This paper briefly describes statistical methods for analyzing imprecise compositional data that might be elicited from approximate measurement or from expert judgments. Two alternative approaches are discussed: Log-ratio transforms and probability-ratio transforms. The first is well-established and the second is under development by the author. The primary focus in this paper is on generalized linear models for predicting imprecise compositional data

Smithson, Michael

English

The Australian National University

Proceedings of Machine Learning Research 103:364–366, 2019 ISIPTA 2019Imprecise Compositional Data Analysis: Alternative Statistical MethodsMichael Smithson MICHAEL.SMITHSON@ANU.EDU.AUResearch School of Psychology, The Australian National University, Canberra, AustraliaAbstractThis paper briefly describes statistical methods foranalyzing imprecise compositional data that might beelicited from approximate measurement or from expertjudgments. Two alternative approaches are discussed:Log-ratio transforms and probability-ratio transforms.The first is well-established and the second is underdevelopment by the author. The primary focus in thispaper is on generalized linear models for predictingimprecise compositional data.Keywords: compositional data, imprecise data, betadistribution, cdf-quantile distribution, general linearmodel, copula1. IntroductionData that must sum to a constant value are known as “com-positional”, and coherent probability assignments are atypical example. Given a composition consisting of K parts,suppose that we have N collections of points in the K-simplex, 0 ≤ pi( ji)ki ≤ 1, for k = 1, . . . ,K and i = 1, . . . ,N,such that for each i they sum to 1 across the k. For the ithcollection, there are Ji points, indexed by the bracketed jisuperscript. These collections may have been derived fromcredal sets, coherent lower and/or upper previsions, or setsof desirable gambles.For the most part, this paper will set aside the details ofthese collections. Instead, our main topic is how to con-nect these collections with regression or generalized linearmodels (GLMs) that treat them as dependent variables. Wefurther assume that the modeler has a set of covariates(whether continuous or categorical) to be used as predictorsin such a GLM.Although methods for statistical analysis of precise com-positional data are well-established, their application toimprecise compositional data does not seem to have beendeveloped. This short paper initiates investigations intothis application. There are two well-established approachesto modeling compositional data: Dirichlet regression andlog-ratio regression. A third, probability-ratio regression, isunder development by the author. We will focus primarilyon comparing the log-ratio transforms and probability-ratiotransforms because these approaches are directly related.The primary focus in this paper is on generalized linearmodels for predicting imprecise compositional data.2. Log-Ratio Transform MethodThe log-ratio transform method [1] maps data from the sim-plex to an unrestricted vector space, via the logit transformof odds-ratios. For instance, suppose the Kth compositionpart is the part of the composition against which we wouldlike to compare the other parts. Then Aitchison’s “additivelog-ratio” transform would yieldη( ji)ki = log((pi( ji)ki1−pi( ji)ki)/(pi( ji)Ki1−pi( ji)Ki)), (1)for k = 1, . . . ,K−1.The η( ji)ki are considered as continuous random variableson the real line, and therefore may be analysed with ap-propriate statistical methods for such variables, such asmulti-level linear regression with multivariate Gaussianerrors. Predictors in these regression models may be in-troduced at either the collection level, the compositionalpart level, or even at the ji level (e.g., for distinguishingbetween lower and upper probabilities).The log-ratio framework enjoys several attractive prop-erties that account for its popularity. Chief among these aresubcompositional coherence and permutation invariance.Subcompositional coherence means that the inferential out-comes of an analysis of any subcomposition should remainthe same for that analysis in the entire composition. Per-mutation invariance guarantees that outcomes remain thesame regardless of the ordering of the components in acomposition.Although it is straightforward to use, the log-ratio frame-work has important limitations. First, it is unable to ex-tend to non-Gaussian distributions without adding moreparameters (e.g., via skew-normal distributions). Second,dispersion is routinely ignored in the log-ratio framework,despite its obvious relevance to doubly-bounded and sum-constrained random variables. Third, the logit transform isunsuited for dealing with zeroes or ones in the data. Thepopular approach of adding a small constant to the zeroesor subtracting it from the ones introduces difficulties re-garding the arbitrary choice of its magnitude and sensitivityto that choice.© 2019 M. Smithson.IMPRECISE COMPOSITIONAL DATA ANALYSIS: ALTERNATIVE STATISTICAL METHODS3. Probability-Ratio Transform MethodThe limitations imposed by the assumption of multivariatenormality, on the other hand, motivate us to seek alterna-tives to the log-ratio transform. We therefore now considerprobability-ratios. Rather than taking logs of relative odds,we take the corresponding relative probabilities and modelthem. Turning once again to our example with the Kthcategory as the base, the relevant probability ratios areν( ji)ki = pi( ji)ki/(pi( ji)ki +pi( ji)Ki), (2)for k = 1, . . . ,K−1. Any of the log-ratio transforms has aprobability-ratio analogue, so this approach inherits bothsubcompositional coherence and permutation invariance.Moreover, the probability-ratio approach includes struc-tures not found in the log-ratio literature, such as the stick-breaking construction (See [6] for a demonstration). It isless clear at the present stage of development whetherdistance-based techniques (e.g., certain types of clusteranalysis) employed in the log-ratio framework translateinto the