Common Principal Components Model Research Articles

Quantitative genetic variation in Daphnia: temporal changes in genetic architecture.

Nonadditive genetic variation and genetic disequilibrium are two important factors that influence the evolutionary trajectory of natural populations. We assayed quantitative genetic variation in a temporary-pond-dwelling population of Daphnia pulex over a full season to examine the role of nonadditive genetic variation and genetic disequilibrium in determining the short-term evolutionary trajectory of a cyclic parthenogen. Quantitative traits were influenced by three factors: (1) clonal selection significantly changed the population mean phenotype during the course of the growing season; (2) sexual reproduction and recombination led to significant changes in life-history trait means and the levels of expressed genetic variation, implying the presence of substantial nonadditive genetic variation and genetic disequilibrium; and (3) Egg-bank effects were found to be an important component of the realized year-to-year change. Additionally, we examined the impact of genetic disequilibria induced by clonal selection on the genetic (co)variance structure with a common principal components model. Clonal selection caused significant changes in the (co)variance structure that were eliminated by a single bout of random mating, suggesting that a build-up of disequilibria was the primary source of changes in the (co)variance structure. The results of this study highlight the complexity of natural selection operating on populations that undergo alternating phases of sexual and asexual reproduction.

QUANTITATIVE GENETIC VARIATION IN DAPHNIA: TEMPORAL CHANGES IN GENETIC ARCHITECTURE

Nonadditive genetic variation and genetic disequilibrium are two important factors that influence the evolutionary trajectory of natural populations. We assayed quantitative genetic variation in a temporary-pond-dwelling population of Daphnia pulex over a full season to examine the role of nonadditive genetic variation and genetic disequilibrium in determining the short-term evolutionary trajectory of a cyclic parthenogen. Quantitative traits were influenced by three factors: (1) clonal selection significantly changed the population mean phenotype during the course of the growing season; (2) sexual reproduction and recombination led to significant changes in life-history trait means and the levels of expressed genetic variation, implying the presence of substantial nonadditive genetic variation and genetic disequilibrium; and (3) Egg-bank effects were found to be an important component of the realized year-to-year change. Additionally, we examined the impact of genetic disequilibria induced by clonal selection on the genetic (co)variance structure with a common principal components model. Clonal selection caused significant changes in the (co)variance structure that were eliminated by a single bout of random mating, suggesting that a build-up of disequilibria was the primary source of changes in the (co)variance structure. The results of this study highlight the complexity of natural selection operating on populations that undergo alternating phases of sexual and asexual reproduction.Corresponding Editor: D. Waller

The Decomposition of the Behrens-Fisher Statistic in q-Dimensional Common Principal Component Submodels

Takemura (1985, Multivariate Analysis VI, ed. P. R. Krishnaiah, 583-597, Elsevier, Amsterdam) presented a decomposition of Hotelling's T2-statistic into analogues of univariate Student-t variates along the principal component axes of the (pooled) sample covariance matrix. In this paper the idea is extended to the heteroscedastic situation where an analogous decomposition of the Behrens-Fisher statistic is considered when the nature of the heteroscedasticity between the two samples can be described by a common principal component (CPC) model, or more particularly a q-dimensional CPC subspace model.

PHYLOGENETIC ANALYSIS OF PHENOTYPIC COVARIANCE STRUCTURE. I. CONTRASTING RESULTS FROM MATRIX CORRELATION AND COMMON PRINCIPAL COMPONENT ANALYSES

Applications of quantitative techniques to understanding macroevolutionary patterns typically assume that genetic variances and covariances remain constant. That assumption is tested among 28 populations of the Phyllotis darwini species group (leaf-eared mice). Phenotypic covariances are used as a surrogate for genetic covariances to allow much greater phylogenetic sampling. Two new approaches are applied that extend the comparative method to multivariate data. The efficacy of these techniques are compared, and their sensitivity to sampling error examined. Pairwise matrix correlations of correlation matrices are consistently very high (> 0.90) and show no significant association between matrix similarity and phylogenetic relatedness. Hierarchical decomposition of common principal component (CPC) analyses applied to each clade in the phylogeny rejects the hypothesis that common principal component structure is shared in clades more inclusive than subspecies. Most subspecies also lack a common covariance structure as described by the CPC model. The hypothesis of constant covariances must be rejected, but the magnitudes of divergence in covariance structure appear to be small. Matrix correlations are very sensitive to sampling error, while CPC is not. CPC is a powerful statistical tool that allows detailed testing of underlying patterns of covariation.

Multiple group principal component analysis

Common Principal Component Analysis is a generalization of standard principal components to several groups under the rigid mathematical assumption of equality of all latent vectors across groups (i.e., principal component directions), whereas the latent roots are allowed to vary between groups (differing inflations of dispersion ellipsoids). In practice, data that fulfill these strict requirements are relatively rare. Examples from palaeontology are used to illustrate the principles. Compositional data can be made to fit the Common Principal Component (CPC) model by the appropriate logratio covariance matrix.

Rao's Distance between Normal Populations That Have Common Principal Components

SUMMARY A theoretical derivation is provided of Rao's distance between two normal populations that have different means and whose dispersion matrices satisfy the 'common principal component model.' According to this model, the dispersion matrices share the same eigenvectors but are permitted to have arbitrary eigenvalues. This structure is useful for modeling a variety of real data sets, so the theoretical result has important practical implications. It is illustrated on a biometric data set.

