LUNDS UNIVERSITET STATISTISKA INSTITUTIONEN MATS HAGNELL STA102:4 Skrivning i multivariata metoder lördagen den 30 augusti 2003 Förutom Körners tabell- och formelsamling och miniräknare är även läroboken: Marcoulides-Hershberger, Multivariate Statistical Methods, tillåtet hjälpmedel. 1. Betrakta följande matriser: A= `12 6 6 12 och B= 1 2 3 4 5 6. a) Bestäm A*B och B*A! b) Bestäm egenvärden och egenvektorer till A! Visa också att de två egenvektorerna är ortogonala! 2. Vi har 5 observationer på slumpvektorn X =(X 1,X 2,X 3 ): (1, 5, 2), (1, 3, 3), (5, 6, 7), (6, 7, 9), (4, 8, 9). Beräkna medelvärdesvektorn, kovariansmatrisen, den totala SSCP-matrisen och korrelationsmatrisen! 3. Slumpvektorn X =(X 1,X 2,X 3 ) är 3-dimensionellt N(μ, Σ), `6 3 2 där μ =(5, 2, 6) och Σ = 3 10 5. Vad är 2 5 12 sannolikhetsfördelningarna för ( X 2, X 3 ), ( X 1, X 2 ), X 1 respektive X 2?
4. Från ett stickprov om 60 observationer på slumpvektorn X 1 =(X 11,X 12 ), som är 2-dimensionellt N(μ 1, Σ) fås x 1 =( 10, 9 ) och S 1 = `10 3 3 7. Från ett oberoende stickprov om 40 observationer på slumpvektorn X 2 =(X 21,X 22 ), som är 2-dimensionellt N(μ 2, Σ) fås x 2 =( 14, 13 ) och S 2 = `11 4 4 8. Från ett oberoende stickprov om 30 observationer på slumpvektorn X 3 =(X 31,X 32 ), som är 2-dimensionellt N(μ 3, Σ) fås x 3 =( 34, 46 ) och S 3 = `12 3 3 10. a) Skatta μ 1, μ 2, μ 3 och Σ! b) Antag att H 0 : μ 1 =μ 2 =μ 3 = μ är sann. Hur skattas μ? Och vilken sannolikhetsfördelning har då ˆμ? c) Bestäm SSCP-matriserna T, B och W! 5. För ett stickprov om 100 bankanställda mättes de fyra bakgrundsvariablerna Y 1 = EDUC (utbildning, i antal år), Y 2 = AGE (ålder), Y 3 = EXPER (antal år med relevant arbetslivserfarenhet vid anställningen) och Y 4 = SENIOR (nivå på bank-anställningen). Vidare mättes två lönevariabler, X 1 =LCURRENT=ln(nuvarande lön) och X 2 = LSTART= ln(begynnelselön). I appendix finns resultatet från en kanonisk korrelationsanalys mellan bakgrundsvariablerna och lönevariablerna. a) Redovisa kortfattat resultatet av den kanoniska korrelations-analysen! b) Försök att tolka resultatet av den kanoniska korrelationsanalysen! 6. Fortsättning av uppgift 5: Bestäm för de tre variablerna AGE (ålder), EDUC (utbildning) och EXPER (arbetslivserfarenhet) för de 100 bankanställda, om det finns någon skillnad mellan kön! Datautskrifter från proc ANOVA finns i Appendix. Individerna indelades i två grupper efter variabeln SEX (kön), vilken antar två värden, 0= man och 1= kvinna.
7. 250 amerikanska studenter testades på 8 variabler: ABILITY1 = allmän intelligens, ABILITY2 = medelbetyg sista skolåret, ABILITY3 = medelbetyg första högskoleåret, MOTIVN1 = poäng för motivation att prestera 1, MOTIVN2 = poäng för motivation att prestera 2, MOTIVN3 = poäng för motivation att prestera 3, ASPIRN1 = poäng på en allmän utbildningsaspiration och ASPIRN2 = poäng på en allmän yrkesaspiration. Tolka resultatet av en explorativ faktoranalys på dessa variabler och ange de skattade parametrarna! Datautskrifter från proc FACTOR finns i Appendix. 8. Fortsättning på uppgift 7: Under antagandet att en trefaktormodell är lämplig, vill man pröva om faktorladdningarna följer följande mönster: ABILITY1, ABILITY2 och ABILITY3 laddar högt enbart på faktor 1, medan MOTIVN1, MOTIVN2 och MOTIVN3 laddar högt enbart på faktor 2 samt att ASPIRN1 och ASPIRN2 laddar högt enbart på faktor 3! a) Rita den antagna modellen som ett pathdiagram! b) Skriv den antagna modellen som ett ekvationssystem! c) Förklara antalet frihetsgrader i chitvåtestet! d) Dra slutsatser om den antagna modellen (hypotesen) från LISREL-outputen (i Appendix)!
