Inter Item Correlation Matrix (R )

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1 7 1. I have the ability to influence my child s well-being. 2. Whether my child avoids injury is just a matter of luck. 3. Luck plays a big part in determining how healthy my child is. 4. I can do a lot to prevent my child from getting hurt. 5. I can do a lot to prevent my child from getting sick. 6. Whether my child avoids sickness is just a matter of luck. Inter Item Correlation Matrix (R ) Parents Health Belief Scale 7. The things I do at home with my child are an important part of my child s well-being. 8. My child s safety depends on me. 9. I can do a lot to help my child stay well. 10. My child s good health is largely a matter of good fortune. 11. I can do a lot to help my child be strong and healthy. 12. Whether my child stays healthy or gets sick is just a matter of fate. Item Item Item Item Item Item Item Item Item Item Item Item Cronbach s Alpha = 09 Eigenvalues from Factor Analysis of Inter Item Correlation Matrix. Ordinal % Total Cumulative Position Eigenvalues Difference Variance % Variance SCREE PLOT ORDINAL POSITION

2 8 Factor Loadings for Factor Analysis of Parents Health Belief Scale. Unrotated Solution(U ) Reordered Varimax Rotated Solution(V ) Variables Factor 1 Factor 2 h 2 Variables Factor 1 Factor 2 h 2 Item Item Item Item Item Item Item Item Item Item Item Item Item Item Item Item Item Item Item Item Item Item Item Item Eigenvalue SSL % of Common Variance 63.78% 36.22% 100% 521% 47.19% 100% % of Total Variance 372% 218% 59.30% 31.32% 27.99% 59.30% h 2 = Σv 2 = (-.1378) 2 = 643 OR h 2 = Σu 2 = = 643 (4.539/7.116) = 378 (4.539/12) =.3782 (2.577/7.116) =.3622 (2.577/12) =.2148 (3.758/7.116) =.5281 (3.358/7.116) = 719 (3.758/12) =.3132 (3.358/12) =.2799 The factor loadings and communalities can also be expressed through a regression equation. I j = v 1 F 1 + v 2 F 2 with R 2 = h 2. For example for Item 1, I 1 = 673F F 2 ; R 2 = h 2 = 643 Because this solution has orthogonal structure, the regression coefficients are equivalent to correlation coefficients and therefore can be squared and summed to form the communality. Item 7 u 1 =.5370 u 2 =943 Rotation of Factor 2 Item 7 v 1 =.7285 v 2 = Rotation of Factor UNROTATED SOLUTION VARIMAX ROTATED SOLUTION FACTOR ONE

3 The initial two-dimensional solution using Exploratory Factor Analysis is solving the following model: c 1 = b 11 I 1 + b 12 I 2 + b 13 I 3 + b 14 I 4 + b 15 I 5 + b 16 I 6 + b 17 I 7 + b 18 I 8 + b 19 I 9 + b 110 I 10 + b 111 I 11 + b 112 I 12 : c 2 = b 21 I 1 + b 22 I 2 + b 23 I 3 + b 24 I 4 + b 25 I 5 + b 26 I 6 + b 27 I 7 + b 28 I 8 + b 29 I 9 + b 210 I 10 + b 211 I 11 + b 212 I 12 In matrix notation: C = BI, I is not the Identity matrix in this case. Because the s are orthogonal the extraction can occur sequentially. 9 One Two I 1 I 2 I 3 I 4 I 5 I 6 I 7 I 8 I 9 I 10 I 11 I 12 Varimax Rotation keeps the Factors orthogonal (uncorrelated). It is an attempt to simplify the interpretation of the solution by rearranging the loadings, maximizing some weights (moving them toward 1) while also minimizing others (moving them toward 0). Although the loadings are technically not zero, if they are near zero the loading can be ignored for the purposes of interpretation. The solution may be interpreted as follows: Factor 1 Name?? Factor 2 Name?? I 1 I 4 I 5 I 7 I 8 I 9 I 11 I 2 I 3 I 6 I 10 I 12

4 Factor Solution as a Regression Equation 10 Rotated Matrix (V) 1 2 h 2 Item Item Item Item Item Item Item Item Item Item Item Item h 2 =(673) 2 +(-.1378) 2 = 643 The Pattern coefficients are standardized regression weights. The Stucture coefficients are zero-order (bivariate) correlations between each of the 12 variables and the 2 sets of Factor Scores. When the factor solution has employed orthogonal solution, the correlation between factors is zero and the Structure Matrix is identical to the Pattern Matrix, P = S. h 2 =(014) 2 +(-.2257) 2 = 931 I 1 = p 11 F 1 + p 12 F 2 I 1 = 67(F 1 ) -.138(F 2 ) h 2 =R 2 for this Model Model Summary Model R R Square Adjusted R Square Std. Error of the Estimate 1 81a a Predictors: (Constant), F1, F2 Coefficients Collinearity Unstandardized Standardized Coefficients Coefficients 95% Confidence Correlations Statistics Interval for B B Std. Beta t Sig. Lower Upper Zeroorder Error Bound Bound Partial Part Tol. VIF 1 (Constant) F F a Dependent Variable: Item I 9 = p 91 F 1 + p 92 F 2 I 9 = 01(F 1 ) -.226(F 2 ) h 2 =R 2 for this Model Model Summary Model R R Square Adjusted R Square Std. Error of the Estimate 1 32a a Predictors: (Constant), F1, F2 Note: Tolerance values of 1 indicate Coefficients Zero correlation between the 2 Factors Collinearity Unstandardized Standardized Coefficients Coefficients 95% Confidence Correlations Statistics Interval for B B Std. Beta t Sig. Lower Upper Zeroorder Error Bound Bound Partial Part Tol. VIF 1 (Constant) F F a Dependent Variable: Item 9 β -hats & Zero-order are identical due the orthogonal (uncorrelated) solution

5 11 From the Fundamental Theorem of Principal s, the original correlation matrix (R) can be reproduced by R (r) = U p,f U f,p. From the Fundamental Factor Theorem, the original correlation matrix (R) can be reproduced by R (r) = P p,f F f,f P f,p, where P p,f is a matrix of Pattern coefficients and F f,f is the correlation matrix for the factors. The matrix of loadings after Varimax rotation (V ) is a Pattern matrix and has the same variance as the unrotated solution (U ). Moreover, Varimax rotation provides an orthogonal solution meaning the correlation among factors is zero. Thus, the Factor Correlation matrix is an Identity matrix, F f,f = I. Therefore, the original correlation matrix (R) can also be reproduced by R (r) = V p,f V f,p. Reproduced Correlation Matrix: R (r) 12,12 = U 12,2 U 2,12 or R (r) 12,12 = V 12,2 V 2,12 Item Item Item Item Item Item Item Item Item Item Item Item NOTE: The diagonals are equal to the communalities (h 2 ). Residual Matrix (E) = R - R (r) Item Item Item Item Item Item Item Item Item Item Item Item Raw Score Standardized Score Raw Score Standardized Score Alpha = Alpha = Alpha = Alpha = Correlation Alpha Correlation Alpha Correlation Alpha Correlation Alpha Variable with Total if Deleted with Total if Deleted Variable with Total if Deleted with Total if Deleted Item Item Item Item Item Item Item Item Item Item Item Item

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