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RESEARCH REPORT |
1 Dental Public Health, Faculty of Dentistry, and
2 Department of Statistics and Actuarial Science, The University of Hong Kong, 34 Hospital Road, Hong Kong SAR, China;
* corresponding author, mcmwong{at}hkucc.hku.hk
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS & METHODSRESULTSDISCUSSIONREFERENCES |
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KEY WORDS: Bayesian approach • biostatistics • multilevel modeling • WinBUGS • survival data
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS & METHODSRESULTSDISCUSSIONREFERENCES |
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Recording multiple failure times from the same subject is a common practice in dental research. It is obvious that data from the same subject are not independent. Thus, when one is analyzing clustered failure time data, it is important to estimate the intra-cluster association. ‘Multilevel modeling’ (Gilthorpe et al., 2000b; Leyland and Goldstein, 2001) or ‘hierarchical linear modeling’ (Bryk and Raudenbush, 1992) is a class of statistical techniques developed to take into account the intra-cluster dependence in the analysis of clustered data. Analysis of clustered multilevel interval-censored data using the ‘frequentist’ approach for parameter estimation requires tailor-made computer programs. It would be desirable for dental researchers if the clustered multilevel interval-censored data could be analyzed by some software.
Bayesian analysis by the Monte Carlo Markov Chain (MCMC) has been a popular tool for analyzing complex data recently, and it has made its way into the medical and dental arena due to advances in computational and modeling techniques. Basically, Bayesian analysis generates conclusions based on the synthesis of new information from a study (the observed data) and previous knowledge or external evidence from independent sources (priors). By specifying a probability model for the observed data, D, given a set of unknown parameters, 
(unknown quantities that are of interest), and assuming that is random with a prior distribution 
() derived from external evidence, one can make a Bayesian inference concerning via the posterior distribution (|D) (revised distribution of based on the observed data and priors), according to Bayes’ theorem (Bayes, 1763). [For an introduction to Bayesian analysis in dental research, refer to Gilthorpe and co-workers (2000a) and Petrie and co-workers (2003).] Both point and credible interval estimates could be obtained from the posterior distribution. The fundamental difference between the classic ‘frequentist’ confidence interval and the Bayesian credible interval is that, for a long series of 95% confidence intervals, 95% of them should contain the true , whereas there is a 95% probability that the true lies in a 95% credible interval based on a specific prior distribution (Spiegelhalter et al., 2004).
This paper aims to use the Bayesian approach to analyze a set of multilevel clustered interval-censored data from a clinical study to investigate the effectiveness of silver diamine fluoride and sodium fluoride varnish in arresting active dentin caries in Chinese pre-school children.
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MATERIALS & METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS & METHODSRESULTSDISCUSSIONREFERENCES |
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Follow-up examinations were carried out every 6 mos after baseline by the same examiner, who did not know the subjects’ group assignments. Caries was diagnosed at cavitation level and explored with a sharp sickle-shaped probe at the center of the cavity. A tooth surface could be recorded as sound, caries-active, caries-arrested, filled, or missing.
Statistical Analysis
The analysis was performed at the tooth surface level. For each child, 24 tooth surfaces (4 from each of the 6 anterior teeth) were clinically examined, and those surfaces with dentin caries at the baseline examination were included in this analysis. We carried out multilevel modeling to compare the effectiveness of SDF and NaF with or without caries removal in arresting dentin caries, by taking into account the possible dependence due to the clustering effect of the kindergartens and the subject. The time to arrest of caries on a surface, namely, T, was used as the outcome measure of effectiveness. Thus, the shorter the arrest time, the more effective was the agent to arrest dentin caries. Since the children were examined every 6 mos, the arrest time was grouped into (0,6), (6,12), (12,18), (18,24), and (24,30) mos. For carious tooth surfaces that were not arrested at the time of exfoliation, subject drop-out, or at the end of the study, the arrest time was assigned as 0+, 6+, 12+, 18+, 24+, or 30+ mos. The arrest times of the five groups (1 = SDF+caries removal, 2 = SDF, 3 = NaF+caries removal, 4 = NaF, 5 = Control) were compared, and location of residence (1 = urban vs. 0 = rural) and gender (1 = boy vs. 0 = girl) were also included in the model as covariates.
