後續研究建議

雖本文已提出等期間 ECM 鑑定策略，但研究過程中仍有一些值得後續研究。

5.3.1 二階 LGM 一般化 ECM 之鑑定

當二階 LGM 之誤差間不獨立時，此時如何選擇同問項誤差不同期、不同問項誤差但同期、不同問項誤差在不同期設定 ECM，變成一很複雜 ECM 之結構仍值得後續研究。

5.3.2 非等距的空間 ECM 之鑑定

本研究為等距，若非等距或空間 ECM 之鑑定仍為一個可以研究之議題。

5.3.3 單變量與多變量資料結構估計差異

在 HLM (PROC MIXED)採用之資料結構為單變量結構、而 SEM(PROC CALIS) 採用之資料結構為多變量結構(Singer, 1998)，兩者參數估計出來結果相近( Bovaird, 2007 ; Rovine & Molenaar, 2000)，但本研究估計結果在成長因素固定效果參數相近而隨機誤差參數 (level-1/2 ECM)稍許出入且概似函數值(表 4.3)差異很大，均值得後續研究且釐清。

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105

A-1: 第一章附註

[註 A-1.1] ECM (Singer and Willett, 2003, ch7)

本文之ECM 係指 LGM (HLM)中第一層測量誤差或第二層成長因素誤差之共變結構之矩陣(error covariance structural matrix,簡稱 ECM)，該名詞引用 Singer and Willett((2003), Ch7.此有別於計量經濟學 error correlation model (ECM)。

A-2: 第二章附註

[註 A-2.1] LGM 以 HLM 表示時 level-1 及 level-2 誤差之示意圖 (Rabe-Hesketh and

Skrondal,2008, ch4).

在LGM 固定效果係數

β

₀和

β

₁表示整個成長模型截距的平均值及斜率的平均值；隨機效果係數

u 和

_{0 j}

u (即

_{1 j} u向量之元素)表示第 j 個受測者成長模型截距及斜率和整個成長模型截距的平均值及斜率的平均值之偏差(deviation)，用以表示受測者彼此之間成長趨勢之變化；隨機效果

r (即

_jt

r 向量之元素)表示第 t 個時點及第 j 個受測者之量測值偏離本身

成長曲線之誤差。為更清楚

u 、

_{0 j}

u 及

_{1 j}

r 之意義，我們以第 j 個受測者成長模型為

_tj

0 1 0 1

0 0 1 1

( ) ( )

( ) (A.2.1)

jt tj j j tj tj

j j tj tj

y Time u u Time r

u u Time r

β β

= + + + +

設由母體抽出第

j 個受測者之 LGM，就全部受測者而言，整體線性趨勢線(trajectory line)

可表為式(A.2.1)，即圖 A.2.1 線

AB ，該迴歸線對所有受測者( j

∀ )而言均成立。

0 1

( )_t ( _t)

E y

β

β Time

(A.2.2) 今假設抽到第

j 個人設隨機效果之得分為 u 及

₀_j

u ，則式(A.2.2)之條件期望值，即為式

₁_j (A.2.3)

0 1 0 0 1 1

( _jt| _j, _j, _tj) _j ( _j) _tj

E y u u Time

β

u

β

u Time

(A.2.3) 而式(A.2.3)之迴歸線，即為圖 A.2.1 線CE。為易於說明式(A.2.2)及式(A.2.3)起見，我們將線

AB 平行移動 u 大小，相當於固定

₀_j

u 之迴歸線，即為式(A.2.4)及圖 A.2.1 線

₀_j CD

0 0 0 1

( _jt| _j, _jt) _j ( _jt)

E y u Time

β

u

β Time

(A.2.4) 因而CE即為線CD(

AB 往上平移 u )以 C 為圓心旋轉遞增

_{0 j}

u ，因此

_{1 j}

u 及

_{0 j}

u 分別表示每

_{1 j} 一個受測者對整體迴歸線偏差(deviance)程度，故u隨機向量反映受測者彼此間(between

106

ε

隨機誤差向量反映第

j 個受測者本身(within subjects)偏離本身迴歸線

程度。 才有層次(level)之觀念(Raudenbush and Bryk, 2002, Ch2)，而 SEM 為測量模式(measure model)及結構方程(structural equation) (Bollen, 1998, Ch8)各有其要解決之了問題，唯 LGM 在 HLM 及 SEM 其解析式卻相同，如圖 2.1 在 SEM 稱潛在變數(intercept(或 level) 及 slope(shape))為成長因素構念，而在 HLM 稱為 random coefficients，其他變數在不同統計模型之術語其對對映關係如表表 A-2.1 所示。

