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J Dent Res 82(3): 200-205, 2003
© 2003 International and American Associations for Dental Research

RESEARCH REPORT
Clinical

Unification of the "Burst" and "Linear" Theories of Periodontal Disease Progression: A Multilevel Manifestation of the Same Phenomenon

M.S. Gilthorpe1,*, A.T. Zamzuri2, G.S. Griffiths3, I.H. Maddick4, K.A. Eaton3, and N.W. Johnson5

1 Biostatistics Unit, Academic Unit of Epidemiology and Health Services Research, Medical School, University of Leeds, 24 Hyde Terrace, Leeds, LS2 9LN, UK;
2 Dental Training School Malaysia, No 3 Sepoy Lines Road, 10450 Penang, Malaysia;
3 Department of Periodontology &
4 Department of Paediatric Dentistry, Eastman Dental Institute, University College London, 256 Gray’s Inn Road, London, UK; and
5 Department of Oral and Maxillofacial Medicine and Pathology, Guy’s King’s and St Thomas’ Schools of Medicine, Dentistry and Biomedical Sciences, Caldecot Rd, London, UK;

*corresponding author, m.s.gilthorpe{at}leeds.ac.uk


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS & METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
Previously, burst and linear theories for periodontal disease progression were proposed based on different but limited statistical methods of analysis. Multilevel modeling provides a new approach, yielding a more comprehensive model. Random coefficient models were used to analyze longitudinal periodontal data consisting of repeated measures (level 1), sites (level 2), teeth (level 3), and subjects (level 4). Large negative and highly significant correlations between random linear and quadratic time coefficients indicated that subjects and teeth with greater-than-average linear change experienced decelerated variation. Conversely, subjects and teeth with less-than-average linear change experienced accelerated variation. Change therefore exhibited a dynamic regression to the mean at the tooth and subject levels. Since no equilibrium was attained throughout the study, changes were cyclical. When considered as a multilevel system, the "linear" and "burst" theories of periodontal disease progression are a manifestation of the same phenomenon: Some sites improve while others progress, in a cyclical manner.

KEY WORDS: multilevel modeling • hierarchical linear modeling • random coefficients • lifetime cumulative attachment loss • pocket probing depth


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS & METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
A common problem in periodontal research is the complexity of clinical data, since the assumption underpinning most statistical methods is that observations are independent. However, data are usually collected on several different occasions, at several sites around different teeth, within several subjects. Such data are not independent, and the majority of single-level statistical methods (e.g., the t test) become invalid. Most periodontists would comment that there is variation between subjects in their susceptibility to periodontal disease, and that within these subjects there is also considerable variation in tooth and site susceptibility. The ability to analyze periodontal data accounting for all levels of a natural hierarchy is therefore desirable.

Aggregation of information to the subject level has led to the impression that progression of periodontal diseases is a continuous phenomenon, occurring at a relatively constant pace, only varying between population groups (Löe et al., 1978). However, frequent examinations of individual sites, along with stringent criteria for what constitutes disease rather than measurement error, led Socransky et al. (1984) to postulate that periodontal disease progressed through bursts of activity, accompanied by remission, repair, or further activity. This gave rise to two competing theories: the "linear" (continuous-rate) model, where, overall, sites slowly and progressively lose attachment; and the (random) "burst" model, where multiple sites show breakdown within a short period, with periods of remission that might last for months, years, or decades.

It has been argued that measurement error, and the possibility of different disease patterns at different sites, could produce erroneous evidence of burst progression (Ralls and Cohen, 1986; Cohen and Ralls, 1988). Yang et al. (1991) advocated regression methods of analysis, though Allen and Hausmann (1995), using simulated measurements, found model-fitting assessed by the least-squares criterion unreliable. Hujoel and Leroux (1998) found that lack-of-fit methods were unable to resolve the dilemma and that burst sizes of 3-5 mm would be necessary to be reliably distinguished from linear progression—an unlikely clinical scenario. For the "burst" model, statistical methods have utilized the distribution of the sizes of bursts over time as a normal random variable, or as a uniform random variable (Yang et al., 1993). The fit of these models to longitudinal data was assessed (Sterne et al., 1990, 1992; Jeffcoat and Reddy, 1991; Machtei et al., 1993), but findings were not conclusive.

