Estimating peer density effects on oral health for community-based older adults

Population data for adults aged 50 years and older were obtained from the 2010 US Decennial Census [20] and assigned to the centroids of the respective Census blocks (as mapped in Fig. 1). Kernel density estimation was used to produce smoothed peer density surfaces based on the older adult population associated with the centroid of each Census block. Kernel density estimate (KDE) values were then assigned to each participant record in the dataset based upon their geocoded residential location.

The process of kernel density estimation produces a smoothed density surface based upon the distribution of observed values at different locations (i.e., the number of adults aged 50 years and older associated with Census block centroids). In essence, kernel estimators smooth out the contribution of each observed data point over its local neighborhood. The contribution of the i ^th data point x _i (i = 1, …, n) to the estimate at a given location x depends upon how far apart x _i and x are geographically. The extent of this contribution is dependent upon the shape of the kernel function adopted and its bandwidth. If the kernel function is denoted as K(•) and its bandwidth by h, the general KDE function at any point x is calculated as:where x ₁, x ₂, …, x _n are n observed points [27]. The kernel function K(•) is non-negative and integrates to 1. Here, a quadratic form is used for K(•), such that K(u) = ¾(1-u ²) if |u| ≤ 1 and K(u) = 0 if |u| > 1. Since u = (x-x _i )/h, this condition ensures that the quadratic form of the kernel function is applied where the distance between the given point x and the observed point x _i is no greater than the bandwidth h, thus accounting only for the observed points that are within the search window from the given location. The peer density (KDE) estimated by \( \widehat{f}(x) \) in this study has units of adults aged 50 years and older per square mile.

The bandwidth parameter (h) of the kernel function determines the size of the search window in estimating kernel density. Because of its importance in estimating the kernel density surface, 4 different values of bandwidth were used in this study to calculate the density surfaces, i.e., h = 0.25, 0.50, 1.00, and 1.50 miles. These bandwidth values were selected to encompass areas within walking distance from the residence of a participant, such as senior centers and other third places [8] where community-based older adults are likely to socialize [13]. While we expect a bandwidth of 0.25 or 0.50 mile to better represent the typical socializing radius or ambit [28] of older adults in urban areas, larger bandwidths were also employed to assess model robustness, i.e., how sensitive our peer density findings are to this measurement assumption. Figure 2 compares the kernel density surfaces for each of these 4 bandwidth values that distinguish the 4 spatial models used in the subsequent logistic regression analyses of effects on oral health outcomes.

As illustrated in Fig. 2, larger bandwidths generate smoother density estimates over space, as more distant observations are included in the calculation and data observed farther away thereby exert a greater influence on the estimation of kernel density for a given resident. In contrast, smaller bandwidths produce more local variation and distinct clusters. For instance, the kernel density surface created for Model 2 at h = 0.50 mile smooths over the small clusters that appear in Model 1 as the smallest bandwidth surface, but retains greater spatial variation in density than the surfaces for Models 3 and 4 using larger bandwidths.

Logistic regression analyses were performed to examine 2 oral health outcomes, dentition status and self-rated oral health, using 4 spatial models that include the KDE measure from different bandwidth versions of peer density as the main determinant of interest. A summary of the variables used for the generalized adjacent-categories logistic regression of ordinal dentition status and for the standard logistic regression of binary self-rated oral health is presented in Table 1.

The first oral health outcome of interest was dentition status, derived from the presence or absence of teeth as measured by an ElderSmile dentist based upon a complete dentition of 28 teeth. Third molars were excluded from the analyses because they are often missing for reasons other than dental caries or other oral diseases. Being edentulous was defined as having no natural permanent teeth in the mouth or 28 missing teeth [26, 29]. Because having 20 teeth is considered necessary for functional dentition [6, 26, 30], participants with 0 to 8 missing teeth were considered to have functional dentition, and participants with 9 to 27 missing teeth were considered to have limited functional capacity. Therefore, for this study, dentition status was coded as an ordinal variable, where 0 = functional dentition, 1 = limited functional capacity, and 2 = edentulous.

The second oral health outcome of interest was self-rated oral health. On the program intake form, ElderSmile participants rated their oral health as excellent, good, fair, or poor [25]. A weighted kappa statistic demonstrated a significant level of agreement between the self-rated oral health of ElderSmile participants and dentist-rated oral hygiene; in particular, untreated dental caries and severe periodontal inflammation were significantly related to poor self-rated oral health [13, 25]. This appears to be an important cutpoint (i.e., at least fair versus poor) in this community-based sample of older adults based upon the stratified analyses and the goodness of fit of the logistic regression models. It may be that what matters most for underserved older adults in rating their oral health is pain related to untreated disease, rather than aesthetics or oral hygiene, and thus there appears to be a lower cutpoint in the ElderSmile program sample than that found in other population-based samples (i.e., excellent or good versus fair or poor). The 4-category self-rated oral health variable was thus recoded as a binary variable for the analyses, where a value of 0 represents poor self-rated oral health and a value of 1 represents at least fair or better self-rated oral health [13, 25].

