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Predicting Dental Implant Survival by Use of the Marginal Approach of the Semi-parametric Survival Methods for Clustered Observations

^1,4 Department of Oral and Maxillofacial Surgery, Massachusetts General Hospital and Harvard School of Dental Medicine, 55 Fruit Street, Warren 1201, Boston, MA 02114;
¹ Department of Oral Health Policy and Epidemiology, Harvard School of Dental Medicine, 188 Longwood Avenue, Boston, MA 02115; and
^1,2,3 Department of Biostatistics, Harvard School of Public Health, 677 Huntington Avenue, Boston, MA 02115;

*corresponding author, PO Box 67376, Chestnut Hill Station, Chestnut Hill, MA 02467, schuang{at}hsph.harvard.edu

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS & METHODS RESULTS DISCUSSION REFERENCES

 
The analyses of clustered survival observations within the same subject are challenging. This study's purpose was to compare and contrast predicted dental implant survival estimates assuming the independence or dependence of clustered observations. Using a retrospective cohort composed of 677 patients (2349 implants), we applied an innovative analytic marginal approach to produce point and variance estimates of survival predictions given the covariates smoking status, implant staging, and timing of placement adjusted for clustered observations (dependence method). We developed a second model assuming independence of the clustered observations (naïve method). The 95% confidence intervals for survival prediction point estimates given the naive method were 5.9% to 14.3% more narrow than the dependence method estimates, resulting in an increased risk for type I error and erroneous rejection of the null hypothesis. To obtain statistically valid confidence intervals for survival prediction of the Aalen-Breslow estimates, we recommend adjusting for dependence among clustered survival observations.

KEY WORDS: survival predictions • dental implants • clustered data • correlated survival analysis • proportional hazards model • marginal approach • Aalen-Breslow estimator

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS & METHODS RESULTS DISCUSSION REFERENCES

 
In clinical dental research, each patient frequently contributes numerous, potentially correlated observations to a dataset. For example, in periodontal research, one patient may contribute to the database multiple probing depths obtained on multiple teeth at multiple times. The analysis of correlated survival observations is a challenging biostatistical problem. Commonly, assumptions regarding independence of the observations (naïve method) are utilized during survival analyses (Higuchi et al., 1995; Wheeler, 1996; Buser et al., 1997; Brocard et al., 2000; Testori et al., 2001).

Statisticians have significant interest in developing and applying statistical methods and models to address correlated or clustered multivariate survival data (Spiekerman and Lin, 1998). The analytical challenge lies in accounting for the correlation of the observations for valid statistical inferences. To this end, two classes of models have been proposed: (1) proportional hazards frailty and (2) marginal proportional hazards models (Vaida and Xu, 2000). The former approach formulates the dependence structure explicitly. Some investigators had mainly utilized the gamma frailty for the dependence structure of the proportional hazards frailty model discussed by Vaida and Xu (2000). The latter class of models, marginal proportional hazards, does not specify the dependence structure in the model formulation but adjusts for it in the inference by means of sandwich-type estimators (Lee et al., 1992; Lin, 1994; Spiekerman and Lin, 1998).

In dental implant research, multiple implants placed into the same patient produces challenging analytic problems, because the dataset is composed of multiple, correlated observations. To address the issue of correlated, dependent observations, many authors recommended randomly selecting one implant per patient for analysis, resulting in inefficient estimation because not all of the data are used (Haas et al., 1996; Herrmann et al., 1999; Lekholm et al., 1999; Gomez-Roman et al., 2001; Weibrich et al., 2001). The purpose of this study is to compare and contrast two methods for producing survival prediction estimates and their associated confidence intervals for a dataset composed of clustered observations. The first method, commonly used but statistically invalid, assumes independence of clustered observations. The second method is an innovative analytic survival prediction method, i.e., modified Aalen-Breslow estimates, designed to produce statistically valid, efficient prediction models of implant survival for clustered observations using the marginal approach by Spiekerman and Lin (1998). We believe that this is the first report with integrated clinical applications of theoretical clustered survival methodologies to predict dental implant survival.