Several studies have therefore attempted to model periodontal disease progression, though none has satisfactorily overcome the complexity of the data structure. This study proposes the use of multilevel modeling (MLM) to exploit periodontal data hierarchy, delivering a comprehensive model that describes the underlying progression of periodontal disease. Lifetime cumulative attachment loss (LCAL) (Eaton et al., 2001; Griffiths et al., 2001) and pocket probing depth (PPD) are modeled over time, and these models are further developed to consider potential periodontal disease risk factors and confounders.


   MATERIALS & METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS & METHODS
RESULTS
 DISCUSSION
 REFERENCES
 
Periodontal Research Data
Data were taken from a periodontal survey described previously (Griffiths et al., 1992), and included LCAL, PPD, (supragingival) plaque, supra- and subgingival calculus, and gingival bleeding on probing, for 100 white male trainee engineers, aged 16 to 20 yrs, entering a three-year apprenticeship at the UK Royal Air Force school, Halton. The authorities and their ethics committee approved the research protocol, and, following a verbal and written explanation of the study, participants signed a consent form. Clinical measurements were collected from the mesiobuccal, distobuccal, mesiolingual, and distolingual surfaces of all teeth present at enrollment and 12 and 30 mos thereafter. One periodontist (GSG) carried out all probing measurements, while three dental hygienists (involved in training and calibration exercises throughout the study) recorded all other clinical variables. Questionnaires, taken at baseline, recorded tobacco and alcohol consumption. Since only subjects with measurements at each occasion were included in each model, the number of subjects fell due to withdrawals. Four levels were identified: 3 measurement occasions (level 1); 4 sites (level 2); 28 teeth (level 3); and 89 subjects (level 4).

Model Considerations
LCAL and PPD, measured in millimeters, were treated as positive continuous outcomes, since their underlying distributions were sufficiently close to Normal for standard assumptions to be adopted. Two "trajectory" models were sought to describe the values of LCAL and PPD, respectively. A trajectory in this instance was a hypothetical "path" of outcomes across all measurement occasions during the study. Any parameterization of change would suffice, provided that it facilitated the description of non-linear transitions over time and provided an evaluation of how change (linear progression/recession) related to acceleration/deceleration (non-linear progression/recession). With only three measurement occasions, trajectories were described by a quadratic expression of time through two time-dependent covariates: linear time (t) and quadratic time (t2), where t was the duration between initial and subsequent measurements, centered on the second occasion for improved model estimation (Gilthorpe and Cunningham, 2000).

Periodontal "Risk Factors" and Confounders
Initially, multilevel models were developed without any consideration of factors that might potentially affect either outcome. Clinical factors recorded at the time of each inspection reflected oral hygiene and oral health status, and were considered as potential "risk factors" and/or confounders. Calculus, plaque, and bleeding were coded as absent (0) or present (1) for each site. Aggregate variables were constructed, such as the proportion of sites with plaque present within each subject. The proportions of sites with LCAL > 1 mm and PPD > 2 mm were also derived. Subject-based proportions were centered. Subjects were coded as smokers (1) or non-smokers (0) and drinkers/non-drinkers of alcoholic beverages. Tooth type was coded such that central incisors were the reference against which all other teeth were compared.

Some covariates were correlated, thereby giving rise to collinearity, potentially yielding unreliable model coefficients and elevated standard errors (Slinker and Glantz, 1985). Each covariate was modeled separately before being combined in one model. Contrasts between coefficients of these models yielded insight into how robust model estimates were in the presence of multicollinearity. Non-significant covariates were not removed from any model, since variable subset selection is not straightforward (Miller, 1990), and retaining these covariates only affected other parts of the model through collinearity, and this was addressed directly.

Multilevel Models
The basic form of a multilevel model has been described previously within a series of methodological articles (Gilthorpe et al., 2000a,b,c, 2001, 2002; Gilthorpe and Cunningham, 2000; Lewsey et al., 2000, 2001). In brief, a multilevel model is a regression model for data that form a hierarchy, with variation at each level determined separately. Coefficients for explanatory variables may exhibit random variation about their mean estimate (at any level), leading to the inclusion of additional variance and covariance terms. Each covariance provides a measure of correlation between two variance terms. These models, known as random coefficient models, were evaluated for each periodontal outcome, yielding an overall mean trajectory across all sites, teeth, and subjects. All possible changes are summarized in the Fig.

