DATA4207 · Data Analysis in the Social Sciences
Measuring Latent Variables: PCA, Factor Analysis and IRT
Week 7 introduces methods for measuring things no single survey question captures — principal component analysis, factor analysis and item response theory — to reduce variables and detect structure among them. The focus is factor analysis: extracting a latent factor from correlated indicators, reading loadings, and deciding how many factors to keep. The lab builds a quality-of-life measure from World Values Survey data that becomes the dependent variable for Group Project 1.
What this chapter covers
- 01Latent variable / trait: not directly observable, inferred from observed indicators (manifest variables)
- 02Goal: dimension reduction and detecting structure; measuring traits like ideology or authoritarianism
- 03Principal Component Analysis (PCA): orthogonal components as variance-maximising linear combinations; scree plot
- 04Factor Analysis (FA): a measurement model where a latent factor drives the indicators; loadings are the F→Y weights
- 05Reading factor loadings (0-1) and extracting per-respondent factor scores
- 06PCA vs FA: PCA optimises variance/components; FA discovers hidden common themes
- 07Item Response Theory (IRT): latent ability θ → response probability; 1PL/2PL/GRM/3PL; item characteristic curves
- 08Deciding how many factors/components to retain (scree plot, eigenvalues, parallel analysis)
Reading a one-factor solution
- +1State the model: factor analysis assumes a latent factor F drives the four observed indicators, and the loadings are the F→Y association strengths (0-1). Higher loading = the item is more strongly explained by the factor.
- +1Read the loadings: all four are positive and moderate-to-strong (courts 0.82 highest, media 0.48 lowest). A consistent positive pattern means the items hang together on a single coherent dimension of civic trust.
- +1Decide on retention: one factor is reasonable given the coherent positive loadings; confirm with a scree plot / eigenvalues (a sharp drop after the first component supports keeping one). The weak media loading flags that indicator as the loosest fit.
- +1Use it downstream: extract each respondent's factor score (roughly standardised, NA where indicators are missing) and use it as a dependent or independent variable in a later model — exactly how the lab builds a quality-of-life measure for Group Project 1.
Key terms
- Latent variable
- A trait that cannot be measured directly (ideology, wellbeing, authoritarianism) and is inferred from several observed indicators. Latent-variable methods use the correlations among indicators to estimate it.
- Factor analysis
- A measurement model assuming a latent factor drives correlated observed variables; it estimates the factor loadings (item-factor weights) and factor scores, grouping indicators into interpretable dimensions.
- Factor loading
- The strength of association between an item and the factor, from 0 to 1. Larger loadings mean the item is better explained by the factor; a low loading flags a weakly fitting indicator.
- Factor score
- Each respondent's estimated position on the latent factor — the value carried forward as a variable in later models (NA where the underlying indicators are missing).
- Principal Component Analysis (PCA)
- A dimension-reduction method producing orthogonal components that are variance-maximising linear combinations of the original variables. PCA optimises variance captured, whereas factor analysis seeks the hidden common themes.
- Scree plot
- A plot of variance explained per component, used to decide how many components or factors to keep — you look for the 'elbow' where additional components add little.
Measuring Latent Variables: PCA, Factor Analysis and IRT FAQ
What's the difference between PCA and factor analysis?
PCA is pure dimension reduction: it builds orthogonal components that capture as much variance as possible, front-loaded into the first few. Factor analysis is a measurement model: it assumes a latent factor causes the correlations among the observed items and estimates how strongly each item loads on that factor. Use PCA to compress or de-noise variables, and factor analysis to measure an underlying construct like quality of life.
How do I decide how many factors to keep?
Combine evidence: read a scree plot for the elbow, check eigenvalues, optionally run a parallel analysis, and — crucially — use theory to judge whether the retained factors are interpretable and worth including. A one-factor solution is supported when the indicators load consistently and the scree plot drops sharply after the first component.
Why does this week matter for Group Project 1?
Because the lab uses factor analysis on World Values Survey items to build a quality-of-life measure that becomes the dependent variable for Group Project 1 (8%, due Week 8). If you can extract and validate that factor score cleanly, the modelling and write-up that follow are far easier. Confirm the project brief and due date on Canvas.
Can AI help me with factor analysis in DATA4207?
Yes, as a study aid. Sia can explain latent variables, loadings versus scores, PCA versus FA, and IRT, and help you interpret a loadings table and a scree plot step by step. It teaches the method and checks your reasoning; it does not do graded work, and University of Sydney academic-integrity rules apply — confirm on Canvas whether AI is permitted for the project.
Assessment move
Anchor the week on one distinction and one output. The distinction is PCA (variance-maximising components) versus factor analysis (a latent factor driving indicators); rehearse it in a sentence. The output is a loadings table plus factor scores: practise reading loadings, justifying how many factors to keep with a scree plot, and extracting scores to use in a later model. Run the WVS-style quality-of-life example in R so you can reproduce the pipeline for Group Project 1, whose dependent variable is exactly this kind of factor score. Keep IRT at the conceptual level — know that it maps a latent ability to response probability via item characteristic curves. Confirm the group-project brief and timing on Canvas.
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