Ontogeny and Individual Variation: Analysis of Patterned Covariance Matrices with Common Principal Components

Morphometric data from longitudinal growth studies with multiple measurements made in several growth stages on the same specimens confront researchers with difficult statistical problems because traits are correlated both within and across stages. Here, we introduce a statis? tical model especially designed to deal with this complexity. The common principal component (CPC) model for dependent random vectors is based on the assumption that the same pattern underlies both variation within stages and covariation across stages. Thus, a single transformation, when applied to all stages, renders the resulting CPCs uncorrelated not only within but also across stages. Because of these simplifying assumptions, the CPC model greatly reduces the num? ber of parameters to be estimated; it is thus an efficient tool for data reduction. This model is demonstrated using growth of the water strider Limnoporus canaliculatus. The CPCs can be inter? preted as patterns of size variation and contrasts between parts that are common to all stages, although there are minor deviations from the model. The size CPC accounts for most variation in all instars and is therefore an effective measure of overall growth. Moreover, the CPC model clarifies the link between static and ontogenetic variation by including both levels in a joint anal? ysis and can be used to study morphological integration and constraints on the evolution of ontogenies. (Allometry; common principal components; Gerridae; growth; longitudinal data; mul? tivariate morphometrics; size.)

On the Use of Distance in the Taxonomic Study of Critical Plant Groups—Case Studies of Western European Orchidaceae

Four distance coefficients are compared on four data sets composed of samples coming from western European populations of the genera Dactylorhiza, Orchis and Epipactis (Orchidaceae). The performance of the distance coefficients is evaluated through: (a) the quality of clusters obtained by five classical methods (as compared to a priori classification), (b) the Mantel statistic with respect to an a priori distance matrix resulting from previous knowledge, (c) the result obtained with the k-means method, and (d) principal coordinate diagrams. It appears that: (a) the Mahalanobis distance based on the pooled dispersion matrix performs best on the whole; (b) a distance based on the recently developed Common Principal Component model, used with a log transformation, also provides useful information and performs best on the largest data set; (c) the Gölz and Reinhard taxonomic distance, widely used among orchidologists, is attractive for its simplicity, yet good performance and the valuable information it provides, despite its theoretical shortcomings. A brief taxonomic discussion is made on the results obtained for the Dactylorhiza samples, especially about samples whose identification was in doubt.

Letters to the editor

When dealing with several populations, the common principal components (CPC) model assumes equal principal axes but different variances along them. In this paper, a robust log-likelihood ratio statistic allowing to test the null hypothesis of a CPC model versus no restrictions on the scatter matrices is introduced. The proposal plugs into the classical log-likelihood ratio statistic robust scatter estimators. Using the same idea, a robust log-likelihood ratio and a robust Wald-type statistic for testing proportionality against a CPC model are considered. Their asymptotic distributions under the null hypothesis and their partial influence functions are derived. A small simulation study allows to compare the behavior of the classical and robust tests, under normal and contaminated data.

Between-group analysis with heterogeneous covariance matrices: The common principal component model

Analysis of between-group differences using canonical variates assumes equality of population covariance matrices. Sometimes these matrices are sufficiently different for the null hypothesis of equality to be rejected, but there exist some common features which should be exploited in any analysis. The common principal component model is often suitable in such circumstances, and this model is shown to be appropriate in a practical example. Two methods for between-group analysis are proposed when this model replaces the equal dispersion matrix assumption. One method is by extension of the two-stage approach to canonical variate analysis using sequential principal component analyses as described by Campbell and Atchley (1981). The second method is by definition of a distance function between populations satisfying the common principal component model, followed by metric scaling of the resulting between-populations distance matrix. The two methods are compared with each other and with ordinary canonical variate analysis on the previously introduced data set.

Two Generalizations of the Common Principal Component Model

Under the common principal component model the covariance matrices Ψi, of k populations are assumed to have identical eigenvectors, that is, the same orthogonal matrix diagonalizes all Ψi, simultaneously. This paper modifies the common principal component model by assuming that only q out of p eigenvectors are common to all Ψi, while the remaining p–q elgenvectors are specific in each group. This is called a partial common principal component model. A related modification assumes that q eigenvectors of each matrix span the same subspace, a problem that was first considered by Krzanowski (1979). For both modifications this paper derives the normal theory maximum likelihood estimators. It is shown that approximate maximum likelihood estimates can easily be computed if estimates of the ordinary common principal component model are available. The methods are illustrated by numerical examples.

Two generalizations of the common principal component model

SUMMARY Under the common principal component model the covariance matrices Ti of k populations are assumed to have identical eigenvectors, that is, the same orthogonal matrix diagonalizes all Ti simultaneously. This paper modifies the common principal component model by assuming that only q out of p eigenvectors are common to all Ti,, while the remaining p - q eigenvectors are specific in each group. This is called a partial common principal component model. A related modification assumes that q eigenvectors of each matrix span the same subspace, a problem that was first considered by Krzanowski (1979). For both modifications this paper derives the normal theory maximum likelihood estimators. It is shown that approximate maximum likelihood estimates can easily be computed if estimates of the ordinary common principal component model are available. The methods are illustrated by numerical examples.

Asymptotic Theory for Common Principal Component Analysis

Under the common principal component model $k$ covariance matrices $\mathbf{\Sigma}_1,\cdots,\mathbf{\Sigma}_k$ are simultaneously diagonalizable, i.e., there exists an orthogonal matrix $\mathbf{\beta}$ such that $\mathbf{\beta'\Sigma_i\beta = \Lambda_i}$ is diagonal for $i = 1,\cdots, k$. In this article we give the asymptotic distribution of the maximum likelihood estimates of $\mathbf{\beta}$ and $\mathbf{\Lambda}_i$. Using these results, we derive tests for (a) equality of eigenvectors with a given set of orthonormal vectors, and (b) redundancy of $p - q$ (out of $p$) principal components. The likelihood-ratio test for simultaneous sphericity of $p - q$ principal components in $k$ populations is derived, and some of the results are illustrated by a biometrical example.