Principal Components Analysis: Row Scaling and Compositional Data

Abstract

Row scaling is sometimes called normalisation, but this term is also sometimes used for column standardisation, so we will avoid the latter term in this article, to prevent confusion.

Of course, whether this improvement is observed does depend on the structure of the data, but if the difference between samples is primarily due to the relative concentrations or proportions and the amount of sample is not easy to control, row scaling to constant total often results in an improvement. It can be combined with other approaches for column transformation such as standardisation as discussed in the previous article.

If there are only two variables, the simplex is a line. In Figure 4, we illustrate the scores first 2 PCs of the dataset formed by the first two variables from Table 1. We see that after row scaling there is only one non-zero PC. In this case, the position along the line relates to the class membership of each object, although this is not always so and depends on an appropriate choice of variables.

In the case of the data in Table 1, row scaling improves visualisation of the class differences and structure in the data in this case. However, row scaling is not always appropriate. If the absolute values of each variable are known accurately (e.g., the amount of sample extracted can be kept constant or calibrated to a known standard), compositional data lose information. In addition, sometimes there may be one or two very intense variables that are of subsidiary interest; for example, a primary metabolite that is very intense but has little or no relationship to the factors of interest; the proportions will be dominated by this uninteresting factor.

However, row scaling is a common procedure in many areas of chemometrics. There is a significant statistical literature about multivariate compositional data. If the main aim of an analysis is qualitative, for example, to separate groups or find outliers, often some of the more elaborate statistical considerations are of secondary importance. If, however, the data are to be used for statistical inference, such as hypothesis tests or p values or estimation, it is a good idea to look closely at the classical literature in order to best interpret and process compositional data.