Appendix Uppgift 5 The CANCORR Procedure VAR Variables 2 WITH Variables 4 Observations 100 Means and Standard Deviations Standard Variable Mean Deviation Label x1 9.499574 0.441853 LCURRENT x2 8.803613 0.399290 LSTART y1 13.100000 3.365001 EDUC y2 42.677100 12.246945 AGE y3 11.478600 10.133139 EXPER y4 81.440000 10.359517 SENIOR The CANCORR Procedure Correlations Among the Original Variables Correlations Among the VAR Variables x1 x2 x1 1.0000 0.8889 x2 0.8889 1.0000 Correlations Among the WITH Variables y1 y2 y3 y4 y1 1.0000-0.2942-0.2537 0.0544 y2-0.2942 1.0000 0.7304 0.0701 y3-0.2537 0.7304 1.0000-0.0125 y4 0.0544 0.0701-0.0125 1.0000 Correlations Between the VAR Variables and the WITH Variables y1 y2 y3 y4 x1 0.6662-0.3330-0.0988 0.0500 x2 0.6725-0.2336-0.0024-0.0804 The CANCORR Procedure Canonical Correlation Analysis Adjusted Approximate Squared Canonical Canonical Standard Canonical Correlation Correlation Error Correlation 1 0.747607 0.736454 0.044331 0.558916 2 0.377391 0.359534 0.086190 0.142424 Test of H0: The canonical correlations in Eigenvalues of Inv(E)*H the current row and all that follow are zero = CanRsq/(1-CanRsq) Likelihood Approximate Eigenvalue Difference Proportion Cumulative Ratio F Value Num DF Den DF Pr > F 1 1.2671 1.1011 0.8841 0.8841 0.37826344 14.71 8 188 <.0001 2 0.1661 0.1159 1.0000 0.85757598 5.26 3 95 0.0021
Multivariate Statistics and F Approximations S=2 M=0.5 N=46 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.37826344 14.71 8 188 <.0001 Pillai's Trace 0.70133956 12.83 8 190 <.0001 Hotelling-Lawley Trace 1.43321693 16.74 8 131.98 <.0001 Roy's Greatest Root 1.26713949 30.09 4 95 <.0001 NOTE: F Statistic for Roy's Greatest Root is an upper bound. NOTE: F Statistic for Wilks' Lambda is exact. The CANCORR Procedure Canonical Correlation Analysis Raw Canonical Coefficients for the VAR Variables bakgrund1 bakgrund2 x1 LCURRENT 0.9791998913-4.84178399 x2 LSTART 1.4915830235 5.2589297542 Raw Canonical Coefficients for the WITH Variables salary1 salary2 y1 EDUC 0.2712997562 0.0662595864 y2 AGE -0.04076559 0.0481071611 y3 EXPER 0.0529574746 0.0161089242 y4 SENIOR -0.004153311-0.075526405 The CANCORR Procedure Canonical Correlation Analysis Standardized Canonical Coefficients for the VAR Variables bakgrund1 bakgrund2 x1 LCURRENT 0.4327-2.1394 x2 LSTART 0.5956 2.0998 Standardized Canonical Coefficients for the WITH Variables salary1 salary2 y1 EDUC 0.9129 0.2230 y2 AGE -0.4993 0.5892 y3 EXPER 0.5366 0.1632 y4 SENIOR -0.0430-0.7824 The CANCORR Procedure Canonical Structure Correlations Between the VAR Variables and Their Canonical Variables bakgrund1 bakgrund2 x1 LCURRENT 0.9621-0.2729 x2 LSTART 0.9802 0.1982
Variables Correlations Between the WITH Variables and Their Canonical salary1 salary2 y1 EDUC 0.9213-0.0343 y2 AGE -0.3789 0.5880 y3 EXPER -0.0592 0.5468 y4 SENIOR -0.0351-0.7310 Correlations Between the VAR Variables and the Canonical Variables of the WITH Variables salary1 salary2 x1 LCURRENT 0.7192-0.1030 x2 LSTART 0.7328 0.0748 Correlations Between the WITH Variables and the Canonical Variables of the VAR Variables bakgrund1 bakgrund2 y1 EDUC 0.6888-0.0130 y2 AGE -0.2833 0.2219 y3 EXPER -0.0442 0.2064 y4 SENIOR -0.0262-0.2759 The CANCORR Procedure Canonical Redundancy Analysis Raw Variance of the VAR Variables Explained by Their Own The Opposite Canonical Variables Canonical Variables Canonical Variable Cumulative Canonical Cumulative Number Proportion Proportion R-Square Proportion Proportion 1 0.9414 0.9414 0.5589 0.5261 0.5261 2 0.0586 1.0000 0.1424 0.0084 0.5345 Raw Variance of the WITH Variables Explained by Their Own The Opposite Canonical Variables Canonical Variables Canonical Variable Cumulative Canonical Cumulative Number Proportion Proportion R-Square Proportion Proportion 1 0.0852 0.0852 0.5589 0.0476 0.0476 2 0.3768 0.4620 0.1424 0.0537 0.1013 The CANCORR Procedure Canonical Redundancy Analysis Standardized Variance of the VAR Variables Explained by Their Own The Opposite Canonical Variables Canonical Variables Canonical Variable Cumulative Canonical Cumulative Number Proportion Proportion R-Square Proportion Proportion 1 0.9431 0.9431 0.5589 0.5271 0.5271 2 0.0569 1.0000 0.1424 0.0081 0.5352 Standardized Variance of the WITH Variables Explained by Their Own The Opposite Canonical Variables Canonical Variables