107

108 两個點之歐幾理德距離。SP(MATERN)及 SP(MATHSW)表示共變數結構，由 Matern (1986)，

Handcock and Stein (1993), Handcock and Wallis (1994)所提出一系列結構。

K

_v^{代表第二類修正貝} 氏函數(Bessel function), v>0.

109

110

111

112

113

114

115

116

117

ARMA(p,q)之 level-1 ECM 推導 首先定義誤差{

ε

_t}過程為 ARMA(p,q)

1 1 1 1

AR part MA part

, ~ white noise(0, ).

118

The autocovariance function

1 1

119

二. Example for ARMA(2,1) (p=2,q=1)

120

121

122

123

Summarized by matrix

124

三.Example for ARMA(1,2) (p=1,q=2)

1 1 1 1 2 2

125

126

127

128

所以lag-2 autocovariance 為

129

2 2

1 1 1

1 1

( ) ( 1) , 3

, 3

k k

k k k

εε εε ε ε

σ φ σ φ ρ σ ρ σ

ρ φ ρ

′ ′ −

−

= − = = ≥

= ≥

:| | 1

φ

1 <

平穩條件 ,

1 2

2 1

| | 1,

: 1,

1. θ θ θ θ θ

⎧ <

⎪ + <

⎨ ⎪ − <

⎩

可逆條件

130

ARMA(p, q) (autoregressive moving average of order (p, q))

1 1

131

常見ARMA(p,q) stationarity 及 invertibility 條件彙整如下

Model Stationary

conditions

132

133

附錄B: SAS 程式

[B-1]:一階條件式LGM模擬資料程式

/********************************************************************************/

DM 'LOG; CLEAR; output; clear; PGM;recall;ODS; Clear;';

OPTIONS REPLACE notes NODATE PS=58 PAGENO=1 LS=72;

/********************************************************************************/

%MACRO SIMUTOEPH (GA00, GA01, Sigma2_alpha, Sigma2_beta, C_alpha_beta VARE1, VARE2, VARE3, VARE4, CE1E2, CE1E3, CE1E4, CE2E3, CE2E4, CE3E4, Mu_X, Sigma_X, GA10, GA11, N);

%DO runs= 1 %to 1; /***僅模擬一組樣本******/

DM 'LOG; CLEAR; output; clear; ODS; Clear;';

PROC IML; /***呼叫 IML 計算式(2.32)蘊涵平均數向量及式(2.33) 蘊涵共變異數矩陣*****/

LY = { 1 0, 1 1, 1 2, 1 3};

MuEta={&GA00, &GA01};

Mu_X={&Mu_X};

Sigma_X={&Sigma_X};

GA={&GA10, &GA11};

PSI = {&Sigma2_alpha &C_alpha_beta, &C_alpha_beta &Sigma2_beta};

TE = {&VARE1 &CE1E2 &CE1E3 &CE1E4 , &CE1E2 &VARE2 &CE2E3 &CE2E4 , &CE1E3 &CE2E3 &VARE3 &CE3E4 , &CE1E4 &CE2E4 &CE3E4 &VARE4};

COVY = LY*(GA*Sigma_X*GA`+PSI)*LY`+TE;

COVXY = LY*GA*Sigma_X;

COVYX = Sigma_X*GA`*LY`;

COVX = Sigma_X;

UPPER = COVY || COVXY;

LOWER = COVYX || COVX;

COV = UPPER // LOWER; /***式(2.33) 蘊涵共變異數矩陣*****/

MEANY= LY*(MuEta+GA*Mu_X);

MEANX=Mu_X;

MEAN=MEANY//MEANX; /***式(2.32)蘊涵平均數向量 *********/

print COV MEAN;

