1 Compositional data: some challenging problems.- 1.1 Introduction.- 1.2 Geochemical compositions of rocks.- 1.3 Sediments at different depths.- 1.4 Ternary diagrams.- 1.5 Partial analyses and subcompositions.- 1.6 Supervisory behaviour.- 1.7 Household budget surveys.- 1.8 Steroid metabolite patterns in adults and children.- 1.9 Activity patterns of a statistician.- 1.10 Calibration of white-cell compositions.- 1.11 Fruit evaluation.- 1.12 Firework mixtures.- 1.13 Clam ecology.- 1.14 Bibliographic notes.- Problems.- 2 The simplex as sample space.- 2.1 Choice of sample space.- 2.2 Compositions and simplexes.- 2.3 Spaces, vectors, matrices.- 2.4 Bases and compositions.- 2.5 Subcompositions.- 2.6 Amalgamations.- 2.7 Partitions.- 2.8 Perturbations.- 2.9 Geometrical representations of compositional data.- 2.10 Bibliographic notes.- Problems.- 3 The special difficulties of compositional data analysis.- 3.1 Introduction.- 3.2 High dimensionality.- 3.3 Absence of an interpretable covariance structure.- 3.4 Difficulty of parametric modelling.- 3.5 The mixture variation difficulty.- 3.6 Bibliographic notes.- Problems.- 4 Covariance structure.- 4.1 Fundamentals.- 4.2 Specification of the covariance structure.- 4.3 The compositional variation array.- 4.4 Recovery of the compositional variation array from the crude mean vector and covariance matrix.- 4.5 Subcompositional analysis.- 4.6 Matrix specifications of covariance structures.- 4.7 Some important elementary matrices.- 4.8 Relationships between the matrix specifications.- 4.9 Estimated matrices for hongite compositions.- 4.10 Logratios and logcontrasts.- 4.11 Covariance structure of a basis.- 4.12 Commentary.- 4.13 Bibliographic notes.- Problems.- 5 Properties of matrix covariance specifications.- 5.1 Logratio notation.- 5.2 Logcontrast variances and covariances.- 5.3 Permutations.- 5.4 Properties of P and QP matrices.- 5.5 Permutation invariants involving ?.- 5.6 Covariance matrix inverses.- 5.7 Subcompositions.- 5.8 Equivalence of characteristics of ?, ?, ?.- 5.9 Logratio-uncorrelated compositions.- 5.10 Isotropic covariance structures.- 5.11 Bibliographic notes.- Problems.- 6 Logistic normal distributions on the simplex.- 6.1 Introduction.- 6.2 The additive logistic normal class.- 6.3 Density function.- 6.4 Moment properties.- 6.5 Composition of a lognormal basis.- 6.6 Class-preserving properties.- 6.7 Conditional subcompositional properties.- 6.8 Perturbation properties.- 6.9 A central limit theorem.- 6.10 A characterization by logcontrasts.- 6.11 Relationships with the Dirichlet class.- 6.12 Potential for statistical analysis.- 6.13 The multiplicative logistic normal class.- 6.14 Partitioned logistic normal classes.- 6.15 Some notation.- 6.16 Bibliographic notes.- Problems.- 7 Logratio analysis of compositions.- 7.1 Introduction.- 7.2 Estimation of ? and ?.- 7.3 Validation: tests of logistic normality.- 7.4 Hypothesis testing strategy and techniques.- 7.5 Testing hypotheses about ? and ?.- 7.6 Logratio linear modelling.- 7.7 Testing logratio linear hypotheses.- 7.8 Further aspects of logratio linear modelling.- 7.9 An application of logratio linear modelling.- 7.10 Predictive distributions, atypicality indices and outliers.- 7.11 Statistical discrimination.- 7.12 Conditional compositional modelling.- 7.13 Bibliographic notes.- Problems.- 8 Dimension-reducing techniques.- 8.1 Introduction.- 8.2 Crude principal component analysis.- 8.3 Logcontrast principal component analysis.- 8.4 Applications of logcontrast principal component analysis.- 8.5 Subcompositional analysis.- 8.6 Applications of subcompositional analysis.- 8.7 Canonical component analysis.- 8.8 Bibliographic notes.- Problems.- 9 Bases and compositions.- 9.1 Fundamentals.- 9.2 Covariance relationships.- 9.3 Principal and canonical component comparisons.- 9.4 Distributional relationships.- 9.5 Compositional invariance.- 9.6 An application to household budget analysis.- 9.7 An application to clinical biochemistry.- 9.8 Reappraisal of an early shape and size analysis.- 9.9 Bibliographic notes.- Problems.- 10 Subcompositions and partitions.- 10.1 Introduction.- 10.2 Complete subcompositional independence.- 10.3 Partitions of order 1.- 10.4 Ordered sequences of partitions.- 10.5 Caveat.- 10.6 Partitions of higher order.- 10.7 Bibliographic notes.- Problems.- 11 Irregular compositional data.- 11.1 Introduction.- 11.2 Modelling imprecision in compositions.- 11.3 Analysis of sources of imprecision.- 11.4 Imprecision and tests of independence.- 11.5 Rounded or trace zeros.- 11.6 Essential zeros.- 11.7 Missing components.- 11.8 Bibliographic notes.- Problems.- 12 Compositions in a covariate role.- 12.1 Introduction.- 12.2 Calibration.- 12.3 A before-and-after treatment problem.- 12.4 Experiments with mixtures.- 12.5 An application to firework mixtures.- 12.6 Classification from compositions.- 12.7 An application to geological classification.- 12.8 Bibliographic notes.- Problems.- 13 Further distributions on the simplex.- 13.1 Some generalizations of the Dirichlet class.- 13.2 Some generalizations of the logistic normal classes.- 13.3 Recapitulation.- 13.4 The Ad(?,B) class.- 13.5 Maximum likelihood estimation.- 13.6 Neutrality and partition independence.- 13.7 Subcompositional independence.- 13.8 A generalized lognormal gamma distribution with compositional in variance.- 13.9 Discussion.- 13.10 Bibliographic notes.- Problems.- 14 Miscellaneous problems.- 14.1 Introduction.- 14.2 Multi-way compositions.- 14.3 Multi-stage compositions.- 14.4 Multiple compositions.- 14.5 Kernel density estimation for compositional data.- 14.6 Compositional stochastic processes.- 14.7 Relation to Bayesian statistical analysis.- 14.8 Compositional and directional data.- Problems.- Appendices.- A Algebraic properties of elementary matrices.- B Bibliography.- C Computer software for compositional data analysis.- D Data sets.- Author index.

