What is one key assumption underlying Factor Analysis?

• A. All variables must have equal variance
• B. Variables must have a binary outcome
• C. There must be linear relationships among variables
• D. All variables should be categorical

Answer: (C)

Factor analysis is a statistical method used to identify underlying relationships between variables within a dataset. One key assumption of this technique is that there must be linear relationships among the variables involved in the analysis. This means that the model presumes that the correlations between the observed variables can be captured through linear combinations of the factors.

When linear relationships are assumed, the factor analysis can effectively reduce the dimensionality of the data while preserving as much of the variance as possible. This is pivotal because if the relationships among the variables were non-linear, the factors extracted might not accurately represent the structure inherent in the data, potentially leading to misleading conclusions.

In contrast to this, the other options do not accurately reflect the fundamental assumptions of factor analysis. Equal variance among variables is not a requirement; rather, it's often desirable to have diverse variances to adequately capture different dimensions. Factors can be derived from continuous variables rather than solely binary outcomes, thus negating the necessity for a binary outcome across variables. Also, while categorical variables can be part of factor analysis, the assumption does not mandate that all variables be categorical, as continuous variables are commonly used in practice.

Understanding these assumptions is essential for appropriately applying factor analysis and interpreting its results in the context of multivariate data analysis.