Political Science 552

Transcription

1 Political Science 55 Facto and Pincial Comonents Path : Wight s Rules 4 v 4 4 4u R u R v 4. Path may ass though any vaiable only once on a single tavese. Path may go backwads, but not afte going fowad. Path may ass though only one double-headed aow Alying Wight s Rules 4 v 4 4 4u R u R v

2 Examle -.70 FT D PID FT R FT Reagan FT D FT R PID FT R -.70 PID FT Reagan Examle, continued FT D PID FT R FT Reagan (.9.48) + ( ) + ( ) (.48.0) + ( ) + ( ) + ( ) 4 4 Facto Model F a F b 4 5 R R R R 4 R 5 a a a a 0 4 5

3 Facto Model, Coelated Factos F a F b 4 5 R R R R 4 R 5 a a a a + aab b 4 aab 4b Pincial Comonents C a C b 4 5 R R R R 4 R 5 Identification F a F b 4 5 R R R R 4 R 5

4 Identification, continued Model Model F a F b F a F b 4 R R R R R R R 4 Path and Test-Retest Models R B Y () Y () Thee knowns:,, and Six unknowns: ()(), ()B, (), a, (), b, e a e b Assume equal eliabilities: () () a b Thee-Wave Panel Model R B R C 0 unknowns 6 knowns Y () Y () Y () Assume equal eliabilities: () () () e a e b e c a b c e Leaves only 6 unknowns 4

5 Wight s Rules and Panel Model R B R C ( )() ( )() Y () Y () Y () e a e b e c ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) Solving Thee Wave Panel ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) e Exloatoy Facto Facto stuctue (the fom of the model) Unobseved coefficients of that stuctue Measuements of the unobseved vaiables F A F B AB 0 etc. A A A A + + B B B B 4 AB A A B B A AB B etc. 5

6 Detemining the Facto Stuctue Two oblems:. How many undelying factos ae thee?. Ae the factos othogonal o oblique? Detemining numbe of factos:. a ioi (based on theoy o design). using a ule of thumb. by examining the data Extacting Factos Many ossible methods:. Least squaes. Maximum likelihood. othe Pincial Factos:. Extact factos on at a time. Maximize vaiance exlained by each. Result maximizes vaiance exlained by set Pincial Comonents Uncoelated Y Coelated,.0 Y Coelated, <.0 Y 6

7 Rotation to Pincial Comonents Y Y Pincial Comonents as a Poblem in Facto C A AB C B AB U V 0 A B A B U V R U R V Maximize: A + A Subject to the constaints: A A A B + B + + B A B Reca R 0 R C I 0 C P C P n n n n 7

8 Review of Linea Tansfomation W a + by { W } a { } + b { Y} ab {, Y} + k k W C n k n k { C} C' { } C k k Pincial Comonents as Linea Tansfomation let P * { } R C P n n P C P n n 0 0 { } C R C R P R P * * I C Fist Pincial Comonent C * * A A + A * * A * A * A P s should maximize: * { } * Subject to the constaint: * * * * * o 0 ia 8

9 Solution, Ste * W Calculus Review: * * * { } λ d P ( k ) ( d k ) k k d d W * * { } λ Solution, Ste * * { } λ 0 * ( { } λ) 0 ( { } ) 0 λ * det * 0 { } λ 0 Chaacteistic Equation ( { } λ) λ M k λ M k L L L k k M k λ 9

10 Chaacteistic Equation fo x det λ ( { } ) λ det ad bc λ { } λ ( λ)( λ) ( λ)( λ) 0 Chaacteistic Equation fo x continued ( λ)( λ) 0 + λ + λ + λ ( + ) + ( ) 0 λ + λ Rq λq 0 R λ [ ] Examle L P E.707 P * P L * P R P * L R L

