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Factor Analysis
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Factor analytic methods to evaluate scales
 
Under construction (I am still writing this)
 
In most of applied research we use factor analysis for exploratory purposes (to know which variables hang with which variables to form a scale).  However, we also use factor models to evaluate test products, such as student achievement test.  Under the NCLB legislation (No Child Left Behind), there are a lot of efforts in this area.  For example, state authorities need to report to the federal government regarding the validity of sub-scales. 
 
Most basically we use two different factor models.  One model ignores the fact that test items have separate sub-components (e.g., all test items contribute to create a math score).  The other model
 
 
 

Brief Description of Factor Analytic Models
When you create a scale/measure, you also wonder if smaller set of subscales are obtainable.  For example, you create reading scores based on 100 items.  Your theory says that there are five different components in the pool of items, such as ability to reason, vocabulary, etc. 
          How do you know you are justified to think of such separate scales, in addition to/instead of a total scale.  You run two factor models.  One is a simple model that assumes that all of the items conrtibute to the total score (First order factor model).  The other is an ellaborate model that assumes grouping/dimentions among those items (Second order factor model).  You compare chisquare statistics off the two models and see if the difference is statistically significant.  If significant, that means that modeling of the dimensions improves the fit between the model and the data.
 
 
 
Using HLM
 
Using MPLUS
Using SAS PROC NLMIXED
 



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