Matching the Statistical Model to the Research Question for Dental Caries Indices with Many Zero Counts

Abstract

Childhood dental caries, which can be associated with adverse health, social and economic consequences, is a major research focus of oral health scientists. Count regression models for dental caries indices (e.g., DMFS, dmft) are useful for understanding risk factors and describing variations in caries prevalence and severity in cross-sectional dental surveys or in caries increment in prospective epidemiological investigations and in randomized clinical trials. In all these cases, count regression models are used to model a single count outcome for individuals whose responses are assumed to be independent of one another. Extensions of count regression models to clustered or repeated measures outcomes will be briefly covered in the discussion section. The purpose of this article is to describe and illustrate several common choices of count regression models for independently distributed counts.

The oral health research analyst or investigator has several regression models for counts from which to choose. Negative binomial regression [Hilbe, 2011] is sometimes used for modeling dental caries indices because caries counts tend to exhibit overdispersion, i.e., excess variation relative to the Poisson distribution [Grainger and Reid, 1954]. However, as caries rates have declined in populations over time [Campus et al., 2009], distributions of caries counts are increasingly characterized by a large number of zero counts, for which Poisson and negative binomial distributions frequently provide inadequate fits. Zero-inflated Poisson regression (ZIP) models [Mullahy, 1986; Lambert, 1992] were initially applied to dental caries to account for excess zeros [Böhning et al., 1999]. To address excess zeros and overdispersion simultaneously, zero-inflated negative binomial (ZINB) regression [Lewsey and Thompson, 2004] and negative binomial hurdle (NBH) models [Levin et al., 2009] have more recently been applied in the analysis of dental caries count data.

A review of the application of ZIP and ZINB models to dental caries counts published in 2012 found that interpretations provided in published caries research are often imprecise or inadvertently misleading, particularly with respect to failing to discriminate between inference for the class of susceptible persons defined by such models and inference for the sampled population in terms of overall exposure effects [Preisser et al., 2012]. Recommendations were provided to enhance the use as well as the interpretation and reporting of results of count regression models when applied to dental caries. The present article updates the earlier recommendations following the recent introduction of marginalized zero-inflated Poisson (MZIP, Long et al., 2014] and marginalized negative binomial regression models (MZINB, Preisser et al., 2016] that directly specify the effect of exposures and covariates on the marginal (overall) mean instead of the latent class mean of susceptible persons.

Whereas ZIP and ZINB models are often useful in dental research for understanding the mechanisms of disease [Gilthorpe et al., 2009], insufficient emphasis has been given to the effects of caries risk factors on the overall population from which the study sample was drawn [Albert et al., 2014]. Recently, MZIP and MZINB regression models were introduced for the direct estimation of overall exposure effects on a count outcome. With several model classes for the potential analysis of zero-inflated count outcomes, this article aims to provide guidance to researchers in the choice among ZINB, NBH and MZINB models. For this task, an understanding of the different model classes for count outcomes with many zero counts, and the research questions addressed by them, is essential.

In this article, ZINB, NB Hurdle, and MZINB models are applied to two data sets to illustrate how the different models are used to address distinct research questions. The first is a fictional data set of 1,000 observations generated from a ZINB distribution for each of two groups. The second data set is from a three-year double-blind caries incidence trial in Lanarkshire, Scotland conducted from 1988 to 1992 that randomized 4,294 children aged 11–12 years to three active toothpaste formulations to compare their respective anticaries efficacy [Stephen et al., 1994]. The different interpretations provided by the models are emphasized to support the assertion that drawing valid conclusions in caries research relies on matching the statistical analysis to a well-articulated research question [Long et al., 2012].