The Statistical Analysis of Compositional Data

SUMMARY A general class of regression models for ordinal data is developed and discussed. These models utilize the ordinal nature of the data by describing various modes of stochastic ordering and this eliminates the need for assigning scores or otherwise assuming cardinality instead of ordinality. Two models in particular, the proportional odds and the proportional hazards models are likely to be most useful in practice because of the simplicity of their interpretation. These linear models are shown to be multivariate extensions of generalized linear models. Extensions to non-linear models are discussed and it is shown that even here the method of iteratively reweighted least squares converges to the maximum likelihood estimate, a property which greatly simplifies the necessary computation. Applications are discussed with the aid of examples.

https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2517-6161.1980.tb01109.x

Regression Models for Ordinal Data

Improved methods of assessment and research design have established a robust and causal association between stressful life events and major depressive episodes. The chapter reviews these developments briefly and attempts to identify gaps in the field and new directions in recent research. There are notable shortcomings in several important topics: measurement and evaluation of chronic stress and depression; exploration of potentially different processes of stress and depression associated with first-onset versus recurrent episodes; possible gender differences in exposure and reactivity to stressors; testing kindling/sensitization processes; longitudinal tests of diathesis-stress models; and understanding biological stress processes associated with naturally occurring stress and depressive outcomes. There is growing interest in moving away from unidirectional models of the stress-depression association, toward recognition of the effects of contexts and personal characteristics on the occurrence of stressors, and on the likelihood of progressive and dynamic relationships between stress and depression over time-including effects of childhood and lifetime stress exposure on later reactivity to stress.

