E X P L O R E R. R e l e a s e A Program for Common Factor Analysis and Related Models for Data Analysis
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1 E X P L O R E R R e l e a s e 3. 2 A Program for Common Factor Analysis and Related Models for Data Analysis Copyright (c) , J. S. Fleming, PhD Date and Time : 23-Sep-2011, 13:59:56 Number of variables : 11 Command file : wiscr.txt Listing file : wiscr_out.lis L i s t i n g o f C o m m a n d F i l e <data> cases = 232; variables = 11; format = '(16x,11f4.0)'; matrix = raw; file = 'wiscr.dat'; <headers> ''; 'Learning Disabled Children (N = 232)'; ''; <analysis> method = minres; matrix = corr; trans = hc; com = smc; number = 3; wtfile='wiscr_wt.txt'; secord; nsec=1; <details> loval=.195; hival=.349; <labels> 'Info'; 'Similar'; 'Arith'; 'Verbal'; 'Compre'; 'Digit Sp'; 'Pict Cmp'; 'Pict Arr'; 'Blck Des'; 'Obj Assm'; 'Coding'; <plots> trans=3; <output> all; <end>
2 Learning Disabled Children (N = 232) Page 2 Univariate Statistics U n i v a r i a t e S u m m a r y S t a t i s t i c s No. Mean Std. Dev. Skewness Min. Val. Max. Val. Range Name No.Cases Variance Kurtosis Min. Z Max. Z Z-Range Coef.Var. S.E.Mean Skew/S.E. Kurt/S.E. Min.Case Max.Case Info Similar Arith Verbal Compre Digit Sp Pict Cmp Pict Arr Blck Des Obj Assm
3 Learning Disabled Children (N = 232) Page 3 Univariate Statistics No. Mean Std. Dev. Skewness Min. Val. Max. Val. Range Name No.Cases Variance Kurtosis Min. Z Max. Z Z-Range Coef.Var. S.E.Mean Skew/S.E. Kurt/S.E. Min.Case Max.Case Coding
4 Learning Disabled Children (N = 232) Page 4 Univariate Statistics M o s t E x t r e m e Z - S c o r e s No. Case Var. No. Variable Z-value Arith Digit Sp Pict Arr Pict Cmp Pict Cmp Info Info Pict Cmp Obj Assm Compre
5 Learning Disabled Children (N = 232) Page 5 Correlational Analysis C o r r e l a t i o n M a t r i x Info Similar Arith Verbal Compre 1 Info Similar Arith Verbal Compre Digit Sp Pict Cmp Pict Arr Blck Des Obj Assm Coding Digit Sp Pict Cmp Pict Arr Blck Des Obj Assm 6 Digit Sp Pict Cmp Pict Arr Blck Des Obj Assm Coding Coding 11 Coding 1.000
6 Learning Disabled Children (N = 232) Page 6 Latent Roots and Vectors E i g e n v a l u e s o f R No. Value % Trace Cum. % No. Value % Trace Cum. % R is the unaltered correlation matrix. Theta (reliability of first principal component) = Condition number (largest/smallest eigenvalue) = E+02 The trace of a matrix is the sum of it's diagonal elements. If the matrix is Gramian (no eigenvalues less than zero), then the trace represents the total variance. If some of the eigenvalues are negative then percentages are based on just non-negative elements. For this problem the positive trace is: Total iterations for eigenvalues was: 16.
7 E X P L O R E R Learning Disabled Children (N = 232) P a g e 7 E I G E N V A L U E P L O T !...!...!...!...!...!...!...!...!...!...!...!...!...!...!...!...! * E I G E N V A L U E S * -. * *.. *.. * * * -. * *.. * !...!...!...!...!...!...!...!...!...!...!...!...!...!...!...!...! N U M B E R O F F A C T O R S
8 Learning Disabled Children (N = 232) Page 8 Multivariate Statistics M u l t i v a r i a t e S u m m a r y S t a t i s t i c s Multivariate skewness = Approximate z = Multivariate kurtosis = Approximate z = Mean Mahalanobis distance = Generalized variance = E+09 Determinant of R = E-01 Bartlett's test for complete independence of correlations: Chi-square = Degrees of freedom = 55 Approximate z = (Note: Skewness and Kurtosis measures, discussed in Mardia (1972), are expressed as deviations from expected values. Maximum likelihood estimators are used in computing multivariate statistics.)
