Principal component analysis of incomplete data ? A simple solution to an old problem

A long-standing problem in biological data analysis is the unintentional absence of values for some observations or variables, preventing the use of standard multivariate exploratory methods, such as principal component analysis (PCA). Solutions include deleting parts of the data by which information is lost, data imputation, which is always arbitrary, and restriction of the analysis to either the variables or observations, thereby losing the advantages of biplot diagrams. We describe a minor modification of eigenanalysis-based PCA in which correlations or covariances are calculated using different numbers of observations for each pair of variables, and the resulting eigenvalues and eigenvectors are used to calculate component scores such that missing values are skipped. This procedure avoids artificial data imputation, exhausts all information from the data and allows the preparation of biplots for the simultaneous display of the ordination of variables and observations. The use of the modified PCA, called InDaPCA (PCA of Incomplete Data) is demonstrated on actual biological examples: leaf functional traits of plants, functional traits of invertebrates, cranial morphometry of crocodiles and fish hybridization data ? with biologically meaningful results. Our study suggests that it is not the percentage of missing entries in the data matrix that matters; the success of InDaPCA is mostly affected by the minimum number of observations available for comparing a given pair of variables. In the present study, interpretation of results in the space of the first two components was not hindered, however.