Factor analysis

• assumes that the data is generated from a few hidden dimensions and seeks to recover the dimensions
• Mathematical formulation: \[ x_1 = \lambda_{11}f_1 + \lambda_{12}f_2 + ... + \lambda_{1m}f_m + e_1 \\ x_2 = \lambda_{21}f_2 + \lambda_{22}f_2 + ... + \lambda_{2m}f_m + e_2 \\ \vdots\\ x_p = \lambda_{p1}f_2 + \lambda_{p2}f_2 + ... + \lambda_{pm}f_m + e_2 \\ m < p \]

• The error terms are idiosyncratic and are uncorrelated among themselves and with any factor.
• The factors can be uncorrelated or correlated.

Factor analysis via PCA approximation

1. Apply PCA to the dataset
2. Keep the first \(m\) principal components scaled by the square roots of the eigenvalues and treat the rests as idiosyncratic.
3. Rotate the PC factors according to some criterion.

Orthogoal rotation

Jolliffe (2002, 155)

Oblique rotation

Jolliffe (2002, 156)

PCA vs. FA

Correspondence analysis

• Goal: given a count table \(\mathbf{X} = [x_{ij}]\), to find row-weight and column-weight vectors \(\mathbf{r} \in \mathbb{R}^n\) and \(\mathbf{c} \in \mathbb{R}^p\), such that

\[\begin{equation} \begin{aligned} r_i &\propto \sum_{j=1}^p\ c_j \frac{x_{ij}}{x_{i\cdot}}, \\ \text{and } c_j &\propto \sum_{i=1}^n\ r_i \frac{x_{ij}}{x_{\cdot j}}, \end{aligned} \end{equation}\] where \(x_{i\cdot}\) and \(x_{\cdot j}\) are the sums of the \(i\)th row and the \(j\)th column, respectively.

Basic Intuition

• The rows and columns are mutually defined in the same space.
  • A row is represented by its weight on all the columns. (A person’s cultural taste is defined by the genres he/she likes.)
  • A column is represented by the contribution of all the rows. (The taste of a genre is represented by the people who like it.)
  • All positions are estimated at once through matrix decomposition.

Multiple correspondence analysis

• is an extension of CA that applies to multiple categorical variables.
• MCA starts with turning individual-level data into an indicator matrix:

Indicator matrix

Discussion

• What is the distinction between “deliberative” and “non-deliberative” cognition?
• What types of measurement better capture each type of cognition?

We will discuss more about using texts as a method next time.

Goldberg & Stein (2018)

• How could global cultural patterns emerge without close interaction?
• Where does dimensional come from?