9 Learning Disabled Children (N = 232) Page 9 Multivariate Statistics M o s t E x t r e m e C a s e s Mahalanobis No. Case D-Square df D-Sq./df (Note: If multivariate normality holds, D-square is approximately distributed as chi-square in large samples.)
10 Learning Disabled Children (N = 232) Page 10 Partial/Multiple Correlation P a r t i a l C o r r e l a t i o n s Info Similar Arith Verbal Compre 1 Info Similar Arith Verbal Compre Digit Sp Pict Cmp Pict Arr Blck Des Obj Assm Coding Digit Sp Pict Cmp Pict Arr Blck Des Obj Assm 6 Digit Sp Pict Cmp Pict Arr Blck Des Obj Assm Coding Coding 11 Coding (Diagonal values are squared-multiple correlations. Off-diagonals are correlations between each variable pair, with all others partialled out.)
11 Learning Disabled Children (N = 232) Page 11 Partial/Multiple Correlation M e a s u r e s o f S a m p l i n g A d e q u a c y 1 Info Compre Blck Des Similar Digit Sp Obj Assm Arith Pict Cmp Coding Verbal Pict Arr Total MSA = 0.905, median = Kaiser-Myer-Olkin statistic = (Note: This is Kaiser's final (1981) version of the MSA. A value of means that the MSA is unacceptable. KMO is an earlier version of MSA)
12 Learning Disabled Children (N = 232) Page 12 Minres Factor Solution MINRES Iteration Summary Minimization Convergence Iteration Criterion Criterion
13 Learning Disabled Children (N = 232) Page 13 Latent Roots and Vectors F i n a l E i g e n v a l u e s o f R* No. Value % Trace Cum. % No. Value % Trace Cum. % R* is the 'reduced' correlation matrix. The trace of a matrix is the sum of it's diagonal elements. If the matrix is Gramian (no eigenvalues less than zero), then the trace represents the total variance. If some of the eigenvalues are negative then percentages are based on just non-negative elements. For this problem the positive trace is:
14 Learning Disabled Children (N = 232) Page 14 Initial Factor Solution C o m m u n a l i t i e s Difference Unique No. Name Initial Final (Initial - Final) Variances 1 Info Similar Arith Verbal Compre Digit Sp Pict Cmp Pict Arr Blck Des Obj Assm Coding Means: Medians:
15 Learning Disabled Children (N = 232) Page 15 Initial Factor Solution S u m m a r y o f F i t M e a s u r e s Method: Unweighted least-squares Significance test for number of factors: Large sample chi-square = , df = 36 Normal deviate = Moderate sample chi-square = Normal deviate = (Test based on Harman, 1976)
16 Learning Disabled Children (N = 232) Page 16 Initial Factor Solution I n i t i a l L o a d i n g s F 1 F 2 F 3 1 Info Similar Arith Verbal Compre Digit Sp Pict Cmp Pict Arr Blck Des Obj Assm Coding
17 Learning Disabled Children (N = 232) Page 17 Residuals Analysis R e s i d u a l C o r r e l a t i o n s (Diagonal contains unique variances) Info Similar Arith Verbal Compre 1 Info Similar Arith Verbal Compre Digit Sp Pict Cmp Pict Arr Blck Des Obj Assm Coding Digit Sp Pict Cmp Pict Arr Blck Des Obj Assm 6 Digit Sp Pict Cmp Pict Arr Blck Des Obj Assm Coding Coding 11 Coding (Residual correlations are remainders after subtracting reproduced correlations from original correlations. If a covariance matrix was factored then residuals were standardized.)
18 Learning Disabled Children (N = 232) Page 18 Residuals Analysis Distribution of Standardized Residuals (Based on sub-diagonal elements of correlation matrix): Interval Freq. Pct. H i s t o g r a m > ******************************** *********************** < (Note: Each symbol represents 1.00 correlations.) Residual Summary Statistics (for sub-diagonal elements): Minimum value = (between Pict Arr and Digit Sp) Maximum value = (between Pict Arr and Info ) Average (root-mean-square) = 0.031
19 Learning Disabled Children (N = 232) Page 19 Residuals Analysis Largest Residual for Each Variable Pair 1 Info with 8 Pict Arr = Similar with 8 Pict Arr = Arith with 5 Compre = Verbal with 7 Pict Cmp = Compre with 3 Arith = Digit Sp with 8 Pict Arr = Pict Cmp with 4 Verbal = Pict Arr with 6 Digit Sp = Blck Des with 8 Pict Arr = Obj Assm with 6 Digit Sp = Coding with 8 Pict Arr = 0.028
20 Learning Disabled Children (N = 232) Page 20 Initial Factor Solution F a c t o r S c o r e W e i g h t s F 1 F 2 F 3 1 Info Similar Arith Verbal Compre Digit Sp Pict Cmp Pict Arr Blck Des Obj Assm Coding
21 Learning Disabled Children (N = 232) Page 21 Initial Factor Solution C o v a r i a n c e o f S c o r e E s t i m a t e s F 1 F 2 F 3 F F F (Estimated score variances are on the diagonal. Covariances are below the diagonal and correlations are above.) (NOTE: The diagonal values are squared-multiple correlations of factors with variables.) (Factor scores weights are estimated by regression method.)
