Factor Analysis Model
That is,
where
Z -> (nxm) standard score matrix
A -> (mxp) factor pattern matrix
F -> (nxp) factor score matrix
Factor Pattern
The factor pattern matrix, A, is the matrix of coefficients which applied to the factor scores reproduces the standard score matrix.
Factor Structure
S -> (mxp) factor structure matrix
The factor structure matrix, S, is the matrix of correlations between factors and variables. Sometimes called the loading matrix. With orthogonal solutions the structure matrix and pattern matrix are the same.
Factor Correlations
Φ -> (pxp) factor correlation matrix
If the factor analysis solution is orthogonal then Φ = I.
Fun with Math
Substituting into [2]
we get
thus, when Φ = I, S = AI or S = A.
Reproduced Correlations
Variance to be Factored
Total variance = hj2 + bj2 + ej2 Reliability = hj2 + bj2 Communality hj2 = hj2 Uniqueness dj2 = bj2 + ej2 Specificity bj2 = bj2 Error ej2 = ej2
PCA vs FA
The Concept of Simple Structure
Simple Structure Example
Initial Rotated
Solution Solution
I II III F1 F2 F3
var1 X X 0 0 0 X
var2 X X 0 0 0 X
var3 X X 0 0 0 X
var4 X -X 0 0 X 0
var5 X -X 0 0 X 0
var6 X 0 X X 0 0
var7 X 0 -X X 0 0
var8 X 0 -X X 0 0
G B B P P P
Decisions in Factor Analysis
Method of Initial Factor Solution
Estimation of Communalities
Number of Factors to Retain
Scree Plot
Methods of Rotation
Rotating Example
Unrotated Factor Solution
Orthogonal Factor Solution
Oblique Factor Solution
The following example uses data for five socio-economic variables for 12 different locations. the variables are total population, median schooling, total employed, misc. professional services, and median housing value. The data are from Harman (1976).
Sample Size for Factor Analysis
There are a number of different guidelines given in the literature as to the appropriate sample size needed for factor analysis. I was taught that you needed at least 10 times as many observations as variables with a minimum of 200 observations. Pedhazur & Schmelkin (1991) suggest at least 50 observations per factor. Guadagnoli and Velicer (1988) have suggested a minimum sample size of 100 to 200 observations. Tabachnick & Fidell (1996) recommend at least 300 cases. And Comrey and Lee (1992) give the following guide for samples sizes: 50 as very poor, 100 as poor, 200 as fair, 300 as good, 500 as very good, and 1,000 as excellent.
Just remember, as with all statistical rules of thumb, your milage may vary.
Principal Axis Factor Analysis
use http://www.gseis.ucla.edu/courses/data/harman1, clear
factor pop medsch employ profser medhouse, pf fac(2)
(obs=12)
(principal factors; 2 factors retained)
Factor Eigenvalue Difference Proportion Cumulative
------------------------------------------------------------------
1 2.73430 1.01823 0.6225 0.6225
2 1.71607 1.67651 0.3907 1.0131
3 0.03956 0.06409 0.0090 1.0221
4 -0.02452 0.04808 -0.0056 1.0165
5 -0.07261 . -0.0165 1.0000
Factor Loadings
Variable | 1 2 Uniqueness
----------+--------------------------------
pop | 0.62533 0.76621 0.02189
medsch | 0.71370 -0.55515 0.18244
employ | 0.71447 0.67936 0.02800
profser | 0.87899 -0.15846 0.20226
medhouse | 0.74215 -0.57806 0.11505
mat psi = e(Psi)'
mat com = J(rowsof(psi),1,1)
mat com = com - psi
mat colnames com=communalities
mat list com
com[5,1]
communalities
pop .97811334
medsch .81756393
employ .97199928
profser .79774303
medhouse .88495002
rotate
(varimax rotation)
Rotated Factor Loadings
Variable | 1 2 Uniqueness
----------+--------------------------------
