( ) 1.1 Polychoric Correlation Polyserial Correlation Graded Response Model Partial Credit Model Tetrachoric Correlation ( ) 2 x y x y s r 1 x 2

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1 1 (,2007) SPSSver (2002) polychoric correlation coefficient (polyserial correlation coefficient) 3. (1999) M-plus R 1

2 ( ) 1.1 Polychoric Correlation Polyserial Correlation Graded Response Model Partial Credit Model Tetrachoric Correlation ( ) 2 x y x y s r 1 x 2

3 1 X1 X2 2 1: x y a i b j a 0 = b 0 = a s = b r = + y ξ η x ξ x = 1 i f ξ < a 1 x = 2 i f a 1 ξ < a 2 x = 3 i f a 2 ξ < a 3.. x = s i f a s 1 ξ (1) y a 1 ξ η ρ 1 3

4 3 ρ X ρ = ρ = 0.1 4

5 6 ρ = ρ = 0.9 (bivariate normal density function) 4 7 gnuplot 1 f (x, y; rho) = exp 1 2π 1 ρ 2 2(1 ρ 2 xy) (x2 2ρ xy + y 2 ) (2) xy 8 ρ 8 4 ρ x a 1, a 2,..., a s y b 1, b 2,..., b r 5

6 ρ twe-step ML Olsson(1979) 4.1 n i j 1 i = 1, 2,, s, j = 1, 2,, r (i,j) π i j L = C s i r j π n i j i j (3) C l = lnl = ln C + s i=1 n i j ln π i j. (4) x a i i = 0, 1,, s y b j j = 0, 1,, r a 0 = b 0 = a s = b r = + π i j = Φ 2 (a i, b j ) Φ 2 (a i 1, b j ) Φ 2 (a i, b j 1 ) + Φ 2 (a i 1, b j 1 ) (5) Φ 2 ρ bivariate normal distribution function with ρ 4.2 1: ρ, a 1,, a s 1, b 1,, b r 1 l ρ = s i=1 n i j π i j π i j ρ (6) = b m = s i=1 s i=1 n i j π i j π i j (7) n i j π i j π i j b m. (8) 6

7 ϕ 2 Φ 2 (u, v)/ ρ = ϕ 2 (u, v) 6 ρ = π i j ρ = ϕ 2(a i, b j ) ϕ 2 (a i 1, b j ) ϕ 2 (a i, b j 1 ) + ϕ 2 (a i 1, b j 1 ). (9) s i=1 n i j π i j { ϕ2 (a i, b j ) ϕ 2 (a i 1, b j ) ϕ 2 (a i, b j 1 ) + ϕ 2 (a i 1, b j 1 ). (10) 7 0 i k i k + 1 π i j a k π i j = Φ 2 (a k,b j ) Φ 2(a k,b j 1 ) Φ 2(a k,b j ) + Φ 2(a k,b j 1 ) k = i k = i 1 7 i k k = r n i j { Φ2 (a k,b j ) π k j Φ 2(a k,b j 1 ) + b { k+1 j Φ π k+1 j 2 (a k,b j ) + Φ 2(a k,b j 1 ) (11) (12) = ( ) r nk j π k j n { k+1 j Φ2 (a k,b j ) π k+1 j Φ 2(a k,b j 1 ). ϕ 1 Φ 1 { Φ 2 (u, v) (v ρu) = ϕ 1 (u) Φ 1 u (1 ρ 2 ) 1/2 (13) (Tallis,1962,p.346) 7 = ( nk j n ) [ { { ] k+1 j (bk ρa k ) (b j 1 ρa k ) ϕ 1 (a k ) Φ 1 Φ π k j π k+1 j (1 ρ 2 ) 1/2 1. (14) (1 ρ 2 ) 1/2 b m = ( nim n ) [ { { ] im+1 (ai ρb m ) (ai 1 ρb m ) ϕ 1 (b m ) Φ 1 Φ π im π im+1 (1 ρ 2 ) 1/2 1. (15) (1 ρ 2 ) 1/2 10,14, :Two-Step ρ = s i=1 n i j π i j { ϕ2 (a i, b j ) ϕ 2 (a i 1, b j ) ϕ 2 (a i, b j 1 ) + ϕ 2 (a i 1, b j 1 ) = 0. (16) 7

8 a i = Φ 1 1 (P i.) (17) b j = Φ 1 1 (P. j) (18) P i j (i, j) P i. P. j P i. = i k=1 P k j (19) P. j = s j P ik (20) i=1 k=1 5 C ( 1 ) double Normal_Percent(double val){ double b0,b1,b2,b3,b4,b5,b6,b7,b8,y,alpha,sum,v; if( val > 0.5 ){ v = 1 - val; else{ v = val; b0 = *10; b1 = e-1; b2 = e-3; b3 = e-3; b4 = e-5; b5 = e-5; b6 = e-5; b7 = e-7; b8 = e-8; y = -log(4*v*(1-v)); sum = b0 + (b1*y) + (b2*pow(y,2)) + (b3*pow(y,3)) + (b4*pow(y,4)) + (b5*pow(y,5)) + (b6*pow(y,6)) + (b7*pow(y,7)) + (b8*pow(y,8)); 8