Methods

The ZINB, NB Hurdle, and MZINB distributions

In the absence of covariates, there exists equivalent ZINB, NBH and MZINB distributions for count data in the sense that they give identical values for the probabilities P(Y=y) for counts y = 0, 1, 2, etc. The distributions are distinguished only by their parameterization, or, in other words, the feature or features of the overall distribution of the counts that are the focus for a chosen model. Each of the ZINB, NBH and MZINB distributions have two parts so as to account for a greater fraction of zeros than can be accommodated by the negative binomial distribution NB(μ,ϕ) where μ is the mean count and ϕ (where ϕ>0) is the dispersion parameter with larger values of ϕ corresponding to greater variation in the counts (equation A.1 in suppl. file). Definitions of the various distributions for zero-inflated counts are given in the suppl. file.

As an illustrative example, the top half of Figure 1 shows three different representations of a distribution of caries counts with a large number of zeros, as is often encountered in caries research. For example, the counts could be dmft from a population of children on a normal diet to be compared to dmft from children on a high sugar diet shown in the bottom half of the figure. Figure 1(a) depicts a ZINB(ψ, μ, ϕ) distribution (equation A.2 in suppl. file), which presupposes that the population of children under study is a mixture of two latent classes, i.e., hypothetical, unobserved groupings of children. The gray area is the fraction ψ=0.40 of the total area of the bars that represents the subpopulation of “non-susceptible” children with only zero counts who are considered to be not-at-risk for caries due perhaps to some unknown environmental or genetic factors. The black bars show the relative frequencies of caries counts based on a negative binomial distribution with mean μ = 1.00 and overdispersion parameter ϕ=0.50 for the “susceptible” subpopulation of children at risk for dental caries; these probabilities are scaled so that the total area of the black bars is 0.60, the probability of being in the at-risk class. The ZINB distribution assumes that children without caries consist of those children who are susceptible but happened not to have any caries observed (i.e., 0.27 random zeros) and children who are believed to be practically not-at-risk for caries (0.40). The challenges that dental researchers face in understanding ZI models are related to the fact that the composition of the two respective latent classes is a theoretical and mathematical construct such that the specific group membership of any given subject in a study with a zero count is unknown. Moreover, the ZINB parameters ψ and μ provide only indirect information on the population caries prevalence π and mean caries extent ν that are often of interest.

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Figure 1

The top half of the figure presents equivalent representations of counts generated from a negative binomial (NB) distribution with added zeros describing dmft among a fictional population of children on a normal diet: (a) a zero-inflated NB distribution with (ψ=0.40, μ=1.00, ϕ=0.50); (b) a NB hurdle distribution with (π=0.33, μ=1.0, ϕ=0.50); (c) a marginalized zero-inflated NB distribution with (ψ=0.40, ν=0.60, ϕ=0.50). The bottom half presents equivalent representations of counts for dmft among children on a high sugar diet: (d) a zero-inflated NB distribution with (ψ=0.10, μ=3.00, ϕ=0.50); (e) a NB hurdle distribution with (π=0.76, μ=3.00, ϕ=0.50); (f) a marginalized zero-inflated NB distribution with (ψ=0.10, ν=2.70, ϕ=0.50).

An alternative representation to the ZINB distribution is provided by the negative binomial hurdle distribution NBH(π, μ, ϕ) (equation A.3 in suppl. file) shown in Figure 1(b) for children on a normal diet. The height of the gray bar is 0.67, which is the probability of not having caries. The black area of Figure 1(b) is the prevalence π=0.33 and corresponds to the group of children with any caries, whose count distribution is characterized with a zero-truncated negative binomial distribution scaled so that the sum of probabilities for non-zero counts when scaled equals π. The prevalence π cannot be greater than the probability of not being an excess zero, 1 − ψ (proportion of total area of bars that is black in Figure 1(a)). The NBH parameter π relates to the ZINB(ψ, μ, ϕ) parameters,

π = (1 − ψ)(1 − g(0|μ, ϕ)]

where g(0| μ, ϕ)is the probability of a zero count under the negative binomial probability function NB(μ, ϕ). As in the ZINB distribution, the NBH parameter μ is the mean of the untruncated NB distribution, not of the black zero-truncated area in Figure 1(b). So while Figure 1(b) provides a graphical representation of the NBH data generating mechanism in equation (A.3), its connection to μ is not readily apparent. Rather, the mean caries count among children with any caries τ = E(Y|Y>0) is given by

τ = μ/[1 − g(0|μ, ϕ)].

