**Statistical Models for Dental Caries Data**

David Todem

*Division of Biostatistics, Department of Epidemiology and Biostatistics, Michigan State University, East Lansing, MI, USA* 

#### **1. Introduction**

92 Contemporary Approach to Dental Caries

Skartveit, L., Knut, AS., Myklebust, S., &Tveit, AB. (1990). Effect of TiF4 solutions on bacterial growth in vitro and on tooth surfaces. *Acta Odontol Scand,* 48, 169-174. Smidt, A., Weller, D., Roman, I., & Gedalia, J. (1998). Effect of bleaching agents on microhardness and surface morphology of tooth enamel. *Am J Dent,* 11, 83-85. Sorvari, R., Meurman, JH., Alakuijala, P., & Frank, RM. (1994). Effect of fluoride varnish and

Spalding, M., Taveira, LA., & de Assis, GF. (2003). Scanning electron microscopy study of

Sydney, GB., Barletta, FB., & Sydney, RB. (2002). *In vitro* analysis of effects of heat used in

Tanizawa, Y. (2005). Reaction characteristics of a tooth-bleaching agent containing H2O2 and

Ten Cate, JM., & Arends, J. (1977). Remineralization of artificial enamel lesions in vitro.

Tezel, H., Atalayin, C., Erturk, O., & Karasulu, E. (2011). Susceptibility of enamel treated

Tezel, H., Ergücü, Z., & Söğüt, Ö. (editted to be published). The effects of titanium tetrafluoride and sodium fluoride on calcium loss from bleached enamel. Tezel, H., Ertaş, SÖ., Özata, F., Dalgar, H., & Korkut, Z. (2007). Effect of bleaching agent on calcium loss from the enamel surface. *Quintessence Int,* 38, 4, 339-347. Titley, K., Torneck, CD., & Smith, DC. (1998). The effect of concentrated hydrogen peroxide

Türkun, M., Sevgican, F., Pehlivan, Y., & Aktener, BO. (2002). Effects of 10% carbamide

Van Rijkom, H., Ruben, J., Vieira, A., Huysmans, MC., Truin, GJ., & Mulder, J. (2003).

Wetter, NU., Barroso, MCS., & Pelino, JEP. (2004). Dental bleaching efficacy with diode laser

Zalkind, M., Arwaz, JR., Goldman, A., & Rotstein, I. (1996). Surface morphology changes in

Zhang, C., Wang, X., Kinoshita, JI., Zhao, B., Toko, T., & Kimura, Y. (2007). Effect of KTP

and LED irradiation: an *in vitro* study. *Lasers Surg Med,* 35, 254-258.

microscopy study. *Endod Dent Traumatol*, 12, 82-88.

*of Dentistry*, 2011, Article ID 953835, 8 pages doi:10.1155/2011/953835 Tezel, H., Ergücü, Z., & Önal, B. (2002). Effects of topical fluoride agents on artificial enamel

lesion formation in vitro. *Quintessence Int,* 33, 347-352.

with 10% carbamide peroxide. *J Esthet Restor Dent*, 15, 3, 154-64.

dental bleaching on human dental enamel. *Braz Dent J,* 13, 166-169.

chemical states of incorporated fluorine. *Int J Cosm Sci,* 56, 121-134.

dental enamel surface exposed to 35% hydrogen peroxide: alone, with saliva, and

NaF: in vitro study of crystal structure change in treated hydoxyapatite and

with bleaching agents to mineral loss after cariogenic challenge. *International Journal* 

solutions on the surface morphology of human tooth enamel. *J Endodont,* 14, 69-74.

peroxide on the enamel surface morphology: a scanning electron microscopy study.

Erosion-inhibiting effect of sodium fluoride and titanium tetrafluoride treatment in

human enamel, dentin and cementum following bleaching: A scanning electron

laser irradiation, diode laser, and LED on tooth bleaching: a comparative study.

solution on enamel in vitro. *Caries Res,* 28, 227-32.

*Caries Research*, 11(5), 277-86

*J Esthet Restor Dent*, 14, 4, 238-44.

vitro. *Eur J Oral Sci,* 111, 253-257.

*Photomed Laser Surg,* 25, 91-95.

Tooth decay is ubiquitous among humans and is one of the most prevalent oral diseases. Although this condition is largely preventable, more than half of all adults over the age of eighteen present early signs of the disease, and at some point in life about three out of four adults will develop the disease. Tooth decay is also common among children as young as five and remains the most common chronic disease of children aged five to seventeen years. It is estimated that tooth decay is four times more prevalent than asthma in childhood (Todem, 2008). Tooth decay and its correlates such as poor oral health place an enormous burden on the society. Poor oral health and a propensity to dental caries have been related to decreased school performance, poor social relationships and less success later in life. It is estimated that about 51 million school hours per year are lost in the U.S. alone because of dental-related illness. In older adults, tooth decay is one of the leading causes of tooth loss which has a dramatic impact on chewing ability leading to detrimental changes in food selection. This, in turn, may increase the risk of systemic diseases such as cardiovascular diseases and cancer.

The etiology of dental caries is well established. It is a localized, progressive demineralization of the hard tissues of the crown and root surfaces of teeth. The demineralization is caused by acids produced by bacteria, particularly mutans Streptococci and possibly Lactobacilli, that ferment dietary carbohydrates. This occurs within a bacteria-laden gelatinous material called dental plaque that adheres to tooth surfaces and becomes colonized by bacteria. Thus, dental caries results from the interplay of three main factors over time: dietary carbohydrates, cariogenic bacteria within dental plaque, and susceptible hard tooth surfaces. Dental caries is also a dynamic process since periods of demineralization alternate with periods of remineralization through the action of fluoride, calcium and phosphorous contained in oral fluids.