/*************利用RANDNORMAL模擬多變量常態分配資料***************************/

SERIES = RANDNORMAL( &N, Mean, Cov ); /*Multinormal distribution*********/

134

CREATE DATA_Sim_ARH1_SEM FROM series[COLNAME={Y1 Y2 Y3 Y4 X }];

APPEND FROM series;

RUN;

/*********************************************************************************/

PROC CORR COV DATA=DATA_Sim_ARH1_SEM OUTP=ARH1_data noprint;

RUN;

proc print data=ARH1_data;

title'ARH(1)_data';

run;

/*********************************************************************************/

PROC PRINT DATA=DATA_Sim_ARH1_SEM /*NOOBS*/;

title'DATA_Sim_ARH(1)_SEM';

RUN;

/********************************************************************************/

%END;

%MEND SIMUTOEPH;

/**************計算ECM為ARH(1)**************************************************/

%MACRO compute (GA00, GA01, Sigma2_alpha, Sigma2_beta, C_alpha_beta, VARE1, VARE2 VARE3, VARE4, MuX, SigmaX, GA10, GA11, RHO, VN );

%let SQRTVARE1=%sysfunc(sqrt(&VARE1));

%let SQRTVARE2=%sysfunc(sqrt(&VARE2));

%let SQRTVARE3=%sysfunc(sqrt(&VARE3));

%let SQRTVARE4=%sysfunc(sqrt(&VARE4));

%let COVE1E2=%sysevalf(&RHO*&SQRTVARE1*&SQRTVARE2);

%let COVE1E3=%sysevalf(&RHO*&RHO*&SQRTVARE1*&SQRTVARE3);

%let COVE1E4=%sysevalf(&RHO*&RHO*&RHO*&SQRTVARE1*&SQRTVARE4);

%let COVE2E3=%sysevalf(&RHO*&SQRTVARE2*&SQRTVARE3);

%let COVE2E4=%sysevalf(&RHO*&RHO*&SQRTVARE2*&SQRTVARE4);

%let COVE3E4=%sysevalf(&RHO*&SQRTVARE3*&SQRTVARE4);

/***********************************************************************************/

%SIMUTOEPH (&GA00, &GA01, &Sigma2_alpha, &Sigma2_beta, &C_alpha_beta, &VARE1, &VARE2, &VARE3, &VARE4, &COVE1E2, &COVE1E3,

&COVE1E4, &COVE2E3, &COVE2E4, &COVE3E4, &MuX, &SigmaX,

&GA10, &GA11, &VN);

%MEND compute;

data overall;

run;

/****設定表4.2 一階LGM 母體參數值***********************************************/

/* (GA00, GA01, Sigma2_alpha, Sigma2_beta, C_alpha_beta, VARE1, VARE2, VARE3, VARE4, MuX, SigmaX, GA10, GA11, RHO, VN ); */

title 'Case 1:ECM as ARH(1), sample size=300';

%compute (10, 4, 15, 10, 7, 36, 25, 49, 64, 0, 1 , 4, 6, 0.7, 300 );

/*********************************************************************************/

135

[B-2]:一階條件式LGM示例程式

/************************************************************************************/

DM 'LOG; CLEAR; output; clear; PGM;recall;ODS; Clear;';

OPTIONS REPLACE notes NODATE PS=58 PAGENO=1 LS=120;

/************************************************************************************/

DATA DATA_Sim_ARH1_SEM;

input ID $ Y1 Y2 Y3 Y4 X;

PROC CORR COV DATA=DATA_Sim_ARH1_SEM OUTP=ARH1_data noprint ; VAR Y1 Y2 Y3 Y4 X;

RUN;

proc print data=ARH1_data;

title'ARH(1)_data';

run;

/****列印表4.2 樣本共變異數矩陣*****************************************************/

PROC PRINT DATA=DATA_Sim_ARH1_SEM;

Title'DATA_Sim_ARH1_SEM';

RUN;

/****多變量資料結構(SEM方法)轉成單變量資料結構(HLM方法)****************************/

DATA DATA_Sim_ARH1_HLM;

SET DATA_Sim_ARH1_SEM;

PROC PRINT DATA=DATA_Sim_ARH1_HLM;