Canonical Variable Cumulative Canonical Cumulative Number Proportion Proportion R-Square Proportion Proportion 1 0.2493 0.2493 0.5589 0.1393 0.1393 2 0.2951 0.5443 0.1424 0.0420 0.1813 The CANCORR Procedure Canonical Redundancy Analysis Squared Multiple Correlations Between the VAR Variables and the First M Canonical Variables of the WITH Variables M 1 2 x1 LCURRENT 0.5173 0.5279 x2 LSTART 0.5370 0.5426 Squared Multiple Correlations Between the WITH Variables and the First M Canonical Variables of the VAR Variables M 1 2 y1 EDUC 0.4744 0.4746 y2 AGE 0.0802 0.1295 y3 EXPER 0.0020 0.0445 y4 SENIOR 0.0007 0.0768 Uppgift 6 The ANOVA Procedure Class Level Information Class Levels Values x3 2 0 1 Number of observations 100 Dependent Variable: y1 EDUC Sum of Source DF Squares Mean Square F Value Pr > F Model 1 15.133005 15.133005 1.34 0.2497 Error 98 1105.866995 11.284357 Corrected Total 99 1121.000000 R-Square Coeff Var Root MSE y1 Mean 0.013500 25.64290 3.359220 13.10000 Source DF Anova SS Mean Square F Value Pr > F x3 1 15.13300493 15.13300493 1.34 0.2497 The ANOVA Procedure Dependent Variable: y2 AGE Sum of Source DF Squares Mean Square F Value Pr > F
Model 1 383.63863 383.63863 2.60 0.1101 Error 98 14465.14063 147.60348 Corrected Total 99 14848.77926 R-Square Coeff Var Root MSE y2 Mean 0.025836 28.46777 12.14922 42.67710 Source DF Anova SS Mean Square F Value Pr > F x3 1 383.6386254 383.6386254 2.60 0.1101 Dependent Variable: y3 EXPER Sum of Source DF Squares Mean Square F Value Pr > F Model 1 577.52942 577.52942 5.90 0.0169 Error 98 9587.84098 97.83511 Corrected Total 99 10165.37040 R-Square Coeff Var Root MSE y3 Mean 0.056813 86.17047 9.891163 11.47860 Source DF Anova SS Mean Square F Value Pr > F x3 1 577.5294239 577.5294239 5.90 0.0169 The ANOVA Procedure Bonferroni (Dunn) t Tests for y1 Alpha 0.05 Error Degrees of Freedom 98 Error Mean Square 11.28436 Critical Value of t 1.98447 Minimum Significant Difference 1.3507 Harmonic Mean of Cell Sizes 48.72 NOTE: Cell sizes are not equal. Means with the same letter are not significantly different. Bon Grouping Mean N x3 A 13.4310 58 0 A A 12.6429 42 1 The ANOVA Procedure Bonferroni (Dunn) t Tests for y2 Alpha 0.05 Error Degrees of Freedom 98 Error Mean Square 147.6035 Critical Value of t 1.98447 Minimum Significant Difference 4.8849 Harmonic Mean of Cell Sizes 48.72
NOTE: Cell sizes are not equal. Means with the same letter are not significantly different. Bon Grouping Mean N x3 A 44.979 42 1 A A 41.010 58 0 The ANOVA Procedure Bonferroni (Dunn) t Tests for y3 Alpha 0.05 Error Degrees of Freedom 98 Error Mean Square 97.83511 Critical Value of t 1.98447 Minimum Significant Difference 3.977 Harmonic Mean of Cell Sizes 48.72 NOTE: Cell sizes are not equal. Means with the same letter are not significantly different. Bon Grouping Mean N x3 A 13.524 58 0 B 8.655 42 1 The ANOVA Procedure Multivariate Analysis of Variance Characteristic Roots and Vectors of: E Inverse * H, where H = Anova SSCP Matrix for x3 E = Error SSCP Matrix Characteristic Characteristic Vector V'EV=1 Root Percent y1 y2 y3 0.44473491 100.00 0.00700427-0.01231246 0.01658119 0.00000000 0.00 0.02929665 0.00219377-0.00295436 0.00000000 0.00 0.00937033 0.00621759 0.00355072 MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall x3 Effect H = Anova SSCP Matrix for x3 E = Error SSCP Matrix S=1 M=0.5 N=47 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.69216851 14.23 3 96 <.0001 Pillai's Trace 0.30783149 14.23 3 96 <.0001 Hotelling-Lawley Trace 0.44473491 14.23 3 96 <.0001 Roy's Greatest Root 0.44473491 14.23 3 96 <.0001