22 Learning Disabled Children (N = 232) Page 22 Oblique Primary Factors Oblique Transformation Summary: Method = Hypercon Power = Normalization = Yes
23 Learning Disabled Children (N = 232) Page 23 Oblique Primary Factors Row - N o r m a l i z e d L o a d i n g s F 1 F 2 F 3 1 Info Similar Arith Verbal Compre Digit Sp Pict Cmp Pict Arr Blck Des Obj Assm Coding
24 Learning Disabled Children (N = 232) Page 24 Oblique Primary Factors T a r g e t M a t r i x F 1 F 2 F 3 1 Info Similar Arith Verbal Compre Digit Sp Pict Cmp Pict Arr Blck Des Obj Assm Coding 0 0 1
25 Learning Disabled Children (N = 232) Page 25 Oblique Primary Factors O b l i q u e T r a n s f o r m a t i o n M a t r i x
26 Learning Disabled Children (N = 232) Page 26 Oblique Primary Factors H y p e r c o n F a c t o r P a t t e r n F 1 F 2 F 3 1 Info Similar Arith Verbal Compre Digit Sp Pict Cmp Pict Arr Blck Des Obj Assm Coding Sums of Squared Loadings: F 1 F 2 F
27 Learning Disabled Children (N = 232) Page 27 Oblique Primary Factors F a c t o r C o r r e l a t i o n s F 1 F 2 F 3 F F F
28 Learning Disabled Children (N = 232) Page 28 Oblique Primary Factors F a c t o r S t r u c t u r e F 1 F 2 F 3 1 Info Similar Arith Verbal Compre Digit Sp Pict Cmp Pict Arr Blck Des Obj Assm Coding Sums of Squared Loadings: F 1 F 2 F (Elements of the structure matrix are correlations of the variables with the factors. Column sums of squares are measures of the factor's 'total' contribution, irrespective of the other factors.)
29 Learning Disabled Children (N = 232) Page 29 Oblique Reference Factors R e f e r e n c e S t r u c t u r e F 1 F 2 F 3 1 Info Similar Arith Verbal Compre Digit Sp Pict Cmp Pict Arr Blck Des Obj Assm Coding Sums of Squared Loadings: F 1 F 2 F (Elements of the reference structure matrix are semipartial correlations. Column sums of squares are measures of the factor's 'unique' contribution to the total variance.)
30 Learning Disabled Children (N = 232) Page 30 Oblique Reference Factors F a c t o r L o a d i n g P l o t s F # A. 8. C.. 11 #. T O.. R F A C T O R 1 For oblique factors: correlation = 0.575; angle = deg. (Note: Entries are variable indexes. Pounds [#] denote overlapping values. Reference structure coefficients - proportional to factor pattern - are plotted.)
31 Learning Disabled Children (N = 232) Page 31 Oblique Reference Factors F a c t o r L o a d i n g P l o t s F A. 6. C T # O.. R F A C T O R 1 For oblique factors: correlation = 0.049; angle = deg. (Note: Entries are variable indexes. Pounds [#] denote overlapping values. Reference structure coefficients - proportional to factor pattern - are plotted.)
32 Learning Disabled Children (N = 232) Page 32 Oblique Reference Factors F a c t o r L o a d i n g P l o t s F A.. 6. C T # # O.. R F A C T O R 2 For oblique factors: correlation = 0.253; angle = deg. (Note: Entries are variable indexes. Pounds [#] denote overlapping values. Reference structure coefficients - proportional to factor pattern - are plotted.)
