7 Ordered Categorical Response Models
In several chapters so far we have addressed modeling a binary response variable, for instance gingival bleeding upon periodontal probing, which has only two expressions, yes or no (or 1, 0). Many kinds of response variables are ordered categorical. For instance, the Silness & Löe plaque index, which assesses the amount of supragingival plaque at a certain tooth surface adjacent to the gingival margin, comprises clinically defined situations on a scale of scores from 0 to 3 (Table 7.1).
6 January 2015, 11:12 am.
Last modified January 6, 2015.
My MLwiN user manual using periodontal data has been completed but may be supplemented with further models if applicable.
5 Logistic Models for Binary and Binomial Responses
In the previous chapters, continuous response variables had been considered in various variance components, random intercept, random coefficient, time series, and multivariate time series multilevel models. In this chapter, we want to look at binary or binomial (proportion) responses. We will mainly focus on the logit link function. As usual, we start with a single-level model and extend this to appropriately consider the three-level hierarchical structure. We also explore contextual effects here. Significance testing and model interpretation using odds ratios and variance partition coefficients are discussed.
5.1 Description of the Example Data Set
The data for an example are stored in an EXCEL file (bop_pli01.xlsx). The binary response variable here is presence or absence of bleeding on probing (BOP) at gingival units in 50 students at Kuwait University. All had plaque-induced gingival disease.
29 October 2014 @ 6:17 am.
Last modified Otober 29, 2014.
After a long hiatus due to other projects, final exams and, well, some vacation, the next chapter of my multilevel modeling manual was overdue, but here it comes.
MULTILEVEL MODELING OF PERIODONTAL DATA
Manual version 2014.1
3 Time Series Models
In the two previous chapters, observations have been modeled which had been made at teeth which were nested in subjects. Now, consider that observations had been repeated in a longitudinal way, for example after therapy. For that purpose, we want to define a lower level (one) as repeated measures, or occasion, while teeth and subjects would define levels two and three, respectively.
Consider, for instance changes of the entire gingival unit after the implantation of a bio-resorbable membrane for surgical root coverage employing the principle of guided tissue regeneration. Apart from achievable root coverage, the implantation of the membrane and concomitant coronal advancement of the mucoperiosteal flap leads to an immediate increase in thickness of gingiva and later in width of keratinized tissue. Of course, the mucogingival border is displaced but may re-establish itself later in its original position. Note that these alterations at teeth are non-linear and, since wound healing proceeds and regenerated tissue matures over time, non-monotonic (see example (C) in Fig. 13.1 in Rasbash et al. 2012). And they certainly depend on the subject. So, we have a three-level structure with a sample of subjects in which we study gingival dimensions at teeth, which have been surgically treated, over time. The data had been analyzed by multilevel modeling in some detail by Müller (2008).
In this chapter we want to introduce the data set which had been created in a longitudinal study (Müller et al. 2000d) and set up several multilevel models further elaborating the basic time series model.
3.1 Repeated Measures Data on Gingival Dimensions after Surgical Root Coverage
The data set consists of observations which have been made in 14 patients who had presented with a large variety of recession types at altogether 31 teeth. They had been treated according to the principles of guided tissue regeneration employing a bio-resorbable membrane. Surgical root coverage consisted, after periosteal dissection, of a coronally advanced flap which was secured with sling sutures. Several preoperative clinical parameters and intrasurgical observations were assessed. Patients were followed up for 1 year, and re-examinations of the clinical situation were carried out after 3, 6, 9 and 12 months.
14 August 2014 @ 8:01 am.
Last modified August 14, 2014.
For those who are interested, here comes chapter 2 of my manual for MLwiN 2.30 using exclusively own periodontal data. As mentioned earlier, this is still work in progress and any comments are welcome.
MULTILEVEL MODELING OF PERIODONTAL DATA
Manual version 2014.1
2 Variance Components
In the previous chapter, the question had been asked whether gingival thickness, as measured mid-buccally at each tooth, was related to width of buccal gingiva. Several models had been built in a stepwise approach: a model ignoring the subject level, a two-level random intercept model, and a two-level random coefficient model. It could be shown that each model fitted the collected data better than the previous one. Eventually it turned out that the influence of gingival width on gingival thickness, if any, was low in general but depended significantly on the subject.
Gingival dimensions, i.e. its width and thickness, have long been related to the so-called periodontal phenotype, which appears to be a characteristic of a given subject. Gingival dimensions show great intra- and interindividual variation which is associated with tooth type and shape, and which is also mainly genetically determined. Our group has used cluster analysis of gingival appearance at upper anterior teeth and the respective shapes of the teeth of data collected in two independent samples of young adults (Müller and Eger 1997, Müller et al. 2000b) to study the periodontal phenotype in some detail. As cluster analysis is largely explorative, and external validity is questionable, the typical hierarchical structure of data collected in a third sample acquired in dental students was analyzed by multilevel modeling (Müller and Könönen 2005) with the specific aim to determine subject variation of gingival thickness, a supposed important part of the periodontal phenotype.
A most reasonable way to set up any multilevel model is to start with the basic variance components, or null, model without any covariates. The model should then be build up to increasing complexity by adding possible covariates which are assumed to have an influence on the response variable, and then checking whether they have substantial and/or significant fixed and random coefficients.