TABELL 1, RAYKOV-MARCOULIDES The FACTOR Procedure Initial Factor Method: Principal Components Prior Communality Estimates: ONE Eigenvalues of the Covariance Matrix: Total = 4.48 Average = 0.56 Eigenvalue Difference Proportion Cumulative 1 2.10961304 1.29468787 0.4709 0.4709 2 0.81492517 0.39231667 0.1819 0.6528 3 0.42260851 0.12078490 0.0943 0.7471 4 0.30182361 0.01359870 0.0674 0.8145 5 0.28822491 0.07180270 0.0643 0.8788 6 0.21642221 0.03982580 0.0483 0.9271 7 0.17659641 0.02681025 0.0394 0.9666 8 0.14978615 0.0334 1.0000 2 factors will be retained by the MINEIGEN criterion. The FACTOR Procedure Initial Factor Method: Principal Components Scree Plot of Eigenvalues 2.5 ˆ 1 2.0 ˆ E i g 1.5 ˆ e n v a l u e 1.0 ˆ s 2 0.5 ˆ 3 4 5 6 7 8 0.0 ˆ Šƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒ 0 1 2 3 4 5 6 7 8 Number The FACTOR Procedure
Initial Factor Method: Principal Components Factor Pattern Factor1 Factor2 ABIL1 0.62184 0.51942 ABIL2 0.65722 0.56314 ABIL3 0.61354 0.55406 MOTIV1 0.71970-0.03540 MOTIV2 0.76111-0.10694 MOTIV3 0.77680-0.02785 ASP1 0.64181-0.54036 ASP2 0.63515-0.56757 Variance Explained by Each Factor Factor Weighted Unweighted Factor1 2.10961304 3.71106067 Factor2 0.81492517 1.52148733 Final Communality Estimates and Variable Weights Total Communality: Weighted = 2.924538 Unweighted = 5.232548 Variable Communality Weight ABIL1 0.65647998 0.45000000 ABIL2 0.74906446 0.56000000 ABIL3 0.68341866 0.45000000 MOTIV1 0.51921777 0.55000000 MOTIV2 0.59071598 0.66000000 MOTIV3 0.60419030 0.61000000 ASP1 0.70391002 0.58000000 ASP2 0.72555082 0.62000000 The FACTOR Procedure Initial Factor Method: Maximum Likelihood Prior Communality Estimates: SMC ABIL1 ABIL2 ABIL3 MOTIV1 MOTIV2 MOTIV3 ASP1 ASP2 0.49569986 0.53344470 0.49180562 0.38686685 0.43177771 0.47040307 0.48590254 0.48332279 Preliminary Eigenvalues: Total = 7.2537368 Average = 0.9067171 Eigenvalue Difference Proportion Cumulative 1 6.10801210 4.12656273 0.8421 0.8421 2 1.98144937 1.65454769 0.2732 1.1152 3 0.32690168 0.39507443 0.0451 1.1603 4 -.06817275 0.07215149-0.0094 1.1509 5 -.14032424 0.06532390-0.0193 1.1315 6 -.20564814 0.07147959-0.0284 1.1032 7 -.27712773 0.19422577-0.0382 1.0650 8 -.47135350-0.0650 1.0000 2 factors will be retained by the NFACTOR criterion. Iteration Criterion Ridge Change Communalities 1 0.2063919 0.0000 0.1219 0.57818 0.65537 0.60938 0.42496 0.46448 0.49261 0.57638 0.57734 2 0.2047790 0.0000 0.0162 0.58419 0.65771 0.60998 0.41548 0.45088 0.48138 0.58996 0.59350 3 0.2045252 0.0000 0.0070 0.58411 0.65784 0.60993 0.41205 0.44642 0.47717 0.59696 0.59944 4 0.2044856 0.0000 0.0028 0.58408 0.65766 0.60987 0.41081 0.44470 0.47550 0.59926 0.60226 5 0.2044797 0.0000 0.0011 0.58402 0.65760 0.60986 0.41030 0.44401 0.47490 0.60035 0.60315 6 0.2044788 0.0000 0.0004 0.58401 0.65757 0.60985 0.41012 0.44376 0.47465 0.60068 0.60358
Convergence criterion satisfied. Significance Tests Based on 250 Observations Pr > Test DF Chi-Square ChiSq H0: No common factors 28 789.7989 <.0001 HA: At least one common factor The FACTOR Procedure Initial Factor Method: Maximum Likelihood Significance Tests Based on 250 Observations Pr > Test DF Chi-Square ChiSq H0: 2 Factors are sufficient 13 49.9269 <.0001 HA: More factors are needed Chi-Square without Bartlett's Correction 50.915226 Akaike's Information Criterion 24.915226 Schwarz's Bayesian Criterion -20.863766 Tucker and Lewis's Reliability Coefficient 0.895596 Squared Canonical Correlations Factor1 Factor2 0.88227114 0.73798578 Eigenvalues of the Weighted Reduced Correlation Matrix: Total = 10.31068 Average = 1.288835 Eigenvalue Difference Proportion Cumulative 1 7.49409362 4.67750688 0.7268 0.7268 2 2.81658674 2.32943622 0.2732 1.0000 3 0.48715052 0.38438271 0.0472 1.0472 4 0.10276781 0.10202390 0.0100 1.0572 5 0.00074391 0.06672027 0.0001 1.0573 6 -.06597636 0.08489371-0.0064 1.0509 7 -.15087007 0.22294613-0.0146 1.0363 8 -.37381620-0.0363 1.0000 Factor Pattern Factor1 Factor2 ABIL1 0.67535-0.35763 ABIL2 0.71458-0.38332 ABIL3 0.67432-0.39388 MOTIV1 0.62503 0.13929 MOTIV2 0.63802 0.19137 MOTIV3 0.67425 0.14132 ASP1 0.54861 0.54753 ASP2 0.53715 0.56135 The FACTOR Procedure Initial Factor Method: Maximum Likelihood Variance Explained by Each Factor Factor Weighted Unweighted Factor1 7.49409362 3.26329242 Factor2 2.81658674 1.12087231 Final Communality Estimates and Variable Weights Total Communality: Weighted = 10.310680 Unweighted = 4.384165