33 Learning Disabled Children (N = 232) Page 33 Oblique Primary Factors I n d e x e s o f F a c t o r S i m p l i c i t y 1 Info Compre Blck Des Similar Digit Sp Obj Assm Arith Pict Cmp Coding Verbal Pict Arr Total IFS = 0.941; median = Loadings in the interval (-.10, +.10) = 16 ( 48.5%) Loadings in the interval (-.15, +.15) = 20 ( 60.6%) Loadings in the interval (-.20, +.20) = 20 ( 60.6%) Hyperplane count (Kaiser-Cerny, 1978) = 20 ( 60.6%)
34 Learning Disabled Children (N = 232) Page 34 Oblique Primary Factors S o r t e d L o a d i n g s F 1 F 2 F 3 5 Compre Info Similar Verbal Arith Digit Sp Obj Assm Blck Des Pict Cmp Pict Arr Coding (Note: Values above were considered highly salient; values below were set to zero.)
35 Learning Disabled Children (N = 232) Page 35 Factor Scale and Item Analysis F a c t o r S c a l e S u m m a r y S t a t i s t i c s Factor scales are defined by items with salient loadings above Scale fit indexes are: SFI = 1 - SSQ(nonsalient)/SSQ(total). Average (RMS) of scale items: Average (RMS) of nonscale items: Scale fit indexes: F 1 F 2 F F 1 F 2 F F 1 F 2 F Overall Scale Fit Index = Bentler's Matrix Simplicity Measure = Lorenzo-Seva's Factor Simplicity Index = 0.510
36 Learning Disabled Children (N = 232) Page 36 Factor Scale and Item Analysis F a c t o r A n a l y s i s o f S c a l e I t e m s Scale based on Factor 1 Commun- Unique- No. Name F 1 ality ness 1 Info Similar Arith Verbal Compre Digit Sp Sum of squares: McDonald's omega: Total iterations = 3
37 Learning Disabled Children (N = 232) Page 37 Factor Scale and Item Analysis F a c t o r A n a l y s i s o f S c a l e I t e m s Scale based on Factor 2 Commun- Unique- No. Name F 2 ality ness 7 Pict Cmp Pict Arr Blck Des Obj Assm Sum of squares: McDonald's omega: Total iterations = 3
38 Learning Disabled Children (N = 232) Page 38 Factor Scale and Item Analysis F a c t o r S c a l e C o r r e l a t i o n s F 1 F 2 F 3 F F F (Note: These correlations are for scales formed by simple summation of the items which define them. If no values define a scale, then is printed)
39 Learning Disabled Children (N = 232) Page 39 Factor Scale and Item Analysis Coefficient alpha for scale items (standardized): F 1 F 2 F Coefficient alpha for scale items (raw): F 1 F 2 F (Note: alpha cannot be computed for scales with less than two items. In this case, a value of zero is printed.)
40 Learning Disabled Children (N = 232) Page 40 Factor Scale and Item Analysis I t e m A n a l y s i s f o r F a c t o r S c a l e s Scale Based on Factor 1: Item-Wtd. Total Alpha without item: No. Name Correlation Raw Standardized 1 Info Similar Arith Verbal Compre Digit Sp Mean item-total r = Standardized alpha= Standard error = Large sample Z = % C.I. = ( 0.797, 0.865)
41 Learning Disabled Children (N = 232) Page 41 Factor Scale and Item Analysis I t e m A n a l y s i s f o r F a c t o r S c a l e s Scale Based on Factor 2: Item-Wtd. Total Alpha without item: No. Name Correlation Raw Standardized 7 Pict Cmp Pict Arr Blck Des Obj Assm Mean item-total r = Standardized alpha= Standard error = Large sample Z = % C.I. = ( 0.621, 0.753)
42 Learning Disabled Children (N = 232) Page 42 Oblique Primary Factors F a c t o r S c o r e W e i g h t s F 1 F 2 F 3 1 Info Similar Arith Verbal Compre Digit Sp Pict Cmp Pict Arr Blck Des Obj Assm Coding
43 Learning Disabled Children (N = 232) Page 43 Oblique Primary Factors C o v a r i a n c e o f S c o r e E s t i m a t e s F 1 F 2 F 3 F F F (Estimated score variances are on the diagonal. Covariances are below the diagonal and correlations are above.) (NOTE: The diagonal values are squared-multiple correlations of factors with variables.) (Factor scores weights are estimated by regression method.)