2 May 2014 @ 5:18 pm.
Last modified May 2, 2014.
As promised, below comes the first chapter of a manual for MLwiN which uses exclusively my own periodontal data. Much of this has been published over the years. As you may see on the list of contents in the beginning of the pdf below, this is still work in progress but those who are interested (and in possession of the software) may now contact me via email for getting access to respective EXCEL files and may go through respective analyses.
The manual is very much based, but yet not exhaustive, on a respective Users’s Guide to MLwiN by the Centre of Multilevel Modelling at Bristol University which was written by late Professor J. Rasbash and his coworkers in 2012. So, until the current work has been finished, the more interested new applicant of multilevel modeling is referred to Rasbash et al. (2012). Anyway, I hope that what I have written so far is useful. And, since more chapters are being uploaded soon, stay tuned!
MULTILEVEL MODELING OF PERIODONTAL DATA
Manual version 2014.1
How clinically collected data are properly statistically analyzed very much depends on its structure. Periodontal and other dental data are usually manifold observations which are made in one oral cavity. For instance, in order to describe the overall periodontal situation in a certain cohort, (i) sites (or gingival units) around (ii) teeth within (iii) patients or subjects are considered by using metric, ordinal, or binary variables. Then, observations may be (iv) repeated in a longitudinal way. This is a typical hierarchical situation with lower (occasions, sites) and upper levels (teeth, subjects). In clinical trials, a further (higher) level is present when patients are assigned to different centers.
A suitable armamentarium for the study of fixed (estimates of covariates) and random effects (variances and covariances) is multilevel modeling which has been applied to dental research data for long (Sterne 1988, Albandar and Goldstein 1992, Gilthorpe et al. 2000). Whereas the methods are well-known and have now been implemented in major statistical software packages such as SAS, STATA, R, even SPSS (and many others; for a comprehensive review of software programs and packages that are designed or can be used for multilevel analyses see de Leeuw and Kreft (2001)), major and somewhat revealing obstacles for applying them has long been at least twofold: a perceived (by clinicians) unwillingness of common biostatisticians to make themselves familiar with the more sophisticated methods of multilevel modeling, which are otherwise rarely used in medicine; and the simple fact that their application by clinical scientists, if not of most other statistical methods (Tu and Gilthorpe 2012), is vehemently discouraged by some biostatisticians.
The easy-to-apply special software MLwiN has been developed more than a decade ago, and the program has been applied in a considerable number of papers in dentistry; see, for instance, Gilthorpe et al. (2000), Ciantar et al. (2005), Müller (2008, 2009a), Müller et al. (2006), Müller and Stadermann (2006), Müller and Barrieshi-Nusair (2010), Tomasi et al. (2007), Fransson et al. (2010). Usually insights into complex data structure are revealing. Since a respective manual by the Centre of Multilevel Modelling in Bristol (Rasbash et al. 2012) explicitly uses examples and data sets from the social sciences, the aim of the present tutorial is to give a rather non-technical description of the basic principles of multilevel modeling using exclusively periodontal datasets which have been collected over the past ten years in order to further promote the correct statistical analysis of frequently hierarchically organized dental data.
1.1 The Problem
As Rasbash et al. (2012) commence in the introduction of the latest MLwiN manual, “In the social, medical and biological sciences multilevel or hierarchical structured data are the norm and they are also found in many other areas of application.”
Whereas any statistical model should explicitly recognize a hierarchical structure when it is present, and data structure is expected to be commonly hierarchical in dentistry and, in particular, periodontology, there are essentially two traditional approaches to data analysis.
1.2 Traditional Solutions
1.2.1 Site-specific analysis disregarding the subject
This approach, which can mainly be traced in scientific papers in Periodontology well up into the mid- or end-1980s, has vehemently been condemned by biostatisticians (Imrey 1986). As fact of the matter, clustered or hierarchical observations made in a certain subject are not independent, which is a fundamental assumption required for most statistical hypothesis testing. For instance, measures of periodontal disease within an oral cavity of a given patient are more alike than observations across oral cavities of other patients or subjects. By ignoring the subject level, standard errors of regression coefficients will inevitably be underestimated with grave consequences for hypothesis testing.
1.2.2 Aggregate analysis
By far the most common approach is, therefore, aggregating observations at the subject level. As an example, consider the cohort of 127 young adults with gingivitis where the association between presence or absence of supragingival dental plaque (a biofilm constantly forming on tooth surfaces, which can and should be removed regularly by toothbrushing) and gingival bleeding on probing (BOP) with a periodontal probe exerted with a more or less defined pressure (a sign for gingival inflammation caused, according to common sense, by dental plaque) had been assessed (Müller et al. 2000a).
In a subject-level, aggregate analysis one could have a look at the correlation between the proportion of tooth surfaces covered by plaque in each subject and the proportion of respective gingival units bleeding on probing. As an example, Fig. 1.1 displays results of such an analysis.
Ordinary regression was used to assess the relationship between the two variables. What might be stunning is the considerable scatter of data pairs representing the subjects. Correspondingly, the correlation between the two sets of proportions was only a moderate with Pearson’s r of 0.54.
31 March 2014 @ 12:55 pm.
Last modified March 31, 2014.