Next, a third equivalent representation of the frequency count distribution is shown in Figure 1(c), which displays the overall, or marginalized, zero-inflated negative binomial distribution for caries counts with a mean of all the counts denoted v= E(Y). The mean of MZINB (ψ, ν, ϕ) distribution is defined by a transformation of the two parameters of the ZINB distribution, ν = (1 − ψ)μ, which for children on a normal diet is 0.60 × 1.00 = 0.60. Note that ν ≤ μ so that mean caries count in the overall population (mean for the black area in Figure 1(c)) cannot be greater than the mean count of the susceptible population (mean for the black area in Figure 1(a)). Figure 1(c) emphasizes the overall population from which the study sample was drawn. For example, in a cross-sectional study, ν is caries severity or extent. The characterization shown in Figure 1(c) of a single overall distribution for the caries counts views the mixture distribution model in Figure 1(a) as a convenient explanation for a distribution of counts with excess zeros [Mwalili et al., 2008]. Finally, Figures 1(d), 1(e) and 1(f) show analogous representations of a distribution for the number of caries in a fictional population of children on a high sugar diet. Our interest is in comparing the distribution of dmft of children in the high sugar to normal diet groups.

Results

Discussion

Three types of two-part models for counts with many zeros were described and applied to two datasets with emphasis given to choosing the class of model to match the research question. The example with fictional data illustrated the possibility of large numerical differences in the different types of rate ratios and odds ratios that may be estimated for exposure effects according to the model chosen. In the Lanarkshire caries trial, treatment effect estimates were similar across the models, despite distinct interpretations among some of the effects.

The similarity of ZINB and MZINB model estimates of incidence rate ratios summarizing the toothpaste comparisons in the Lanarkshire caries trial are the result of small predicted probabilities of excess zeros (ranging from 1% for children with calculus and high baseline caries who received SMFP to 17% for children without calculus and low baseline caries who received NaF) and the fact that estimates for toothpaste effects in the excess zero model were small. If estimates for toothpaste effects in the excess zero model had been zero, then the IRRs from ZINB and MZINB for the toothpaste effect would have been identical; see equation (4) of Preisser et al. [2012]. Conversely, when there are moderate to large proportions of excess zeros, and when the exposure effects of interest are strong in both model parts, the marginalized zero-inflated count regression models can give notably different estimates compared to traditional zero-inflated models as demonstrated in the fictional data example.

Even though MZINB gave improved fits relative to NB regression, the two models produced similar estimates of overall incidence rate ratios (θ_M) in both data examples. In a simulation study based on the Lanarkshire clinical trial, MZINB gave only slightly improved power over NB when empirical standard errors were used in the latter to account for variance misspecification through failure to model the excess zeros [Preisser et al., 2014b]. However, another simulation study found that MZINB gave less biased overall exposure effects, improved Type I error and coverage of 95% confidence intervals closer to the nominal level than NB regression in models with continuous covariates having skewed (log-normal) distributions [Preisser et al., 2016] .

Hurdle models are used for count data when the proportion of individuals with any caries is of interest, which could be prevalence in a cross-sectional study or incidence in a prospective study or trial. They may also be used to compare the truncated-at-zero means between groups through post-modeling calculations, which was illustrated in the fictional data example. It would also be possible to estimate a parameter akin to θ_τ using appropriate adjustments for covariates by either fixing covariates at their mean values or by employing an average-predicted-value methods [Albert et al., 2014].