44 Learning Disabled Children (N = 232) Page 44 Second-Order Analysis Second-order Factor Analysis Summary: Model: Common Factor Analysis Number of factors: 1
45 Learning Disabled Children (N = 232) Page 45 Latent Roots and Vectors E i g e n v a l u e s o f R No. Value % Trace Cum. % No. Value % Trace Cum. % R is the first-order factor correlation matrix. The trace of a matrix is the sum of it's diagonal elements. If the matrix is Gramian (no eigenvalues less than zero), then the trace represents the total variance. If some of the eigenvalues are negative then percentages are based on just non-negative elements. For this problem the positive trace is: Total iterations for eigenvalues was: 6.
46 E X P L O R E R Learning Disabled Children (N = 232) P a g e 46 E I G E N V A L U E P L O T !...!...!...!...!...!...!...!...!...!...!...!...!...!...!...!...! E I. *. G E N V A L U. *. E S * !...!...!...!...!...!...!...!...!...!...!...!...!...!...!...!...! N U M B E R O F F A C T O R S
47 Learning Disabled Children (N = 232) Page 47 Latent Roots and Vectors I n i t i a l E i g e n v a l u e s o f R* No. Value % Trace Cum. % No. Value % Trace Cum. % R* is the 'reduced' correlation matrix. The trace of a matrix is the sum of it's diagonal elements. If the matrix is Gramian (no eigenvalues less than zero), then the trace represents the total variance. If some of the eigenvalues are negative then percentages are based on just non-negative elements. For this problem the positive trace is: *** Warning: Negative eigenvalues found. Matrix is non-grammian.
48 Learning Disabled Children (N = 232) Page 48 Initial Factor Solution Common factor extraction summary: Method = Principal Axes Iteration Summary: Iteration Criterion Pct. Meeting Criterion *** Warning - Improper value for communality on iteration 14. Communality for item 2 was: E+01. The solution for the previous iteration was instead retained.
49 Learning Disabled Children (N = 232) Page 49 Latent Roots and Vectors F i n a l E i g e n v a l u e s o f R* No. Value % Trace Cum. % No. Value % Trace Cum. % R* is the 'reduced' correlation matrix. The trace of a matrix is the sum of it's diagonal elements. If the matrix is Gramian (no eigenvalues less than zero), then the trace represents the total variance. If some of the eigenvalues are negative then percentages are based on just non-negative elements. For this problem the positive trace is:
50 Learning Disabled Children (N = 232) Page 50 Second-Order Analysis S e c o n d - O r d e r F a c t o r P a t t e r n G 1 1 Factor Factor Factor (NOTE: Rows are first-order factors. G1, G2, etc., are second-order factors.)
51 Learning Disabled Children (N = 232) Page 51 Communalities C o m m u n a l i t i e s Difference Unique No. Name Initial Final (Initial - Final) Variances 1 Factor Factor Factor Means: Medians:
52 Learning Disabled Children (N = 232) Page 52 Second-Order Analysis S c h m i d - L e i m a n F a c t o r P a t t e r n G 1 F 1 F 2 F 3 1 Info Similar Arith Verbal Compre Digit Sp Pict Cmp Pict Arr Blck Des Obj Assm Coding Sums of Squared Loadings: Proportion of variance: Reliabilities based on all items: McDonald's omega (higher-order) = McDonald's omega (total) = Alpha (higher-order) = Alpha (total) = 0.766
53 Learning Disabled Children (N = 232) Page 53 Residuals Analysis R e s i d u a l C o r r e l a t i o n s (Diagonal contains unique variances) F 1 F 2 F 3 F F F (Residual correlations are remainders after subtracting reproduced correlations from original correlations. If a covariance matrix was factored then residuals were standardized.)
54 Learning Disabled Children (N = 232) Page 54 Residuals Analysis Distribution of Standardized Residuals (Based on sub-diagonal elements of correlation matrix): Interval Freq. Pct. H i s t o g r a m > **************** ******** < (Note: Each symbol represents 0.12 correlations.) Residual Summary Statistics (for sub-diagonal elements): Minimum value = (between Factor 3 and Factor 1) Maximum value = (between Factor 3 and Factor 2) Average (root-mean-square) = 0.042
55 Learning Disabled Children (N = 232) Page 55 Residuals Analysis Largest Residual for Each Variable Pair 1 Factor 1 with 3 Factor 3 = Factor 2 with 3 Factor 3 = Factor 3 with 1 Factor 1 =
L i s t i n g o f C o m m a n d F i l e
E X P L O R E R R e l e a s e 3. 0 A Program for Common Factor Analysis and Related Models Copyright (c) 1990-2011, J. S. Fleming, PhD Date and Time : 12-Feb-2011, 10:30:27 Number of variables : 5 Command
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