A longitudinal study of the probability of developing malocclusion in children using a Bayesian analysis

Introduction

This study evaluated the probability of developing malocclusions in mixed dentition.

Methods

A longitudinal study was conducted with 598 children (aged 5 years) in deciduous dentition. The children were followed for 3 years until mixed dentition (aged 8 years). Overjet, overbite, and transversal relations were evaluated. Bayesian models were used to analyze the data and estimate the parameters.

Results

The parameter θ was used for the distributions, indicating the probability of presenting a given condition with a credibility index (ICr) of 95%. After 3 years of follow-up, 121 children were reevaluated. The results showed that children have a high probability of malocclusion in mixed dentition. There was a higher probability of developing an increased overjet in the mixed dentition of 20.5% (ICr 95%, 13.6-28.4) to 48.3% (ICr 95%, 39.1-57.7) and a higher probability of having a normal overbite in the deciduous dentition and a lower probability in the mixed dentition (ICr 95%, 9.2-21.3).

Conclusions

Considering the probabilistic model of Bayesian analysis, children with normal overjet in the deciduous dentition may show an increased overjet in the mixed dentition. Concerning overbite, children may present an anterior open bite during the transition between deciduous and mixed dentition, as well as self-correction of deep overbite in mixed dentition. Furthermore, they may present a posterior crossbite during the mixed dentition when there is a normal transverse relationship in the deciduous dentition.

Highlights

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We assessed the probability of developing malocclusion in the primary dentition for mixed dentition.
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The probability analysis used Bayesian statistical methods, a pioneering study in the field.
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Our findings indicate a higher likelihood of developing increased overjet in the mixed dentition compared to the primary dentition.
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Children are more inclined to exhibit a reduced overbite in mixed dentition than in the primary dentition.
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Bayesian analysis can be employed to assess the likelihood of malocclusion.

Malocclusions can be observed at different stages of occlusion development because of genetic and environmental etiologic factors. In this context, childhood is the most active occlusal and facial development stage, with changes in dentitions, facial growth, and dimensional alterations of the dental arches. Most interceptive orthodontic procedures are performed in this stage.

The prevalence of malocclusion in the primary dentition is considered high, ^, and the transition to permanent dentition should be monitored during the mixed dentition stage to determine the need and opportunity for orthodontic treatment. At this stage, preventive and interceptive orthodontics can minimize esthetic, social, and psychological impacts, in addition to improving the quality of life of children.

Most studies have used frequency distribution and association analyses to present data on malocclusion in children and factors that can influence occlusal changes. No studies have analyzed occlusion using methods, such as Bayesian analysis, that consider the scenario in which malocclusion appears because of its multifactorial etiology. Bayesian analysis facilitates decision-making by directly calculating the probability of various hypotheses or scenarios using the available information about that change. In addition, it allows for more robust estimates in situations with minimal data—a difficulty found in the analysis of longitudinal data because of sample loss over time. ^, ^, This study is the first to use Bayesian analysis to determine the probability of developing malocclusion in childhood.

Considering the uncertainty in the development of malocclusion, from its establishment to spontaneous self-correction, this analysis can contribute to decision-making for early and preventive treatment of alterations. This study longitudinally evaluated the probability of developing malocclusion during the transition from deciduous to mixed dentition using Bayesian analysis.

Material and methods

This study was approved by the Ethics in Research with Human Committee (no. 31459420.5.0000.5418) and followed the STROBE (STrengthening the Reporting of OBservational studies in Epidemiology). Parents of children studying in public schools were invited to participate voluntarily with their children. Those who agreed signed the Informed Consent Form, and the children verbally consented to their participation.

This longitudinal observational study was developed to assess the probability of developing malocclusion in the transition from primary dentition to mixed dentition during a 3-year follow-up. The study included 598 five-year-old children enrolled in public schools in the interior of São Paulo, Brazil. The allocation of children aged 5 years across administrative areas within each municipal district was based on data from the Municipal Department of Education. The sample was stratified according to the administrative sectors within the district. In the initial phase, 14 schools were chosen using a simple randomization procedure. Similarly, children were selected for inclusion in the sample during the second phase using another simple randomization procedure. Bayesian analyses do not include sample calculation.

During the initial assessment (T0) only children in primary dentition were included. Children with previous or current orthodontic or orthopedic treatment and with mental and/or physical disabilities that made oral examination impossible were excluded from the study.

All children initially assessed at T0 were contacted for follow-up when they had transitioned into mixed dentition (T1). Children who reported current or past orthodontic treatment, situations in which parental or guardian authorization for examination was not obtained, nonresponsive subjects, and those who could not be located through the Municipal Education Network were excluded. A total of 121 children were reevaluated for the study at T1. The Figure shows a flowchart for the sample distribution.