Variable Communality Weight ABIL1 0.58400456 2.40391244 ABIL2 0.65756543 2.92030740 ABIL3 0.60984939 2.56312386 MOTIV1 0.41006742 1.69526442 MOTIV2 0.44368804 1.79777594 MOTIV3 0.47458971 1.90348016 ASP1 0.60075696 2.50424864 ASP2 0.60364322 2.52256711 The FACTOR Procedure Rotation Method: Varimax Orthogonal Transformation Matrix 1 2 1 0.76267 0.64678 2-0.64678 0.76267 Rotated Factor Pattern Factor1 Factor2 ABIL1 0.74639 0.16405 ABIL2 0.79292 0.16983 ABIL3 0.76904 0.13574 MOTIV1 0.38661 0.51049 MOTIV2 0.36283 0.55861 MOTIV3 0.42284 0.54387 ASP1 0.06428 0.77242 ASP2 0.04660 0.77555 Variance Explained by Each Factor Factor Weighted Unweighted Factor1 5.53735422 2.24344105 Factor2 4.77332614 2.14072368 Final Communality Estimates and Variable Weights Total Communality: Weighted = 10.310680 Unweighted = 4.384165 Variable Communality Weight ABIL1 0.58400456 2.40391244 ABIL2 0.65756543 2.92030740 ABIL3 0.60984939 2.56312386 MOTIV1 0.41006742 1.69526442 MOTIV2 0.44368804 1.79777594 MOTIV3 0.47458971 1.90348016 ASP1 0.60075696 2.50424864 ASP2 0.60364322 2.52256711 The FACTOR Procedure Initial Factor Method: Maximum Likelihood Prior Communality Estimates: SMC ABIL1 ABIL2 ABIL3 MOTIV1 MOTIV2 MOTIV3 ASP1 ASP2 0.49569986 0.53344470 0.49180562 0.38686685 0.43177771 0.47040307 0.48590254 0.48332279 Preliminary Eigenvalues: Total = 7.2537368 Average = 0.9067171 Eigenvalue Difference Proportion Cumulative 1 6.10801210 4.12656273 0.8421 0.8421 2 1.98144937 1.65454769 0.2732 1.1152 3 0.32690168 0.39507443 0.0451 1.1603 4 -.06817275 0.07215149-0.0094 1.1509 5 -.14032424 0.06532390-0.0193 1.1315
6 -.20564814 0.07147959-0.0284 1.1032 7 -.27712773 0.19422577-0.0382 1.0650 8 -.47135350-0.0650 1.0000 3 factors will be retained by the NFACTOR criterion. Iteration Criterion Ridge Change Communalities 1 0.0734940 0.0000 0.2873 0.59551 0.67840 0.60566 0.45964 0.54259 0.59252 0.53150 0.77064 2 0.0658402 0.1250 0.2294 0.59675 0.67775 0.60467 0.46487 0.55159 0.57905 0.43129 1.00000 3 0.0599169 0.0313 0.0544 0.60577 0.67434 0.60499 0.47409 0.57324 0.55288 0.48564 1.00000 4 0.0598821 0.0000 0.0038 0.60577 0.67420 0.60489 0.47418 0.57365 0.55019 0.48944 1.00000 5 0.0598818 0.0000 0.0004 0.60585 0.67413 0.60487 0.47428 0.57369 0.54982 0.48968 1.00000 Convergence criterion satisfied. Significance Tests Based on 250 Observations Pr > Test DF Chi-Square ChiSq H0: No common factors 28 789.7989 <.0001 HA: At least one common factor H0: 3 Factors are sufficient 7 14.5812 0.0418 HA: More factors are needed The FACTOR Procedure Initial Factor Method: Maximum Likelihood Chi-Square without Bartlett's Correction 14.910567 Akaike's Information Criterion 0.910567 Schwarz's Bayesian Criterion -23.739659 Tucker and Lewis's Reliability Coefficient 0.960193 Squared Canonical Correlations Factor1 Factor2 Factor3 1.0000000 0.8611106 0.5577677 Eigenvalues of the Weighted Reduced Correlation Matrix: Total = 7.4612284 Average = 1.06588977 Eigenvalue Difference Proportion Cumulative 1 Infty Infty 2 6.19997350 4.93871843 0.8310 0.8310 3 1.26125507 1.09589646 0.1690 1.0000 4 0.16535861 0.06472223 0.0222 1.0222 5 0.10063638 0.04712539 0.0135 1.0357 6 0.05351099 0.11847197 0.0072 1.0428 7 -.06496098 0.18958418-0.0087 1.0341 8 -.25454516-0.0341 1.0000 Factor Pattern Factor1 Factor2 Factor3 ABIL1 0.20825 0.71421-0.22890 ABIL2 0.16971 0.77783-0.20076 ABIL3 0.13252 0.74820-0.16582 MOTIV1 0.35962 0.48690 0.32846 MOTIV2 0.39082 0.48811 0.42742 MOTIV3 0.43904 0.51023 0.31100 ASP1 0.65036 0.15036 0.21007 ASP2 1.00000 0.00000 0.00000