Data were analyzed within the statistical package MLwiN (Rasbash et al., 2000), and model assumptions were confirmed by analysis of residuals (Goldstein, 1995). Model fit and improvement were assessed by examination of the -2 Log Likelihood statistic (McCullagh and Nelder, 1989). For nested models (i.e., where covariates are only added or subtracted, but not both), changes in this parameter followed a chi-squared distribution, with the number of degrees of freedom equal to the number of covariates added or subtracted.


   RESULTS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS & METHODS
 RESULTS
DISCUSSION
 REFERENCES
 
Longitudinal Variation
Variance and covariance terms reduced across teeth and subjects with the inclusion of potential risk factors and confounders (Table 1Go), though there was consistently more overall variation between occasions than between sites, teeth, and/or subjects. Adding covariates improved model fit. The random part of each model determined inter-correlations among the intercept and both time-dependent covariates, across each level. With only four sites, three random terms were feasible at the site level (including the random intercept), yet quadratic random coefficients always converged to zero in every model considered, because differences across sites of the same tooth were too small to be detected as anything other than disparities in linear change.


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  		Table 1. Multilevel Random Coefficient Models for LCAL and PPD: Excluding and Including Clinical Covariates
 
Correlations involving the intercept differed between models, whereas correlations between both time-dependent covariates were similar (Table 2Go). These correlation structures facilitated important inferences to be made regarding longitudinal patterns of change. The large negative and highly significant correlations observed between the linear and quadratic time coefficients indicated that subjects and teeth with greater-than-average linear change experienced decelerated variation. Conversely, subjects/teeth with less-than-average linear change experienced accelerated variation. Therefore, changes in both outcomes exhibited a dynamic regression to the mean trajectory at the tooth and subject levels. Since no equilibrium was attained throughout the study, changes were thus cyclical.


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  		Table 2. Correlations of the Random Coefficients for LCAL and PPD Random Covariate Models with and without "Risk Factors" Included
 
The Role of Potential "Risk Factors" and Confounders
Alcohol consumption had no significant influence on either outcome, and smoking was only borderline significant in one model (Table 3Go). Both outcomes consistently increased from central incisor to 2nd molar, and sites experienced a worse outcome if they fluctuated with respect to LCAL/PPD throughout the study (i.e., they were non-constant), with change being predominantly progression. Subject-based measures indicated that the number of sites present within each subject had no significant association with either outcome. Subjects with a greater-than-average proportion of sites with LCAL > 1 mm and PPD > 2 mm experienced elevated levels of disease. When both covariates were considered simultaneously, sites with LCAL > 1 mm were more strongly associated with elevated LCAL, and sites with PPD > 2 mm were more strongly associated with elevated PPD.


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  		Table 3. Multilevel Random Coefficient Models for LCAL and PPD: the Role of Potential Risk Factors
 
If subjects had a greater-than-average proportion of sites with either subgingival calculus or bleeding upon probing, they experienced elevated periodontal breakdown when considered separately. In a combined covariate model, subjects experienced reduced periodontal breakdown for higher levels of these conditions (though not always significantly so). A reversal of covariate effects was also observed for proportions of supragingival calculus and plaque, though coefficients did not attain significance in the combined models.


   DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS & METHODS
 RESULTS
 DISCUSSION
REFERENCES
 
Patterns of Periodontal Disease Progression
Although several studies have tried to overcome the complexity of periodontal data, the key development offered by MLM is the ability to respect hierarchy along with the capacity to model complex variation. Variation within the data was far more informative than mean trajectories—hence the reason trajectory parameterization necessitated meaningful interpretation of relative changes. It should not be inferred that either outcome followed a quadratic path, and it would be erroneous to attach meaning to the individual trajectories. The complexity of many thousands of site-level trajectories yielded time-dependent covariate correlations that indicated how patterns of disease were cyclical across teeth and subjects (Table 2Go). Inclusion of potential risk factors and confounders did not affect these inferences. Patterns of disease transition were similar across teeth and subjects apart from the magnitude of oscillation, which was less across subjects as a direct consequence of information being summarized through averaging.