In addition to the KDEs described above as the spatially-derived determinants of interest, all other covariates used in the analyses were derived from the ElderSmile program intake form. Peer density (KDE) and age (in years) were treated as continuous covariates. Of the other covariates, 3 were coded as binary: gender (0 = male, 1 = female); whether or not they have Medicaid coverage (1 = has Medicaid coverage, 0 = does not have Medicaid coverage); and whether or not they have private dental insurance (1 = has private dental insurance, 0 = does not have private dental insurance). The 3 remaining covariates were treated as categorical: race/ethnicity (0 = Hispanic, 1 = non-Hispanic White, 2 = non-Hispanic Black, 3 = Other); education level (0 = primary school, 1 = high school, 2 = college); and smoking status (0 = current smoker, 1 = former smoker, 2 = never smoked). The distribution of racial/ethnic identities among ElderSmile participants reflects the predominantly Hispanic (largely Dominican and Puerto Rican) and African American communities served by the program in the neighborhoods of northern Manhattan and the Bronx. Because most participants (57.2%) identified as Hispanic, this group was selected as the reference category in specifying the effects of race/ethnicity in the models. Of the 3 education levels, the greatest proportion of participants (37.8%) indicated primary school as their highest level of educational attainment.

When analyzing datasets with partial missing information, researchers either ignore all of the available information for participants with missing values (which results in loss of information and introduces bias into the findings) or elect to use all of the available information at hand by first imputing the missing data and then analyzing the imputed versions of the dataset. Percentages of missing values for the variables used in the logistic regression models are listed in Table 1 above. Note the high rates of missing data for certain variables such as smoking status (24.8%). Imputation of these missing values (rather than exclusion of these records) ensures that the sample of participants analyzed in this study is representative of the population of participants in the ElderSmile program. Missing values were imputed using a Bayesian statistical technique known as multiple imputation [21, 31], which has been used in other health research [32, 33].

A foundational assumption for multiple imputation is that data are missing at random:where M denotes that one observation is missing, Y _obs denotes the observed values, and Y _mis denotes the missing values in the target data set. Under this assumption, the probability that an observation is missing depends on the whole population only through the observed value Y _obs . It may be derived that any inference on θ depends only on the observed data likelihood:where θ is the potential parameter for the data distribution [34]. By Bayesian inference, the posterior distribution of θ depends on:where π is the prior distribution of θ [35]. Multiple imputation draws values from the posterior predictive distribution of Y _mis , P(Y _mis | Y _obs ) instead of from Y _mis [36]. Denoting P(Y _mis | Y _obs , θ) as the conditional predictive distribution of Y _mis , the posterior predictive distribution of Y _mis may be written as:

Different models for multiple imputation are appropriate for different variable types. For a continuous target variable (e.g., participant age in years), predictive mean matching is used to impute missing values. In this model, the 5 most highly correlated variables are used as predictors, denoted X ₁, …, X ₅. The linear model:is thus fitted using observations with observed values for the variable Y and its covariates X ₁, …, X ₅,where ϵ ∼ N(0, σ ²). By specifying the prior distribution of the parameters β = (β ₀, …, β ₅) ^T and σ, β ^∗ and σ ^∗may be drawn from their posterior distributions. Hence, the predicted value for each observed y _l may be obtained by:

Then the imputation value of a missing y _j will be:where \( {\left({\widehat{\mu}}_j-{\widehat{\mu}}_k\right)}^2\le {\left({\widehat{\mu}}_j-{\widehat{\mu}}_l\right)}^2 \) for all observed l, \( {\widehat{\mu}}_j \) is the predicted mean of Y and y _k is the observed value of Y for the kth observation [37].

Imputation using logistic regression is appropriate if the target variable Y is binary (e.g., gender). Define p as the probability that Y = 1. Consider the logistic regression model:where X ₁, …, X ₅ are the top 5 highly correlated variables for target variable Y and \( logit\ p=\ln \left(\frac{p}{1-p}\right) \). New parameters\( {\boldsymbol{\beta}}^{\ast }={\left({\beta}_0^{\ast },\dots, {\beta}_5^{\ast}\right)}^T \) are drawn from the posterior distribution of β. Then, for an observation with missing Y _l and covariates x _1l , …, x _5l , the expected probability that Y _l  = 1 is:where \( {\widehat{\mu}}_l={\beta}_0^{\ast }+{\beta}_1^{\ast }{x}_{1l}+\dots +{\beta}_5^{\ast }{x}_{5l} \). The imputed value is then drawn based on the estimated probability \( {\widehat{p}}_l \) [21].

Imputation using polytomous regression is needed if the target variable Y is categorical or ordinal, e.g., the 3-category measure of smoking status. Define probability p ₀, p ₁ and p ₂ as follows:

Next, consider the multinomial logit model:

where X = (X ₁, …, X ₅) ^T is the vector containing the 5 most highly correlated variables for Y as predictors. The estimated parameters \( {a}_0^{\ast },{a}_1^{\ast },{\boldsymbol{b}}_0^{\ast },{\boldsymbol{b}}_1^{\ast } \) are drawn from the posterior distributions. These parametric estimates are then used to calculate the estimated probabilities of each possible outcome (i.e., 0 = current smoker, 1 = former smoker, and 2 = never smoked) for each missing Y. The imputed value for each missing Y is then randomly drawn based upon these estimated probabilities [38].