Variance Explained by Each Factor Factor Weighted Unweighted Factor1 2.10414211 1.98752143 Factor2 6.19997350 2.43319150 Factor3 1.26125507 0.55161137 The FACTOR Procedure Initial Factor Method: Maximum Likelihood Final Communality Estimates and Variable Weights Total Communality: Weighted = 9.565371 Unweighted = 4.972324 Variable Communality Weight ABIL1 0.60585543 2.53712836 ABIL2 0.67412648 3.06868602 ABIL3 0.60486812 2.53080838 MOTIV1 0.47428232 1.90215882 MOTIV2 0.57367684 2.34570905 MOTIV3 0.54980774 2.22133286 ASP1 0.48970737 1.95954702 ASP2 1.00000000 Infty The FACTOR Procedure Rotation Method: Varimax Orthogonal Transformation Matrix 1 2 3 1 0.06997 0.22997 0.97068 2 0.86257 0.47483-0.17467 3-0.50108 0.84950-0.16514 Rotated Factor Pattern Factor1 Factor2 Factor3 ABIL1 0.74532 0.19257 0.11520 ABIL2 0.78340 0.23782 0.06202 ABIL3 0.73774 0.24488 0.02533 MOTIV1 0.28056 0.59292 0.20979 MOTIV2 0.23421 0.68474 0.22352 MOTIV3 0.31500 0.60743 0.28569 ASP1 0.06995 0.39941 0.57034 ASP2 0.06997 0.22997 0.97068 Variance Explained by Each Factor Factor Weighted Unweighted Factor1 5.17851588 1.95606644 Factor2 3.32014276 1.55540769 Factor3 1.06671203 1.46085017 Final Communality Estimates and Variable Weights Total Communality: Weighted = 9.565371 Unweighted = 4.972324 Variable Communality Weight ABIL1 0.60585543 2.53712836 ABIL2 0.67412648 3.06868602 ABIL3 0.60486812 2.53080838 MOTIV1 0.47428232 1.90215882 MOTIV2 0.57367684 2.34570905 MOTIV3 0.54980774 2.22133286 ASP1 0.48970737 1.95954702 ASP2 1.00000000 Infty
L I S R E L 8.51 BY Karl G. J reskog & Dag S rbom The following lines were read from file C:\DATA\SK\mult\03augu8.LS8: TABELL 1, RAYKOV-MARCOULIDES, MED LISREL DA NI=8 NO=250 LA ABILITY1 ABILITY2 ABILITY3 MOTIVN1 MOTIVN2 MOTIVN3 ASPIRN1 ASPIRN2 CM 0.45 0.32 0.56 0.27 0.32 0.45 0.17 0.20 0.19 0.55 0.20 0.21 0.18 0.30 0.66 0.19 0.25 0.20 0.30 0.36 0.61 0.08 0.12 0.09 0.23 0.27 0.22 0.58 0.11 0.10 0.07 0.21 0.25 0.27 0.39 0.62 MO NX=8 NK=3 LX=FU,FI TD=DI,FREE PH=SY,FREE LK ABILITY MOTIVATN ASPIRATN VA 1.00 PH(1,1) PH(2,2) PH(3,3) FREE LX(1,1) LX(2,1) LX(3,1) FREE LX(4,2) LX(5,2) LX(6,2) LX(7,3) LX(8,3) PATH DIAGRAM OU ND=4 MI TABELL 1, RAYKOV-MARCOULIDES, MED LISREL Number of Input Variables 8 Number of Y - Variables 0 Number of X - Variables 8 Number of ETA - Variables 0 Number of KSI - Variables 3 Number of Observations 250 TABELL 1, RAYKOV-MARCOULIDES, MED LISREL Covariance Matrix ABILITY1 ABILITY2 ABILITY3 MOTIVN1 MOTIVN2 MOTIVN3 -------- -------- -------- -------- -------- -------- ABILITY1 0.4500 ABILITY2 0.3200 0.5600 ABILITY3 0.2700 0.3200 0.4500 MOTIVN1 0.1700 0.2000 0.1900 0.5500 MOTIVN2 0.2000 0.2100 0.1800 0.3000 0.6600 MOTIVN3 0.1900 0.2500 0.2000 0.3000 0.3600 0.6100 ASPIRN1 0.0800 0.1200 0.0900 0.2300 0.2700 0.2200 ASPIRN2 0.1100 0.1000 0.0700 0.2100 0.2500 0.2700 Covariance Matrix ASPIRN1 ASPIRN2 -------- --------
ASPIRN1 0.5800 ASPIRN2 0.3900 0.6200 TABELL 1, RAYKOV-MARCOULIDES, MED LISREL Parameter Specifications LAMBDA-X ABILITY MOTIVATN ASPIRATN -------- -------- -------- ABILITY1 1 0 0 ABILITY2 2 0 0 ABILITY3 3 0 0 MOTIVN1 0 4 0 MOTIVN2 0 5 0 MOTIVN3 0 6 0 ASPIRN1 0 0 7 ASPIRN2 0 0 8 PHI ABILITY MOTIVATN ASPIRATN -------- -------- -------- ABILITY 0 MOTIVATN 9 0 ASPIRATN 10 11 0 THETA-DELTA ABILITY1 ABILITY2 ABILITY3 MOTIVN1 MOTIVN2 MOTIVN3 -------- -------- -------- -------- -------- -------- 12 13 14 15 16 17 THETA-DELTA ASPIRN1 ASPIRN2 -------- -------- 18 19 TABELL 1, RAYKOV-MARCOULIDES, MED LISREL Number of Iterations = 5 LISREL Estimates (Maximum Likelihood) LAMBDA-X ABILITY MOTIVATN ASPIRATN -------- -------- -------- ABILITY1 0.5188 - - - - (0.0388) 13.3724 ABILITY2 0.6148 - - - - (0.0425) 14.4650 ABILITY3 0.5215 - - - - (0.0387) 13.4614