The evaluation of the severity of tooth decay is often performed at the tooth surface level. According to the World Health Organization, both the shape and the depth of a carious lesion at the tooth surface level can be scored on a four-point scale, D1 to D4. Level D1 refers to clinically detectable enamel lesions with non-cavitated surfaces; D2 for clinically detectable cavities limited to the enamel; D3 for clinically detectable lesions in dentin; and finally D4 for lesions into the pulp. Despite these detailed tooth-level data, most epidemiological studies often rely on the decayed, missing and filled (DMF) index,

Statistical Models for Dental Caries Data 95

parameter the mean. This mean is often related to potential explanatory variables using a log link function. Specifically, let Y define the outcome variable and X the set of explanatory variables. A Poisson regression model for the mean is defined as E�Y|X� = e����, where α and β are the intercept and the regression parameter vector associated with X. The

One major restriction of the Poisson regression model is that its mean is equal to its variance. For dental caries data, however, it is not uncommon for the variance to be much greater than the mean. For such data, a negative binomial regression model has been advocated as an alternative to Poisson regression models. It is typically used when the variability in the data cannot be properly captured by Poisson regression models. The negative binomial model is a conjugate mixture distribution for count data (Agresti, 2002). It is entirely specified by two parameters, its mean and the overdispersion parameter. Similarly to the Poisson regression model, the mean is related to potential explanatory variables using a log link function. However, the probability mass function of Y is given by:

���

where μx = E�Y|X� = e���� is the conditional mean which depends on covariates, and κ is the

unknown and estimated from data to evaluate the extent of overdispersion in the data. When κ tends to zero, the negative binomial model converges to a Poisson process (Agresti,

The presence of an upper bound for possible values taken by DMF scores suggests a model based on the binomial rather than the Poisson distribution (Hall, 2000). Data are then viewed as being generated from a binomial process with m trials and success probability π�. Here m represents the maximum number of teeth or tooth surfaces in the mouth susceptible to decay, and π� the probability for a tooth or tooth surface to present a sign of decay. The

intercept and the regression parameter vector associated with X. One should note however that Poisson and negative binomial distributions provide a reasonable approximation to the

Dental caries data with excess zeros are common in statistical practice. For example, in young children, DMF scores generally generate an excessive number of zeros in that many children do not experience dental caries. This is typically due to a short exposure time to caries development. The limitations of Poisson and negative binomial regression models to analyze such data are well established (see, for example, Lambert, 1992; and Hall, 2000). One approach to analyze count data with many zeros is to use zero-inflated models. This class of

 �1 − <sup>κ</sup>��

μ� + ��

 �� �1−π�����, y = 0,1, … m,

�

, y = 0,1, …

�������, with α and β being the

�. Parameter κ is typically

 � � μ� + κ��� ����� , y=0,1,…, where μx = E(Y|X)

probability mass function of Y is given by: P�Y=y|X� <sup>=</sup> �������

is the conditional mean which depends on covariates.

P�Y=y|X� <sup>=</sup> Г�y+κ���

2002).

binomial model is given by:

Г�κ���Г�y+1�

P�Y=y|X� <sup>=</sup> Г�m+1�

binomial distribution in dental caries research.

Г�m−y+1�Г�y+1�

where the success probability is related to covariates as �� <sup>=</sup> �����

overdispersion parameter. This distribution has variance μ� + κμ�

developed in the 1930s by Klein *et al.* (see for example Klein and Palmer, 1938). This index is applied to all the teeth (DMFT) or to all surfaces (DMFS), and represents the cumulative severity of dental caries experience for each individual. These scores have well documented shortcomings regarding their ability to describe the intra-oral distribution of dental caries (Lewsey and Thomson, 2004). But they continue to be instrumental in evaluating and comparing the risks of dental caries across population groups. Most importantly, they remain popular in dental caries research for their ability to conduct historical comparisons in population-based studies.

Statistical analysis of dental caries data relies heavily on the research question under study. These questions can be classified into two groups. The first group represents questions that can be answered using mouth-level outcomes generated using aggregated scores such as the DMF index. The second group refers to questions that necessitate the use of tooth or toothsurface level outcomes. A very important issue to address for the data analyst is the modeling strategy to adopt for the response variable under investigation. Broadly, two fairly different views are advocated. The first view, supported by large-sample properties, states that normal theory should be applied as much as possible, even to non-normal data such as counts (Verbeke and Molenberghs, 2000). This view is strengthened by the notion that, normal models, despite being a member of the generalized linear models (GLIM), are much further developed than any other GLIM (e.g. model checks and diagnostic tools), and that they enjoy unique properties (e.g., the existence of closed form solutions, exact distributions for test statistics, unbiased estimators, etc...). Although this is correct in principle, it fails to acknowledge that normal models may not be adequate for some types of data. As an example, the abundance of zeros in DMF scores rules out any attempt to use normal models, such as linear models, even after a suitable transformation. While a transformation may normalize the distribution of nonzero response values, no transformation could spread the zeros (Hall, 2000). A different modeling view is that each type of outcome should be analyzed using tools that exploit the nature of the data. For dental data, features to be accommodated include the discrete nature of the data (count responses for mouth-level data and binary response for intra-oral data), the abundance of zeros for example in the DMF/S scoring, and the clustering in intra-oral responses. The clustering of participants as a result of the study design is another important feature.

This chapter reviews common statistical parametric models to answer questions that arise in dental caries research, with an eye to discerning their relative strengths and limitations. Missing data problems arising in caries dental reasrch will also be discussed but touched on briefly.