probability-ratio framework. Smithson (2019) iden-tifies conditions under which Euclidean distances in theunit hypercube between probability-ratios do not have thesame rank-order as their log-ratio Euclidean distance coun-terparts.The ν( ji)ki may be modeled using any distribution whosesupport is the open unit interval (0,1). Many flexible two-parameter distributions are available, such as the beta dis-tribution and the CDF-quantile family [7]. GLMs may beconstructed following the same structure as presented in [7],although these are not GLMs in the formal sense becausethe distributions employed here are not members of the ex-ponential family. In fact, if necessary, a different marginaldistribution can be specified for each probability-ratio com-ponent. Note that all of this added flexibility in the marginaldistribution models is achieved with the same number ofparameters as in the log-ratio setup, i.e., two-parametermarginal distributions and the same regression models. Fur-thermore, the inclusion of the CDF-quantile distributionfamily enables flexible quantile regression models that areunavailable in the log-ratio framework.Dispersion typically is not modeled with covariates inthe log-ratio literature, although this could be done witha bit of extra work. However, explicit dispersion submod-els are readily accommodated in both beta-distribution [8]and CDF-quantile distribution regression models [7]. Pre-liminary work using real data-sets has already indicatedthat modeling dispersion adds important insights to com-positional data analysis and can correct location modelmisspecification [6].The dependency structure may be modeled via random-effects models for the marginal parameters [9]. Alterna-tively, the dependency structure can be modeled withcopulae or copula vines [5]. The conventional two-stagemaximum-likelihood estimation procedure for this enablesthe marginal distributions to be modeled separately fromthe dependency structure.Finally, the zeroes problem can be dealt with via hurdlemodels (in the log-ratio as well as probability-ratio setting).Shou and Smithson [5] have implemented hurdle models intheir cdfquantreg package. Given that the beta regressionGLM and many of the CDF-quantile distributions’ GLMsuse link functions that are defined only on (0,1), a hurdlemodel rather than a zero-inflated model is technically therelevant alternative.4. Dirichlet MethodsDirichlet regression models are a natural and popular choicefor modeling compositional data [4]. These models havetwo main limitations. First, the Dirichlet distribution’smarginal distributions are beta distributions sharing thesame precision parameter, so all parts of the compositionmust have the same submodel for their precisions. Thislimits its ability to model multivariate heteroskedasticity.On the other hand, a probability-ratio model with betamarginals can incorporate a unique precision submodel foreach marginal distribution. Second, a single Dirichlet dis-tribution can model only negative associations among thevariables. Although this restriction may be relaxed whencovariates are modeled or other kinds of mixture modelsare employed [3], the probability-ratio approach does nothave this limitation even for the single-distribution case.Promising imprecise Dirichlet process models [2] havebeen developed and applied to non-parametric Bayesianversions of hypothesis tests such as Wilcoxon tests. A sys-tematic comparison between log-ratio, probability-ratio,and imprecise Dirichlet approaches to modeling imprecisecompositional data has yet to be completed and so is be-yond the scope of this paper.In conclusion, we have briefly surveyed practical statisti-cal methods for analyzing the kinds of imprecise compo-sitional data that may arise from measurement or expertjudgments, focusing on two related methods: The well-established log-ratio approach, and a new “probability ra-tio” approach. Much remains to be done in evaluating theirrespective merits, for instance their relative sensitivitiesto noise or other sources of imprecision. The probability-ratio approach shows promise in overcoming some of thelimitations of the log-ratio method, and both methods cancomplement one another.References[1] John Aitchison. The statistical analysis of composi-tional data. Journal of the Royal Statistical Society.Series B (Methodological), pages 139–177, 1982.365IMPRECISE COMPOSITIONAL DATA ANALYSIS: ALTERNATIVE STATISTICAL METHODS[2] Alessio Benavoli, Giorgio Corani, Francesca Mangili,Marco Zaffalon, and Fabrizio Ruggeri. A bayesianwilcoxon signed-rank test based on the dirichlet pro-cess. In International conference on machine learning,pages 1026–1034, 2014.[3] Rafiq H Hijazi and Robert W Jernigan. Modellingcompositional data using dirichlet regression models.Journal of Applied Probability & Statistics, 4(1):77–91,2009.[4] Marco J Maier. Dirichletreg: Dirichlet regression forcompositional data in r. Research Report Series, 125,2014.[5] Yiyun Shou and Michael Smithson. cdfquantreg: An rpackage for cdf-quantile regression. Journal of Statis-tical Software, 88:1–30, 2019.[6] Michael Smithson. A new approach to compositionaldata analysis. The Australian National University, Can-berra, Australia, 2019.[7] Michael Smithson and Yiyun Shou. Cdf-quantile dis-tributions for modelling random variables on the unitinterval. British Journal of Mathematical and Statisti-cal Psychology, 70(3):412–438, 2017.[8] Michael Smithson and Jay Verkuilen. A better lemonsqueezer? maximum-likelihood regression with beta-distributed dependent variables. Psychological meth-ods, 11(1):54, 2006.[9] Jay Verkuilen and Michael Smithson. Mixed andmixture regression models for continuous bounded re-sponses using the beta distribution. Journal of Educa-tional and Behavioral Statistics, 37(1):82–113, 2012.366

Imprecise compositional data analysis: Alternative statistical methods

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Imprecise compositional data analysis: Alternative statistical methods

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