Part 1

Introduction to Factor Analysis

• A statistical technique distinct from other methods.

• Analyzes individual items forming a scale of psychological constructs rather than composite scores.

• Aims to find coherent clusters of correlated items that are relatively independent from others.

Key Purpose and Process of Factor Analysis

• Aims for parsimony by summarizing numerous items into a few components while explaining maximum variance in the correlation matrix.

• Seeks to identify underlying structures (latent variables) represented by items regarding psychological constructs.

Example of Graduate Student Readiness

• Aspects include: motivation, intellectual ability, academic performance, family history, health.

• Items for survey responses run through factor analysis to investigate correlations.

• Patterns revealed help identify how items cluster into factors reflecting dimensions of graduate readiness.

• Poor items not aligning with dimensions may be identified for deletion.

Applications of Factor Analysis

• Extensively used in psychology for:

  • Testing dimensionality of psychometric scales.

  • Development of new scales from a large set of items to a smaller set.

• Example: Starting with 40 items to assess readiness reduced to 20 items after analysis.

Cautionary Note

• Factor analysis measures correlation but doesn’t ascertain if remaining items accurately reflect their dimensions.

• Critical evaluation is essential after performing factor analysis.

Exploratory vs. Confirmatory Factor Analysis

• Exploratory factor analysis (EFA) is not hypothesis testing; requires multiple alterations to arrive at a solution.

• Confirmatory factor analysis (CFA) tests hypothesis fit and is done with structured software like AMOS, not SPSS.

• This course focuses on EFA for foundational understanding.

Starting Point of Factor Analysis

• The correlation matrix (also termed R matrix) is the initial step—investigating correlations between items.

Case Study: Popularity Construct

• Variables in survey: talk, social skills, interest, self-talk, selfish, liar.

• Factor analysis reduces R matrix into smaller dimensions reflecting sociability and consideration.

Factor Loadings

• Graphing factor loadings helps visualize relationships between items and factors.

• Factor loadings represent strength of the relationship between an item and the respective factor, akin to a Pearson correlation.

• Loadings range from -1 to +1; higher is better.

• Ideal scenario: one factor strongly represents items correlating with it while being weakly related to others.

Guidelines for Factor Loadings

• Ideal loadings: 0.4 or higher on its factor, while 0.3 or lower on others.

• Can distinguish factors needed summarizing correlation patterns.

Types of Factor Analysis

• Two main distinctions:

  • Exploratory Factor Analysis (EFA).

  • Confirmatory Factor Analysis (CFA) (not covered in this unit).

• EFA types: Principal Components Analysis (PCA) and Principal Axis Factoring.

Principal Components Analysis (PCA)

• Used by researchers focused on observable data.

• Aims to explain maximum variance with a transformed dataset of linear components.

Principal Axis Factoring

• Used by researchers interested in latent psychological constructs.

• Aims to explain common variance among items, disregarding unique variance.

• PCA treats total variance fully, while Factor Analysis addresses only shared (common) variance.

Commonality in Factor Analysis

• Communality: Proportion of variance an item shares with other items.

  • Range: 0 (no common variance) to 1 (all variance shared).

• In PCA, initial commonalities are set at 1 (100% shared).

• In Factor Analysis, commonalities are lower due to accounting for unique variance first.

Graphical Representation of Variance

• Visual illustrations help to convey shared and unique variances across items, demonstrating commonality.

Output from SPSS in Factor Analysis

• Displays initial and after extraction commonalities as criteria for evaluating items.

• High commonality indicates good representation; low commonality flags potential problematic items.

Mathematical Representation in Factor Analysis

Factor Analysis Equations

• Factor analysis predicts scores based on measured items from underlying factors.

Principal Components Analysis Equation

• Describes how items contribute to a linear model and establishes component loadings.

Running Factor Analysis

• Initially produces as many factors as original items, then retains minimum number required for maximum variance.

• Both methods produce tables of loadings detailing item-factors relationships.

Utilizing Factor Scores

• Factor scores depict composite scores for individual factors in place of raw item scores.

• Various techniques exist for calculating these scores—Anderson Rubin method is recommended for unbiased results in SPSS.

Final Reading Recommendations

• Chapter 18 in Andy Field's textbook on Exploratory Factor Analysis.

• Chapter 9 in Mertler's textbook “Advanced Multivariate Statistical Methods”.