Since the arrest times were not totally independent, 2 additive random effects were included in the model, to account for the clustering effects of the carious tooth surfaces in the same child’s mouth, and of children attending the same kindergarten, namely, Bj (j = 1, 2, ... , 367) and Ck (k = 1, 2, ... , 8), respectively. The random effects Bj and Ck were assumed to follow the N(0,
2child) and N(0,2school) distributions, respectively. Finally, conditional on the random effects, the arrest times Tijk (i = 1, 2, ... , njk) are independent and assumed to follow a Weibull distribution with survivor P(Tijk > t | Bj, Ck) and hazard 
ijk(t) functions
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and
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respectively, where
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and X are the observed covariates, such as group allocation. In this study, all the covariates X are coded as 1 or 0, indicating the presence or absence of a certain characteristic or treatment. Typically, a positive regression coefficient ß corresponds to a higher risk of the failure being observed among those with the associated characteristic, relative to those without. Alternatively, it is natural to report a more intuitive measure, namely, the relative risks [RR = exp(ß), RR > 1 indicates a higher risk of failure]. In this study, a positive ß or RR > 1 corresponds to a higher chance of arrest of active dentin caries and thus expects a shorter arrest time.
The shape parameter r characterizes the shape of the distribution (r > 1 for increasing failure rate; r < 1 for decreasing failure rate; and r = 1 for constant failure rate). With the above model, the intra-cluster correlation between the logarithmic arrest times from the same child and from children attending the same kindergarten can be estimated by [2school + 2child] / [2school + 2child + 2/6] and 2school / [2school + 2child + 2/6], respectively (Lindeboom and Van Den Berg, 1994).
With the arrest time Tijk being interval-censored in the interval (t1, t2), conditioned on the random effects Bj and Ck, the contribution to the likelihood can be expressed as
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‘Unconditioning’ the random effects is very often an intractable task in the interval-censored set-up, particularly in multilevel modeling (with more than one random effect). Hence, the Bayesian approach with MCMC algorithms was adopted, and the analysis was carried out with the software WinBUGS, version 1.3, in which Gibbs’ sampler was used for the generation of samples (Spiegelhalter et al., 1999). A three-level model was considered, with tooth surfaces as level 1, children as level 2, and kindergartens as level 3. In the estimation of the parameters, the first 5000 simulations were treated as burn-ins and discarded, while the estimation was based on the next 10,000 simulations. Non-informative priors were adopted in this analysis, since we did not want to impose any prior beliefs on the effects of the treatments. A graphed presentation of the model used in the analysis and the model statements used for the programming are shown in the Appendix for technical reference (readers could skip this without loss of continuity).
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS & METHODSRESULTSDISCUSSIONREFERENCES |
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).
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), concluded that the correlation between the arrest times of children from the same school was negligible [2school = 0.025; 95% credible interval = (0.001, 0.151), Table 2]. However, the clustering effect among the arrest times from the same child was very strong [2child = 2.394; 95% credible interval = (1.822, 3.066), Table 2], and the estimated intra-cluster correlation coefficient among the arrest times of different caries surfaces from the same child was 0.596. In the following discussions, the effects are considered statistically insignificant if the 95% credible intervals for ß include 0 or, equivalently, the 95% credible interval for RR includes 1, and vice versa. Effects of location of residence and gender on the arrest times were statistically insignificant (Note 1 in Table 2). The two agents (SDF and NaF) with caries removal and SDF without caries removal significantly shortened the arrest time of the dentin caries compared with the controls (Note 2 in Table 2). When soft caries was removed prior to the application of the test agents, the arrest times could further be significantly shortened (Note 3 in Table 2). When the two agents were compared, SDF had shorter arrest times than NaF, both with and without prior caries removal (Note 4 in Table 2). As an illustration, the survival curves of the arrest times of the five groups for a girl living in the rural area are shown in the Fig. The median times for a caries-active surface to become arrested when SDF was applied, with and without caries removal, were 7.0 and 12.5 mos, respectively. The median arrest times when NaF was applied, with and without caries removal, were 14.5 and 21.0 mos, respectively, while that in the controls was 30 mos (Fig.).