pop | 0.01319 0.98891 0.02189
medsch | 0.90415 0.00911 0.18244
employ | 0.13701 0.97633 0.02800
profser | 0.78689 0.42255 0.20226
medhouse | 0.94068 0.00887 0.11505
rotate, promax(2)
(promax rotation)
Rotated Factor Loadings
Variable | 1 2 Uniqueness
-------------+--------------------------------
pop | -0.06530 0.99749 0.02189
medsch | 0.91274 -0.06808 0.18244
employ | 0.06080 0.97421 0.02800
profser | 0.76140 0.35944 0.20226
medhouse | 0.94966 -0.07145 0.11505
estat common
Correlation matrix of the promax(3) rotated common factors
----------------------------------
Factors | Factor1 Factor2
-------------+--------------------
Factor1 | 1
Factor2 | .1623 1
----------------------------------
rotate, promax(3)
(promax rotation)
Rotated Factor Loadings
Variable | 1 2 Uniqueness
----------+--------------------------------
pop | -0.09090 1.00505 0.02189
medsch | 0.92105 -0.10087 0.18244
employ | 0.03670 0.97716 0.02800
profser | 0.75785 0.33428 0.20226
medhouse | 0.95833 -0.10556 0.11505
estat common
Correlation matrix of the promax(3) rotated common factors
----------------------------------
Factors | Factor1 Factor2
-------------+--------------------
Factor1 | 1
Factor2 | .2204 1
----------------------------------
rotate, promax(4)
(promax rotation)
Rotated Factor Loadings
Variable | 1 2 Uniqueness
-------------+--------------------------------
pop | -0.10777 1.01050 0.02189
medsch | 0.92625 -0.11581 0.18244
employ | 0.02078 0.98049 0.02800
profser | 0.75527 0.32365 0.20226
medhouse | 0.96375 -0.12112 0.11505
estat common
Correlation matrix of the promax(4) rotated common factors
----------------------------------
Factors | Factor1 Factor2
-------------+--------------------
Factor1 | 1
Factor2 | .2507 1
----------------------------------
The correlation between the factors increases as the promax power increases. promax(1) is the same as varimax. promax(3) is the default. Powers between 2 and 4 are recommended and non-integer powers can be used. You can try several different powers to get the solution that is most interpretable. In this example, the interpretations are all about the same.
Remember, with oblique rotations you can get loadings greater than one.
Iterated Principal Axis Factor Analysis
factor pop medsch employ profser medhouse, ipf fac(2)
(obs=12)
(iterated principal factors; 2 factors retained)
Factor Eigenvalue Difference Proportion Cumulative
------------------------------------------------------------------
1 2.75653 1.01187 0.6124 0.6124
2 1.74466 1.71387 0.3876 1.0000
3 0.03079 0.03118 0.0068 1.0068
4 -0.00039 0.03002 -0.0001 1.0068
5 -0.03041 . -0.0068 1.0000
Factor Loadings
Variable | 1 2 Uniqueness
----------+--------------------------------
pop | 0.63002 0.79452 -0.02819
medsch | 0.70064 -0.52408 0.23444
employ | 0.69731 0.67102 0.06349
profser | 0.88077 -0.14704 0.20262
medhouse | 0.77891 -0.60568 0.02645
mat psi = e(Psi)'
mat com = J(rowsof(psi),1,1)
mat com = com - psi
mat colnames com=communalities
mat list com
com[5,1]
communalities
pop 1.0281865
medsch .76556374
employ .93651122
profser .7973836
medhouse .97355023
rotate
(varimax rotation)
Rotated Factor Loadings
Variable | 1 2 Uniqueness
----------+--------------------------------
pop | 0.01056 1.01394 -0.02819
medsch | 0.87483 0.01557 0.23444
employ | 0.13944 0.95764 0.06349
profser | 0.78595 0.42387 0.20262
medhouse | 0.98669 -0.00090 0.02645
rotate, promax(3)
(promax rotation)
Rotated Factor Loadings
Variable | 1 2 Uniqueness
----------+--------------------------------
