Interdependence Analysis

Interdependence Analysis

Establishing an association between factors A and B generally requires less evidence than establishing that A causes B. A causal claim must address alternative explanations, including other factors that could influence both. Thus, within the conceptual realm of co-occurrence, A can cause B or there could be another, unaccounted for factor C that impacts both A and B – in either case, the general conclusion regarding the association between A and B remains the same.

In the context of statistical analyses of data, a widely used expression of co-occurrence is correlation, which is a quantity measuring a possible two-way linear association between variables. Correlation describes the behavior of pairs of variables in terms of two distinct dimensions of interdependence: the direction and the strength of the association. The former can be either positive, which is when an increase in one variable is associated with an increase in another, or negative, which is when an increase in one variable is associated with a decrease in the other; the latter can range from no linear association to an exact linear relationship. It is important to note that perfect association can be either positive or negative – perfect positive correlation means the variables follow an exact increasing straight-line relationship; perfect negative correlation means they follow an exact decreasing one. Their changes need not have the same magnitude or units. Commonly expressed on a standardized scale, perfect negative correlation is denoted by a coefficient of '−1', perfect positive correlation is denoted by a coefficient of '1' (with '+' being the implied qualifier) while no correlation is denoted by '0'. Given its standardized nature, correlation coefficients enable quick and easy strength of association comparisons.

There have been numerous types of correlation coefficients developed in the course of the past several decades, though most are based on, or derived from the two main types of correlation coefficients: Pearson's product moment and Spearman's rank correlation. Pearson's product moment correlation, typically denoted by 'r', measures linear association between quantitative variables. Computing it does not require normal distributions, although some inferential procedures make distributional assumptions. Extreme values can exaggerate or dampen the measured association. An alternative to Pearson's correlation, Spearman's rank correlation coefficient, denoted either by the Greek letter ρ (rho) or by r s, is nonparametric (meaning, it does not assume variables to be normally distributed) and it is appropriate to use when one or both variables are at least ordinal or continuous but skewed; it uses ranks, so the magnitude of an extreme value has less influence than it can have on Pearson’s coefficient.

Although the notion of correlation is commonly used to describe pairwise co-occurrence, the broader idea of variable interdependence is not limited to just bivariate associations. In fact, one of the important data curation considerations is dimensionality reduction, which is the process of decreasing the number of variables in a dataset by obtaining a set of principal components. Exemplified by the widely used methods of principal component analysis and factor analysis,24 which are statistical algorithms designed to reduce the original set of variables into smaller sets of components or factors that account for most of the information found in the original variables, dimensionality reduction makes use of multivariate, or many variables, interdependence. In a more general sense, multivariate interdependence analysis is not limited to just the normally distributed, continuous variables – for instance, the method of principal component analysis has been extended to also include categorical variables (aptly named as categorical principal component analysis or CATPCA).