	MATERIALS & METHODS

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Methods
The research methods have been described in detail in other studies (Chuang et al., 2001, 2002; Vehemente et al., 2002). Briefly, we conducted a retrospective cohort study composed of patients who had one or more Bicon^® implants placed between May, 1992, and July, 2000. We reviewed patient charts to obtain information regarding exposures associated with implant failures grouped into the following categories: demographic, health status, anatomic, implant- and abutment-specific, anticipated restoration, peri-operative chemotherapy, reconstructive, and operator. The major outcome variable of interest was implant failure, defined as the removal of the implant for any reason (Dental Implant Clinical Research Group, 1997). We estimated survival time by computing the difference in time (months) between implant placement and implant explantation or the date of the last follow-up visit for patients whose implants had not been removed. This study was reviewed and approved by the Human Research Committee.

Statistical Issues
In this article, we considered the survival prediction problem for the Cox proportional hazards model (Cox, 1972). Specifically, let T be the failure time, and let Z be the corresponding covariate vector. Also, let S_Z(•) be the survival function of T given Z. The Cox model can be written as log(-log(S_Z(t)) = h(t) + ß^TZ, which is a member (special case) of the general linear transformation model g(S_Z(t)) = h(t) + ß^TZ for link function g (x) = log(-log(x)), where h(t) is a completely unspecified strictly increasing function and ß is a p x 1 vector of unknown regression coefficients. (Cheng et al., 1995, 1997). We propose pointwise procedures for S_z0(t), the "t-year" (or "t-months") survival probability of future patients with a given covariate vector z₀ with clustered implant observations. The survival prediction method proposed by Spiekerman and Lin (1998) was integrated and applied to predict survival probabilities of correlated dental implants for a given set of covariates via the Aalen-Breslow estimators. The set of covariates selected to be used in this study, i.e., current tobacco use, implant staging, and timing of implant placement, was previously identified by statistically valid and efficient methods that adjusted for clustered observations (Chuang et al., 2002). Databases were stored in Epiinfo 2000 (Centers for Disease Control and Prevention, Atlanta, GA, USA) and SAS-PC 8.0 (Cary, NC, USA) files. Advanced statistical computations for clustered survival predictions were utilized by S-plus (Version 3.4, 1996) with special functions and coding with programming in the Unix environment provided by our co-authors (Tian and Wei).

Statistical Notation
As discussed by Spiekerman and Lin (1998), if each patient consists of L_i implants, the marginal hazard function for the lth implant of the ith patient is related to the corresponding possibly time-dependent covariate vectors Z_il(t) by

The above set-up parallels the idea in the Lee et al. (1992) and Lin (1994) methods.

For i = 1,...,N, and l = 1,...,L_i, let T_il and C_il be the failure and censoring times with respect to the lth implant of the ith patient, and Z_il = (Z_1il,...,Z_pil)^T be the corresponding (and possibly time-varying) covariate vector. The marginal distribution of T_il is related to Z_il through model (*). Define T_i = {T_il; l = 1,...,L_i}, with C_i and Z_i defined similarly. Suppose that (T_i, C_i, Z_i) (i = 1,...,N) are independent, identically distributed (i.i.d.), and that T_i is independent of C_i conditional on Z_i. The patients are allowed to have different sizes of dental implants (i.e., L_i, which denotes that the cluster sizes of dental implants in the same patient can vary). We define X_il = minimum (T_il, C_il) and _il = 1 (T_il C_il), where 1(•) is the indicator function.

Under the independence working assumption, the "quasi-partial likelihood" for ß₀ is:

and

L(ß) is the partial likelihood function, the observed information matrix is I( ) = -²logL(ß)/ß², given that ß = , n is the total number of dental implants, and N is the total number of patients.

To make survival predictions and the variance-covariance matrix of the survival function S_z0(t), we need to derive the Aalen-Breslow type estimators for ₀(t), which is the baseline cumulative hazard function. The variance-covariance matrix for the survival function also needs to be derived and computed. [For statistical methodologies with vigorous mathematical details on the convergence and asymptotic properties of these estimators, please refer to Spiekerman and Lin (1998, pp. 1167-1169).]

