Quantitative Methods in Cultural Industries: Latent Variables and Factor Analysis

Latent Variables Analysis

• Often, direct measurement of concepts like intelligence, satisfaction, preferences, political attitude, socio-economic status, happiness, or quality of life is not possible.

• Indirect Measurement: Concepts and characteristics that provide information on the concepts of interest are measured instead.

• Multidimensional Analysis: Concepts may have several dimensions that cannot be directly measured; each dimension is a latent variable.

• Several manifest (indicator) variables are measured to gather information on the latent variables.

• Intelligence Example:

  • Manifest variables include scores in a battery of tests (verbal, mathematical, spatial).

• Socio-Economic Status Example:

  • Manifest variables include income measurement at different levels, occupation measurement, and level of education measurement.

Approaches to Latent Variable Analysis

• Proxy Variable: Using a single observed variable as a substitute for the latent construct.

• Index or Composite Measure: Combining multiple observed variables into a single score.

• Latent Variable Model as Factor Analysis: A statistical method to uncover underlying latent variables from a set of observed variables.

• Measuring Participation in the Arts:

  • Participation in the art is a latent construct.

  • SPPA (Survey of Public Participation in the Arts) uses nine numerical variables.

  • Multiple linear regression analysis is applied as a proxy approach.

  • Examples of variables:

    • PTC1Q1B: Number of live jazz performances in the last 12 months.

    • PTC1Q2B: Number of live Latin, Spanish, or salsa music performances in the last 12 months.

    • PTC1Q3B: Number of live classical music performances in the last 12 months.

    • PTC1Q4B: Number of live opera performances in the last 12 months.

    • PTC1Q5B: Number of live musical stage plays in the last 12 months.

    • PTC1Q6B: Number of live non-musical stage plays in the last 12 months.

    • PTC1Q7B: Number of live ballet performances in the last 12 months.

    • PTC1Q8B: Number of live dance (non-ballet) performances in the last 12 months.

    • PEC1Q10A: Number of visits to art museums or galleries in the last 12 months.

  • A composite measure of participation can be the sum of the nine numerical variables.

Factor Analysis

• Interdependency Technique: Multivariate technique involving the joint analysis of several manifest variables to:

  • Assess associations among them.

  • Identify patterns of association.

• How Factor Analysis Works:

  • Identification of groups of variables.

    • High association within the group.

    • Low association between groups.

  • Each group of variables represents a pattern of association, corresponding to a latent dimension.

  • For each group, a new variable is generated to measure the latent dimension.

• Factor analysis is typically run on numerical or ordinal variables (treated as numerical).

• Association is measured as correlation.

• Aims of Factor Analysis:

  • Data summarization: Stop at the identification of patterns.

  • Data reduction: Use fewer new variables in further analysis.

• Manifest Variables:

  • PTC1Q1B Number of live jazz performances last 12 months

  • PTC1Q2B Number of live Latin, Spanish, or salsa music performances last 12 months

  • PTC1Q3B Number of live classical music performances last 12 months

  • PTC1Q4B Number of live opera performances last 12 months

  • PTC1Q5B Number of live musical stage plays last 12 months

  • PTC1Q6B Number of live non musical stage plays last 12 months

  • PTC1Q7B Number of live ballet performances last 12 months

  • PTC1Q8B Number of live dance (non-ballet) performances last 12 months

  • PEC1Q10A Number of visits to art museum or gallery last 12 months

• Aims of measuring participation in art using Factor Analysis:

  • Identification of patterns of association and respondents profiles.

  • Data reduction by generating a smaller number of variables.

Stages in Factor Analysis

I. Data screening
II. Extraction of factors
III. Interpretation and rotation
IV. Generation of new variables

I. Data Screening

• Work on standardized variables, where each variable has a mean equal to 0 and a variance equal to 1.

• Aims of data screening are to:

  • Assess association among the variables.

  • Measure the strength of the association using correlation.

  • Determine if the variables are highly inter-correlated (partial correlations)

• Measuring Pairwise Correlation

  • Pearson's correlations.

  • Partial correlations.

