and the top of the educational or income hierarchy. If there is no inequality, RII assumes a value of 1.0. The further the value of RII from 1.0, the higher the level of inequality. RII assumes only positive values, with values larger than one indicating a concentration of the indicator among the advantaged and values smaller than one indicating a concentration of the indicator among the disadvantaged. If there is no inequality, SII takes the value of zero. Greater absolute values indicate higher levels of inequality. Positive values indicate higher coverage in the advantaged subgroups, and negative values indicate higher coverage in the disadvantaged subgroups . RCI and ACI were estimated based on the methodology described by Clarke et al. . The RCI provides a measure of the relative differences among education/ income groups. Considering that the RCI was based on morbidity measure (tooth loss), a negative RCI indicates inequality favoring higher income groups. The ACI quantified the absolute differences in health between education and income groups, and this index is not affected by whether it is measured concerning health or morbidity. As RII/SII are mathematically related to RCI/ACI, they will produce the same rank ordering of health inequality between groups but will differ in scale.
Association between tooth loss and socioeconomic indicators
Multilevel Poisson regression procedures with unstructured covariances matrix were used to model the two-level structure of individuals (level 1) nested within municipalities (level 2). Multilevel techniques of analyses provide the overall relationship between the individual, compositional factors, and oral health (fixed part), and the variation between areas that cannot be accountable for such factors (random-intercept parameter). Besides, it is possible to assess the variation in certain individual relationships between municipalities (random slope parameters) and the interaction between individual and contextual characteristics (cross-level interactions). A five-step sequential modeling strategy was adopted: i) model 1(empty model): a model without the inclusion of any covariates, in which the variance in tooth loss is inspected between municipalities. A significant random intercept variance indicates the presence of unexplained differences in tooth loss between municipalities. The Wald test evaluated the significance of random intercept, and the Median Rate Ratio measured the heterogeneity among municipalities, according to Austin et al. (2017) . There is no variation between municipalities if the MRR is 1.0, but the higher the MRR, the greater the area-level variation. ii) model 2 (random intercept, fixed effect): considers all the individual-level variables in the fixed part. This model assessed the association between tooth loss and income and education adjusted for covariables at the individual level. The variation between municipalities is allowed for, conditional on the individual, compositional factors. iii) model 3: as model 2, but including the municipalities-level variables. The proportional change in variance (PCV) was calculated according to Merlo et al. (2005) , using the following formula: PCV = (variance model 1 – variance model 2)/variance model 1. iv) model 4 (random slope and random intercept model): as model 2, but the model allows both the intercept and slope to be random parameters. Therefore, each municipality has its intercept and slope in which the variability from the overall intercept and slope can be investigated with the addition of individual and municipality variables and their interactions (cross-level interactions). Random slopes for education and income were considered. The comparison of goodness fit between model 2 and model 4 was performed using the LR-test. v) model 5: as model 4, including the municipalities-level variables, in which the cross-level interactions between education/income and HDI were considered. The interaction term represents the change in the slope of education/income on tooth loss across municipalities when HDI changed from low to high. The estimated mean of missing teeth according to individual socioeconomic variables for municipalities with high and low HDI was demonstrated using a graph of the predicted model.
Procedures for complex sample design were used to calculate the proportions of adults according to investigated variables, the age- and sex-adjusted estimates of the outcomes, SII, RII and ACI. Statistical analyses were performed using STATA version 15.0 (StataCorp LP, College Station, Texas, USA).
The Brazilian National Council of Ethics in Research approved the SBBrasil 2010 study, protocol no. 15498, January 7, 2010.