11第十一章階層線性模式.DOC

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1 (student-level) (personal-level) ( ) (school-level) (organization-level) ( ) 1. (disaggregation) (estimated standard errors) (type one error). (aggregation) (within-group) (1997) (hierarchical linear model, HLM) HLM (covariance components models) (Dempster, Rubin, & Tsutakawa, 1981; Longford, 1987, 1993) (multilevel linear models) (Hox, 43

2 ; Goldstein, 1987, 1995) (mixed-effects models) (random-effects models) (Laird & Ware, 198) (random coefficient regression models) (Rosenberg, 1973) hierarchical linear models HLM HLM Lindley Smith (197)Smith (1973) (Bayesian estimation of linear models) Dempster, Laird, Rubin (1977) EM HLM Strenio, Weisberg, Bryk (1983) HLM (growth) Mason, Wong, Entwistle (1983) HLM Laird Ware (198) HLM (meta-analysis) (Draper, 1995) HLM BMDP 5V SAS MIXED (Wang, 1997)SPSS 11 MIXED GENMODHLMMLnVARCL Kreft, de Leeuw, Kim (1990) HLM MLn VARCL 1995 Journal of Educational and Behavioral Statistics HLM HLM HLM VARCL HLM HLM

3 45 X Y ( (SES) (academic achievement) ) X ( X X ) Y Yi = + 1( Xi Xi) + ri, ri ~ N(0, ) ( 11-1) 0 σ 11-1 ( X X ) =0 Yˆ = 0 ) 1( X X X Y ( 1X ) S S + S S = ( 11-) Y ( X X ) YX Y X YX 1( X X ) = = = 1X S( X X ) S X + S X S X ( 1 ) L L 1 : : Y i1 Y i = = ( X 1 i1 ( X X. 1) + r i i1 X. ) + r i ( 11-3) 11-3 L 1 = L ( 01 = 0 11 = 1 ) Y L 1 =L 0 X X 11-1L = L 1 1 (LT ) ( L1 L ) LT

4 46 1 L 010 L ( ) 1 Y L 1 L 0 X X L1 L ( ) LT Y = ( X X. ) + r, r ~ N(0, σ ) = 1,,3,... n ( 11-4) E( E( 0 1 Cov( ) ) 0 0 1,,, 1 Var( Var( ) = τ 01, 0 1 ) = τ ) = τ ρ( 11 0, 1 τ ) = 01 τ τ HLM r γ τ γ 1 τ 11

5 47 τ 01 0, 1 W ( ) W 0, W W + u + u 1 0 ( 11-5) u 0 u 1 (random effect) γ,..., γ 11 (fixed effect) HLM 11-6 Y W ( X X. ) W ( X X. ) + u + u ( X X. ) r ( 11-6) E( r ) = 0, Var( r ) = σ u, E u u =, Var 0 u 0 1 τ = τ 10 τ τ = T 11-6 (least square method) (independent) (normal) (constant variance) u1 ( X X. ) r u 0 u + u u 0 u 1 ( X X ) 11-6 HLM LS OLS (ordinary least squares) ML (maximum likelihood) (iteration) X, W (centering) ( ) X 1. (X ). (centering around the grand mean) ( X X.. )3. (centering around the group mean) ( X X. )W 1. (W ). ( W W. ) X

6 48 W 0 X W ( ) 0 X W HLM ( ) X (centering around the group mean) (X - X. ) HLM (general model) ( X ) HLM Y = = X + Q q= 1 1 q + X q X + r r q X q + r ~ N(0, σ ) ( 11-7) 11-7 Q+1 ( 1 Q ) Q+1 q = rq 0 + rq 1W1 + rq W r S = r + r W + u q0 q s= 1 qs s q qs q W S q + u q ( 11-8) ( q = 0,1,...,Q) Q+1 W ( S = 1,..., S ) q S q +1 1 Var ( uq) = τ qq; Cov( uq, uq ) = τ qq q HLM (Bryk & Raudenbush, 199; p.) 1. ~ N(0, σ ); r. Cov ( X q, r ) = 0; 3. u = ( u 0,..., u ) ~ N( 0, T); q 4. Cov ( W s, u q ) = 0; S q