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS & METHODSRESULTSDISCUSSIONREFERENCES |
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Recently, in dental research, several approaches have been proposed for handling clustered survival data with exact failure times (Chuang et al., 2002a,b; Gilthorpe et al., 2002), or for handling clustered interval-censored data (Härkänen et al., 2000, 2002; Hannigan et al., 2001; Bogaerts et al., 2002). Both the ‘frequentist’ and the Bayesian approaches have been used, different models (frailty vs. marginal) have been suggested, and different software packages (SAS, S-plus) have been recommended. To our knowledge, this is the first study in dental research to use the software package WinBUGS for analyzing multilevel (clustered) interval-censored data, and to report the correlations among the failure times. Multilevel modeling in terms of multivariate frailty can also be applied if the data structure is much more complicated—for instance, multi-stage clustering or nested design in a randomized controlled trial.
The Bayesian approach rests on an essentially subjective interpretation of the observed data in the light of external evidence, judgment, and past experiences (i.e., the informative priors) and then to derive the conclusion in a manner that fits naturally with the clinical decision-making process (Spiegelhalter et al., 1994). It is well-known that turning informally expressed opinion into a mathematical prior distribution is perhaps the most difficult aspect of Bayesian analysis and therefore should be introduced with caution (Spiegelhalter, 2001). In situations where informative priors are unavailable, or to provide a kind of ‘objective’ Bayesian analysis free from subjectivity, non-informative priors can be adopted, as in this study. Bayesian inference has several advantages over the ‘frequentist’ approaches, particularly in the flexibility of model-building for complex data. Moreover, for many models, ‘frequentist’ inference can be obtained as a special case of Bayesian inference with the use of non-informative priors (Ibrahim et al., 2001). The Bayesian approach enables us to make exact inference based on the posterior distribution for any sample size, whereas the ‘frequentist’ approach relies heavily on the large sample approximation, and there is always the issue of whether the sample size is large enough for the approximation to be valid (Ibrahim et al., 2001). There is a danger that the additional complexity of Bayesian methods could lead to improper data analysis if it is not used correctly. In addition, software for implementation of Bayesian methods is still limited in user-friendliness (Spiegelhalter et al., 2004).
Bayesian inference Using Gibbs Sampling (BUGS or WinBUGS) is a piece of freely available computer software for the Bayesian analysis of complex statistical models using Markov chain Monte Carlo (MCMC) methods (Spiegelhalter et al., 1999). It is reasonably easy to use and comes with a wide range of examples (Spiegelhalter et al., 1996a, b). However, much technical statistical knowledge is required for it to be used correctly.
With the abovementioned advantages and the availability of the software WinBUGS, analysis of clustered multilevel interval-censored data is made possible and simple. In conclusion, the annual application of silver diamine fluoride to caries lesions, and caries removal before the application, were found to have shortened the arrest time.
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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Received November 13, 2003; Last revision April 28, 2005; Accepted May 31, 2005
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REFERENCES |
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TOPABSTRACTINTRODUCTIONMATERIALS & METHODSRESULTSDISCUSSIONREFERENCES |
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Bayes T (1763). An essay towards solving a problem in the doctrine of chances. Phil Trans R Soc 53:370–418.
Bogaerts K, Leroy R, Lesaffre E, Declerck D (2002). Modelling tooth emergence data based on multivariate interval-censored data. Stat Med 21:3775–3787.[ISI][Medline]
Bryk AS, Raudenbush S (1992). Hierarchical linear models: applications and data analysis methods. Newbury Park: Sage Publications.
Chu CH, Lo EC, Lin HC (2002). Effectiveness of silver fluoride and sodium fluoride varnish in arresting dentin caries in Chinese pre-school children. J Dent Res 81:767–770.