pop | -0.09906 1.03143 -0.02819
medsch | 0.89062 -0.09031 0.23444
employ | 0.03850 0.95844 0.06349
profser | 0.75574 0.33634 0.20262
medhouse | 1.00650 -0.12066 0.02645
estat common
Correlation matrix of the promax(3) rotated common factors
----------------------------------
Factors | Factor1 Factor2
-------------+--------------------
Factor1 | 1
Factor2 | .2225 1
----------------------------------
Maximum Likelihood Factor Analysis
factor pop medsch employ profser medhouse, ml fac(2)
(obs=12)
Factor analysis/correlation Number of obs = 12
Method: maximum likelihood Retained factors = 2
Rotation: (unrotated) Number of params = 9
Schwarz's BIC = 26.0449
Log likelihood = -1.84039 (Akaike's) AIC = 21.6808
Beware: solution is a Heywood case
(i.e., invalid or boundary values of uniqueness)
--------------------------------------------------------------------------
Factor | Eigenvalue Difference Proportion Cumulative
-------------+------------------------------------------------------------
Factor1 | 2.13887 -0.22952 0.4745 0.4745
Factor2 | 2.36839 . 0.5255 1.0000
--------------------------------------------------------------------------
LR test: independent vs. saturated: chi2(10) = 60.63 Prob>chi2 = 0.0000
LR test: 2 factors vs. saturated: chi2(1) = 2.50 Prob>chi2 = 0.1135
(tests formally not valid because a Heywood case was encountered)
Factor loadings (pattern matrix) and unique variances
-------------------------------------------------
Variable | Factor1 Factor2 | Uniqueness
-------------+--------------------+--------------
pop | 1.0000 -0.0000 | 0.0000
medsch | 0.0098 0.9000 | 0.1900
employ | 0.9725 0.1179 | 0.0404
profser | 0.4389 0.7892 | 0.1844
medhouse | 0.0224 0.9600 | 0.0779
-------------------------------------------------
faform /* Available from ATS via the Internet */
Factor Loadings in Canonical Form
1 2
pop 0.62151 0.78340
medsch 0.71109 -0.55170
employ 0.69679 0.68853
profser 0.89106 -0.14672
medhouse 0.76602 -0.57911
mat psi = e(Psi)'
mat com = J(rowsof(psi),1,1)
mat com = com - psi
mat colnames com=communalities
mat list com
com[5,1]
communalities
pop .99999969
medsch .81003767
employ .95956448
profser .81555395
medhouse .92206406
rotate
Factor analysis/correlation Number of obs = 12
Method: maximum likelihood Retained factors = 2
Rotation: orthogonal varimax (Horst off) Number of params = 9
Schwarz's BIC = 26.0449
Log likelihood = -1.84039 (Akaike's) AIC = 21.6808
Beware: solution is a Heywood case
(i.e., invalid or boundary values of uniqueness)
--------------------------------------------------------------------------
Factor | Variance Difference Proportion Cumulative
-------------+------------------------------------------------------------
Factor1 | 2.38260 0.25793 0.5286 0.5286
Factor2 | 2.12467 . 0.4714 1.0000
--------------------------------------------------------------------------
LR test: independent vs. saturated: chi2(10) = 60.63 Prob>chi2 = 0.0000
LR test: 2 factors vs. saturated: chi2(1) = 2.50 Prob>chi2 = 0.1135
(tests formally not valid because a Heywood case was encountered)
Rotated factor loadings (pattern matrix) and unique variances
-------------------------------------------------
Variable | Factor1 Factor2 | Uniqueness
-------------+--------------------+--------------
pop | 0.0145 0.9999 | 0.0000
medsch | 0.9000 -0.0033 | 0.1900
employ | 0.1320 0.9706 | 0.0404
profser | 0.7955 0.4274 | 0.1844
medhouse | 0.9602 0.0085 | 0.0779
-------------------------------------------------
Factor rotation matrix
