探索性因子分析EFAFactor Analysis

1、EXPLORATORY FACTORANALYSIS (EFA),Kalle Lyytinen ranging between -1.0 and +1.0. The closer the value is to 1.0, positive or negative, the stronger the relationship between the factor and the item. Loadings can be both positive or negative.,Factor Loadings: Note the direction of the arrows; the factor

2、s are thought to influence the indicators, not vice versa. Each item is being predicted by the factors.,V1,V2,V3,V4,Factor 1,Factor 1,Exploratory Factor Analysis,Errors in Measurement: Each of the indicator variables has some error in measurement. The small circles with the indicate the error. The e

3、rror is composed of we know not what or are not measured directly. These errors in measurement are considered the reliability estimates for each indicator variable.,V1,V2,V3,V4,Factor 1,Factor 1,Exploratory Factor Analysis,Multi-Indicator Approach,A multiple-indicator approach reduces the overall ef

4、fect of measurement error of any individual observed variable on the accuracy of the results A distinction is made between observed variables (indicators) and underlying latent variables or factors (constructs) Together the observed variables and the latent variables make up the measurement model,Co

5、nceptual Model,Negative Affect,This model holds that thereare two uncorrelated factorsthat explain the relationshipsamong the six emotion variables,Variables Factor (Observed) (Latent),Measurement Model,*The loading is a data-driven parameter that estimates the relationships (correlation) between an

6、 observed item and a latent factor.,Data Matrix must have sufficient number of correlations Variables must be inter-related in some way since factor analysis seeks the underlying common dimensions among the variables. If the variables are not related each variable will be its own factor! Rule of thu

7、mb: substantial number of correlations greater than .30 Metric variables are assumed, although dummy variables may be used (coded 0,1). The factors or unobserved variables are assumed to be independent of one another. All variables in a factor analysis must consist of at least an ordinal scale. Nomi

8、nal data are not appropriate for factor analysis.,Assumptions of Factor Analysis,Quick Quips about Factor Analysis,How many cases? Rule of 1010 cases for every item; rule of 100 number of respondents should be the larger of (1) 5 times number of variables or (2) 100. How many variables do I need to

9、FA? More the better (at least 3) Is normality of data required? Nope Is it necessary to standardize one variables before FA? Nope Can you pool data from two samples together in a FA? Yep, but must show they have same factor structure.,Two statistics on the SPSS output allow you to look at some of th

10、e basic assumptions. Kaiser-Meyer-Olkin (KMO) Measure of Sampling Adequacy, and Bartletts Test of Sphericity Kaiser-Meyer-Olkin Measure of Sampling Adequacy generally indicates whether or not the variables are able to be grouped into a smaller set of underlying factors. That is, will data factor wel

11、l? KMO varies from 0 to 1 and should be .60 or higher to proceed (can us .50 more lenient cut-off) High values (close to 1.0) generally indicate that a factor analysis may be useful with your data. If the value is less than .50, the results of the factor analysis probably wont be very useful.,Tests

12、for Basic Assumptions,Kaiser-Meyer-Olkin (KMO),Marvelous - - - - - - .90s Meritorious - - - - - .80s Middling - - - - - - - .70s Mediocre - - - - - - - .60s Miserable - - - - - - .50s Unacceptable - - - below .50,KMO Statistics: Interpreting the Output,In this example, the data support the use of fa

13、ctor analysis and suggest that the data may be grouped into a smaller set of underlying factors. What does Bartletts Test of Sphericity explore?,Correlation Matrix,Bartletts Test of Sphericity Tests hypothesis that correlation matrix is an identity matrix. Diagonals are ones Off-diagonals are zeros

14、Significant result indicates matrix is not an identity matrix.,Bartletts Test of Sphericity,Bartletts Test of Sphericity compares your correlation matrix to an identity matrix An identity matrix is a correlation matrix with 1.0 on the principal diagonal and zeros in all other correlations. So clearl

15、y you want your Bartlett value to be significant as you are expecting relationships between your variables, if a factor analysis is going to be appropriate! Problem with Bartletts test occurs with large ns as small correlations tend to be statistically significant so test may not mean much!,Two Extr

16、action Methods,Principal Component Analysis Considers all of the available variance (common + unique) (places 1s on diagonal of correlation matrix). Seeks a linear combination of variables such that maximum variance is extractedrepeats this step. Use when there is concern with prediction, parsimony

17、and knows specific and error variance are small. Results in orthogonal (uncorrelated factors) Principal Axis Factoring (PFA) or Common Factor Analysis Considers only common variance (places communality estimates on diagonal of correlation matrix). Seeks least number of factors that can account for t