The Aalen-Breslow-type estimator for ₀(t) is:

The variance of the Aalen-Breslow type estimators adjusted for clustered observations for ₀(t) is:

where

We are interested in constructing pointwise confidence intervals for ₀(t), and the corresponding survival function S₀(t) = e^-₀(t), for a given time point t. Then, the corresponding confidence intervals for ₀(t) and S₀(t) are

and

respectively, where (t) = exp(- ((·),t)).

To make inferences about the cumulative hazard and survival functions with a given covariate vector Z₀, one can simply replace Z_il–Z₀, i=1,...,N, l=1,...,L_i in the original dataset and obtain the confidence intervals for the underlying cumulative hazard and survival function with this modified dataset.

	RESULTS

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The study patients were composed of 677 subjects who had 2349 dental implants placed. On average, 3.5 implants (range 1-22) were placed per patient. There were 137 implants that failed. There were 57 smokers (10.3%, N = 554). Out of 2349 implants, 339 (14.4%) were placed in one stage, and 243 (10.3%) were placed immediately after tooth extraction. The details of the descriptive statistics were described by Chuang et al. (2002).

We previously identified the variables statistically associated with implant survival: timing of implant placement (delayed vs. immediate), current tobacco use (yes or no), and implant staging (one- or two-stage) (Chuang et al., 2002). For patients who had a delayed procedure, did not smoke, and underwent a two-stage implant procedure (best-case scenario), the predicted one- and five-year survival rates were 97.2% and 93.4%, respectively. For patients who had an immediate implant placed, smoked, and underwent a one-stage procedure (worst-case scenario), the predicted survival rates at one and five years were 58.5% and 27.6%, respectively. The details of various combinations are shown in Table 1. As an example of producing valid Aalen-Breslow variance of the predicted one-year survival point estimates, we used the cohort of non-smoking patients having immediate implants placed in one stage. The predicted one- and five-year survival point estimates were 83.7%. and 65.2%, respectively. The 95% confidence intervals were 76.5% to 91.6% when computed by the naïve method and 73.3% to 94.2% when computed by the dependence method. The 95% confidence interval computed by the dependence method was 5.9% wider (absolute difference) than the confidence interval computed by the naïve method (Table 2). For the five-year predicted survival estimate, the 95% confidence intervals were 51.2% to 83.0% computed by the naïve method and 42.2% to 88.2% computed by the dependence method. The 95% confidence interval computed by the dependence method was 14.2% wider (absolute difference) than the confidence interval computed by the naïve method. The dependence method for computing variance produced an adjusted standard error that was larger and with wider confidence intervals than with the naïve method, illustrating the positive, strongly correlated association of the survival of implants within the same subject given specific important predictors (Table 2). The results of the above analyses suggest that, in the setting of clustered observations, the naïve method variance estimate was invalid. As such, the choice of method for variance estimation is an important consideration when survival predictions are performed.

	DISCUSSION

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To produce statistically valid prediction models adjusted for clustered failure time data, we applied the method described by Spiekerman and Lin (1998). Based on that method, patients who were non-smokers, had implants placed in two stages, and did not have immediate implants had the best survival estimates (one- and five-year survival estimates were 97.2% and 93.4%, respectively). Patients who were smokers and had implants placed immediately in one stage had the worst survival estimates (one- and five-year survival estimates were 58.5% and 27.6%, respectively).

To compare and contrast how modifying the assumptions of implant observations (independent or dependent) affected the predicted survival point and variance estimates, we utilized a common method for analyzing implant survival (Higuchi et al., 1995; Wheeler, 1996; Buser et al., 1997; Brocard et al., 2000; Testori et al., 2001), ignoring the issue of dependency among implant observations from the same subject (naïve model). In the second model, we utilized all implants but adjusted for correlation among implants from the same subject for survival predictions (dependent model). When comparing the two analytic strategies, we found that the one- and five-year survival point estimates were similar, but the variance estimates were drastically different. The 95% confidence intervals for the naïve model were narrower, by approximately 5.8% to 14.2%, than the dependent model. In the setting of clustered observations, we believe that the standard error estimates calculated by the Spiekerman and Lin (1998) method is the most statistically valid and efficient choice for analysis of correlated survival predictions in implant research, because it accounts for the correlation of implants within subjects.