VAR I ID TIME WAVE Y X;

136 TITLE'DATA_Sim_ARH(1)_HLM';

RUN;

QUIT;

/************************************************************************************/

/*****Level-1 ECM=UN 供穩態性檢定***************************************************/

PROC TCALIS METHOD=ML data=DATA_Sim_ARH1_SEM MAXFUNC=1500 MAXITER=1500 OUTRAM=OUT_RAM OUTEST=OUT_EST PRINT;

LINEQS

Y1 = F_alpha + 0 F_beta + E1, Y2 = F_alpha + 1 F_beta + E2, Y3 = F_alpha + 2 F_beta + E3, Y4 = F_alpha + 3 F_beta + E4,

F_alpha =GA00(10) intercept + GA10(4) X+D0, F_beta = GA01(4) intercept + GA11(6) X+D1;

STD

E1 =VARE1(36), E2 =VARE2(25), E3 = VARE3(49), E4 = VARE4(64), D0 = VARD0(15), D1 =VARD1(10);

COV

E2 E1 = COVE2E1(21), E3 E2 = COVE3E2(24.5), E4 E3 = COVE4E3(39.2), E3 E1 = COVE3E1(20.58), E4 E2 = COVE4E2(19.6), E4 E1 = COVE4E1(16.46), D0 D1= CD0D1(7);

VAR

Y1 Y2 Y3 Y4 X;

/***** ECM 採SIMTESTS 穩態性測試(表4.3及4.2.5.3)*************************************/

SIMTESTS

ERR_STATIONARY_TEST =[VAREQ_1 VAREQ_2 VAREQ_3 COVLag1EQ_1 COVLag1EQ_2 COVLag2EQ ];

/*SAS Programming statements for defining the parametric functions********************/

VAREQ_1 = VARE1-VARE2; VAREQ_2 = VARE1-VARE3; VAREQ_3 = VARE1-VARE4;

COVLag1EQ_1=COVE2E1-COVE3E2; COVLag1EQ_2=COVE2E1-COVE4E3;

COVLag2EQ=COVE3E1-COVE4E2;

TITLE 'ECM_UN_SEM';

RUN;

QUIT;

/************************************************************************************/

/**表4.4 步驟1, 檢定H0:MT=MS, 當MT=TOEPH (1) 時所需卡方值及自由度之程式************/

PROC TCALIS METHOD=ML data=DATA_Sim_ARH1_SEM MAXFUNC=1500 MAXITER=1500 OUTRAM=OUT_RAM OUTEST=OUT_EST PRINT;

LINEQS

Y1 = F_alpha + E1, Y2 = F_alpha + 1 F_beta + E2, Y3 = F_alpha + 2 F_beta + E3, Y4 = F_alpha + 3 F_beta + E4,

137 F_alpha =GA00(10) intercept + GA10(4) X+D0, F_beta = GA01(4) intercept + GA11(6) X+D1;

STD

E1 =VARE1(36), E2 =VARE2(25), E3 = VARE3(49), E4 = VARE4(64), D0 = VARD0(15), D1 =VARD1(10);

COV

D0 D1= CD0D1(7);

VAR

Y1 Y2 Y3 Y4 X;

TITLE 'ECM_UN(1)_SEM';

RUN;

QUIT;

/*************************************************************************************/

/**表4.4 步驟1:虛無假設 H0:MT=MS, MT=TOEPH (1), MS=UN之卡方差異檢定程式************/

DATA Chisquare_Diff_Prob_TOEPH1_UN;

Saturated_M_UN_Chisquare=5.1700;

Saturated_M_UN_DF=1;

Restricted_M_TOEPH1_Chisquare=30.9590;

Restricted_M_TOEPH1_DF=7;

ChisquareDIFF=Restricted_M_TOEPH1_Chisquare-Saturated_M_UN_Chisquare;

DF_DIFF=Restricted_M_TOEPH1_DF-Saturated_M_UN_DF;

ChiSquareProb=1-PROBCHI(ChisquareDIFF, DF_DIFF,0); output;

RUN;

/*************************************************************************/

PROC PRINT DATA= Chisquare_Diff_Prob_TOEPH1_UN;

title'SCDT_TOEPH(1)_UN_SEM';

