PCA与FA英文ppt

PCA/FA
• While often used similarly, PCA and FA are distinct from one another • Principal Components Analysis
▫ Extracts all the factors underlying a set of variables ▫ The number of factors = the number of variables ▫ Completely explains the variance in each variable
▫ Creating linear combinations ▫ PCA especially falls more along the line of what we‟ve already been doing
• What we do have different from previous methods is that there is no IV/DV distinction
Questions
• Which factors account for the most variance? • How well does the factor structure fit a given theory? • What would each subject‟s score be if they could be measured directly on the factors? • What does each factor mean? • What is the percentage of variance in the data accounted for by the factors? (FA or how much by the most notable components in PCA)
es only the shared variance
Error is estimated apart from shared variance
FA vs. PCA conceptually
• FA produces factors; PCA produces components • Factors cause variables; components are aggregates of the variables • The underlying causal model is fundamentally distinct between the two
Principal Components and Factor Analysis
Principal components
Intro
• In many disciplines we study phenomena or constructs that cannot be directly measured
▫ Just a single set of variables
Summary
• PCA goal is to analyze variance and reduce the observed variables • PCA reproduces the R matrix perfectly • PCA – the goal is to extract as much variance with the least amount of factors • PCA gives a unique solution • FA analyzes covariance (communality) • FA is a close approximation to the R matrix • FA – the goal is to explain as much of the covariance with a minimum number of factors that are tied specifically to assumed constructs • FA can give multiple solutions depending on the method and the estimates of communality
• FA
▫ FA distributes common variance into orthogonal factors ▫ FA recognizes measurement error and true factor variance
FA vs. PCA
• In some sense, PCA and FA are not so different conceptually than what we have been doing since multiple regression
PC1 = b11X1 + b21X2 + … + bk1Xk PC2 = b12X1 + b22X2 + … + bk2Xk PCf = b1fX1 + b2fX2 + … + bkfXk
• Common factor model
X1 = λ1ξ + δ1 X2 = λ2ξ + δ2 Xf = λfξ + δf
Questions
• Three general goals: data reduction, describe relationships and test theories about relationships • How many interpretable factors exist in the data? • How many factors are needed to summarize the pattern of correlations?
Basic Concept
• If two items are highly correlated
▫ They may represent the same phenomenon ▫ If they tell us about the same underlying variance, combining them to form a single measure is reasonable for two reasons
Alcohol Use X1 Marijuana Use X2 Hard Drug Use X3 Distress Y1 Self-Esteem Y2 Powerlessness Y3
Substance Use F1
Psychosocial Functioning F2
Example of PCA-FA each with 3 main (bold-faced) loadings and each with 3 inconsequential (dashed-line) loadings.
Parsimony Reduction in Error
• Suppose one is just a little better than the other at representing this underlying phenomena? • And suppose you have 3 variables, or 4, or 5, or 100? • FACTOR ANALYSIS (general) looks for the phenomena underlying the observed variance and covariance in a set of variables. • These phenomena are called “factors” or “principal components.”
• Because of this, we‟ll want to “reduce” them to a smaller set of factors
Purpose of PCA/FA
• To find underlying latent constructs ▫ As manifested in multiple items/variables • To assess the association between multiple factors • To produce usable scores that reflect critical aspects of any complex phenomenon • As an end in itself and a major step toward creating error free measures
▫ Some do not consider PCA as part of the FA family*
FA
PCA
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Contrasting the underlying models*
• PCA
▫ Extraction is the process of forming PCs as linear combinations of the measured variables as we have done with our other techniques
▫ Items are representations of underlying or latent factors.
We want to know what these factors are
▫ We have an idea of the phenomena that a set of items represent (construct validity).