MOTIVN1 - - 0.5114 - - (0.0451) 11.3369 MOTIVN2 - - 0.5935 - - (0.0486) 12.2059 MOTIVN3 - - 0.5953 - - (0.0462) 12.8854 ASPIRN1 - - - - 0.6143 (0.0489) 12.5524 ASPIRN2 - - - - 0.6348 (0.0506) 12.5460 PHI ABILITY MOTIVATN ASPIRATN -------- -------- -------- ABILITY 1.0000 MOTIVATN 0.6364 1.0000 (0.0545) 11.6767 ASPIRATN 0.2764 0.6810 1.0000 (0.0735) (0.0541) 3.7628 12.5793 THETA-DELTA ABILITY1 ABILITY2 ABILITY3 MOTIVN1 MOTIVN2 MOTIVN3 -------- -------- -------- -------- -------- -------- 0.1809 0.1821 0.1781 0.2885 0.3077 0.2557 (0.0228) (0.0273) (0.0227) (0.0322) (0.0368) (0.0330) 7.9359 6.6736 7.8455 8.9654 8.3634 7.7576 THETA-DELTA ASPIRN1 ASPIRN2 -------- -------- 0.2026 0.2170 (0.0397) (0.0424) 5.1076 5.1197 Squared Multiple Correlations for X - Variables ABILITY1 ABILITY2 ABILITY3 MOTIVN1 MOTIVN2 MOTIVN3 -------- -------- -------- -------- -------- -------- 0.5981 0.6749 0.6043 0.4754 0.5337 0.5809 Squared Multiple Correlations for X - Variables ASPIRN1 ASPIRN2 -------- -------- 0.6507 0.6500
Goodness of Fit Statistics Degrees of Freedom = 17 Minimum Fit Function Chi-Square = 20.5813 (P = 0.2456) Normal Theory Weighted Least Squares Chi-Square = 18.8933 (P = 0.3347) Estimated Non-centrality Parameter (NCP) = 1.8933 90 Percent Confidence Interval for NCP = (0.0 ; 16.7808) Minimum Fit Function Value = 0.08266 Population Discrepancy Function Value (F0) = 0.007604 90 Percent Confidence Interval for F0 = (0.0 ; 0.06739) Root Mean Square Error of Approximation (RMSEA) = 0.02115 90 Percent Confidence Interval for RMSEA = (0.0 ; 0.06296) P-Value for Test of Close Fit (RMSEA < 0.05) = 0.8435 Expected Cross-Validation Index (ECVI) = 0.2285 90 Percent Confidence Interval for ECVI = (0.2209 ; 0.2883) ECVI for Saturated Model = 0.2892 ECVI for Independence Model = 3.2814 Chi-Square for Independence Model with 28 Degrees of Freedom = 801.0588 Independence AIC = 817.0588 Model AIC = 56.8933 Saturated AIC = 72.0000 Independence CAIC = 853.2305 Model CAIC = 142.8011 Saturated CAIC = 234.7726 Normed Fit Index (NFI) = 0.9743 Non-Normed Fit Index (NNFI) = 0.9924 Parsimony Normed Fit Index (PNFI) = 0.5915 Comparative Fit Index (CFI) = 0.9954 Incremental Fit Index (IFI) = 0.9954 Relative Fit Index (RFI) = 0.9577 Critical N (CN) = 405.2190 Root Mean Square Residual (RMR) = 0.01145 Standardized RMR = 0.02022 Goodness of Fit Index (GFI) = 0.9814 Adjusted Goodness of Fit Index (AGFI) = 0.9606 Parsimony Goodness of Fit Index (PGFI) = 0.4634 TABELL 1, RAYKOV-MARCOULIDES, MED LISREL Modification Indices and Expected Change Modification Indices for LAMBDA-X ABILITY MOTIVATN ASPIRATN -------- -------- -------- ABILITY1 - - 0.0022 0.0875 ABILITY2 - - 0.0032 0.0667 ABILITY3 - - 0.0112 0.3235 MOTIVN1 0.2437 - - 0.0158 MOTIVN2 0.9586 - - 0.3869 MOTIVN3 0.2459 - - 0.5156 ASPIRN1 0.1102 0.1102 - - ASPIRN2 0.1102 0.1102 - -
Expected Change for LAMBDA-X ABILITY MOTIVATN ASPIRATN -------- -------- -------- ABILITY1 - - 0.0025 0.0114 ABILITY2 - - 0.0034 0.0110 ABILITY3 - - -0.0057-0.0218 MOTIVN1 0.0318 - - 0.0091 MOTIVN2-0.0692 - - 0.0493 MOTIVN3 0.0341 - - -0.0553 ASPIRN1 0.0171 0.0549 - - ASPIRN2-0.0176-0.0567 - - No Non-Zero Modification Indices for PHI Modification Indices for THETA-DELTA ABILITY1 ABILITY2 ABILITY3 MOTIVN1 MOTIVN2 MOTIVN3 -------- -------- -------- -------- -------- -------- ABILITY1 - - ABILITY2 0.0666 - - ABILITY3 0.0119 0.0211 - - MOTIVN1 0.1084 0.3015 2.0899 - - MOTIVN2 1.1907 1.2764 0.4039 0.0790 - - MOTIVN3 1.1272 1.6582 0.0034 0.1786 0.4922 - - ASPIRN1 2.8122 1.4707 0.3890 1.7992 2.0867 7.4815 ASPIRN2 4.3029 0.7913 1.8809 1.1302 1.2263 4.6694 Modification Indices for THETA-DELTA ASPIRN1 ASPIRN2 -------- -------- ASPIRN1 - - ASPIRN2 - - - - Expected Change for THETA-DELTA ABILITY1 ABILITY2 ABILITY3 MOTIVN1 MOTIVN2 MOTIVN3 -------- -------- -------- -------- -------- -------- ABILITY1 - - ABILITY2 0.0083 - - ABILITY3-0.0029-0.0047 - - MOTIVN1-0.0060-0.0108 0.0263 - - MOTIVN2 0.0212-0.0237-0.0123-0.0081 - - MOTIVN3-0.0194 0.0255-0.0011-0.0120 0.0232 - - ASPIRN1-0.0290 0.0226 0.0108 0.0302 0.0354-0.0645 ASPIRN2 0.0372-0.0171-0.0245-0.0247-0.0280 0.0527 Expected Change for THETA-DELTA ASPIRN1 ASPIRN2 -------- -------- ASPIRN1 - - ASPIRN2 - - - - Maximum Modification Index is 7.48 for Element ( 7, 6) of THETA-DELTA