Assessment of dental occlusion included overjet, overbite, and posterior crossbite. The absence or presence of malocclusion was also assessed. ^,

The overjet assessment considered the relationship between the maxillary and mandibular incisors in the horizontal direction, with no distance between the maxillary and mandibular incisors defined as normal overjet (0 mm). Increased overjet was recorded when the distance was >2 mm, and decreased overjet was recorded at <0 mm. The overbite assessment considered the relationship between the maxillary and mandibular incisors in the vertical direction. The normal overbite was defined when the maxillary incisors overlapped the mandibular incisors vertically by 2 mm. An anterior open bite was recorded without contact between the anterior teeth when the posterior teeth were occluded. The posterior crossbite considered the maxillary molars occluded in lingual relationship with the mandibular molars in centric occlusion. The abnormal relationship was considered posterior crossbite. Children were diagnosed with normal occlusion when all conditions were normal. If exhibiting ≥1 of the previously described conditions, patients were classified as having malocclusion. ^,

A calibrated examiner (P.R.S.) performed clinical examinations and recorded the malocclusion. Before the survey, the calibration was performed on a group of 30 five-year-old children. Theoretical and clinical training and calibration exercises were organized for a total of 36 hours under the supervision of a reference examiner (P.R.S.). After the reproducibility process, it was noted that the lowest agreement index measured among all evaluated occlusal conditions was a kappa of 0.92 (with a confidence interval of 0.81-1.00), indicating perfect agreement, as per the classification proposed by Landis and Koch (1977).

Statistical analysis

Bayesian models were used to study the longitudinal behavior of the orthodontic variables overjet, overbite, posterior crossbite, and malocclusion of 121 children in deciduous dentition (T0) and mixed dentition (T1). Bayes’ theorem can be described as:

<SPAN role=presentation tabIndex=0 id=MathJax-Element-1-Frame class=MathJax style="POSITION: relative" data-mathml='P(A|B)=P(B|A)×P(A)P(B)’>𝑃(𝐴|𝐵)=𝑃(𝐵|𝐴)×𝑃(𝐴)𝑃(𝐵)P(A|B)=P(B|A)×P(A)P(B)
P(A|B)=P(B|A)×P(A)P(B)

P(A|B) (probability of A given B) represents the conditional probability that event A will happen, given that event B has occurred, just as P(B|A) represents the conditional probability that event B will happen, given that event A occurred. P(A) represents the probability of event A happening, and P(B) is used as a normalization constant to guarantee that the sum of the probabilities equals 1.

For parameter estimation by Bayesian analysis, A is related to a parameter of the distribution, whereas B is related to the obtained data. The probability of a parameter assuming a specific value is calculated given that certain data were obtained (also called the posterior). When looking for the parameter’s value that maximizes this probability, it is possible to find an estimate for it. As this equation usually boils down to a complex integral, Markov chain Monte Carlo (MCMC) methods obtain the parameters that maximize the probability to posteriori.

In the models, the Bernoulli distribution was considered for binary variables (crossbite and malocclusion) and the categorical distribution (a generalization of the Bernoulli distribution) for categorical variables with >2 levels (overjet and overbite). The Bernoulli distribution was also used to analyze the dichotomous variables.

The θ parameter was used for the Bernoulli and categorical distributions, indicating the probability of presenting a certain level. For example, a Bernoulli distribution with 0 = absence of malocclusion and 1 = presence of malocclusion may show θ = 0.50 if there is a 50% chance that a sampled subject will have a malocclusion. In contrast, θ = 0.75 would indicate a 75% chance that a subject will have a malocclusion. For binary variables, the values for θ were sampled from a beta distribution, with parameters α = 1 and β = 1 (before distribution). For categorical variables with ≥2 levels, θ received a Dirichlet before distribution with every parameter equal to 1.

Models were also applied considering that there could be differences in the parameters of populations from different groups (in this instance, subjects were grouped by the variables year, gender, income, and race).

To estimate the parameters using methods of MCMC, the package used RStan (Stan Development Team, 2019) for statistical R software (R Core Team, Vienna, Austria). In all instances, 4 chains with 6000 iterations each were used, so the first 1000 were disregarded in the final analysis because it is a period in which the algorithm is still adjusting the estimate (warm-up), which may be very far from the real. ^, The convergence of the MCMC chains was assessed by visual analysis of trace plots, posterior density plots, autocorrelation plots, and also by the R-hat and effective sample size values.

Results

For all variables in this study, the MCMC chains presented good convergence, with R-hat values close to 1 and effective sample size values close to the actual sample size.

Table I shows the sample distribution in T0 (deciduous dentition) and T1 (mixed dentition), whereas Table II presents the frequency and estimate of the parameter θ (95% credibility interval [ICr]) of the categorical variables (overjet, overbite, crossbite, and presence of malocclusion) in the T0 and T1. Children were more likely to develop increased overjet in T1. The probability increased from 20.5% (ICr 95%, 13.6-28.4) to 48.3% (ICr 95%, 39.1-57.7). The anterior open bite increased in T1 from 15.5% (ICr 95%, 9.6-22.4) to 34.2% (ICr 95%, 26.1-42.7), with a significantly higher probability of developing this malocclusion. Considering the 3-year follow-up, there was no significant difference in the odds of crossbite and malocclusion. In T0 and T1, the probability of crossbite was low (T0, 9.8%; T1, 13.8%) in the T1, whereas the probability of malocclusion was high, 70.7% in the T0 and 82.0% in the T1.

Table I

Descriptive analysis of the sample at T0 and T1