While cyclical change might be described as regression to the mean (RTM) (Blomqvist, 1987; Egelberg, 1989), results at the tooth and subject levels were not an artefact of measurement error. At the site level, the consistently negative correlation between the intercept and linear-time indicates that sites with initially higher-than-average scores experienced reduced linear change, such that some sites improved while others deteriorated at a much slower rate. This could be due to RTM. However, for higher levels of the hierarchy, RTM yields much less influence due to the averaging of site-level errors across teeth and subjects. While minor discrepancies in the correlations estimated might occur due to measurement error, the inference that change is oscillatory depends only upon the correlations’ sign.

The proposed theory needs verification for other cohorts, especially for more severely diseased subjects. It is necessary to validate the principle that disease patterns are cyclical, i.e., verifying the sign of time-dependent covariate correlations.

The Role of Covariates
Determination of potential risk factors and confounders is important for our understanding of disease prevention and treatment. After accounting for these factors, there remained residual variation, indicating that not all potentially important factors had been included. Trajectories for both outcomes were attenuated to reveal less improvement once risk factors were included. The effect of covariates was similar, though changes in the time-dependent coefficients differed between models, since overall changes in each outcome initially differed.

Not all risk factors were informative. Alcohol consumption had no impact on either outcome, and tobacco smoking demonstrated only a modest protective effect for PPD. Along with the usual difficulties with self-reported variables, these factors were recorded only at baseline, so there was no knowledge of changing behaviors. The impacts of smoking and alcohol therefore remain ambiguous.

Multicollinearity was observed through inconsistent subject-based coefficients between models that considered covariates in isolation and in combination (Table 3Go). These covariates were aggregated variables, derived from site-based measures. Collinearity was greatest for the two most-correlated aggregate measures: LCAL > 1 mm and PPD > 2 mm. There were sizeable correlations between aggregates and their site-based measures. However, it is an advantage of MLM that measures related through aggregation can be combined in the same model, as illustrated by site-based covariates yielding robust coefficient estimates when subject-based aggregate coefficients were included (Table 3Go). It is erroneous to place any meaningful interpretation on the subject-based covariate coefficients—an important observation given that subject-level (aggregate) covariates are often used in periodontal studies where less appropriate statistical methods are adopted.

Sites that experienced fluctuating LCAL/PPD (non-constant sites) were expected to experience greater disease on average, because many sites at baseline had little or no disease; for these sites, there was only one direction to change. Disease was progressively worse from anterior to posterior sites, while supragingival calculus moderated the progression of LCAL/PPD, mainly because supragingival calculus accumulated more in the lower anterior teeth.

The Role of Multilevel Modeling
The most powerful feature of MLM is the facility to investigate the underlying complexity of hierarchical systems, simultaneously modeling fixed effects and complex variation. Methods that accommodate hierarchy but fail to model variation explicitly (e.g., Generalized Estimating Equations) could not explore periodontal data in this manner. MLM provided a novel insight into the dynamic hierarchical system of periodontal breakdown and its progression, illustrating complex patterns of disease variation that had not been described by other methodologies. Many concepts attributed to the "linear" and "burst" models of periodontal disease progression are satisfied when examined in a multilevel context, since aspects of both theories have validity at different levels of the hierarchical system.



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  		Figure. A summary of all the possible changes in either periodontal outcome across the study period. The nine charts represent all possible transitions across three time points, where the black dots represent hypothetical outcome values at each measurement occasion. Outcomes may remain constant (chart 1) or, after no initial change, may exhibit change from occasions two to three (charts 2, 3) or, after initial progression, there may be no change (chart 4), further progression (chart 5), or improvement (chart 6), and, similarly, following initial improvement (charts 7 to 9).

 

   ACKNOWLEDGMENTS
 
This work was supported by funding from the British Medical Research Council and the UK government Higher Education Funding Council for England (HEFCE). The authors thank the staff at the former Medical Research Council Dental Research Unit and the staff of the Institute for Dental Health Training, RAF Halton, for their assistance in collection of the data. We also thank the apprentices who took part in this study and the Director of the Royal Air Force Dental Branch for his permission to carry out the study and publish its results.

Received July 5, 2001; Last revision October 30, 2002; Accepted November 7, 2002


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 DISCUSSION
 REFERENCES
 
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This Article
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