18、he common variance (correlation) of a set of variables. PAF is only analyzing common factor variability; removing the uniqueness or unexplained variability from the model. Called Principal Axis Factoring (PFA). PFA preferred in SEM cause it accounts for co-variation, whereas PCS accounts for total v

19、ariance,Methods of Factor Extraction,Principal-axis factoring (PAF) diagonals replaced by estimates of communalities iterative process continues until negligible changes in communalities,What is a Common Factor?,It is an abstraction, a hypothetical construct that affects at least two of our measurem

20、ent variables. We want to estimate the common factors that contribute to the variance in our variables. Is this an act of discovery or an act of invention?,What is a Unique Factor?,It is a factor that contributes to the variance in only one variable. There is one unique factor for each variable. The

21、 unique factors are unrelated to one another and unrelated to the common factors. We want to exclude these unique factors from our solution.,Comparison of Extraction Models,PCA vs. PAF Factor loadings and eigenvalues are a little larger with Principal Components One may always obtain a solution with

22、 Principal Components Often little practical difference FYIOther less-used Extraction Methods (Image, alpha, ML ULS, GLS factoring),Principal Components Extraction,A communality (C) is the extent to which an item correlates with all other items. Thus, in PCA extraction method when the initial commun

23、alities are set to 1.0, then all of the variability of each item is accounted for in the analysis. Of course some of the variability is explained and some is unexplained. In PCA with these initial communalities set to 1.0, you are trying to find both the common factor variance and the unique or erro

24、r variance.,Principal Components Extraction,Statisticians have indicated that assuming that all of the variability of the items whether explained or unique can be accounted for in the analysis is flawed and definitely should not be used in an exploratory factor model. Some researchers suggest PAF as

25、 the appropriate method for factor extraction using EFA. In PAF extraction, the amount of variability each item shares with all other items is determined and this value is inserted into the correlation matrix replacing the 1.0 on the diagonals. As a result, PAF is only analyzing common factor variab

26、ility; removing the uniqueness or unexplained variability from the model.,Factor Rotation: Orthogonal,Varimax (most common) minimizes number of variables with high loadings (or low) on a factormakes it possible to identify a variable with a factor Quartimax minimizes the number of factors needed to

27、explain each variable. Tend to generate a general factor on which most variables load with med to high valesnot helpful for research Equimax combination of Varimax and Quartimax Q&A: Why use rotation method? Rotation causes factor loading to be more clearly differentiatednecessary to facilitate inte

28、rpretation,Non-orthogonal (oblique),The real issue is you dont have a basis for knowing how many factors there are or what they are much less whether they are correlated! Researchers assume variables are indicators of two or more factors, a measurement model which implies orthogonal rotation. Direct

29、 oblimin (DO) Factors are allowed to be correlated. Diminished interpretability Promax Computationally faster than DO Used for large datasets,Oblique Rotation,The variables are assessed for the unique relationship between each factor and the variables (removing relationships that are shared by multi

30、ple factors) The matrix of unique relationships is called the pattern matrix. The pattern matrix is treated like the loading matrix in orthogonal rotation.,Decisions to be made,EXTRACTION: PCA vs PAF ROTATION: Orthogonal or Oblique (non-orthogonal),Procedures for Factor Analysis,Multiple different s

31、tatistical procedures exist by which the number of appropriate number of factors can be identified. These procedures are called Extraction Methods. By default SPSS does PCA extraction This Principal Components Method is simpler and until more recently was considered the appropriate method for Explor

32、atory Factor Analysis. Statisticians now advocate for a different extraction method due to a flaw in the approach that Principal Components utilizes for extraction.,What else?,How many factors do you extract? One convention is to extract all factors with eigenvalues greater than 1 (e.g. PCA) Another

33、 is to extract all factors with non-negative eigenvalues Yet another is to look at the scree plot Number based on theory Try multiple numbers and see what gives best interpretation.,Eigenvalues greater than 1,Scree Plot,Three Factor Solution,Criteria For Retention Of Factors,Eigenvalue greater than

34、1 Single variable has variance equal to 1 Plot of total variance - Scree plot Gradual trailing off of variance accounted for is called the scree. Note cumulative % of variance of rotated factors,Interpretation of Rotated Matrix,Loadings of .40 or higher Name each factor based on 3 or 4 variables with highest loadings. Do not expect perfect conceptual fit of all variables.,Loading size based on sample (from Hair et al 2010 Table 3-2),What else?,How do you know when the factor structure is good? When it makes sense and has a (relatively) simple and clean structur