RUN;

/*************************************************************************************/

/**表4.4 步驟2, 檢定H0:MT=MS,當MT=TOEPH (2) 時所需卡方值及自由度之程式**************/

PROC TCALIS METHOD=ML data=DATA_Sim_ARH1_SEM MAXFUNC=1500 MAXITER=1500 OUTRAM=OUT_RAM OUTEST=OUT_EST PRINT;

LINEQS

Y1 = F_alpha + E1, Y2 = F_alpha + 1 F_beta + E2, Y3 = F_alpha + 2 F_beta + E3, Y4 = F_alpha + 3 F_beta + E4,

F_alpha =GA00(10) intercept + GA10(4) X+D0, F_beta = GA01(4) intercept + GA11(6) X+D1;

STD

E1 =VARE1(17.54), E2 =VARE2(9.46), E3 = VARE3(23.35), E4 = VARE4(31.4), D0 = VARD0(15), D1 =VARD1(10);

COV

E2 E1 = COVE2E1(3.96), E3 E2 = COVE3E2(4.57), E4 E3 = COVE4E3(8.3069),

138 D0 D1= CD0D1(7);

PARAMETERS RHO (0.3);

COVE2E1=SQRT(VARE2)*SQRT(VARE1)*RHO;

COVE3E2=SQRT(VARE3)*SQRT(VARE2)*RHO;

COVE4E3=SQRT(VARE4)*SQRT(VARE3)*RHO;

BOUNDS

-1.<RHO<1.;

VAR

Y1 Y2 Y3 Y4 X;

TITLE 'ECM_TOEPH(2)_SEM';

RUN;

QUIT;

/*************************************************************************************/

/**表4.4 步驟2:虛無假設 H0:MT=MS, MT=TOEPH (2), MS=UN之卡方差異檢定程式************/

DATA Chisquare_Diff_Prob_TOEPH2_UN;

Saturated_M_UN_Chisquare=5.1700;

Saturated_M_UN_DF=1;

UnRestricted_M_TOEPH2_Chisquare=18.9406;

UnRestricted_M_TOEPH2_DF=6;

ChisquareDIFF=Unrestricted_M_TOEPH2_Chisquare-Saturated_M_UN_Chisquare;

DF_DIFF=Unrestricted_M_TOEPH2_DF-Saturated_M_UN_DF;

ChiSquareProb=1-PROBCHI(ChisquareDIFF, DF_DIFF,0); output;

RUN;

/************************************************************/

PROC PRINT DATA= Chisquare_Diff_Prob_TOEPH2_UN;

title'SCDT_TOEPH(2)_UN_SEM';

RUN;

/************************************************************************************/

/**表4.4 步驟3, 檢定H0:MT=MS, 當MT=CSH 時所需卡方值及自由度之程式******************/

PROC TCALIS METHOD=ML data=DATA_Sim_ARH1_SEM MAXFUNC=1500 MAXITER=1500 OUTRAM=OUT_RAM OUTEST=OUT_EST PRINT;

LINEQS

Y1 = F_alpha + E1, Y2 = F_alpha + 1 F_beta + E2, Y3 = F_alpha + 2 F_beta + E3, Y4 = F_alpha + 3 F_beta + E4,

F_alpha =GA00(10) intercept + GA10(4) X+D0, F_beta = GA01(4) intercept + GA11(6) X+D1;

STD

E1 =VARE1(36), E2 =VARE2(25), E3 = VARE3(49), E4 = VARE4(56), D0 = VARD0(15), D1=VARD1(10);

COV

E2 E1= COVE2E1(21), E3 E2 = COVE3E2(24.5), E4 E3 = COVE4E3(31.0),

139

E3 E1 = COVE3E1(20.58), E4 E2 = COVE4E2(15.484), E4 E1 = COVE4E1(13.02), D0 D1= CD0D1(7);

PARAMETERS RHO (0.7);

COVE2E1=SQRT(VARE2)*SQRT(VARE1)*RHO;

COVE3E1=SQRT(VARE3)*SQRT(VARE1)*RHO;

COVE3E2=SQRT(VARE3)*SQRT(VARE2)*RHO;