7 49 5. Cov( r, u q ) = 0 HLM HLM (full model) (intercepts- and slopes-as- outcomes) 0 (submodels) (Bryk & Raudenbush, 199) (one-way ANOVA with random effects) = 0 Y = 0 r, r ~ N(0, σ ) ( 11-9) γ 0 01 = ( 11-10) 0 + u Y γ + r ( 11-11) = + u γ u 0 ( ) r ( ) ) ( u 0

8 430 Var( Y ) Var( u + r = τ + σ ρ = τ /( τ + ) ρ = 0 ) σ (intraclass correlation coefficient) (cluster effect) τ ) ( τ + ) ( σ ρ ANCOVA (random-intercept model) Var Y ) τ + ( σ (means-as-outcomes regression) ( 11-1) 0 01W + u ( W ) Y γ + r ( 11-13) = 01 W + u u 0 = 0 γ 01W γ u 0 (residual)u 0 ( τ ) W 0 (conditional variance) (Bryk & Raudenbush, 199, p.18) u 0 = 0 γ (u 0 ) W

9 (one-way ANCOVA with random effects) ) Y = ( X X..) + r ( ( X ) ( X X.. ) ( 11-5) γ γ 11 u u 0 ( 11-15) ( 1 ) γ ( 1 ) 0 0 (pooled within-group regression) X. ) 10 ( 0 = Y γ 10( X. X ( X..) (adusted mean) (..) ) Y γ + r ( 11-16) = 10( X X..) + u Var( ) = σ r ( X ) ANCOVA (W ) ANCOVA u 0 Y γ + r ( 11-17) = 01W 10( X X..) + u0

10 43 ANCOVA ANCOVA (classical fixed-effect ANCOVA) = H 0 : Var ) = τ 0 1 γ 10 ( 1 11 = (random coefficients regression model) u + u 0 1 ( 11-18) u 0 τ τ Var = = T u1 τ10 τ11 u 0 u 1 T (unconditional variance-covariance matrix) γ γ var( var( 0 1 ) = var( ) = var( 0 1 γ γ 10 ) = var( u ) = var( u 0 1 ) = τ ) = τ 11 ( 11-19) τ τ Y ( X X. ) + r ( 11-0) 10 ( X X. ) + u0 + u1 u u ( X X. ) 1 r

11 r = Y 0 1 ( X X. ) r Y = ( r ) X ( ) (a model with nonrandomly varying slopes) 11-4 u W W + u 0 ( 11-1) Y γ + r ( 11-) = 01W 10( X X. ) 11W ( X X. ) + u ( 1 ) (W ) Bryk Raudenbush (199) HLM uˆ

12 Y = ( X X. ) + r 01W ( γ u, 1 W + u Y W X X. ) + W ( X X. ) + u + u ( X X. ) + r Y = 0 + r 0 + u 0 Y γ + u + r = 0 ρ = τ /( τ + ) σ Y = 0 + r 0 01W + u0 Y γ + W + u + r = γ 01 0 Y = ( X X..) u 0, 1 10 Y 10( X X..) + u0 + r Y = ( X X. ) + r 0 + 0, u u 1 Y + X X. ) + u + u ( X X. ) + r γ 10 ( 0 1 r ( r r )/ r Y = ( X X. ) + r u0, 1 10 W 11 Y γ W ( X X. ) W ( X X. + u + r = ) 0 W

13 HLM (1984) 0 ( ) ( T ) 40 (Bryk & Raudenbush, 199) (1994, 1995) ( ) ( ) HLM/L 5.05 HLM (Bryk & Raudenbush, Congdon, 1994) Bryk & Raudenbush (199, ) ( )

14 HLM (identification number ) HLM SSM HLM ASCII ( Windows ) SASSPSSSYSTAT HLM SPSS HLM 1. SPSS ( ) IDSESACH CITY1CITY ID ID 101 ( ) 1 ( ID)31 3 ( ) 1. (aggregate) Data Aggregate Variable(s) Break Variable(s) ID Create new data file HLM_.SAV