/pdf/stress-and-depression-2zp4i7xmjt.pdf

Stress and Depression

Objective To qualitatively review the literature of the past decade covering the epidemiology, clinical characteristics, natural course, biology, and other correlates of early-onset major depressive disorder (MDD) and dysthymic disorder (DD). Method A computerized search for articles published during the past 10 years was made and selected studies are presented. Results Early-onset MDD and DD are frequent, recurrent, and familial disorders that tend to continue into adulthood, and they are frequently accompanied by other psychiatric disorders. These disorders are usually associated with poor psychosocial and academic outcome and increased risk for substance abuse, bipolar disorder, and suicide. In addition, DD increases the risk for MDD. There is a secular increase in the prevalence of MDD, and it appears that MDD is occurring at an earlier age in successive cohorts. Several genetic, familial, demographic, psychosocial, cognitive, and biological correlates of onset and course of early-onset depression have been identified. Few studies, however, have examined the combined effects of these correlates. Conclusions Considerable advances have been made in our knowledge of early-onset depression. Nevertheless, further research is needed in understanding the pathogenesis of childhood mood disorders. Toward this end, studies aimed at elucidating mechanisms and interrelationships among the different domains of risk factors are needed.

Childhood and adolescent depression: a review of the past 10 years. Part I.

Background It is not clear whether cortisol or dehydroepiandrosterone (DHEA) hypersecretion increases the risk for major depression in the presence of undesirable life events.

Aims To determine whether there is a specific pattern of psychoendocrine factors that predicts the onset of major depressive disorder.

Method 180 adolescents (73 boys, 107 girls) at high risk for psychopathology were assessed for cortisol, DHEA, depressive symptoms, life events and psychiatric disorder at entry and 12 months later.

Results Major depression was predicted for both genders by the additive effects of: higher depressive symptoms; personal disappointments and losses only in the month before onset; one or more daily levels of cortisol at 08.00 h or DHEA at 20.00 h greater than the 80th percentile of the daily mean.

Conclusions A subgroup of adolescents may carry a physiological risk for major depression which may be either of genetic and/or earlier psychosocial origin.

/pdf/recent-life-events-cortisol-dehydroepiandrosterone-and-the-rro7dyz0md.pdf

Recent life events, cortisol, dehydroepiandrosterone and the onset of major depression in high-risk adolescents.

The association between basal cortisol, dehydroepiandrosterone (DHEA), its sulphate (DHEAS) and major depression was investigated in 8- to 16-year-olds. Eighty-two subjects with major depression, 25 non-depressed psychiatric cases and 40 community controls were systematically assessed for current mental state and hormone levels at 08.00, 12.00 and 20.00 h, assayed from salivary samples collected over a 48 h period. The average mean of the two time points was compared between the three groups. Evening cortisol hypersecretion and morning DHEA hyposecretion were significantly, and independently, associated with major depression. High evening cortisol (> 0·594 ng/ml) and low morning DHEA (< 0·200 ng/ml) identified subgroups of depressives with different types of adrenal hormone dysregulation. The association between high evening cortisol or low morning DHEA and MDD was not affected by either age or gender.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0033291700034644

Adrenal secretion during major depression in 8- to 16-year-olds, I. Altered diurnal rhythms in salivary cortisol and dehydroepiandrosterone (DHEA) at presentation.