Svar till skrivning i multivariata metoder.den 30 augusti 2003: 36 54 72 1) a) B*A är inte definerad; A*B = 54 72 90 0.7071 b) Egenvärden är 18 och 6. Egenvektorer är 0.7071 respektivive 0.7071 0.7071. 3.4 5.3 3.35 7.0 2) Medelvärdesvektorn blir 5.8, kovariansmatrisen 3.35 3.7 5.5 och 6.0 7.0 5.5 11.0 1 0.757 0.917 21.2 13.4 28.0 korrelationsmatrisen 0.757 1 0.862, T=(n-1)S= 13.4 14.8 22.0. 0.917 0.862 1 28.0 22.0 44 2 3) Bivariat eller tvådimensionellt normalfördelad med väntevärdesvektor 6 och kovariansmatris 10 5 5 12 ; bivariat eller tvådimensionellt normalfördelad med 5 6 3 väntevärdesvektor 2 och kovariansmatris 3 10 ; X 1 är N(5, 6 ) och X 2 är N(2, 10 ). 10 4)a) 9, 14 34 10.76 3.31 13 respektive 46 och 3.31 7.99. 1 1 1 b) ˆμ är N(, Σ ( + + )) =N(, Σ 0.15) 60 40 30 1367 420 c) W = ( ni 1) Si = 420 1015, 3 3 3 1 11963 18683 B= ni( xi x)( xi x) = nixixi ( nixi)( nix i ) = i= 1 n i= 1 i= 1 18683 29303 och slutligen 13330 19103 T=B+W= 19103 30318
5) H 0 : P 12 =0 förkastas ty P<0.001 i testen. Det finns 2 signifikanta kanoniska korrelationskoefficienter, R 1 * = 0.7476 och R 2 * = 0.3774. Dessa två förklarar 55.9% respektive 14.2% av variationen i R 12. Tillhörande kanoniska variabelpar har koefficienter a 1 =(0.979, 1.496), b 1 =(0.271, -0.041, 0.053, -0.004) respektive a 2 =(-4.842, 5.259), b 2 =(0.066, 0.048, 0.016, -0.076). Motsvarande standardiserade storheter blir a 1 =(0.433, 0.596), b 1 =(0.913, -0.499, 0.537, -0.043) respektive a 2 =(-2.139, 2.010), b 2 =(0.233, 0.589, 0.163, -0.782). Vi ser också att de kanoniska variablernas korrelationer med respektive X- och Y-variabler är (0.962, 0.980), (0.689, -0.283, -0.044, -0.026)och (-0.279, 0.198), (-0.013, 0.222, 0.206, -0.276). Så en enkel tolkning av den standardiserade U 1 är att den är approximativt lika med summan av X 1 och X 2, dvs. lönenivå, medan en enkel tolkning av den standardiserade U 2 är att den är approximativt lika med differensen mellan X 2 och X 1, dvs. lönehöjning. Vi ser också att den standardiserade V 1 har positivt tecken för Y 1 och Y 3 men negativt tecken för Y 2 dvs. lönenivån beror positivt av utbildning och erfarenhet men negativt av åldern. Medan den standardiserade V 2 har positivt tecken för Y 2 men negativt tecken för Y 4, dvs. lönehöjning beror positivt av åldern men negativt av en högre ställning. 6) H 0 : 1 = 2 förkastas ty P<0.001 i alla 4 multivariata testen. Univariat finner vi att P=0.2497 för Y 1 =EDUC, P=0.1101 för Y 2 =AGE och P=0.0169 för Y 3 =EXPER. Så det finns signifikant skillnad mellan män och kvinnor, men endast i erfarenhet. Skillnaden utgörs av att männen har större erfarenhet än kvinnorna, 13.5 respektive 8.7. 7) Scree-plott ger att lämpligt antal faktorer är tre. Dock ger egenvärdeskriteriet två faktorer, men med två faktorer blir den roterade lösningen svår att tolka. Vi ser också att H 0 : 3 faktorer räcker har P-värde 0.042. De roterade faktorladdningarna för trefaktorlösningen är 0.75 0.19 0.12 0.78 0.24 0.06 0.74 0.25 0.03 0.28 0.59 0.21 0.23 0.69 0.22, dvs. den 1:a faktorn definieras av X 1 -X 3,. medan den 2:a faktorn 0.32 0.61 0.29 0.07 0.40 0.57 0.07 0.23 0.97 definieras av X 4 - X 6 och 3:e faktorn av X 7 och X 8. Så här har vi ett exempel på simple structure. Kommunaliterna blir 0.61, 0.67, 0.61, 0.47, 0.57, 0.55, 0.49 respektive 1.00, dvs. högst för X 1 -X 3, mer måttliga för X 4 -X 7 och väl högt för X 8.
8) a) Se diagram i läroboken! b) y 1 = λ 11 η 1 + ε 1 y 2 = λ 21 η 1 + ε 2 y 3 = λ 31 η 1 + ε 3 y 4 = λ 42 η 2 + ε 4 y 5 = λ 52 η 2 + ε 5 y 6 = λ 62 η 2 + ε 6 y 7 = λ 73 η 3 + ε 7 y 8 = λ 83 η 3 + ε 8 c) Antalet element i S=p*(p+1)/2=8*9/2=36 och antalet parametrar = antal faktorladdningar + antal specifika varianser + antal faktorkorrelationer = 8 + 8 +3 = 19 så antalet fg = 36 19 = 17 d) Chitvå = 18.89 med 17 frihetsgrader, vilket ger P = 0.335, så hypotesen förkastas inte på 5 %-nivån. Även andra anpassningsmått såsom RMSEA, NFI och GFI är hyfsade enligt de vanliga tumreglerna i boken. Vi ser att det största modifikationsindexet är 7.48 ( för TD(7,6) ), vilket inte är så stort.