One strength of the factor analytic measurement model is the ability to "purify" the measures by distinguishing between variance related to the common factor(s) and variance due to measurement error. The question then arises, can the researcher use the measurement model to assign scores to individual cases or respondents on the factors? For example, suppose a school headmaster develops a model of "teaching effectiveness,"based on data collected from the school's teachers. Can the headmaster then use the model's parameter estimates, and the manifest variable scores, to derive "factor scores" for the purpose of grading teaching performance?

Bollen indicates that one way to approach the problem (although other approaches exist) is to set up a hypothetical regression of the common factors () on the manifest variables:

 

One may then estimate the "B" regression parameter in the usual way, as the ratio of the covariance of and X over the variance of X. This leads to the formula:

 

where is the covariance matrix of the common factors, is the matrix of loadings, is the model-implied covariance matrix of the manifest variables, and ^ indicates that the matrices are based on estimated parameters.

Factor scores are problematic for several reasons. Besides the existence of competing methods for deriving the scores, the principle of factor indeterminacy suggests that different rotations of the factor solution could lead to different factor scores. While the different sets of factor scores that result may be highly correlated, there is no guarantee that the relative positions of the cases or individuals on the factor score continuum will be identical.

In addition, the measurement model indicates that the manifest variables are functions of both the common factors and the measurement error terms. The factor score equation constructs the factor scores as weighted composites of the manifest variables. Thus, the factor scores are also influenced by measurement error. For this reason, a factor score should not be considered a perfect measure of the factor itself.

This dilemma highlights one of the relative strengths of partial least squares (PLS). In PLS, the "latent variables" are weighted composites of the manifest variables. So the PLS approach leads directly to explicit factor scores. Some researchers and practitioners find this to be one of the most attractive features of PLS.

[此贴子已经被作者于2008-2-16 22:52:34编辑过]

Encyclopedia of Biostatistics
Factor Scores
Standard Article
Ralph B. D'AGOSTINO SR1 and Heidy K. Russell1
1Boston University, Boston, MA, USA
Copyright © 2005 John Wiley & Sons, Ltd. All rights reserved.
 
Abstract
After a set of interpretable factor loadings is determined, a next possible step is to compute factor scores. Exact factor scores can be obtained for a principal components analysis model. Methods that compute factor scores in the common factor analysis model are also presented. In general, there is limited value in the use of factor scores, for even though the factors are interpretable, the estimated factor scores are not well determined.

 

[此贴子已经被作者于2008-2-16 22:49:29编辑过]

Hi

Does anyone know how to quickly calculate factor scores for cases?

I've used principal components analysis in JMP to identify factors (which
are meaningful) as well as finding the rotated components.  JMP kindly
provides the Std Score Coefficients, (they are the same as Factor Score
coefficients in SPSS I believe), which are then used to calculate factor
scores for cases.  Is there an easy way to do this?

Right now I'm creating new columns and writing the formula for each factor
score.  There surely is an easier way?

(One other question which I'm not certain of:  Do I need to standardize the
values of the original cases before calculating the factor scores?  If so,
is this done automatically in some way or do I need to do that?)

Appreciate your help...

1.如果是主成分分析，综合得分是自己算的，即factor做完之后（得选在factor anaylsis界面选中scores中的display factor score……才会出来这个矩阵）因子载荷矩阵下面那个带score的的表格就是计算主成分得分的系数矩阵，然后将原始数据标准（用描述性统计分析就能直接得到）化后的结果带入方程式（方程系数就是系数矩阵，这个过程得自己算），得到各个主因子的综合的得分，若要计算综合得分，则需要在写一个方程式，Y=Y1*a+y2*b……，y1,y2……为各个主成分得分，a,b……为各个主成分的发差贡献率，在特征值那表里头了。最后得到Y即为综合得分。
2.要是因子分析算因子得分就简单了，直接在scores中选中save as variable，那么在表格中直接就会多出来一列得分变量了，即为因子得分。
    在说说他俩区别吧，主成分就一个用途，那就是排序，比如算哪个城市发展的好可以用，主成分没有含义。因子分析可以分析出来各个因子代表什么，比如算影响各个城市发展的主要因素是什么。共同点就是在SPSS中操作的过程是一样。

因子分析可以确定权重可以看一下https://bbs.pinggu.org/thread-10814155-1-1.html

因子得分是因子分析的结果，代表了原始变量在新识别出的因子上的投影或贡献程度。每个因子得分都是一个数值，反映了观测值在这特定因子上的表现。它们是通过线性组合原始变量计算得出的，每个因子得分系数矩阵（也称为载荷矩阵）描述了每个变量对各个因子的影响。

因子得分的主要用途包括：

1. **简化数据**：通过少数几个因子代替大量的原始变量，减少数据的复杂性。
2. **解释因素结构**：分析哪些原始变量对特定因子影响最大，帮助理解因子的含义。
3. **建模和预测**：使用因子得分作为新的独立变量进行回归或其他统计建模，以改进模型性能或简化模型。
4. **比较不同观测值**：通过比较各观测值的因子得分，可以了解它们在因子上的相对位置。

在实际应用中，因子得分可能需要通过旋转（如主成分旋转或最大方差旋转）来优化，以便更好地解释各个因子。同时，由于因子得分是估计值，有时可能需要借助因子得分回归或其他方法来获得更准确的估计。

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