• Measuring Inter-correlation

  • MSA (Measure of Sampling Adequacy): For the i-th variable, it's calculated as: MSAi = \frac{\sum{j \neq i} r{ij}^2}{\sum{j \neq i} r{ij}^2 + \sum{j \neq i} r_{ij \cdot partial}^2}

    • Where r{ij} is the Pearson correlation between variable i and variable j, and r{ij \cdot partial} is the partial correlation.

  • MSA for all variables: Kaiser-Meyer-Olkin (KMO) MSA. This is the same computation, but for all variables. → is the data fit “adequately” for factor analysis

• Bartlett’s Test:

  • Null hypothesis: all Pearson’s correlations equal to 0.

  • Alternative hypothesis: at least one Pearson’s correlation is different from zero.

  • Test statistic: measure of distance between observed correlation matrix and identity matrix (e.g., 9147.205).

  • Significance (p-value).

II. Factors Extraction

Analytical Stage:

• Selection of extraction method

• Selection of the number of factors

• Production of initial solution

Approaches: the extraction methods

• PCA – Principal Components Analysis

• PAF – Principal Axis Factoring

• Similar results

PCA (Principal Component Analysis):

• Descriptive technique.

• Used for data reduction to replace p correlated numerical variables by a smaller number of uncorrelated variables. correlated → uncorrelated smaller number

Total variance:

• Indicator variables are standardized.

• Total variance is a proxy of the total amount of information.

• Total variance is the sum of the variables' variances and equals the total number of variables.

How PCA work:

• Given p manifest variables, PCA generates p not correlated variables.

• The total variance of the components = total variance of manifest variables = p.

• Each component is a weighted average (linear combination) of the original variables.

• Components are ranked with respect to their variance.

• The first component has the largest variance and explains the largest proportion of the total variance.

• The second component has the second largest variance.

Data reduction:

• p components contain the same amount of information as the original variables.

• The first components, which account for the largest amount of information, are kept.

• The remaining components are discarded without significant loss of information.

Data summarization:

• The first components allow understanding the underlying structure in the data and identifying patterns in the data.

Intuition of Computation

• Linear algebra computation on correlation matrix.

• Eigenvalue: variance of the component.

• Eigenvector: vector of weights of the linear combinations.

Selection of the Number of Components

• Select few components to retain as much as possible of the total amount of information.

• Retain as much as it is possible of the total amount of information

Methods

• Kaiser’s criterion: components with variance greater than 1.

• Fix the percentage of information to retain.

• Inspection of the scree plot.

Component as Linear Combination

• First component = \beta1 X1 + \beta2 X2 + … + \betap Xp

Component Loading.

• For each component

• Component loadings

• Component loading

• Rescaled weight of the linear combination

• Rescaled larger coefficient for the most important components.

Component loading

*   Weight multiplied by the component standard deviation
*   Each loading is a correlation between the indicator and the component

Component Interpretation

*   Interpretation based on loadings.
*   Each loading = indicator’s contribution to component

Readable Solution

*   For each component, few indicators with high loadings (absolute value) and remaining indicators with very small loading (absolute value)

III. Components Rotation

• Stages:
  I. Data screening
  II. Extraction of factors
  III. Interpretation and rotation of the factors
  IV. Generation of new variables

• Second analytical stage is for clarification of the underlying structure

• Readable component matrix Geometric transformation of the components

• Types of Rotation:

  • Orthogonal Rotation: components remain uncorrelated

  • Oblique Rotation: components are correlated

• Varimax Rotation: = Orthogonal

  • Orthogonal rotation

  • High loadings for a smaller number of components

  • Low loadings for the rest

IV. Generation of New Variables

• New variables become weighted averages of the indicators Weights such that the highest weights indicators with the highest loadings the new variables are standardized

PCA: Component Interpretation and scores

• Factor’s extraction

• Principal Components Method

• Selection and interpretation of the components

• Component Loading-> Each loading correlation between the indicator and the component

Principal Axis Factoring

*   Intution of the computation
*   Factor model with 4 factors  32% total variability explainedFactor INTERPRETATION