This link has been bookmarked by 30 people . It was first bookmarked on 16 Oct 2006, by Abhijit.
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01 Sep 15
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The requisite property for a variable to function as a general factor g is that any partial correlation between any two observed variables, partialing out g, is zero.
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09 Aug 11
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02 Jun 10
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24 Apr 10
Mostafa HusseinI don't understand a single word. I think the main purpose behind bookmarking this page is to try to look intelligent on delicious.
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28 Mar 10
George Bradford"Factor analysis includes both component analysis and common factor analysis. More than other statistical techniques, factor analysis has suffered from confusion concerning its very purpose. This affects my presentation in two ways. First, I devote a long section to describing what factor analysis does before examining in later sections how it does it. Second, I have decided to reverse the usual order of presentation. Component analysis is simpler, and most discussions present it first. However, I believe common factor analysis comes closer to solving the problems most researchers actually want to solve. Thus learning component analysis first may actually interfere with understanding what those problems are. Therefore component analysis is introduced only quite late in this chapter."
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25 Mar 10
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12 Apr 09
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06 Mar 09
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it is used to study the patterns of relationship among many dependent variables, with the goal of discovering something about the nature of the independent variables that affect them, even though those independent variables were not measured directly.
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But it would be very difficult to discover even a very clear and simple factor structure with fewer than about 50 cases, and 100 or more cases would be much preferable for a less clear structure.
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In factor analysis it is perfectly okay to have many more variables than cases. In fact, generally speaking the more variables the better, so long as the variables remain relevant to the underlying factors.
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Readers familiar with factor analysis will be surprised to find no mention of Kaiser's familiar eigenvalue rule or Cattell's scree test. Both rules are mentioned later, though as explained at that time I consider both rules obsolescent.
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m the number of factors.
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I recommend an alternative approach. This approach was once impractical, but today is well within reach. Perform factor analyses with various values of m, complete with rotation, and choose the one that gives the most appealing structure.
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Actually PCA often provides a good approximation to common factor analysis
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The central concept in PCA is representation or summarization. Suppose we want to replace a large set of variables by a smaller set which best summarizes the larger set
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Henry Kaiser suggested a rule for selecting a number of factors m less than the number needed for perfect reconstruction: set m equal to the number of eigenvalues greater than 1. This rule is often used in common factor analysis as well as in PCA
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An alternative method called the scree test was suggested by Raymond B. Cattell. In this method you plot the successive eigenvalues, and look for a spot in the plot where the plot abruptly levels out. Cattell named this test after the tapering "scree" or rockpile at the bottom of a landslide. One difficulty with the scree test is that it can lead to very different conclusions if you plot the square roots or the logarithms of the eigenvalues instead of the eigenvalues themselves, and it is not clear why the eigenvalues themselves are a better measure than these other values
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31 Oct 08
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08 Oct 08
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15 Jul 08
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factor analysis looks for the factors which underlie the variables.
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The central concept in PCA is representation or summarization. Suppose we want to replace a large set of variables by a smaller set which best summarizes the larger set.
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Each component's eigenvalue is called the "amount of variance" the component explains. The major reason for this is the eigenvalue's definition as a weighted sum of squared correlations. However, it also turns out that the actual variance of the component scores equals the eigenvalue. Thus in PCA the "factor variance" and "amount of variance the factor explains" are always equal. Therefore the two phrases are often used interchangeably, even though conceptually they stand for very different quantities.
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Henry Kaiser suggested a rule for selecting a number of factors m less than the number needed for perfect reconstruction: set m equal to the number of eigenvalues greater than 1. This rule is often used in common factor analysis as well as in PCA.
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However, Kaiser's major justification for the rule was that it matched pretty well the ultimate rule of doing several factor analyses with different numbers of factors, and seeing which analysis made sense. That ultimate rule is much easier today than it was a generation ago, so Kaiser's rule seems obsolete.
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An alternative method called the scree test was suggested by Raymond B. Cattell. In this method you plot the successive eigenvalues, and look for a spot in the plot where the plot abruptly levels out.
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For each eigenvalue L, define S as the sum of all later eigenvalues plus L itself. Then L/S is the proportion of previously-unexplained variance explained by L. For instance, suppose that in a problem with 7 variables the last 4 eigenvalues were .8, .2, .15, and .1. These sum to 1.25, so 1.25 is the amount of variance unexplained by a 3-factor model. But .8/1.25 = .64, so adding one more factor to the 3-factor model would explain 64% of previously-unexplained variance. A similar calculation for the fifth eigenvalue yields .2/(.2+.15+.1) = .44, so the fifth principal component explains only 44% of previously unexplained variance.
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The question, "Do these two groups have the same factor structure?" is actually quite different from the question, "Do they have the same factors?" The latter question is closer to the question, "Do we need two different factor analyses for the two groups?"
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13 May 08
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21 Mar 08
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19 Feb 08
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2. Proportion of variance explained = eigenvalue / sum of eigenvalues
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18 Jan 08
smilex3mdFactor Analysis Richard B. Darlington
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12 Mar 07
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27 Dec 06
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