COVE4E1=SQRT(VARE4)*SQRT(VARE1)*RHO;

COVE4E2=SQRT(VARE4)*SQRT(VARE2)*RHO;

COVE4E3=SQRT(VARE4)*SQRT(VARE3)*RHO;

BOUNDS

-1.<RHO<1.;

VAR

Y1 Y2 Y3 Y4 X;

TITLE 'ECM_CSH_SEM';

RUN;

QUIT;

/*************************************************************************************/

/**表4.4 步驟3:虛無假設 H0:MT=MS, MT=CSH, MS=UN之卡方差異檢定程式******************/

DATA Chisquare_Diff_Prob_CSH_UN;

Saturated_M_UN_Chisquare=5.1700;

Saturated_M_UN_DF=1;

UnRestricted_M_CSH_Chisquare=18.3761;

UnRestricted_M_CSH_DF=6;

ChisquareDIFF=Unrestricted_M_CSH_Chisquare-Saturated_M_UN_Chisquare;

DF_DIFF=Unrestricted_M_CSH_DF-Saturated_M_UN_DF;

ChiSquareProb=1-PROBCHI(ChisquareDIFF, DF_DIFF,0); output;

RUN;

/*************************************************************************************/

PROC PRINT DATA= Chisquare_Diff_Prob_CSH_UN;

title'SCDT_CSH_UN_SEM';

RUN;

/************************************************************************************/

/**表4.4 步驟4, 檢定H0:MT=MS, 當MT=ARH(1) 時所需卡方值及自由度之程式***************/

PROC TCALIS METHOD=ML data=DATA_Sim_ARH1_SEM MAXFUNC=1500 MAXITER=1500 OUTRAM=OUT_RAM OUTEST=OUT_EST PRINT;

LINEQS

Y1 = F_alpha + E1, Y2 = F_alpha + 1 F_beta + E2, Y3 = F_alpha + 2 F_beta + E3, Y4 = F_alpha + 3 F_beta + E4,

F_alpha =GA00(10) intercept + GA10(4) X+D0, F_beta = GA01(4) intercept + GA11(6) X+D1;

STD

140

E1 =VARE1(36), E2 =VARE2(25), E3 = VARE3(49), E4 = VARE4(64), D0 = VARD0(15), D1 =VARD1(10);

COV

E2 E1 = COVE2E1(21), E3 E2 = COVE3E2(24.5), E4 E3 = COVE4E3(39.2), E3 E1 = COVE3E1(20.58), E4 E2 = COVE4E2(19.6), E4 E1 = COVE4E1(16.46), D0 D1= CD0D1(7);

PARAMETERS RHO( 0.7) ;

COVE2E1=SQRT(VARE2)*SQRT(VARE1)*RHO;

COVE3E1=SQRT(VARE3)*SQRT(VARE1)*RHO*RHO;

COVE3E2=SQRT(VARE3)*SQRT(VARE2)*RHO;

COVE4E1=SQRT(VARE4)*SQRT(VARE1)*RHO*RHO*RHO;

COVE4E2=SQRT(VARE4)*SQRT(VARE2)*RHO*RHO;

COVE4E3=SQRT(VARE4)*SQRT(VARE3)*RHO;

BOUNDS

-1.<RHO<1.;

VAR

Y1 Y2 Y3 Y4 X;

TITLE 'ECM_ARH(1)_SEM';

RUN;

QUIT;

/***表4.4 步驟4:虛無假設 H0:MT=MS, MT=ARH(1), MS=UN之卡方差異檢定程式**************/

DATA Chisquare_Diff_Prob_ARH1_UN;

Saturated_M_UN_Chisquare=5.1700;

Saturated_M_UN_DF=1;

UnRestricted_M_ARH1_Chisquare=11.0760;

UnRestricted_M_ARH1_DF=6;

ChisquareDIFF=Unrestricted_M_ARH1_Chisquare-Saturated_M_UN_Chisquare;

DF_DIFF=Unrestricted_M_ARH1_DF-Saturated_M_UN_DF;

ChiSquareProb=1-PROBCHI(ChisquareDIFF, DF_DIFF,0); output;

RUN;