--------------------------------
| Factor1 Factor2
-------------+------------------
Factor1 | 0.0145
Factor2 | 0.9999 -0.0145
--------------------------------
rotate, promax(3)
Factor analysis/correlation Number of obs = 12
Method: maximum likelihood Retained factors = 2
Rotation: oblique promax (Horst off) Number of params = 9
Schwarz's BIC = 26.0449
Log likelihood = -1.84039 (Akaike's) AIC = 21.6808
Beware: solution is a Heywood case
(i.e., invalid or boundary values of uniqueness)
--------------------------------------------------------------------------
Factor | Variance Proportion Rotated factors are correlated
-------------+------------------------------------------------------------
Factor1 | 2.49044 0.5525
Factor2 | 2.22530 0.4937
--------------------------------------------------------------------------
LR test: independent vs. saturated: chi2(10) = 60.63 Prob>chi2 = 0.0000
LR test: 2 factors vs. saturated: chi2(1) = 2.50 Prob>chi2 = 0.1135
(tests formally not valid because a Heywood case was encountered)
Rotated factor loadings (pattern matrix) and unique variances
-------------------------------------------------
Variable | Factor1 Factor2 | Uniqueness
-------------+--------------------+--------------
pop | -0.0886 1.0153 | 0.0000
medsch | 0.9171 -0.1091 | 0.1900
employ | 0.0341 0.9717 | 0.0404
profser | 0.7662 0.3412 | 0.1844
medhouse | 0.9772 -0.1042 | 0.0779
-------------------------------------------------
Factor rotation matrix
--------------------------------
| Factor1 Factor2
-------------+------------------
Factor1 | 0.1292 0.9963
Factor2 | 0.9916 0.0865
--------------------------------
estat common
Correlation matrix of the promax(3) rotated common factors
----------------------------------
Factors | Factor1 Factor2
-------------+--------------------
Factor1 | 1
Factor2 | .2145 1
----------------------------------
How to Do It
Types of Factor Analysis
use http://www.gseis.ucla.edu/courses/data/api99g
keep if stype==1 /* use only elementary schools */
(1773 observations deleted)
summarize meals ell yr_rnd acs_k3 acs_46 avg_ed full enroll, clear
Variable | Obs Mean Std. Dev. Min Max
-------------+-----------------------------------------------------
meals | 4421 51.88102 31.07313 0 100
ell | 4421 25.19204 22.91157 0 95
yr_rnd | 4421 1.178919 .3833277 1 2
acs_k3 | 4359 19.29571 1.539583 12 31
acs_46 | 4294 28.90452 3.21889 14 50
avg_ed | 4257 2.749298 .7542556 1 5
full | 4420 87.86357 13.35186 13 100
enroll | 4397 426.9616 175.8747 101 1570
univar meals ell yr_rnd acs_k3 acs_46 avg_ed full enroll
-------------- Quantiles --------------
Variable n Mean S.D. Min .25 Mdn .75 Max
-------------------------------------------------------------------------------
meals 4421 51.88 31.07 0.00 24.00 53.00 79.00 100.00
ell 4421 25.19 22.91 0.00 6.00 18.00 40.00 95.00
yr_rnd 4421 1.18 0.38 1.00 1.00 1.00 1.00 2.00
acs_k3 4359 19.30 1.54 12.00 19.00 19.00 20.00 31.00
acs_46 4294 28.90 3.22 14.00 27.00 29.00 31.00 50.00
avg_ed 4257 2.75 0.75 1.00 2.17 2.71 3.26 5.00
full 4420 87.86 13.35 13.00 81.00 92.00 100.00 100.00
enroll 4397 426.96 175.87 101.00 303.00 403.00 523.00 1570.00
-------------------------------------------------------------------------------
corr meals ell yr_rnd acs_k3 acs_46 avg_ed full enroll
(obs=4059)
| meals ell yr_rnd acs_k3 acs_46 avg_ed full enroll
-------------+------------------------------------------------------------------------
meals | 1.0000
ell | 0.7716 1.0000
yr_rnd | 0.3027 0.3158 1.0000
