McPherson (2004)

Three dimensions of social hierarchy:

• Economic capital
• Cultural capital
• Social capital

Why are cultural tastes associated with education/class?

Culture as autonomous fields

Habitus

Homology between the spaces

Homology between the spaces

Cultural capital

• People derive profit from the rarity of their positions in the space
• Producers compete in the field to make distinct products
• Consumers choose products based on their positions in the space.

Correspondence analysis

Problems

Discussion

• What are possible pitfalls of using surveys to study culture?
• What can we social scientists do with the limitations in measurement?

Recap: PCA as a dimension-reduction techinque

• uses linear combinations of variables to find dimensions that explain the maximal variance.
• can be applied to any matrix
• The goal is to find a lower-dimensional representation of the original data and does not rely on any assumption on how the data is generated.

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.

Varimax rotation (orthogonal)

• Goal: makes most orthogal factor loadings either close to zero or far from zero.
• find factor loadings \(b_{ij}\)s that maximize \[ \left(\frac{1}{p}\sum_{j=1}^k \sum_{i=1}^p b^4_{ij} - \sum_{j=1}^k \left(\frac{1}{p}\sum_{i=1}^p b^2_{ij}\right)^2\right). \]

Geometric interpretation

Jolliffe (2002, 155)

Oblique rotation

Jolliffe (2002, 156)

Comparison with PCA

• FA rests on an assumption of how the data is generated
• \(m\) factors still explain the same proportion of total variance as the \(m\) principal components do.
• However, the first factor(s) no longer explain maximal variation.
• The factors may or may not be orthogonal.

PCA vs. FA

When dealing with categorical variables

You can present bivariate association in a crosstab

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.

Correspondence analysis

• The row scores and column scores in a correspondence analysis are reciprocally determined.
• The problem can be solved via a generalized singular value decomposition applied to \(\mathbf{X}\)
• In CA, the total amount of variability is called inertia and is calculated as \(\frac{\chi^2}{N}\).
• Like in PCA, the eigenvalues correspond to the amount of inertia explained by the corresponding dimensions. All dimensions are orthogonal to each other, and the first dimension explains the maximal inertia.

• As in PCA, the solutions can be presented in a bi-plot.
• CA can also be conveniently applied to supplementary columns and rows.

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

Multiple correspondence analysis

• Then, it applies CA to the indicator matrix \(\mathbf{X}\) or the Burt matrix \(\mathbf{X}^\top\mathbf{X}\)
• One caveat: due to the introduction of dummy variables, the total amount of inertia is inflated. The eigenvalues need to be corrected.