The NBH model reported in this article has distinct parameters in its two model parts, and as a result, it gives results for prevalence that are identical to those from ordinary logistic regression. A shared-parameter NBH model, also referred to as zero-altered negative binomial model for logistic regression (ZANB-logist), may be used when interest is in the dichotomized outcome of any caries [Preisser et al., 2014a]. In simulations, ZANB-logist models were shown to give large gains in statistical power relative to ordinary logistic regression for dichotomized counts.

All of the models discussed in this article, which were models for independent data estimated using maximum likelihood methods, have extensions to longitudinal or clustered data. Random effects have been included in ZIP [Hall, 2000], ZINB [Yau et al., 2003], Poisson hurdle [Min and Agresti, 2005] and MZIP (Long et al., 2015] models. Generalized estimating equations (GEE) methodology has been employed for ZIP models [Hall and Zhang, 2004] and ZINB models [Kong et al., 2015] to obtain population-averaged interpretations. A possible extension of GEE methodology would be to the analysis of correlated outcomes with MZIP and MZINB models.

In some situations a reasonable alternative approach to modeling counts with or without excess zeros is to use a one-part cumulative logits model applied to an ordinal outcome created by clumping adjacent count categories together to form a small number (e.g., three, four or five) categories. Such models for cross-sectional and longitudinal data have been applied to patient outcomes in a randomized clinical trial of orthognathic surgery patients [Preisser et al., 2011].

Finally, some general recommendations on the use of MZINB, ZINB, NBH and NB are offered to encourage discussion on the uses and relative merits of the classes of count data models reviewed in this article. Foremost, the choice between a latent class model (ZINB), a model for prevalence (NBH) and a model for the marginal mean (NB, MZINB) should be based on the study’s purpose (Figure 2). When there is specific interest in the processes of disease, the latent class effects of ZINB models may be of interest. When interest is in modeling the probability of any caries, e.g., caries prevalence in a cross-sectional study or caries incidence in a prospective study, logistic regression or NBH models may be of interest. In most other settings, modeling the marginal mean will likely be of interest. In dental surveys, for example, zero-inflated marginalized models often correspond to study goals that are basically descriptive in nature. Additionally, MZIP and MZINB models as well as other members of a general class of marginal mean models for zero-inflated counts [Todem et al., 2016] should find wide use in epidemiological studies, and in some clinical trials such as the caries trial presented here. However, MZIP and MZINB models are prone to convergence problems to a degree shared by ZIP and ZINB models, which may preclude their use in settings where analyses must be prespecified such as for confirmatory randomized clinical trials in pharmaceutical regulatory environments [Long et al., 2014]. In such settings, Poisson or negative binomial regression with empirical variance estimation (for robustness against violation of variance assumptions) may be a better choice than two-part models for their simplicity and relative stability (i.e. higher convergence rates) of their computational algorithms. Moreover, the practical performance in terms of statistical bias and efficiency of Poisson and NB (one-part) models as shown in simulation studies may be nearly as good as correctly specified two-part models [Preisser et al., 2016].

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Figure 2

Flowchart for count regression model choice. ZIP and ZINB models are chosen when interest in latent classes for the study of disease processes. Otherwise, MZIP and MZINB, like Poisson regression and negative binomial regression, have use for random samples in population-based oral health surveys and epidemiological studies or convenience samples for clinical trials where overall effects of treatment and exposures on marginal means are of interest. Hurdle models are of interest in modeling prevalence in populations or, secondarily (dashed line), at-risk class means.

Conclusions

Investigators are often interested in estimating the effect of exposures and risk factors on the mean dental caries indices or increment in the overall sampled population as opposed to the index/increment for some unobserved subpopulation of individuals believed to be at risk for dental caries. In data with many zero counts (e.g., no dental caries), two-part count models are widely recognized as an effective tool for achieving adequate fits of regression models to count data. This article emphasized the distinct interpretations of NBH, ZINB and MZINB models and argued that the statistical model class chosen should match the study’s purpose. Moreover, illustrative examples showed that regression parameter estimates for exposure effects on the different types of mean counts estimated by the respective models may have substantial differences.

Supplementary Material

Acknowledgments

References