/*******************************************************************/

PROC PRINT DATA= Chisquare_Diff_Prob_ARH1_UN;

title'SCDT_ARH(1)_UN_SEM';

RUN;

/***********************************************************************************/

/***底下程式係以PROC MIXED,進行-2* Log Likelihood ratio檢定 */

/***ECM穩態性及利用SCDT 鑑定ECM */

/************************************************************************************/

/**4.2.5.1 利用概似比檢定ECM穩態性,計算非穩態為UN之-2LLu值****************/

PROC MIXED DATA=DATA_Sim_ARH1_HLM METHOD=ML noclprint MAXFUNC=1500 MAXITER=1500 noinfo covtest noitprint;

CLASS I wave;

MODEL Y=time X time*X / solution ddfm=bw notest;

141 REPEATED wave / subject=I type=UN R RCORR;

RANDOM intercept time / sub=I type=UN G GCORR /*solution*/;

PARMS (14.53, 4.58, 9.28, 32.66, 21.85, 28.92, 23.67, 26.97, 48.84, 22.31, 20.31, 31.72, 40);

TITLE 'ECM_UN_HLM';

RUN;

QUIT;

/*************************************************************************************/

/**4.2.5.1 利用概似比檢定ECM穩態性,計算穩態為TOEP之-2LLr值***************************/

PROC MIXED DATA=DATA_Sim_ARH1_HLM METHOD=ML noclprint MAXFUNC=1500 MAXITER=1500 noinfo covtest noitprint;

CLASS I wave;

MODEL Y=time X time*X / solution ddfm=bw notest;

REPEATED wave / subject=I type=TOEP R RCORR;

RANDOM intercept time / sub=i type=UN G GCORR /*solution*/;

TITLE 'ECM_TOEP_HLM';

RUN;

QUIT;

/*************************************************************************************/

/***4.2.5.1 計算-2LLr-(-2LLu)之卡方值,自由度為UN參數TOEP參數個數差=10-4=6**************/

DATA Minu_2_LRT_TOEPH_UN;

Minu_2_logLikelihood_UN=7795.6;

Minu_2_logLikelihood_TOEP=7849.0;

Minu_2_LRT=Minu_2_logLikelihood_TOEP-Minu_2_logLikelihood_UN;

DF_DIFF=6;

ChiSquareProb=1-PROBCHI(Minu_2_LRT, DF_DIFF,0); output;

RUN;

/*****************************************************************/

PROC PRINT DATA= Minu_2_LRT_TOEPH_UN;

title'Stationarity TEST_HLM';

RUN;

/*************************************************************************************/

/*****表4.4 步驟1, 計算TOEPH(1)之-2LLr配適值程式**************************************/

在文檔中成長模型第一層次誤差共變異結構之鑑定準則 (頁 114-200)

5.3.1 二階 LGM 一般化 ECM 之鑑定

5.3.2 非等距的空間 ECM 之鑑定

5.3.3 單變量與多變量資料結構估計差異

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A-1: 第 一 章 附 註

[註 A-1.1] ECM (Singer and Willett, 2003, ch7)

A-2: 第 二 章 附 註

[註 A-2.1] LGM 以 HLM 表示時 level-1 及 level-2 誤差之示意圖 (Rabe-Hesketh and

β

β

u 和

u (即

r (即

r 向量之元素)表示第 t 個時點及第 j 個受測者之量測值偏離本身

u 、

u 及

r 之意義，我們以第 j 個受測者成長模型為

( ) ( )

( ) (A.2.1)

β β

β β

= + + + +

= + + + +

j 個受測者之 LGM，就全部受測者而言，整體線性趨勢線(trajectory line)

AB ，該迴歸線對所有受測者( j

E y

β

β Time

j 個人設隨機效果之得分為 u 及

u ，則式(A.2.2)之條件期望值，即為式

E y u u Time

β

u

β

u Time

AB 平行移動 u 大小，相當於固定

u 之迴歸線，即為式(A.2.4)及圖 A.2.1 線

E y u Time

β

u

β Time

AB 往上平移 u )以 C 為圓心旋轉遞增

u ，因此

u 及

u 分別表示每

ε

A-1: 第一章附註

A-2: 第二章附註