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# MVChap03 - STA 4107/5107 Chapter 3 Factor Analysis 1 Key...

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STA 4107/5107 Chapter 3 Factor Analysis 1 Key Terms Please review and learn these terms. 2 What is Factor Analysis? Factor analysis is an interdependence technique (see chapter 1) that primarily uses metric variables, however, non-metric variables can be included as “dummy variables. The basic idea is to describe the set of P variables X 1 , X 2 , · · · , X P in our data as linear combinations of a smaller number of factors , and in the process get a better understanding of the relationships between our variables. Factor analysis is based on a statistical model, unlike principal components (which we will cover later). Mathematically, factor analysis can be represented thusly: x 1 = l 11 F 1 + l 12 F 2 + · · · + l 1 m F m + ² 1 x 2 = l 21 F 1 + l 22 F 2 + · · · + l 2 m F m + ² 2 . . . x p = l p 1 F 1 + l p 2 F 2 + · · · + l pm F m + ² p Here, the X i are the original variables, the l ij are called the loadings and the F j are the common factors , and the ² i are the unique factors . In factor analysis, the loadings or weights are chosen to maximize the correlation between the variable and the factor. Our motive for studying factor analysis in this course is that we will extend the concept to structural equation modeling , which is also called confirmatory factor analysis near the end of the term. 3 Some History and Examples Factor analysis was developed at the turn of the 20 th century by psychologist Charles Spear- man, who hypothesized that a person’s score on a wide variety of tests of mental ability– mathematical skill, vocabulary, other verbal skills, artistic skills, logical reasoning ability, etc.–could all be explained by one underlying “factor” of general intelligence that he called g . From a data set of test scores of boys in preparatory school, he noticed that any two rows in the table of correlations were approximately proportional across the different variables. So for example, Classics and English in the table below have the ratios . 83 . 67 . 70 . 64 . 66 . 54 . 63 . 51 . Upon observing this, he hypothesized that it was due to a common factor, g . 1

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Classics French English Mathematics Pitch Music Classics 1.0 .83 .78 .70 .66 .63 French .83 1.0 .67 .67 .65 .57 English .78 .67 1.0 .64 .54 .51 Mathematics .70 .67 .64 1.0 .45 .51 Pitch .66 .65 .54 .45 1.0 .40 Music -.63 .57 .51 .51 .40 1.0 It was an interesting idea, but it turned out to be wrong. Today the College Board testing service operates a system based on the idea that there are at least three important factors of mental ability–verbal, mathematical, and logical abilities–and most psychologists agree that many other factors could be identified as well. 3.1 Examples 1. Consider various measures of the activity of the autonomic nervous system–heart rate, blood pressure, etc. Psychologists have wanted to know whether, except for random fluctuation, all those measures move up and down together–the “activation” hypothesis.
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