acs_k3 | -0.0251 0.0275 0.0016 1.0000
acs_46 | -0.0274 0.0077 0.0522 0.2788 1.0000
avg_ed | -0.8392 -0.6818 -0.2842 -0.0193 0.0288 1.0000
full | -0.5145 -0.5146 -0.2592 0.0344 -0.0304 0.4036 1.0000
enroll | 0.1984 0.3092 0.5125 0.1374 0.2017 -0.1645 -0.2696 1.0000
factor meals ell yr_rnd acs_k3 acs_46 avg_ed full enroll, pf
(obs=4059)
Factor analysis/correlation Number of obs = 4059
Method: principal factors Retained factors = 4
Rotation: (unrotated) Number of params = 26
--------------------------------------------------------------------------
Factor | Eigenvalue Difference Proportion Cumulative
-------------+------------------------------------------------------------
Factor1 | 2.82884 2.08409 0.8482 0.8482
Factor2 | 0.74475 0.44056 0.2233 1.0715
Factor3 | 0.30419 0.23495 0.0912 1.1627
Factor4 | 0.06924 0.14128 0.0208 1.1834
Factor5 | -0.07204 0.04088 -0.0216 1.1618
Factor6 | -0.11292 0.07997 -0.0339 1.1280
Factor7 | -0.19289 0.04100 -0.0578 1.0701
Factor8 | -0.23389 . -0.0701 1.0000
--------------------------------------------------------------------------
LR test: independent vs. saturated: chi2(28) = 1.3e+04 Prob>chi2 = 0.0000
Factor loadings (pattern matrix) and unique variances
---------------------------------------------------------------------
Variable | Factor1 Factor2 Factor3 Factor4 | Uniqueness
-------------+----------------------------------------+--------------
meals | 0.8969 -0.2277 0.0614 0.0141 | 0.1398
ell | 0.8240 -0.0428 0.0292 -0.0927 | 0.3097
yr_rnd | 0.4422 0.3837 -0.2378 0.0969 | 0.5913
acs_k3 | 0.0214 0.2651 0.3521 0.0123 | 0.8051
acs_46 | 0.0303 0.3445 0.2947 -0.0304 | 0.7926
avg_ed | -0.8195 0.2249 -0.1193 -0.1402 | 0.2440
full | -0.5728 -0.0453 0.0832 0.1734 | 0.6328
enroll | 0.3856 0.5497 -0.1052 0.0157 | 0.5378
---------------------------------------------------------------------
/* don't display small loadings */
factor meals ell yr_rnd acs_k3 acs_46 avg_ed full enroll, pf blanks(.35)
(obs=4059)
Factor analysis/correlation Number of obs = 4059
Method: principal factors Retained factors = 4
Rotation: (unrotated) Number of params = 26
--------------------------------------------------------------------------
Factor | Eigenvalue Difference Proportion Cumulative
-------------+------------------------------------------------------------
Factor1 | 2.82884 2.08409 0.8482 0.8482
Factor2 | 0.74475 0.44056 0.2233 1.0715
Factor3 | 0.30419 0.23495 0.0912 1.1627
Factor4 | 0.06924 0.14128 0.0208 1.1834
Factor5 | -0.07204 0.04088 -0.0216 1.1618
Factor6 | -0.11292 0.07997 -0.0339 1.1280
Factor7 | -0.19289 0.04100 -0.0578 1.0701
Factor8 | -0.23389 . -0.0701 1.0000
--------------------------------------------------------------------------
LR test: independent vs. saturated: chi2(28) = 1.3e+04 Prob>chi2 = 0.0000
Factor loadings (pattern matrix) and unique variances
---------------------------------------------------------------------
Variable | Factor1 Factor2 Factor3 Factor4 | Uniqueness
-------------+----------------------------------------+--------------
meals | 0.8969 | 0.1398
ell | 0.8240 | 0.3097
yr_rnd | 0.4422 0.3837 | 0.5913
acs_k3 | 0.3521 | 0.8051
acs_46 | | 0.7926
avg_ed | -0.8195 | 0.2440
full | -0.5728 | 0.6328
enroll | 0.3856 0.5497 | 0.5378
---------------------------------------------------------------------
(blanks represent abs(loading)<.35)
/* parallel analysis for eigenvalues
compare the eigenvalues of the factor analysis
with eigenvalues of randomly generated variables
to assist in determing the number of factors.