SUMMARY The sum of k independent and identically distributed (0, 1) variables has a binomial distribution. If the variables are identically distributed but not independent, this may be generalized to a two-parameter distribution where the k variables are assumed to have a symmetric joint distribution with no second- or higher-order "interactions". Two distinct generalizations are obtained, depending on whether the "multiplicative" or "additive" definition of "interaction" for discrete variables is used. The multiplicative generalization gives rise to a two-parameter exponential family, which naturally includes the binomial as a special case. Whereas with a beta-binomially distributed variable the variance always exceeds the corresponding binomial variance, the "additive" or "multiplicative" generalizations allow the variance to be greater or less than the corresponding binomial quantity. The properties of these two distributions are discussed, and both distributions are fitted, successfully, to data given by Skellam (1948) on the secondary association of chromosomes in Brassica.

Two Generalizations of the Binomial Distribution

Background. An investigation of the association between
 diurnal changes in cortisol and DHEA levels, or in the cortisol/DHEA ratio at five different time points
 at presentation, and the occurrence of undesirable life events (losses, dangers to self and others, disappointments)
 during follow-up, and the outcome of major depression at 36 weeks were
 investigated. Methods. Psychosocial and endocrine assessment of a consecutive
 cohort ( N =68) of 8- to 16-year-old subjects with first episode major
 depression reassessed 12 months after presentation using a repeat measures
 design. Results. Higher cortisol/DHEA ratios at 20.00 or 24.00
 h predicted persistent major depression. Basal levels of either hormone alone or cortisol/DHEA ratios at the
 other three time points (08.00, 12.00 or 16.00 h) did not. High cortisol/DHEA
 ratios (i.e. values greater than the 60th percentile) at both evening points (20.00 and 24.00 h) also predicted the occurrence
 of subsequent disappointing life events but no other category of undesirable
 event. Both high evening cortisol/DHEA ratio at 20.00 h and one or more severely disappointing life events between presentation
 and follow-up predicted persistent major depression: 86% of subjects with both of these
 factors were still depressed at 36 weeks whereas 81% with neither factor were not. Conclusions. The finding that it is depressed subjects with
 high cortisol/DHEA ratios at presentation who are specifically at risk for subsequent disappointing
 life events suggests a putative role for these adrenal steroids in abnormal cognitive or emotional processes
 associated with disturbed interpersonal behaviour.

Adrenal steroid secretion and major depression in 8- to 16-year-olds, III. Influence of cortisol/DHEA ratio at presentation on subsequent rates of disappointing life events and persistent major depression

Background There is little information on whether patterns of steroids precede and are associated with depressive onset.

Aims To establish whether there is an association between salivary cortisol and or dihydroepiandrosterone (DHEA) levels and depression independent of psychosocial risk.

Method Two subgroups of adolescents in the community at high ( n =181) and low ( n =65) risk for psychopathology were interviewed for recent psychiatric disorder at entry and again at 12 months. Salivary samples (08.00 and 20.00 h) for hormone estimations and self-reports on current mood and cognitive style were obtained at both assessments.

Results Neither hormone was associated with risk status, current mood or cognitive style at entry. Of 31 onsets of major depression that occurred over the next 12 months, 30 came from the high-risk group but were not associated with any particular pattern of risk. Increased negative mood and feelings and DHEA (08.00 h) hypersecretion at entry were associated with subsequent major depression.

Conclusions Both negative mood and feelings and alterations in adrenal steroid function precede the onset of first-episode major depression in adolescents. Variation in levels of hormones may arise from more distal origins than recent life events and current ongoing difficulties.

/pdf/first-episode-major-depression-in-adolescents-affective-4b2sdsyw8b.pdf

P. M. E. Altham

Papers

Recent life events, cortisol, dehydroepiandrosterone and the onset of major depression in high-risk adolescents.

Adrenal secretion during major depression in 8- to 16-year-olds, I. Altered diurnal rhythms in salivary cortisol and dehydroepiandrosterone (DHEA) at presentation.

Two Generalizations of the Binomial Distribution

Adrenal steroid secretion and major depression in 8- to 16-year-olds, III. Influence of cortisol/DHEA ratio at presentation on subsequent rates of disappointing life events and persistent major depression

First-episode major depression in adolescents. Affective, cognitive and endocrine characteristics of risk status and predictors of onset.