15 SES_1 ( SES_AVG) SES SESACH_1 ( ACH_AVG) ACH ( ) 4. HLM File New Stat package input SSM/MDM HLM ( ) ( SASSPSS SYSTAT)

16 SPSS Browse HLM_1.SAV Choose Variable ( ID ) SSM 6. Browse HLM_.SAV Choose Variable ( ID ) SSM

17 Save Response File SSM Chen.ssm Make SSM SSM Check Stats ( ) Done HLM (1). (). Y = 0 + r, r ~ N(0, σ ) ( 11-3) ( 11-4) 0 + u 0

18 440 Υ i 0 γ 40 σ ( ) τ ( u 0 ) ( ) 1. ACH Outcome Variable ( ). 3. Run Analysis Save Run the model shown 4.

19 γ χ p u r % (.860), (.860) ( , 15.37) τ.01 (τ ) ( σ ) ρˆ ρ ˆ = τˆ /(ˆ τ + ˆ ) ( 11-5) σ =306.63/( ) = %

20 44 ˆ _ λ = reliability ( Y ) = τˆ /[ˆ τ + ( σˆ / n )] ( 11-6) λˆ = Σλ ˆ/ J =.914 ( 11-7) (1) ( ) ( ) ( ) (). 40 ( ) ) 11-4Y X X X ( ) σ =Var (Y ) γ γ 1 u 0 τ u 1 τ 11 10

21 SES add variable group centered ( ). SES 3.

22 t p γ γ 10 χ p u u 1 r t =[ $σ ( ANOVA)$σ ( ) ]/ $σ ( ANOVA) = ( ) / % τˆ , df=401=39, χ =50.084, τ ˆ =0.150, df=39, 11 χ =65.90, (1). ().

23 ( 11-8) 0 01( W W.) + u0 W W. 01 γ u 0 τ τ Var( 0 W W.) 1. SES Delete variable from model ( ). Level- Vars SES_AVG add variable grand centered ( ) 3. SES_AVG

24 t p γ γ χ p u r (t=6.977, p<.01) =[ $u ( ANOVA) $u ( ) ] / $u ( ANOVA) =( )/

25 = % ( τ = , df = 38, χ , = p<.1) ρ ˆ = τˆ /(ˆ τ + ˆ ) =9.01/( )=.196 σ (1). ( ) (). (3) ( W ( W 1 1 W 1.) W.) 0 1 W W W W u + u ) W W. 01 γ W γ W γ u 0 τ 11 γ γ 1 13 γ u 1 τ τ 11 11

26 448 Var( ( W 1 W 1.), W, W 3, ) 1 1. ACH Outcome Variable ( ). SES add variable group centered ( ) 3. Level- Vars SES_AVG add variable grand centered ( ) 4.

27 Level- Vars SES_AVG add variable grand centered ( ) 6 CITY_1 CITY_ Add variable uncentered ( ) 6. 7

28 t p γ γ γ γ γ γ γ γ χ p u u 1 r

29 451 u 0 (( )/ =0.733) u 0 χ u 1 u 1 (( )/0.150=.1533) ( ) % ( 6.56%) Bryk Raudenbush (199)

30 45 ( 73.5%)

64 Application of HLM 6 ㈠㈡㈢㈣ covariance component models Dempster et al., 1981 multilevel linear models Hox, 1994 mixedeffects models Laird & Ware,

64 Application of HLM 6 ㈠㈡㈢㈣ covariance component models Dempster et al., 1981 multilevel linear models Hox, 1994 mixedeffects models Laird & Ware, 63 階層線性模式於追蹤研究之應用 -- 以子宮切除婦女之術後初期症狀困擾為例李靜芳溫福星 * 階層線性模式為目前處理多層次資料時最佳統計方法之一, 而追蹤研究資料則屬多層次資料結構, 欲瞭解同一群受試者在不同時間點重複