*/
fapara, seed(123456789) /* Available from ATS via the Internet */
(obs=4421)
Parallel Analysis for Eigenvalues
Eigen Random Dif
c1 2.8288 0.0646 2.7642
c2 0.7448 0.0407 0.7041
c3 0.3042 0.0257 0.2785
c4 0.0692 0.0202 0.0490
c5 -0.0720 -0.0167 -0.0553
c6 -0.1129 -0.0338 -0.0792
c7 -0.1929 -0.0417 -0.1512
c8 -0.2339 -0.0468 -0.1870
rotate, varimax fact(3) blanks(.35)
Factor analysis/correlation Number of obs = 4059
Method: principal factors Retained factors = 4
Rotation: orthogonal varimax (Horst off) Number of params = 26
--------------------------------------------------------------------------
Factor | Variance Difference Proportion Cumulative
-------------+------------------------------------------------------------
Factor1 | 2.57781 1.66607 0.7729 0.7729
Factor2 | 0.91173 0.52349 0.2734 1.0463
Factor3 | 0.38825 0.31901 0.1164 1.1627
Factor4 | 0.06924 . 0.0208 1.1834
--------------------------------------------------------------------------
LR test: independent vs. saturated: chi2(28) = 1.3e+04 Prob>chi2 = 0.0000
Rotated factor loadings (pattern matrix) and unique variances
---------------------------------------------------------------------
Variable | Factor1 Factor2 Factor3 Factor4 | Uniqueness
-------------+----------------------------------------+--------------
meals | 0.9226 | 0.1398
ell | 0.7920 | 0.3097
yr_rnd | 0.5731 | 0.5913
acs_k3 | 0.4327 | 0.8051
acs_46 | 0.4158 | 0.7926
avg_ed | -0.8570 | 0.2440
full | -0.5128 | 0.6328
enroll | 0.6391 | 0.5378
---------------------------------------------------------------------
(blanks represent abs(loading)<.35)
Factor rotation matrix
--------------------------------------------------
| Factor1 Factor2 Factor3 Factor4
-------------+------------------------------------
Factor1 | 0.9400 0.3410 0.0119 0.0000
Factor2 | -0.3117 0.8443 0.4359 0.0000
Factor3 | 0.1386 -0.4134 0.8999 0.0000
Factor4 | 0.0000 0.0000 0.0000 1.0000
--------------------------------------------------
rotate, promax fact(3) blanks(.35)
Factor analysis/correlation Number of obs = 4059
Method: principal factors Retained factors = 4
Rotation: oblique promax (Horst off) Number of params = 26
--------------------------------------------------------------------------
Factor | Variance Proportion Rotated factors are correlated
-------------+------------------------------------------------------------
Factor1 | 2.74173 0.8220
Factor2 | 1.66111 0.4980
Factor3 | 0.48782 0.1463
Factor4 | 0.06924 0.0208
--------------------------------------------------------------------------
LR test: independent vs. saturated: chi2(28) = 1.3e+04 Prob>chi2 = 0.0000
Rotated factor loadings (pattern matrix) and unique variances
---------------------------------------------------------------------
Variable | Factor1 Factor2 Factor3 Factor4 | Uniqueness
-------------+----------------------------------------+--------------
meals | 0.9492 | 0.1398
ell | 0.7535 | 0.3097
yr_rnd | 0.6371 | 0.5913
acs_k3 | 0.4472 | 0.8051
acs_46 | 0.4222 | 0.7926
avg_ed | -0.9137 | 0.2440
full | -0.4183 | 0.6328
enroll | 0.6794 | 0.5378
---------------------------------------------------------------------
(blanks represent abs(loading)<.35)
Factor rotation matrix
--------------------------------------------------
| Factor1 Factor2 Factor3 Factor4
-------------+------------------------------------
Factor1 | 0.9793 0.6754 -0.0530 0.0000
Factor2 | -0.1914 0.6827 0.6330 0.0000
Factor3 | 0.0652 -0.2789 0.7723 0.0000
Factor4 | 0.0000 0.0000 0.0000 1.0000