9 sum = sum * y; alpha = sqrt(sum); if( val < 0.5 ){ alpha = -1 * alpha; return alpha; 1 ( Encyclopedia of Statistical Sciences;Polychoric and Polyserial Correlations ) A B C aˆ 1 = Φ 1 (111/227) = Φ 1 (0.489) = aˆ 2 = Φ 1 (198/227) = Φ 1 (0.872) = ˆ b 1 = Φ 1 (92/227) = Φ 1 (0.754) = ˆ b 2 = Φ 1 (214/227) = Φ 1 (0.942) = (ϕ 2 ;probability density function;pdf) (Φ 2 ;cumulative distribution function;cdf) 2,1994) //( ) double N(double z) { double b1 = ; double b2 = ; 9

10 double b3 = ; double b4 = ; double b5 = ; double p = ; double c2 = ; if (z > 6.0) { return 1.0; ; // this guards against overflow if (z < -6.0) { return 0.0; ; double a=fabs(z); double t = 1.0/(1.0+a*p); double b = c2*exp((-z)*(z/2.0)); double n = ((((b5*t+b4)*t+b3)*t+b2)*t+b1)*t; n = 1.0-b*n; if ( z < 0.0 ) n = n; return n; // // double f( double x, double y, double aprime, double bprime,double rho ){ double r = aprime * ( 2 * x - aprime ) + bprime *( 2 * y - bprime ) + 2 * rho * ( x - aprime ) * ( y - bprime ); return exp(r); double sgn( double x) { // sign function if (x>=0.0) return 1.0; return -1.0; // double binormalint(double a, double b, double rho){ double val; if( a <= 0 && b <= 0 && rho <= 0 ){ double aprime = a/(sqrt(2 * ( 1 -rho * rho ))); double bprime = b/(sqrt(2 * ( 1- rho * rho ))); 10

11 long double A[15],B[15]; A[0] = E-02; A[1] = E-01; A[2] = E-01; A[3] = E-01; A[4] = E-01; A[5] = E-01; A[6] = E-02; A[7] = E-02; A[8] = E-03; A[9] = E-04; A[10]= E-05; A[11]= E-06; A[12]= E-08; A[13]= E-10; A[14]= E-14; B[0] = E-02; B[1] = E-01; B[2] = E-01; B[3] = E-01; B[4] = E-01; B[5] = E+00; B[6] = E+00; B[7] = E+00; B[8] = E+00; B[9] = E+00; B[10]= E+00; B[11]= E+00; B[12]= E+00; B[13]= E+00; B[14]= E+00; long double sum = 0; for( int i = 0 ; i < 15 ; i++ ){ for( int j = 0; j < 15 ; j++ ){ sum = sum + A[i] * A[j] * f( B[i], B[j],aprime,bprime,rho); 11

12 sum = sum * ( sqrt( rho * rho )/ PI ); return (double)sum; else if( a* b * rho <= 0.0 ){ if( ( a <= 0.0 ) && ( b >= 0.0 ) && ( rho >= 0.0 )){ return N(a) - binormalint(a,-b,-rho); else if (( a >= 0.0) && ( b <= 0.0 ) && ( rho >= 0.0)){ return N(b) - binormalint(-a,b,-rho); else if (( a >= 0.0) && ( b >= 0.0 ) && ( rho <= 0.0)){ return N(a) + N(b) binormalint(-a,-b,rho); else if ( a * b * rho >= 0.0 ){ double denum = sqrt( a* a -2 * rho * a * b + b * b); double rho1 = (( rho * a - b ) * sgn(a))/denum; double rho2 = (( rho * b - a ) * sgn(b))/denum; double delta = ( 1.0-sgn(a)*sgn(b))/ 4.0; return binormalint( a,0.0,rho1)+binormalint(b,0.0,rho2)-delta; return -99.9; douuble A[4] ={ , , , ; double B[4] ={ , , , ; double sum = 0; for( int i = 0 ; i < 4 ; i++ ){ for( int j = 0; j < 4 ; j++ ){ sum = sum + A[i] * A[j] * f( B[i], B[j],aprime,bprime,rho); 1 ρ CDF ( 5 π i j ) 16 12

13 I j π i j ρ f (x) < ϵ ( ) (1992) 6 1. Hull,J. Options, futures, and other derivative securities ( ), AMOS EQS CALIS 3. Olsson,U Maximum Likelihood Estimation of the Polychoric Correlation Coefficient. Psychometrika,44(4), UNIX

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