The approach to multiple imputation employed in this study involved constructing m = 10 different datasets by estimating each missing observation m times, performing data analysis on each of the imputed datasets, and combining the results from the analysis of each dataset. The combined results, derived from the separate analyses of the 10 multiple-imputed datasets, were then used as a basis for statistical inference.

In this study, multiple imputation was performed for explanatory variables, whereas complete cases were used for the outcome variables of dentition status and self-rated oral health. This approach was adopted due to the nature of the outcome variables and to address the potential concern among certain researchers that imputation of missing outcomes produces overly optimistic standard errors. For self-rated oral health, missingness is likely a distinct outcome (i.e., someone who actually does not know how to assess their health status) rather than an unobserved one (i.e., someone who has an assessment in mind but does not want to report it). For dentition status, the rationale against imputation rests on the stronger assumption that the missingness is likely to be completely at random given that it occurs due to participants opting out of a dental examination rather than survey item nonresponse (which is the nature of missingness for the explanatory variables). Given the statistical concern, though, that listwise deletion of cases due to missingness can introduce estimation bias [39], sensitivity analyses were performed to assess robustness relative to analyses in which all variables, including the 2 outcome variables, were imputed and those in which only complete cases were included (without imputation). Overall, the results of these auxiliary analyses (available from the authors upon request) are qualitatively similar in terms of the effects of peer density and other covariates.

After construction of the kernel density surfaces and multiple imputation of missing data for the explanatory variables, logistic regression analyses were performed to estimate the effects of peer density and other sociodemographic characteristics on the oral health outcomes of dentition status and self-rated oral health. For each of these 2 outcome variables, a set of 4 logistic regression models was estimated with specifications that differ in the measure of peer density included, where the KDE surface was constructed using 4 distinct bandwidth parameter h values as follows: 0.25 mile (Model 1), 0.50 mile (Model 2), 1.00 mile (Model 3), and 1.50 miles (Model 4). These 4 spatial models provide alternative estimates of the effect of peer density based on different characterizations of a participant’s social network, with Model 1 including the most local measure and Model 4 including the most dispersed measure (see Fig. 2).

Generalized adjacent-categories logistic regression (GACLR) models [40] were developed to estimate the effects of peer density and other covariates on dentition status, as indicated by 3 ordered categories (0 = functional dentition or 0-8 missing teeth; 1 = limited functional capacity or 9-27 missing teeth; and 2 = edentulous or 28 missing teeth) [6, 26, 30]. The general form of the GACLR model is:where X ₁ , X ₂ , …, X _q are the set of covariates used in the model for comparing the odds of the category (j + 1) relative to its adjacent category j of the ordinal outcome Y; the parameters β _1j , β _2j , …, β _qj are the corresponding regression coefficients, and β _0j denotes the intercept. As in standard logistic regression, the exponential transform of the regression coefficient, exp.(β _1j ), denotes the odds ratio (OR) of the covariate X ₁ for comparing the category (j + 1) relative to its adjacent category j of the ordinal outcome Y, and so on. The GACLR model was chosen over other alternatives for analyzing ordinal outcome variables for 2 reasons. First, the GACLR model does not assume the restrictive proportional odds assumption [40], thereby allowing for parametric differences in the effects of the predictions on the relative probabilities of different adjacent-outcome pairs (i.e., different for limited functional capacity relative to functional dentition than for edentulous relative to limited functional capacity). Second, given the importance of being able to calculate incidence rates, the GACLR model is easier to interpret since it directly estimates the effects of the predictors on the OR of each adjacent-outcome pair rather than requiring these to be derived via nonlinear functions that necessitate estimation of standard errors using simulation or resampling methods.

Standard logistic regression models were developed for self-rated oral health as the second oral health outcome of interest, using the binary outcome of poor vs. fair or better self-rated oral health, with the same set of 4 spatial models and covariates as considered in case of the ordinal dentition status outcome. All analyses were conducted in the statistical software R version 3.1.2 (https://​cran.​r-project.​org/​) using the package vector generalized linear and additive models (VGAM). All appropriate institutional review board and Health Insurance Portability and Accountability Act safeguards were followed.

While the results of this study confirm an earlier report of an association between peer density and oral health [13], more research is needed to fully explicate the causal mechanisms underlying this relationship. Further investigation is also needed to confirm and explicate the associations revealed between oral health and Medicaid coverage, race/ethnicity, gender, smoking, and education. Finally, it may also be useful to consider whether the effects of peer density vary with race/ethnicity and education. As such, this study informs ongoing research by the study team that seeks to explore these mechanisms using dynamic modeling and simulation of social networks to investigate their potential influence on oral health equity.