--------------------------------------------------
predict f1 f2 f3
(regression scoring assumed)
Scoring coefficients (method = regression; based on promax(3) rotated factors)
------------------------------------------------------
Variable | Factor1 Factor2 Factor3 Factor4
-------------+----------------------------------------
meals | 0.52599 0.06864 -0.20729 -0.07627
ell | 0.19849 0.20361 0.03846 -0.27115
yr_rnd | 0.02016 0.32112 -0.03006 0.11481
acs_k3 | -0.00472 0.02699 0.31607 0.00345
acs_46 | -0.00596 0.06858 0.31676 -0.02782
avg_ed | -0.23978 0.03342 -0.05150 -0.43295
full | -0.06266 -0.12827 0.04101 0.21073
enroll | 0.03506 0.37733 0.18618 0.04654
------------------------------------------------------
corr f1 f2 f3
(obs=4059)
| f1 f2 f3
-------------+---------------------------
f1 | 1.0000
f2 | 0.6289 1.0000
f3 | -0.1909 0.2558 1.0000
The three factors can be interpreted as follows. Factor 1 seems to reflect socioeconomic variables.
Factor 2 appears to be related to the size of the population in the school neighborhoods, while
Factor 3 is concerned with classroom size.Stata 9 allows for the following methods for initial factor extraction:
pf principal-axis factor analysis; the default
pcf principal-components factor analysis
ipf iterated principal-axis factor analysis
ml maximum-likelihood factor analysis
The following options are allowed with the factor command: factors(#) maximum number of factors to be retained
mineigen(#) minimum value of eigenvalues to be retained
citerate(#) communality re-estimation iterations (ipf only)
The factor commands has the following post-estimation procedures: estat anti anti-image correlation and covariance matrices
estat common correlation matrix of the common factors
estat factors AIC and BIC model selection criteria for different numbers of
factors
estat kmo Kaiser-Meyer-Olkin measure of sampling adequacy
estat residuals matrix of correlation residuals
estat rotatecompare compare rotated and unrotated loadings
estat smc squared multiple correlations between each variable and the rest
estat structure correlations between variables and common factors
estat summarize estimation sample summary
loadingplot plot factor loadings
rotate rotate factor loadings
scoreplot plot score variables
screeplot plot eigenvalues
The following factor rotation procedures are available in Stata 9 using the rotate command: varimax varimax (orthogonal only); the default
vgpf varimax via the GPF algorithm (orthogonal only)
quartimax quartimax (orthogonal only)
equamax equamax (orthogonal only)
parsimax parsimax (orthogonal only)
entropy minimum entropy (orthogonal only)
tandem1 Comrey's tandem 1 principle (orthogonal only)
tandem2 Comrey's tandem 2 principle (orthogonal only)
promax[(#)] promax power # (implies oblique); default is promax(3)
oblimin[(#)] oblimin with gamma=#; default is oblimin(0)
cf(#) Crawford-Ferguson family with kappa=#, 0<=#<1
bentler Bentler's invariant pattern simplicity
oblimax oblimax
quartimin quartimin
target(Tg) rotate towards matrix Tg
partial(Tg W) rotate towards matrix Tg, weighted by matrix W
The rotate command has the following options: orthogonal restrict to orthogonal rotations; default, except with promax()
oblique allow oblique rotations
rotation_methods rotation criterion
normalize rotate Horst normalized matrix
horst synonym for normalize
factors(#) rotate # factors or components; default all
components(#) synonym for factors()
Ed231A Page
UCLA Department of Education
Phil Ender, 16nov05, 15oct05, 29Jan98