The marginal modeling methodology as described by Wei et al. (1989), Lee et al. (1992), Lin (1994), and Spiekerman and Lin (1998) is likely to increase its popularity for the analysis of clustered multivariate survival data. Lin (1994) also presented a general theoretical statistical methodology for analyzing such data structure, an idea analogous to that of Liang and Zeger (1986), without censoring in longitudinal data analysis. Lin's approach formulates the marginal distributions of multivariate failure times with the familiar Cox proportional hazards models, while leaving the nature of dependence among related failure times completely unspecified. The baseline hazards functions for the marginal models may be identical or different. Generalized estimating equations investigated by Lin (1994) from the marginal approach for the regression parameters revealed consistent and asymptotically normal estimators, and robust variance-covariance estimators are constructed to account for the intra-class correlation.

Further simulation results by Lin (1994) demonstrated that the large-sample approximations are adequate for practical use, and that ignoring the intra-class correlation could yield rather misleading variance estimators which are similar in our investigation. The theoretical work by Spiekerman and Lin (1998) provided additional modeling capabilities by allowing for separate baseline hazard functions among different strata and imposing the same baseline hazard function within each stratum. Second, it provides a rigorous asymptotic theory for the estimation of the regression parameters, filling several important gaps in the existing proofs for the Lee et al. (1992) and Lin (1994) method. Third, it establishes the asymptotic properties of the Aalen-Breslow-type estimators for the cumulative baseline hazard functions and develops the corresponding inference procedures. We have utilized these new ideas, methodologies, and analytic techniques, which will facilitate further research and applications of statistical methodology and methods for analyzing multivariate dental failure time data, such as dental implant research.

In summary, clustered survival observations are frequently encountered in many different areas of patient-oriented dental research. Commonly, the key issue of dependence structure of clustered observations is ignored in analyses, resulting in statistically invalid estimates and inflated type I, risking erroneous rejection of the null hypotheses. Additionally, most current methods to adjust for clustered survival observations, i.e., randomly selecting one observation per patient, result in inefficient estimation. The methods used in this study produce statistically valid and efficient estimates. In their current state, however, these survival methods are resource-intensive, requiring high-level programming and methodological biostatistical expertise with intensive computer time. Regardless, in the setting of correlated survival observations, we recommend adjusting for the correlation of the observations to provide statistically valid variance of the Aalen-Breslow estimator under investigation to predict survival for a given set of covariates. Future efforts are focused on making these important analytic methods more accessible to the average patient-oriented researcher. Predicting survival estimation based on clustered multivariate data continues to be an important, challenging, but under-investigated, biostatistical problem in patient-oriented dental research. Future research efforts will focus on the application of mixed-effects (frailty) and accelerated failure time (AFT) survival models to datasets composed of clustered survival observations.

	ACKNOWLEDGMENTS

 
This research is supported in part by Oral and Maxillofacial Surgery Foundation (OMSF) Clinical Investigation Training Fellowship (SKC), Dentist Scientist Award NIH/NIDCR K16 DE000275 (SKC), Howard Hughes Medical Institute Pre-Doctoral Fellowship in the field of Biostatistics (LT), NIH/NCI grant R01 CA56844 (LJW), Mid-Career Investigator Award in Patient-Oriented Research, NIH/NIDCR K24 DE000448 (TBD), and the Oral and Maxillofacial Surgery Research Fund, Massachusetts General Hospital (SKC, TBD). This manuscript partially fulfilled the doctorate degree (DMSc) requirements at the Harvard University Faculty of Medicine for the first author (SKC). We also thank Ms. Valerie Vehemente for her assistance in data collection. The authors recognize the clinicians and staff of the Implant Dentistry Centre at the Faulkner Hospital, Boston, MA, for their cooperation in this study and their free and unfettered access to patient records.

Received January 14, 2002; Last revision August 7, 2002; Accepted September 10, 2002

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