11 since and Poeties R I C * * CP then C P R C P R P * * I λ T[ R] Facto vs. Pincial Comonents Numbe of factos less than k Vaiance divided between that which is common and that which is unique athe than exlaining all vaiance as in Pincial Comonents The ootion of vaiance in a vaiable attibutable to the common factos is called the communality Residual aths no longe zeo as in Pincial Comonents Pincial Facto Aoach Diagonal of R is elaced with an estimate of the communalities Some numbe of factos less than the numbe of vaiables is extacted We otate the solution to obtain moe meaningful esults A egession-like aoach must be used to tansfom the obseved vaiables into unobseved factos because the tansfomation matix is no longe squae

12 Stes in Exloatoy Facto Obtain initial estimate of communality and modify R Extact initial solution and examine esults Detemine numbe of factos Imove estimate of communality Extact an unotated solution based on initial solution Rotate solution to imove inteetability Obtain facto scoe coefficients and use them to estimate unobseved vaiables (facto scoes) Estimating Communality a ioi Lagest single coelation squaed Squaed multile coelation coefficient Fom initial solution (PC) detemine ootion of vaiance exlained by chosen numbe of factos Equal to sum of squaes of coefficients fo the vaiable Iteate with any of the above Detemining Numbe of Factos Kaise citeion: eigenvalues> Rank of adjusted coelation matix Scee lot

13 Unotated Facto Matix Equal to nomalized eigenvecto matix Columns of eigenvecto matix Divided by the sum of the squaes of the columns Multilied by the squae oot of the coesonding eigenvalues Fist facto will be a geneal facto Most vaiables will load on fist facto Othe factos will have lowe loadings Fist (Geneal) Facto Examle of Unotated, Two Facto Solution

14 Rotation Rotation Citeia Pocustes Simle stuctue Each vaiable load on as few factos as ossible Numbe of vaiables loading highly on any one facto should be minimized Eliminate geneal factos Thustone simle stuctue Othogonal Rotation Quatimax Vaimax Othomax Biquatimax 4

15 Nonothogonal Rotations Oblimin Pomax Othoblique Degee of coelation among factos Coelation matix among factos (Φ) Rotated Facto Loadings Coelations of otated factos with the vaiables ae the same as ath coefficients in othogonal solutions if 0 Examle of Loadings Unotated Rotated F F F F

16 Oblique Rotations Move axes indeendently Two sets of coefficients Path coefficients Facto Patten Coelations of each facto with each vaiable Coelations in matix Φ if 0 Inteeting Factos Look fo and label clustes of vaiables loading on secific factos Identify the common element among those vaiables S q Reca Z standadized data matix n coelation matix, obseved vaiables R P q facto stuctue matix (coelations) facto atten matix (ath coefficients) F standadized facto scoe matix n q V standadized facto scoe coefficient matix q q Φ q coelation matix, unobseved vaiables 6

17 R ZF Getting Facto Scoes n [ ZF] ( q) n + Ζ n F [ Z F][ Z F] [ Z F] Z Z n Z F F Z R F F S S Φ Facto Regession Equation F ZV + E b y Z ZV Z F ( ) y b ( Z Z) Z F V Z F S n Facto Scoe Solution ( Z Z) Z F V V Z Z Z F n n Z Z R n V R S F Z R S n q n q 7

18 Wiley & Wiley Thee Wave Panel Solution θ θ θ ξ α ξ α ξ ε ε ε ξ θ ξ α θ + θ ξ α θ + ε α ( α θ + θ ) + θ ξ + ε α θ + θ + ε ξ + ε ( α θ + θ ) + θ + ε ξ + ε Wiley & Wiley Vaiances { } { θ} + { ε} { } α { θ} + { θ} + { ε } { } α ( α { θ } + { θ }) + { ε } { } E{ } E{ ( θ + ε)( α θ + θ ε )} { } E{ α θ + θ θ + θ ε + α ε θ + ε θ ε ε } { } E{ α θ } α { } + + Similaly: θ { θ θ } E{ θ ε } E{ ε θ } E{ ε θ } E{ ε ε } 0 because: E { } α { } { } α α { θ } + { } α θ ( ) θ Wiley & Wiley Solution Six Equations in Six Unknowns α { } { } {} ε { } { } α { θ } { } { ε} α { } { θ } { θ } { } ( α { } + { ε} ) { θ } { } α { } + { ε} ( ) 8