Chuang SK, Wei LJ, Douglass CW, Dodson TB (2002a). Risk factors for dental implant failure: a strategy for the analysis of clustered failure-time observations. J Dent Res 81:572–577.
Chuang SK, Tian L, Wei LJ, Dodson TB (2002b). Predicting dental implant survival by use of the marginal approach of the semi-parametric survival methods for clustered observations. J Dent Res 81:851–855.
Fleming TR, Lin DY (2002). Survival analysis in clinical trials: past developments and future directions. Biometrics 56:971–983.
Gilthorpe MS, Maddick IH, Petrie A (2000a). Introduction to Bayesian modelling in dental research. Community Dent Health 17:218–221.[ISI][Medline]
Gilthorpe MS, Maddick IH, Petrie A (2000b). Introduction to multilevel modelling in dental research. Community Dent Health 17:222–226.[ISI][Medline]
Gilthorpe MS, Mayhew MT, Bulman JS (2002). Multilevel survival analysis of amalgam restorations amongst RAF personnel. Community Dent Health 19:3–11.[Medline]
Hannigan A, O’Mullane DM, Barry D, Schäfer F, Roberts AJ (2001). A re-analysis of a caries clinical trial by survival analysis. J Dent Res 80:427–431.
Härkänen T, Virtanen JI, Arjas E (2000). Caries on permanent teeth: a non-parametric Bayesian analysis. Scand J Stat 27:577–588.
Härkänen T, Larmas MA, Virtanen JI, Arjas E (2002). Applying modern survival analysis methods to longitudinal dental caries studies. J Dent Res 81:144–148.
Ibrahim JG, Chen MH, Sinha D (2001). Bayesian survival analysis. New York: Springer.
Leyland AH, Goldstein H, editors (2001). Multilevel modeling of health statistics. Chichester: John Wiley & Sons.
Lindeboom M, Van Den Berg GJ (1994). Heterogeneity in models for bivariate survival: the importance of the mixing distribution. J R Stat Soc Series B 56:49–60.
Lindsey JC, Ryan LM (1998). Tutorial in biostatistics methods for interval-censored data. Stat Med 17:219–238; erratum 18:890, 1999.[ISI][Medline]
Lo EC, Chu CH, Lin HC (2001). A community-based caries control program for pre-school children using topical fluorides: 18-month results. J Dent Res 80:2071–2074.
Petrie A, Bulman JS, Osborn JF (2003). Further statistics in dentistry. Part 9: Bayesian statistics. Br Dent J 194:129–134.[ISI][Medline]
Prentice RL, Gloeckler LA (1978). Regression analysis of grouped survival data with application to breast cancer data. Biometrics 34:57–67.[ISI][Medline]
Spiegelhalter DJ (2001). Bayesian methods for clustered randomized trials with continuous responses. Stat Med 20:435–452.[ISI][Medline]
Spiegelhalter DJ, Freedman LS, Parmer MKB (1994). Bayesian approaches to randomized trials. J R Stat Soc Series A 157:357–387.
Spiegelhalter DJ, Thomas A, Best NG, Gilks WR (1996a). BUGS examples. Vol. 1. Version 0.5. Cambridge: MRC Biostatistics Unit.
Spiegelhalter DJ, Thomas A, Best NG, Gilks WR (1996b). BUGS examples. Vol. 2. Version 0.5. Cambridge: MRC Biostatistics Unit.
Spiegelhalter DJ, Thomas A, Best NG (1999). WinBUGS Version. 1.2 user manual. Cambridge: MRC Biostatistics Unit.
Spiegelhalter DJ, Abrams KR, Myles JP (2004). Bayesian approaches to clinical trials and health-care evaluation. West Sussex: John Wiley & Sons.
SPSS Inc. (2002). SPSS 11.5 advanced models. Chicago: SPSS Inc.
Venables WN, Ripley BD (1999). Modern applied statistics with S-Plus (3rd ed.). New York: Springer.
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